







A tangent to a curve 
20171022 

From Jasem:
Suppose that
f(x)=(3x3)^1/2.
(A) Find an equation for the tangent line to the graph of f(x) at x=2
(B) Find all values of xx where the tangent line is horizontal, and enter them as a commaseparated list (e.g., 2,3,6). If there are none, enter none.
Values of x Answered by Penny Nom. 





Salary plus commission 
20171016 

From Natasha: Veronika works at Future Shop and earns ₱10.50/h plus 6% commission on sales. Last week Veronika worked 40 hours. What was Veronika’s weekly gross salary if her total sales were ₱2050? Answered by Penny Nom. 





The distance between the origin and a moving point on a graph 
20171016 

From Paulina: Find the rate of change of the distance between the origin and a moving point on the graph of y=x^2 +1 if dx/dt=2 centimeters per second Answered by Penny Nom. 





The volume of a cone without calculus 
20171002 

From Akash: How to find the volume of a cone without the knowledge of calculus? Answered by Penny Nom. 





Simultaneous equations with fractions 
20170909 

From Farah: Hi, my name is Farah. I hope you can help me with this question . X/2 + g/5= 3 and 2g  f = 10 Answered by Penny Nom. 





Simultaneous equations 
20170902 

From keto: xy=2,x^2+xy=12 Answered by Penny Nom. 





Simultaneous equations with fractions 
20170602 

From Jamal: 1/x + 1/y =5 and 1/y  1/x =1 Answered by Penny Nom. 





Forming a cone from a circle 
20170415 

From Tasha: A sector of a circle subtends an angle of 216 degrees at the centre, If this sector is used to form a cone of vertical height ,8cm, calculate the base radius of the cone Answered by Penny Nom. 





Is the square root of 2 plus the square root of 2 irrational? 
20170329 

From haya: how can i prove the the square root of 2 plus the square root of 2 is an irrational number? Answered by Penny Nom. 





Differentiate y = x^x^x 
20170319 

From Nafis: differentiate y = x^x^x Answered by Penny Nom. 





The average rate of change of cot(t) 
20170118 

From Brianna: Hello!
It's been a while since I've taken a math course, and I'm stuck on a problem in my calculus course.
The question is this:
Find the average rate of change of the function over the given interval.
h(t)=cot(t) a) [5pi/4, 7pi/4] Answered by Penny Nom. 





Five equations 
20170116 

From Muhammed: A x 4 = E
B ÷ 4 = E
C + 4 = E
D  4 = E
___________
& A + B + C + D = 100
what is the value of E Answered by Penny Nom. 





Simultaneous equations with fractions 
20161221 

From Kimi: I am stuck on this,can someone please help me????
1/2x+1/3y=11
8x+2/5y=102
Can someone please explain how to solve questions like these??
Or else I will never learn...
Thank you Answered by Penny Nom. 





Two concentric circles 
20161221 

From shrestha: Two concentric circles have radii of 14 cm and 7 cm respectively. Find the area of space between them. Answered by Penny Nom. 





The area and radius of a circle 
20161219 

From india: How do you find the radius of a circle if only the area is given to you? Answered by Penny Nom. 





Is every rectangle a rhombus? 
20161128 

From Robert: not every rectangle is a rhombus.true or false? if false please leave a short explanation. thank you:) Answered by Penny Nom. 





Simultaneous equations 
20161125 

From Rohan: x2/3 + y  1/4 = 13/12
2x/2 + 3 + y/3 = 11/6 Answered by Penny Nom. 





Radius angle and arc length 
20161124 

From pavidthra: Length or arc 11 and angle of subtended 45.need to find a radius Answered by Penny Nom. 





Volume of liquid remaining in a tilted cylinder 
20161108 

From Brian: I am trying to determine the amount of a liquid remaining in a 55 gallon drum when it is tilted at 45 degrees and the liquid level is low enough so that the liquid does not completely cover the bottom of the drum.
Your help is greatly appreciated. Answered by Harley Weston. 





A circle formed from bending a wire 
20161016 

From Deszaree: The same piece of a wire is bent to form a circle
calculate:
a, the radius of the circle
b, the area of the circle Answered by Penny Nom. 





2,0061234...484950 
20160924 

From Mimi: Compute the following:
2,0061234…484950 Answered by Penny Nom. 





The distance around a warehouse 
20160921 

From Chuck: How far do I walk when I walk around a 751,000 sq. ft. warehouse? Answered by Penny Nom. 





Integrate (x^2  4x + 4) ^4/3 
20160915 

From Ifah: Hi i have questions please answer
Integral 2 sampai 3 dari (x²  4x + 4) ^4/3 dx Answered by Penny Nom. 





The radius of a coffee cup 
20160912 

From Brett: What is inside radius, in centimeters of a coffee cup if it holds 350g of coffee when filled to a depth of 9.5 cm? Assume coffee has the same density of water, 1.00g/cm3. A numeric value is expected and not an expression. Answered by Penny Nom. 





A suspension bridge 
20160818 

From sai: The main cable of a suspension bridge has the shape of a parabola. The cables are strung from the top of two towers, 61 metres apart, each 15.25 metres high above the roadway. The cable is 1.5 metres above the roadway at the point that is directly between the towers.
h(x)=a(x30.5)^2+1.5
where a is the constant and its value to be determined.
a) determine the value of the constant a by using that the fact that height of the suspension cable, h(x) is 15.25 metres at each tower. Hence, write the updated model equation. Answered by Penny Nom. 





The modulus of a complex number 
20160729 

From Cheyenne: There's a question on my Summer Assignment I cant figure out. Here it is:
Find the absolute Value of the complex number. 5i
Please help? Answered by Penny Nom. 





Two equations with fractions 
20160722 

From kanesh: p/2+q/3=3
p/4+2q/3=3 Answered by Penny Nom. 





Simultaneous equations 
20160625 

From sena: 2x/3+3y/2=1
4x5y=22
simultaneous equation Answered by Penny Nom. 





Implicit differentiation 
20160606 

From Pranay: Is a circle x^2+y^2=2 a function? If it is not a function,
why is it possible to do implicit differentiation on it?
Thanks. Answered by Penny Nom. 





The volume of a truncated rectangular pyramidal pond 
20160513 

From Paul: How do you calculate a partially filled truncated rectangular pyramid if you always know the bottom rectangle, the maximum height top rectangle perimeter, but have a varying height. Similar to filling up a pond you know the current height and dimensions at the max rectangle how do you calculate it half full i.e. 10x30 outside perimeter with a 2x8 base and a max height of 6ft how do you calculate it at 3ft without remeasuring the top perimeter.
Thanks,
Paul Answered by Harley Weston. 





A pair of equations with fractions 
20160511 

From tiya: hello, i want to know how to solve this question.
m/6+2n/3=6
m/10=2n/5=2 Answered by Penny Nom. 





Covering a site with 6 inches of rock 
20160426 

From Carla: if a pump house that measures 20'x16' sits in the middle of a site that is 50'x60' and you apply crushed rock to the site at depth of 6", how many yards will you need to haul? Answered by Penny Nom. 





Filling a hole with 14 tons of rock 
20160408 

From Barry: If I had 14 tons of inch and one half sized aggregate rock, how large of a square or rectangular shaped hole would I need to hold that amount? Answered by Penny Nom. 





Maximizing the area of a two lot region 
20160403 

From yousef: A man wishes to enclose two separate lots with 300m of fencing. One lot is a square and the other a rectangle whose length is twice its width. Find the dimensions of each lot if the total area is to be a minimum. Answered by Penny Nom. 





Solve for x and y 
20160227 

From ntshidi: Y=1/2x+4and1/4x6 Answered by Penny Nom. 





A bonus of 8% of sales 
20160222 

From Barghavi: A man earns $325 per week plus an additional 8% on any sales over $500.
If he makes $6,250 in sales, how much money in total does he earn that week?
thank you! Answered by Penny Nom. 





Two equations in two unknowns 
20160213 

From Anumba: 4x + 2y = 4
7x  y = 11 Answered by Penny Nom. 





A chord of a circle, the central angle and the radius 
20160126 

From Nishan: If chord length is given along with angle then how to calculate the radius. Answered by Penny Nom. 





A system of linear equations 
20160124 

From kareem: my name is kareem
and i am a student i have a math puzzle and i tried to solved it but it always have same mistake
xy=9
x+z=12
zn=14
y+n=2 Answered by Penny Nom. 





A Max/Min problem with an unknown constant 
20160117 

From Guido: Question:
The deflection D of a particular beam of length L is
D = 2x^4  5Lx^3 + 3L^2x^2
where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection. Answered by Penny Nom. 





Integration of dx/(x^2+1)^3 
20160107 

From Ishank: Integration of dx/(x^2+1)^3 Answered by Penny Nom. 





A relative maximum and a relative minimum 
20151228 

From kemelo: show for the following function f(x)=x+1/x has its min value greater than its max value Answered by Penny Nom. 





Constructing simultaneous linear equations 
20151228 

From Deborah: Linda thinks of a twodigit number. The sum of the digits is 8. If she reverses the digits, the new number is 36 greater than her original number. What was Linda's original number?
Thank you! Answered by Penny Nom. 





A bus trip 
20151215 

From Michelle: I do not know what equation(s) should be used in order to obtain the answer(s) to this word problem.
A bus was rented for a trip, the cost was dived equally amongst the passengers. During the trip, someone mentioned that if there were 9 more passengers, they would have paid each 5$ less. Another mentioned that if there were 6 less passengers, they would have each paid 5$ more. What is the cost of renting the bus and how much must each passenger pay. Answered by Penny Nom. 





Linear equations in two variables 
20151213 

From priya: I have problem in solving these equations please help me today itself very urgent:
I)2x+y=y
II)pie*x+y=9 Answered by Penny Nom. 





A tangent line to a parabola 
20151202 

From pei: Given that the line y=mx5 is a tangent to the curve y=2x^2+3 find the positive value of M. Answered by Penny Nom. 





The diagonal of a rhombus 
20151114 

From Om: In a rhombus ABCD, angle A=60° and side AB=6 cm. Then diagonal BD is ? Answered by Penny Nom. 





2.236... 
20151013 

From Ann: 2.236...
What is the most specific category of numbers does this fall into? Rational or Irrational? Does the .... mean that it repeats? Answered by Harley Weston. 





Why is the area of square not conserved when it changes to a rhombus? 
20150628 

From shubham: Why is the area of square not conserved when it changes to a rhombus, both have equal sides still rhombus have less area than square.?? Answered by Penny Nom. 





1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 x 0 + 1 = ? 
20150618 

From Sharon: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 x 0 + 1 = ?
I got 1 as my answer despite BODMAS making it 12 because logic tells
me I ought to place brackets around the first set of repeated addition. Could you
please clarify this for me? Thank you 😊 Answered by Harley Weston. 





A tangent to y = x^3 
20150531 

From Brayden: Show that a tangent line drawn to the curve y=x^3 at the point (d,f (d)), where d>0, forms a right triangle with the x and y axes in quadrant 4 whose area is (2/3)d^4. Answered by Penny Nom. 





Two lorries approaching an intersection 
20150515 

From Nuraini: Two straight roads intersect at the right angles. Lorry A, moving on one of the roads,
Approaches the intersection at 50mi/h and lorry B, moving on the other roads, approaches the intersection at 20mi/h.
At what rate is the distance between the lorry changing when A is 0.4 mile from the intersection and B is 0.3 mile from the intersection? Answered by Penny Nom. 





A calculus optimization problem 
20150514 

From Ali: Given an elliptical piece of cardboard defined by (x^2)/4 + (y^2)/4 = 1. How much of the cardboard is wasted after the largest rectangle (that can be inscribed inside the ellipse) is cut out? Answered by Robert Dawson. 





The number of possible musical notes using an nkey instrument 
20150504 

From Farihin: Lets say that i have keys, and each key is for notes of a musical instrument,
So i wanted to find out the number of notes i can get for a certain number keys,
of course in the form of an equation. Notes can use as many keys, it can use 1, or 2, or 3, or even 100.
Notes in real life is not as such, but ignore reality.
I tried doing this but i can't seem to find a formula for it.
For example, i have 4 keys, say A, B, C, and D.
so, for notes that uses one key are 4, which is A, B, C, and D themselves.
for notes that uses two keys are 6,
AB, AC, AD, BC, BD and CD.
for notes that uses three keys are 4,
ABC, ABD, ACD and BCD.
lastly for notes that uses all four keys is 1, ABCD.
So, the total will be 4+6+4+1=15#
The nth term for the first equation is n, the second is [(n^2)n]/2
the third and the fourth, i don't know but the final answer should be like,
n + [(n^2)n]/2 + [3rd] + [4th]
Sorry for the long question though... Answered by Penny Nom. 





The method of elimination 
20150501 

From oreanna:
Question from oreanna, a student:
How do u solve 2x+9y=3
7x4y=25 in elimination Answered by Penny Nom. 





The volume of a sphere 
20150430 

From Cassidy: How do you find the radius of a sphere that has volume 36pI? Answered by Penny Nom. 





Constructing a box of maximum volume 
20150414 

From Margot: I need to do a PA for maths and I'm a bit stuck.
The PA is about folding a box with a volume that is as big as possible. The first few questions where really easy but then this one came up.
8. Prove by differentiating that the formula at 7 does indeed give you the maximum volume for each value of z. Answered by Penny Nom. 





A word problem with fractions 
20150409 

From Lorraine: If the numerator of a certain fraction is doubled and the denominator is increased by 1, the fraction becomes 1/2.
If the numerator of the original faction is squared and the denominator is decreased by 2, the fraction becomes equal to 1.
Let x be the numerator and let y be the denominator of the original fraction.
Write down two simultaneous equation in x and y.
Solve these equations to find two possible values for the given fraction. Answered by Penny Nom. 





The area of the ring between two concentric circles 
20150408 

From Conner: The area of the ring between two concentric circles is 25pi/2 square inches. The length of a chord of the larger circle tangent to the smaller circle is? Answered by Robert Dawson. 





Is a rhombus a square? 
20150324 

From Justin: Is a rhombus a square? Answered by Penny Nom. 





Extraneous solutions 
20150307 

From Emily: I have a question about Extraneous Solutions, Because I was recently researching to figure out
on how to determine that a solution is extraneous and many of the answers talked about how if a answer is negative
that it should always be extraneous but now I found out that positive solutions can also be considered as extraneous solutions
so that is why I am really confused Extraneous Solution's.
I would really appreciate it if you could clearly explain to me how to determine an extraneous solution from a normal solution.
Thanks,
Emily Answered by Chris Fisher. 





The radius of a cylinder 
20150226 

From Rose: Hi. I want to know how to find the radius of cylinder.
When I have height (35cm) and area of the curved surface(880sq.cm).
I know the formula is 2πr(h+r). But I can't understand how to find it's radius.
Please help me. I really need your help. Answered by Penny Nom. 





Two equations with fractions 
20150226 

From Pulane: Hi math centre I've been trying to solve these equations for days now please help
(6/x)(1/y)=4
(9/x)+1=(2/y)
Please help me solve them simultaneously
Thank you Answered by Penny Nom. 





f(x)=(x^21)/(x1) 
20150221 

From Ahmed: Is f(x)=[(x^21)/(x1) and x=2 at x=1] differentiable at x=1 ? Why ? Answered by Penny Nom. 





Two equations 
20150216 

From nigel: 2x+1/2y=1
6x3/2y=21 Answered by Penny Nom. 





The center and radius of a circle 
20150206 

From ariana: I need to find the center and the radius of this circle. I don't know how to put 9/2
than square it into a fraction.
x^2 + 2x + y^2 +9y=5 Answered by Penny Nom. 





128/(16)/(2) 
20150128 

From jackie: 128/(16)/(2) I was wondering if you can show me how to work this question out Answered by Harley Weston. 





Is a square a trapezoid? 
20150126 

From Katie: Can a trapezoid sometimes be a square? Answered by Penny Nom. 





Rates, percentages and units 
20141230 

From Kenneth: Hello:
If percentages have no units, why are some percentages called rates, as in interest rate, or
perhaps a tax rate of 7% as an example? A rate has units of different quantities.
I thank you for your reply. Answered by Robert Dawson. 





4 card hands 
20141102 

From Ronaldo: How many Mus hands (combination of 4 cards allowing repetitions)
can be made from a deck containing 8 types of cards?
RRRR, RRRC, RRRS, RRR7, RRR6, RRR5, RRR4, RRRA,
RRCC, RRCS, RRC7, RRC^, RRC5, RRC4, RRCA,
RRSS, RRS7, etc.
I think 330. If so, what is the formula? Answered by Penny Nom. 





A linear system 
20141014 

From pheter: 4/x  1/y = 3 .... equation (1)
6/x  2/y = 5 .... equation (2) Answered by Penny Nom. 





How does pir^2 = 1/4pid^2? 
20141014 

From al: Hi I cant work out the algebra. How does pir^2 = 1/4pid^2 Thanx Answered by Penny Nom. 





Two equations in x and y 
20140925 

From seyilogo: solve y=2x  3 and (4x  2y) / x + y = 1 simultaneously Answered by Harley Weston. 





Continuity on a closed interval 
20140921 

From Pragya: The trouble I'm having is as follows :
a continuous function is most of the times defined on a closed interval,
but how is it possible to define it on a closed interval ,because to be continuous at endpoints of the interval the function's
limit must exist at that endpoint,for which it has to be defined in its neighborhood,but we don't know anything about whether the function is always defined in the neighborhood.
Please help... Answered by Penny Nom. 





Two equations in two variables 
20140918 

From Susan: (28x + 36y)  [20000  .75(28x + 36y) + 60000] = 5000
x + y = 10000
solving two equations involving variables Answered by Penny Nom. 





A tangent to a curve passing through a point not on the graph 
20140915 

From Aquilah: For the curve y = x2 + 3x, find the equations of all tangent lines for this graph
that also go through the point (3, 14). Answered by Penny Nom. 





Water usage in Ames 
20140829 

From Abigail: The City of Ames water treatment plant treats 7 million gallons of water per day on
average to meet water demands. Hypothetically, if water is withdrawn from a
groundwater system with an area of 51.6 square miles
(approximate area of Ames and surrounding area),
what is the depth of water in inches that would have to be added to
the groundwater system annually to replace the water that is withdrawn by the city? Answered by Penny Nom. 





A car passing a bus 
20140824 

From Athish: the driver of the car is travelling at 36 km/hr and spots a bus 80 m ahead of him after 1 hr the bus is 120 m behind the car what is the speed of the bus Answered by Penny Nom. 





Filling three holes with stones 
20140820 

From mark: how many tonnes of hardcore/crushed stone would it take to fill
1 hole 9ft diameter 5ft deep
and 2 holes both 3ft diameter and 5ft deep Answered by Penny Nom. 





The sum of the first 50 terms of an arithmetic progression 
20140726 

From Joshua: Hello ...my is Joshua...I'm a grade 11 student...I got a question
Calculate the sum of the first 50 terms of an arithmetic progression: 112:98:84 Answered by Penny Nom. 





The method of elimination 
20140705 

From leo: please explain how can i solve this problem
3x6y=38
6x9y=44
using elimination and simultaneous method thank you :) Answered by Penny Nom. 





Differentiate ln[x(2x4)^1/2] 
20140628 

From Igwe: If y=In[x(2x4)^1/2],find dy/dx at x=3 Answered by Penny Nom. 





Simultaneous equations 
20140620 

From rana: solve the simultaneous equations
a)3x=7y
12y=5x1 Answered by Penny Nom. 





The derivative of sin(x) 
20140426 

From Lucky: f(x)=Sin(x), by first principle its f'(x)...show me how to solve such problem. Answered by Penny Nom. 





Simultaneous equations with fractions 
20140419 

From Maryam: I looked at your example of simultaneous equations with fractions and applied it to my question from an educate exam papers but I couldn't get it to work. The question is:
x/8  y = 5/2
3x + y/3 = 13 Answered by Penny Nom. 





A frustum of a pyramid with a square base 
20140418 

From tuba: a pyramid has a base of 10 m and is 15 m high.what is the volume? if 6m is removed from top what is the volume of the remaining frustum? Answered by Penny Nom. 





The area bounded by the Xaxis and y=x^(2)4 from 5 to 0 
20140415 

From Lexie: Determine the area that is bounded by the following curve and the xaxis on the interval below. (Round your answer to three decimal places)
y=x^(2)4, 5 ≤ x ≤ 0
The answer is 32.333 but I have no idea how to get there. Answered by Penny Nom. 





A tangent of the curve (x/a)^n+(y/b)^n =2 
20140415 

From sudhir: the equation of tangent of the curve (x/a)^n+(y/b)^n =2. at(a,b) is Answered by Penny Nom. 





The locus of a point 
20140404 

From srishti: A point P moves such that the difference between its distance from the origin and from the axis of x is always a constant c . what is the locus of the point? Answered by Penny Nom. 





A cable around the Earth 
20140313 

From fikile: By how much must an equatorial cable be extended in order that it runs 1meter above the ground? Answered by Penny Nom. 





A parabolic suspension bridge 
20140311 

From jeffrey: the towers of a parabolic suspension bridges 200 meter long are 40 meter high and the lowest point of the cable is 10 meter above the roadway.Find the vertical distance from the roadway to the cable at 50 meter from the center. Answered by Penny Nom. 





What are the possible lengths of the hypotenuse? 
20140307 

From audrey: The three sides of a right angles triangle measure x2, x+5, and 2x1 in length.
What are the possible lengths of the hypotenuse?
... I'm doing the equation c2=a2+b2 and subbing in the numbers but nothing makes sense Answered by Penny Nom. 





Simultaneous fractional equations 
20140215 

From benjamin: hi math central. benjamin here. during class, i had problem with this topic. normally i wont have problems with math but this topic i just too hard for me. please help i am having exam and test next week on this topic
here is the question:
using substitution method, solve the simultaneous equation.
(x+1)/(y+2)=0.5
(x2)/(y1)=1/3 Answered by Penny Nom. 





The volume of a frustum 
20140202 

From mike: volume of frustum R23", r 18", h 16" Answered by Penny Nom. 





Conics 
20140201 

From Kassidy: Hey, I have searched through all the questions about conics and how
people use them in the real world, but none of them were very specific
on how they are applied and the process, why it's so important etc.
I have a project due asking these questions and it's been very difficult
finding the right answer, if you could name jobs, how they are use and
specifically applied that would be greatly appreciated. Answered by Penny Nom. 





Two nonlinear equations 
20140126 

From Naryn: (1÷x) + (1÷y) = (7÷12)
xy = 12 Answered by Penny Nom. 





An inequality 
20140125 

From LANELL: this is a problem to solve: 1/3 + 2/7 >=x/21  part of the answer is (oo)
not exactly that similarit is on a calculator as a symbol sure you know what it is I am talking about the x will be a number Answered by Penny Nom. 





25% profit 
20140102 

From Finn: Hello,
The question is all about buyandsell business.
Problem:
Pencil  $6 for whole sale price
$8 if I sell the item
How do I get the 25% profit? (you can change the whole sale price and the retail price[if i sell the item])
if I buy the pencil at 24 pieces and sell it at 24 pieces. Answered by Penny Nom. 





The popcorn box problem 
20131107 

From Dave: We know that calculus can be used to maximise the volume of the tray created when cutting squares from 4corners of a sheet of card and then folding up.
What I want is to find the sizes of card that lead to integer solutions for the size of the cutout, the paper size must also be integer. EG 14,32 cutout 3 maximises volume as does 13,48 cutout 3.
I have done this in Excel but would like a general solution and one that does not involve multiples of the first occurence, as 16, 10 cutout 2 is a multiple of 8,5 cutout 1. Answered by Walter Whiteley. 





Substitution type simultaneous equations 
20131103 

From Kayla: I am having problems with substitution type simultaneous equations, when the variable you are substituting is a algebraic one:
y=x^23x+4 and 3x2y=1
I have rearranged 3x2y=1 to get x=(1+2y)/3 but when I substitute this x value into the other equation, I get the wrong answers!
Would appreciate any help! Thank you. Answered by Penny Nom. 





Water flowing out of a tank 
20131103 

From Carolyn: The flow of water out of a hole in a tank is known to be proportional to the square root of the height of water above the hole.
That is,
dV/dt (proportional to) sq root (h)
The tank has a constant crosssectional area A, show that the height of water in the tank is given by
h = ((kt+C)/2)^2
If the tank is 9 metres high, and it takes 5 hours for it to drain from full to half full,
how much longer will we have to wait until it is completely empty? Answered by Penny Nom. 





Extraneous solutions 
20131022 

From tom: i need an equation where x=2 is the correct answer and x=3 is an extraneous solution. can you provide me with such an equation?? Answered by Harley Weston. 





A frustum 
20131012 

From Lily: A cone of height 6in. and radius of base 4in. has its top cut off by a plane parallel to its base and 4in from it.
Find the volume of the remaining frustum.
I have worked out the volume of the entire cone but I don't know how to work out the radius of the top of the frustum.
Thanks Answered by Penny Nom. 





Proportional rates 
20131010 

From Varsha: A province's Ministry of Social services has found that both the number of people needing social assistance and the province's total expenditures on social assistance are proportional to the rate of unemployment. Last August when the provincial unemployment rate was 8.4 %, the province provided assistance to 89,300 individuals at a total cost of 4107.4 million. The forecast unemployment rate for next August is 7.9%. How many people can the province expect to need social assistance next August? What amount should the province budget for social assistance in August? Answered by Penny Nom. 





Maximize the volume of a cone 
20131009 

From Conlan: Hi I am dong calculus at school and I'm stumped by this question:
A cone has a slant length of 30cm. Calculate the height, h, of the cone
if the volume is to be a maximum.
If anyone can help me it would be greatly appreciated.
thanks. Answered by Penny Nom. 





The sum of all whole numbers from 1 to X 
20130906 

From Tim: How do I develop a rule for the sum of all whole numbers from 1 to X when I have no idea how to do this Answered by Penny Nom. 





Ordering crushed stone 
20130903 

From Prakash: Dear Sirs,
I am working in a Soft Landscaping contracting company. If I need to purchase crushed stone with the size 5070mm for $53,429 m^2$ area, how many 20feet containers should I need to order to my suppliers? The 20foot container has internal dimensions 5,897 mm by 2,348 mm by 2,285 mm and the $53,429 m^2$ area is to be covered by 10 cm of stone. Answered by Harley Weston. 





Equal ordinate and abscissa 
20130815 

From sonit: the slope of tangent to the curve y=(4x^2)^1/2 at the point, where the ordinate and abscissa are equal, is Answered by Penny Nom. 





Differentiate x^x  2^sinx 
20130809 

From tarun: derivative of x^x  2^sinx Answered by Penny Nom. 





Practical uses of trigonometry 
20130806 

From tharindu: use of trigonometry Answered by Penny Nom. 





What is the value of 2((i)^(1/2))? 
20130722 

From Delilah: What is the value of 2((i)^(1/2)) ?
i.e. absolute value of 2 multiplied by square root of i. Answered by Penny Nom. 





Simultaneous equations 
20130710 

From Warren: solve this simultaneous equation: xy=4
2x+3y=14 Answered by Penny Nom. 





Water use in a rectangular flush tank 
20130510 

From milo: A rectangular flush tank 22" by 71/4 contains water to depth of 17" how many gallons of water will be saved if a conservation device reduces the capacity to 3/5 of this amount? And reduced to the nearest tenth Answered by Penny Nom. 





A cyclic rhombus 
20130416 

From Marisa: I know that the only rhombus that can be inscribed in a circle is a square, but why is that? I've been racking my brain and the internet for solutions, but have found no logical explainations in relation to the arc degrees and angles. Please help. Answered by Chris Fisher. 





4 linear equations with 3 unknowns 
20130412 

From Marian: how to solve for 3 unknowns in 4 simultaneous equations Answered by Penny Nom. 





Simultaneous equations with fractions 
20130331 

From Terence: 5/x6/y=1
17/x+30/y=16
I been spending whole day to solve this question. Would be very grateful if you can help I try
The denominator value is a equations term which make is simultaneous equations so hard. Answered by Penny Nom. 





Tangents to the curve y = x^3 
20130324 

From Ethan: How many tangent lines to the curve y = x^33 pass through the
point (2, 4)? For each such line, and the exact coordinates of the point of
tangency on the curve. Answered by Penny Nom. 





Extraneous solutions 
20130218 

From Eileen: (5x+4)^1/23x=0 Answered by Penny Nom. 





Related rates 
20130217 

From Ishaak: A hemispherical bowl is filled with water at a uniform rate. When the height of water is h cm the volume is π(rh^21/3 h^3 )cm^3, where r s the radius. Find the rate at which the water level is rising when it is half way to the top, given that r = 6 and the bowl fills in 1 minute. Answered by Penny Nom. 





The continuity of f(x,y)=ln(x^2+y^2) 
20130217 

From anu: the question says we have to find the points in the plane where the function is continuous:
f(x,y)=ln(x^2+y^2) . here we aren't given a particular point (x,y) where we have to check a function's
continuity.
what is to be done if we have to check continuity over the whole domain of the function?
please help . Answered by Harley Weston. 





A word problem involving toys 
20130214 

From sandy: Each boy gets 5 toys.Each girl gets 3 toys.There are 150 pupils.
The boys had 74 more toys than girls.
How many boys?
How many girls? Answered by Penny Nom. 





Simultaneous equations 
20130210 

From Michael: 2P + 1/3V =8
3P  2/V=5 Answered by Penny Nom. 





Integration from 0 to 2pi of 1/(3cos x + 2) dx 
20130204 

From ankit: Integration from 0 to 2pi of 1/(3cos x + 2) dx Answered by Harley Weston. 





Maximize profit 
20130119 

From Chris: A firm has the following total revenue and total cost function.
TR=100x2x^2
TC=1/3x^35x^2+30x
Where x=output
Find the output level to minimize profit and the level of profit achieved at this output. Answered by Penny Nom. 





5 1/2 cubic feet of sawdust 
20130119 

From Vina: I have a bale of sawdust that containes 5 1/2 cubic feet. How many will I need to make a cubic yard? Answered by Penny Nom. 





A triangular island 
20121229 

From Udit: A long time ago Mr Gibson found an island shaped as a triangle with three straight shores of length 3 km,4 km and 5 km. He declared an 'exclusion zone' around his island and forbade anyone to come within 1 km of his shore. What was the area of his exclusion zone? Answered by Penny Nom. 





An integral 
20121216 

From Slavena: integration of (lnx)^2 / x dx Answered by Penny Nom. 





An area bounded by lines 
20121216 

From sidra: find area bounded by functions:
y=x
y=2x
and y=5x Answered by Penny Nom. 





A max/min problem 
20121214 

From bailey: A right angled triangle OPQ is drawn as shown where O is at (0,0).
P is a point on the parabola y = ax – x^2
and Q is on the xaxis.
Show that the maximum possible area for the triangle OPQ is (2a^3)/(27) Answered by Penny Nom. 





The derivative of y = sin (30º + x) 
20121107 

From Saskia: derivative of y = sin (30º + x) Answered by Harley Weston. 





An implicit differentiation problem 
20121026 

From Katie: find y' of x^2y2y^3=3x+2y Answered by Harley Weston. 





How fast is the distance between the aircraft and the car increasing? 
20121024 

From Steven: At a certain instant an aircraft flying due east at 240 miles per hour passes directly over a car traveling due southeast at 60 miles per hour on a straight, level road. If the aircraft is flying at an altitude of .5mile, how fast is the distance between the aircraft and the car increasing 36 seconds after the aircraft passes directly over the car? Answered by Penny Nom. 





A label to cover a plastic cup 
20121023 

From Kevin: I'm trying to make a label to cover the entire outer area or a plastic cup. I know there must be a way to figure out the dimensions needed, but I can't seem to figure it out. The circumference of the bottom of the cup is 21.4cm and the circumference at the top of the cup is 29.8cm. The cup is 14.5cm tall. What should the height of the arc from the plane connecting the two ends of the 21.4cm arc. I attached a diagram where x is the value I'm looking for. I'm guessing there is some simple relationship between the length of a line and the arc needed to turn that line into a perfect circle, but I don't know what it is. Can you figure this out and share it with me? Thanks.
Kevin Answered by Penny Nom. 





Differentiation rules 
20121023 

From Morgan: Use the derivative rules to differentiate each of the following:
1. f(x)=1/x1 2. f(x)= sqrt(x) Answered by Penny Nom. 





A word problem involving a fraction 
20121012 

From Derrick: If the numerator and denominator of a fraction are both decreased by 1 the fraction becomes 2/3. If the numerator and denominator are both increased by 1 the fraction will be 3/4. Find the original fraction. How to do? Answered by Penny Nom. 





The hypotenuse 
20120906 

From Jeevan: how can i find the height and base of a right angle triangle if i have the hypotenuse only ? Answered by Penny Nom. 





A tangent to f(x) = 1/x 
20120904 

From Steven: Consider the graph of the function f(x) = 1/x in the first quadrant, and a line tangent to f at a point P where x = k. Find the slop of the line tangent to f at x = k in terms of k and write an equation for the tangent line l in terms of k. Answered by Penny Nom. 





The length and depth of a chord 
20120816 

From Tim: Can I find the radius of a circle, if I have a chord of 2400mm, and a depth
between the center of the chord and the perimeter of the circle of 150mm ?
Thanks
Tim Answered by Penny Nom. 





Fence post holes 
20120719 

From Gerry: Hello, I'm digging 30 8" dia holes, 5 ft deep for fence posts that are 4"x 4" Can you please help me figure out how much stone dust I should order for all 30 holes. Thanks Answered by Penny Nom. 





A volume of revolution 
20120715 

From Tewodros: Let f(x) = e^x and g(x) = x^1/2 both be defined on [0,1]. Consider the region bounded by f(x), g(x), x = 0, x = 1. Rotate this region about the yaxis and determine the volume using the shell method. Answered by Harley Weston. 





Two cars approach a rightangled intersection 
20120410 

From Michael: Two cars approach a rightangled intersection, one traveling south a 40km/h and the other west at 70km/h.
When the faster car is 4km from the intersection and the other case if 3km from the intersection,
how fast is the distance between the car cars changing? Answered by Penny Nom. 





A maximization problem 
20120409 

From Nancy: After an injection, the concentration of drug in a muscle varies according to a function of time, f(t). Suppose that t is measured in hours and f(t)=e^0.02t  e^0.42t. Determine the time when the maximum concentration of drug occurs. Answered by Penny Nom. 





The spread of a rumor 
20120409 

From Roohi: The function f(t) = a/(1+3e^(bt)) has also been used to model the spread of a rumor. Suppose that a= 70 and b=3 0.2. Compute f(2), the percentage of the population that has heard the rumor after 2 hours. Compute f'(2) and describe what it represents. Compute lim t approaches infinity and describe what it represents. Answered by Penny Nom. 





The period T of a pendulum 
20120327 

From Ashley: The period T of a pendulum is given in terms of its length, l, by T=2pi sqrt(l/g) where g is the acceleration due to gravity(a constant)
a. find dT/dl
b. what is the sign of dT/dl
c. what does the sign of dT/dl tell you about the period of the pendulums? Answered by Penny Nom. 





The derivative of x^(1/2) 
20120114 

From Eric: I have an problem figuring out the derivative of the negative square root of x i.e. x^(1/2) using the first principle.
Could someone please show me?
Thanks in advance! Answered by Harley Weston. 





Lost in the woods 
20120112 

From Liz: I am lost in the woods. I believe that I am in the woods 3 miles from a straight road. My car is located 6 miles down the road. I can walk 2miles/hour in the woods and 4 miles/hour along the road. To minimize the time needed to walk to my car, what point on the road should i walk to? Answered by Harley Weston. 





The radius of a circle 
20120112 

From Janie: Find the radius of a circle knowing that a chord of 24.6 inches has a corresponding arc of 70°. Answered by Penny Nom. 





A volume of revolution 
20120111 

From john: find volume of solid generated by revolving the region in the first quadrant bounded by the curve y squared=x cubed, the line x=4 and the xaxis about the line y=8. The answer in the back of the book is 704 pi divided by5 Answered by Penny Nom. 





A wire spiral 
20120107 

From Pinar: I am trying to help my daughter with very challenging maths questions which sometimes I get stuck
If someone help me with one of them I would appreciate it.
Jane is making a spiral out of wire. She bends the wire after 1 cm, then bends the wire
after 2cm, then 3cm and continues in this manner. After 4 bends she used 15 cm of wire
For each bend how many cm wire was used? After 6 bends how many cm wire will she have used?
How many bends will she have made if she uses 66 cm of wire?
I would appreciate if somebody would help me with this.
Thank you!
Pinar Answered by Penny Nom. 





The volume of a frustum of a cone 
20111224 

From CV: If I know the height, taper, and volume of a frustum cone,
what are its Radii?
Calculating frustum cone volume is straight forward.
v=Pi/3*h*(R*R+r*r+R*r) where;
v=volume;
h=frustum height;
R=major radius;
r=minor radius;
Pi=3.1415926;
t=taper, ('slant angle' where t=0 is a disk)
Here is the tricky part.
Knowing 'h', 't', & 'v';
Calculate one or both of 'R' & 'r'.
Specifically, what is the formula for 'R=' or 'r='? Answered by Penny Nom. 





Water is flowing into a cup 
20111219 

From Tim: A cup has a radius of 2" at the bottom and 6" on the top. It is 10" high. 4 Minutes ago, water started pouring at 10 cubic " per minute. How fast was the water level rising 4 minutes ago? How fast is the water level rising now? What will the rate be when the glass is full? Answered by Penny Nom. 





The circumference and area of a circle 
20111213 

From Mable: A circle that going 70mi across using 22/7 I need the area,radius, and the circumference
and how to set up the steps can you help? Answered by Penny Nom. 





A cube of ice is melting 
20111205 

From Emily: a cube of ice (i.e.) each side is of the same length) is melting at a rate such that the length of each side is decreasing at a rate of 5cm per hour. how fast is the volume of the cube decreasing (in cubic cm per hour) at the instant the length of each side is 25cm? Answered by Penny Nom. 





A suspension bridge 
20111130 

From jennifer: suspension bridges like the golden gate bridge, are used to span large distances.
when the main curved cables are attached to the deck by vertical cables they will end
up in the shape of parabola. assume that we need to build a bridge that spans
2,400 feet. the two towers 165feet tall each where placed at 400feet from either
side. the lowest point of the center of the bridge at 10feet. vertical suspension cables
where placed at 25foot interval. how many feet of cable are needed to connect
the deck to the main cables between the two towers? show all working. Answered by Penny Nom. 





Four carpenters can build eight houses in 10 days. 
20111123 

From Kenneth: Four carpenters can build eight houses in 10 days.
Two carpenters can build how many houses in 15 days? Answered by Penny Nom. 





Water pouring into a conical tank 
20111121 

From Patience: Hi my name is patience and I'm having a problem with this question.
Water pours into a conical tank of semi vertical angle 30 degrees at the rate of 4 cm^3/s, where h is the depth of the water at time t. At what rate is the water rising in the tank when h = 10 cm?
Thank you Answered by Penny Nom. 





Lines tangent to y^2=4x 
20111111 

From Reuchen: Find equations of the lines tangent to y^2=4x and containing (2,1). Answered by Penny Nom. 





A scale drawing 
20111030 

From aretha: i have a scale drawing of a house floor plan, don't understand how
to find the answer. the drawing is 1in: 3ft. need to find the length /
width/and the actual length/width of the living room,kitchen, 3bedrooms
deck,sitting room, entry, and closet Answered by Penny Nom. 





A spherical ball in a conical wine glass 
20111026 

From Jules: A heavy spherical ball is lowered carefully into a full conical wine
glass whose depth is h and whose generating angle (between the axis
and a generator) is w. Show that the greatest overflow occurs when the
radius of the ball is (h*sin(w))/(sin(w)+cos(2w)). Answered by Claude Tardif. 





Implicit differentiation 
20111020 

From Monica: Find dy/dx in terms of x and y, if sin(xy)=(x^2)y. Answered by Penny Nom. 





One central circle and three tangent circles 
20111016 

From Margaret: You have one central circle and three or more circles tangent to the outside of the circle of varying radii. You know the x,y coordinates of the centers of the other circles. If you now remove that central circle (and pretend you never knew where it was), can you calculate its center in x,y coordinates? Answered by Chris Fisher. 





Building a custom range hood 
20111008 

From Bill: I'm building a custom range hood for a customer with special order material that matches
their newly installed cabinets and I need it to be perfect. The hood is basically a pyramid
but the 4th side is the flat wall at the back and a flat, rectangular top. I need to calculate
the bevel and miter of the three sides but I never was very good with geometry functions
(although I am fairly good with other math fields). I either need the calculations from you
at least (shudder) a formula or set of formulas so that I can calculate them myself. Answered by Harley Weston. 





A hemispherical bowl with a lead ball inside 
20110927 

From Jean: "(a) Water is being poured into a hemispherical bowl of radius 3 inch
at the rate of 1 inch^3/s. How fast is the water level rising when the
water is 1 inch deep ?
(b) In (a), suppose that the bowl contains a lead ball 2 inch in
diameter, and find how fast the water level is rising when the ball is
half submerged." Answered by Penny Nom. 





The derivative of f(x) = (x+1)^1/2 
20110905 

From Carla: Find the derivative using the limit process of
f(x) = (x+1)^1/2 Answered by Harley Weston. 





The height of a fluid in a horizontal tank 
20110724 

From jason: Same set up as many others, cylindrical tank on its side, but I am interested in defining the change in volume and/or fluid level as a function of time at a constant volumetric outflow. I plan on hooking a pump to the tank so "gpms' will be constant. I have a couple different sized tanks and pumps so I want a general equation. Thanks for your help. Answered by Harley Weston. 





A line tangent to f(x)=1/x 
20110605 

From Michael: A line tangent to f(x)=1/x in the first quadrant creates a right triangle
with legs the xaxis and the yaxis. Prove that this triangle is always
2 square units regardless of where the point of tangency is. Answered by Penny Nom. 





An exclusion zone around a triangle II 
20110503 

From Aishwarya: A long time ago Mr gibson found an island shaped as a triangle with three straight shores of length 3km, 4km and 5km. He declared an exclusion zone around his island and forbade anyone to come within 1km of his shore. What was the area of his exclusion zone?
This question was answered but did not understand the explanation. Answered by Robert Dawson. 





The radius of a cylinder 
20110427 

From Jazmin: Hi, I don't understand how to find the radius in a cylinder with only the surface area (143.7) and the height (0.8)? I know that the formula is 2pir2+2pirh, but I don't see how to isolate the r? I appreciate your help. Answered by Penny Nom. 





An antiderivative of the square root of (8t + 3) 
20110419 

From Caitlyn: I know how to take an antiderivative. But this one's stumping me. I need it to finish a problem.
What's the antiderivative of the square root of (8t + 3)
~Caitlyn= Answered by Penny Nom. 





Eliminate y 
20110407 

From Lynn: 2x + y = 8
y + 3z =5
z + 2w =1
5w + 3x = 9
Form three equations with y eliminated Answered by Penny Nom. 





Designing a tin can 
20110331 

From Tina: A tin can is to have a given capacity. Find the ratio of the height to diameter if the amount of tin ( total surface area) is a minimum. Answered by Penny Nom. 





A stone is dropped into a lake 
20110324 

From AnneMarie: A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s. Find the rate at which the area within the circle is increasing after 4s. Answered by Penny Nom. 





At what rate is the grain pouring from the chute? 
20110226 

From MJ: Suppose that grain pouring from a chute forms a conical heap in such a way that the height is always 2/3 the radius of the base. At the moment when the conical heap is 3 m high, its height is rising at the rate of 1/2 m/min. At what rate (in m^3/min) is the grain pouring from the chute? Answered by Penny Nom. 





Mathematics and a musical dilemma 
20110119 

From rahul: how is mathematics applied in entertainment? Answered by Harley Weston. 





Integrating ln^3x/x 
20110114 

From ken: y=ln^3x/x from x=1 to x=11 Answered by Penny Nom. 





How do I prove that the quadrilateral is a Rhombus? 
20101216 

From Matthew: Quadrilateral KLMN has vertices K(2,3), L(7,3), M(4,7) and N(1,7). How do I prove that the quadrilateral is a Rhombus?? Answered by Robert Dawson and Penny Nom. 





Simultaneous equations 
20101205 

From ryan:
Question from ryan, a student:
3 4
   = 1 (1)
x y
7 2 11
   =  (2)
x y 12 Answered by Chris Fisher and Stephen La Rocque. 





A 400 gallon drum 
20101104 

From Jerry: Question from Jerry:
I want to build A tank. able to hold 400 gallons of asphalt sealer it will be round . basically like A old fuel oil drum but made with heavy material . math ? was I want tank to hold 400 gallons I think the length of 60" would be perfect. I don't know what diameter or radius of tank needs to be . to make it A 400 Gallon tank Answered by Harley Weston. 





The angles in an mgon and genrealizations 
20101016 

From Michael: Hello:
In answer to a student's question, someone named Penny from
your organization provided a proof that the sum of the interior
angles of a triangle in the plane is pi radians (or 180 degrees).
I am interested (and I'm sure many other people would be too) in
3 potential generalizations of this basic fact in plane geometry: Answered by Walter Whiteley. 





What is the maximum weekly profit? 
20101010 

From Joe: A local artist sells her portraits at the Eaton Mall.
Each portrait sells for $20 and she sells an average of 30 per week.
In order to increase her revenue, she wants to raise her price.
But she will lose one sale for every dollar increase in price.
If expenses are $10 per portrait, what price should be set to maximize the weekly profits?
What is the maximum weekly profit? Answered by Stephen La Rocque and Penny Nom. 





A Taylor polynomial for (lnx)/x 
20100929 

From Dave: I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, 3, 11, 50, 274... Ok, so the negative is an easy fix > (1)^n1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this?
Thanks! Answered by Chris Fisher. 





limit as x approaches a of ((x^(1/2))(a^(1/2)))/(xa)? 
20100929 

From emily: limit as x approaches a of ((x^(1/2))(a^(1/2)))/(xa)? Answered by Penny Nom. 





A limit 
20100927 

From norma: I have a problem like this one but I can get it right. please help me to answer
find the constant a such that the function is continuous on the entire line.
g(x)= {x^2  a^2 / xa if x is not = a
{6 if x = a Answered by Penny Nom. 





Continuity 
20100918 

From Carina: Hi. My name's Carina and I'm currently a sophomore in high school.
I'm having a lot of difficulties in AP Calculus with continuity,
onesided limits, and removable discontinuities. Basically, I have no
idea how to do them or even what they are. I read the lesson but I
still don't get it. Can someone put it in simpler terms so I can
understand how to complete my questions? Thank you! Answered by Robert Dawson. 





Maximizing the volume of a cylinder 
20100831 

From Haris: question: the cylinder below is to be made with 3000cm^2 of sheet metal. the aim of this assignment is to determine the dimensions (r and h) that would give the maximum volume.
how do i do this?
i have no idea. can you please send me a steptostep guide on how t do this?
thank you very much. Answered by Penny Nom. 





A max min problem 
20100819 

From Mark: a rectangular field is to be enclosed and divided into four equal lots by fences parallel to one of the side. A total of 10000 meters of fence are available .Find the area of the largest field that can be enclosed. Answered by Penny Nom. 





The suspension cables of a bridge 
20100729 

From Mike: what is the formula for the suspension cables of a bridge.
The towers are 200 ft above the roadway
The towers are 3400 ft apart
The cable if at 8ft in the middle of the span Answered by Robert Dawson. 





Maximize the floor area 
20100707 

From shirlyn: A rectangular building will be constructed on a lot in the form of a right triangle with legs
of 60 ft. and 80 ft. If the building has one side along the hypotenuse,
find its dimensions for maximum floor area. Answered by Penny Nom. 





A max/min problem 
20100612 

From valentin: What is the maximum area of an isosceles triangle with two side lengths equal to 5 and one side length equal to 2x, where 0 ≤ x ≤ 5? Answered by Harley Weston. 





The capilano suspension bridge 
20100603 

From nida: the capilano suspension bridge in north vancouver is the world's highest footbridge of its kind. the bridge is 140m long . from the ends of the bridge the angles of depression of a point on the river under the bridge are 41 degrees and 48 degrees. how high is the bridge above the river to the nearest metre Answered by Penny Nom. 





Integration of sin^3 (2x) 
20100529 

From ascher: how do you integrate this equation
∫ sin^3 (2x) dx Answered by Robert Dawson and Penny Nom. 





More on a truncated cone 
20100528 

From Mike:
Question from Mike, a parent:
I was reviewing this question and answer:
http://mathcentral.uregina.ca/QQ/database/QQ.02.06/phil1.html
But I have trouble with this part:
Now if we express the radius of the inside circle as r and the outside circle's radius is R, then this means r/R is 911/1728. But earlier we said that the outside radius R is simply w more than the inside radius r, so R = r + 282. That means that r/R = r/(r + 282). Now we can simply solve the equation for r:
r/(r+282) = 911/1728
This means r = 314 mm (with rounding).
Can I get more detail on the method to solve for r?
Thank you,
Mike Answered by Penny Nom. 





An optimization problem 
20100523 

From Marina: Hello, I have an optimization homework assignment and this question has me stumped..I don't even know A hiker finds herself in a forest 2 km from a long straight road. She wants to walk to her cabin 10 km away and also 2 km from the road. She can walk 8km/hr on the road but only 3km/hr in the forest. She decides to walk thru the forest to the road, along the road, and again thru the forest to her cabin. What angle theta would minimize the total time required for her to reach her cabin?
I'll do my best to copy the diagram here:
10km
Hiker_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Cabin
\  /
\  /
f \ 2km /
\  /
theta \___________________________ /
Road Answered by Penny Nom. 





The hypotenuse of a triangle 
20100522 

From linda: find the length of the hypotenuse of a triangle with legs of 12in. and 17in. round to the nearest hundredth Answered by Tyler Wood. 





Extraneous solutions 
20100522 

From Joe:
Question from Joe, a parent:
w+3 2w
   = 1
w21 w1
W2 is = w squared
4
The answer is w= 
3
but have no idea how this was solved. any help is appreciated. Thanks. Answered by Penny Nom. 





The rate of change of y with respect to x 
20100429 

From Tom: I just had a quick calc question about wording that wasn't ever
addressed in class. When the book says "the rate of change of y with
respect to x", should it be considered how fast y is changing in
comparison to x?
I ask because the textbook says that "y is changing 3 times faster than x,
so the rate of change of y with respect to x is 3." I'm use to rate being
like velocity, as in units of distance per units of time. All we're told
in class is that it's the slope of the tangent line, I was hoping you
could clarify for me what exactly is meant by the wording of a "rate of
change of something with respect to something else". More specifically, what
"rate" and "with respect to" mean within this context?
Thanks for your time Answered by Harley Weston. 





A rectangular garden 
20100425 

From Billy: Tanisha wants to make a rectangular garden with a perimeter of 38 feet. What is the greatest area possible that tanisha can make the garden? Answered by Penny Nom. 





Integrate the ((4th root of x^3)+1) dx 
20100412 

From Bridget: integrate the ((4th root of x^3)+1) dx Answered by Tyler Wood. 





The derivative of y=x^x 
20100409 

From David: So, its David, and I was wondering about the derivative of y=x^x. I have often seen it be shown as x^x(ln(x)+1), but when I did it through limits it turned out differently. Here's what I did:
It is commonly know that df(x)/dx of a function is also the limit as h>0 of f(x+h)f(x)/h.
To do this for x^x you have to start with lim h>0 ((x+h)^(x+h)x^x)/h. The binomial theorem then shows us that this is equal to lim h>0 (x^(x+h)+(x+h)x^(x+h1)h+...x^x)/h
This is also equal to lim a>0 lim h>0 (x^(x+a)+(x+h)x^(x+h1)h...x^x)/h.
Evaluating for a=0 you get lim h>0 (x^x+(x+h)x^(x+h1)h...x^x)/h
Seeing as the last 2 terms on the numerator cancel out you can simplify to a numerator with h's is each of the terms, which you can then divide by h to get:
lim h>0 (x+h)x^(x+h1)... which when evaluated for h=0 gives us: x(x^(x1)). This statement is also equal to x^x.
This contradicts the definition of the derivative of x^x that is commonly shown. So, my question is: can you find any flaws in the logic of that procedure? I do not want to be shown how to differentiate x^x implicitly because I already know how to do that. Answered by Robert Dawson. 





A max min problem 
20100406 

From Terry: The vertex of a right circular cone and the circular edge of its base lie on the surface of a sphere with a radius of 2m. Find the dimensions of the cone of maximum volume that can be inscribed in the sphere. Answered by Harley Weston. 





The derivative of cos^3x 
20100406 

From Erson: Find y' of the given function: y = cos^3x. Answered by Harley Weston. 





Sand falling off a conveyer 
20100402 

From Katherine: sand is falling off a conveyer onto a pile at the rate of 1.5 cubic feet per minute. The diameter of the base is approximately twice the altitude. At what rate is the height of the pile changing when it is 10 feet high? Answered by Penny Nom. 





A negative times a negative 
20100325 

From priya: why is minus into minus plus? Answered by Harley Weston. 





A 14 side well house cover 
20100312 

From Kenneth: I am 35 yr I am wanting to build a well house cover. I'm trying to figure out how long the pieces need to be and what angle they need to be for a 4 ft dia with 14 side well house. I would love an answer but would also like to know how to figure it in the future. Thanks Kd Answered by Harley Weston. 





The integral of X^3/the square root of 1x^2 dx 
20100307 

From William: The integral of X^3/the square root of 1x^2 dx. Answered by Harley Weston. 





The volume of a frusta of a hexagonal based pyramid 
20100304 

From sarah: Volume of a frusta of a hexagonal based pyramid Answered by Penny Nom. 





Lissajous curve 
20100303 

From Nikki: I'm interested in information about a particular mathematical figure. My memory is that it is called a "liciju figure", but obviously my spelling of this is incorrect because a google search of this and it's variants has revealed nothing. I believe it's related to the Moebius strip and probably connected with radio waves. It is used as the logo for our national broadcaster (The Australian Broadcasting Corporation) and you see exactly what I'm talking about by going on their website: www.abc.net.au. I have tried contacting them directly, but have received no response in over a month now! Answered by Harley Weston. 





The hypotenuse 
20100227 

From Dannielle: how do you find the hypotenuse if a=8 and b=6? Answered by Penny Nom. 





A square corner 
20100211 

From Trevor: I am building a new house and wish to set it out on site with the use of
profile boards and string. I want to be certain it is correct in terms of
squareness. I have a vague idea that the square on the hypotenuse should
be equal to the sum of the squares of the other two sides.
I get a little lost here and need some help. The building is a rectangle
measuring 40x30 feet to exterior brickwork. I guess that the length
of the hypotenuse should be exactly the square root of the combined
squares of the two sides.
Using the above measurements could you give me calculations from nuts
to soup as to the correct length of the diagonal. And what adjustments
are needed if everthing is not in accord.
Trevor. Answered by Robert Dawson. 





The distance from a chord to an arc 
20100211 

From matt: hello, I have to layout a radius without being able to pull from the center my radius is 3819.53 feet and i have a chord length of 275.59 feet if i broke that up into 25.05 feet sections how would i calculate the lengths from my chord to that radius? Answered by Robert Dawson. 





A tunnel from Toronto to Montreal 
20100125 

From Dave: I want to make a tunnel from Toronto to Montreal (for example)
Something like this
http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/grant1.html

My coordinates are 45.442455,73.861340 (Montreal) and 43.442455, 79.861340 (Toronto)
I need to know how to find arc distance, chord distance and radius
What websites can i find for this subject
Google has many but they are useless (blah blah) websites
LOL
Thanks Answered by Chris Fisher and Robert Dawson. 





The inradius of an icosahedron 
20100121 

From fnavroth: Suppose you have an equilateral triangle.
The area of the triangle is exactly 1200 square centimetres.
Now suppose you have twenty of those triangles.
It's possible to assemble those twenty triangles into a closed
threedimensional shape, a regular polyhedron.
What would be the volume, in cubic centimetres, of the largest sphere
that could fit inside the shape? Answered by Chris Fisher. 





A cone circumscribed about a given hemisphere 
20100119 

From Neven: The cone of smallest possible volume is circumscribed about a given hemisphere. What is the ratio of its height to the diameter of its base?
(G.F.Simmons, Calculus with Analytic Geometry, CH4 Applications of Derivatives) Answered by Chris Fisher. 





A pushbutton padlock 
20100117 

From Vince: Hi,
I have a pushbutton padlock using ten buttons (1234567890).
Five digits must be pushed in order to open the lock.
Each digit can only be used once.
Order is not supposed to matter.
How many different possible combinations? And what are they? Answered by Harley Weston. 





A pair of simultaneous equations 
20100109 

From Yumiko: Solve the following pair of simultaneous equations.
x^2 4x = y^24
3y=2x  3 Answered by Penny Nom. 





A question from a boat builder 
20100101 

From Grant: I am a boat builder, trying to lay out shape of side's elevation.
My question is, how do I define the length of a circle's radius, if I know the chord length
(20 ft) and the segment of the radius between the chord and the circle is known (7 inches)? Answered by Penny Nom. 





Chord length given the length & radius of an arc 
20091231 

From Wayne: Given the length & radius of an arc, is there a formula that will accurately calculate the chord length?
I'm an architectural designer, and would need it explained in layman's terms. Thank you.
Wayne Answered by Penny Nom. 





The volume of a frustum 
20091229 

From dave: I have a frustum top 1.7r bottom .55r and 2.14 h
I have to calculate cement in a silo every week.
I am not very good at maths but i have been adding the top and bottom
to get an average so as to turn it into a cylinder and i come up with
8.5m3 I know that the correct volume is 9.24 m3. Can you tell me why
getting an average width on the top and bottom of a frustum doesn't work.
Thank you. Answered by Chris Fisher. 





f(x)=x+2sinx 
20091212 

From amroziz: for which values of x does the graph of f(x)=x+2sinx have horizontal tangent Answered by Harley Weston. 





How fast is the distance between the two cars decreasing? 
20091208 

From Jenny: Two cares are on a collision course toward point P. The paths of the two cars make a 30 degree angle with each other. The first car is 40 km from P, and traveling toward P at 16 km/hour. The second car is 50 km from P, traveling at 20 km/hour. How fast is the (straight line) distance between the two cars decreasing. (Hint: Law of Cosines) Answered by Harley Weston. 





Solving two equations, one with a square root 
20091123 

From kacie: y = square root of x+3
x4y = 7
im having trouble with this problem...i have to find where they intersect. Answered by Harley Weston. 





The triangle formed by the tangent and the coordinates axes 
20091123 

From Nirmala: Given that y=1/x, x is not equal to zero. Prove that the area of the triangle formed by the tangent and the coordinates axes is 2. Answered by Harley Weston. 





Excluded values 
20091114 

From Janie: I have to State the excluded values for this equation and then solve, but not sure how to do this. Here is the problem
(x+6)/x+3=(3)/(x+3)+2 Answered by Harley Weston. 





f(x)= (e^x) / [(e^x)+(ex^2)] 
20091110 

From natalie: I'm trying to graph the function, f(x)= (e^x) / (e^x)+(ex^2) [e to the x divided by e to the x plus e times x squared] I know that there aren't any vertical asymptotes, but is there a horizontal asymptote? and also, I'm stuck on finding the concavity for this graph. I tried to find f "(x), but it came out to be really long and I am not sure how to find the x values for f "(x) without using a graphic calculator.
thanks,
natalie Answered by Chris Fisher and Harley Weston. 





At what rate are the people moving apart? 
20091101 

From saira: A man starts walking north at 4 ft/s from a point P. 5 minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minute after the woman starts walking ? Answered by Harley Weston. 





A path around a pond 
20091031 

From adeniji: find the area of a concrete path 2m wide surrounding a circular pond 12m in diameter Answered by Penny Nom. 





Painting a dome 
20091030 

From Jessica: A hemispherical dome with a radius of 50 ft will be given a coat of paint .01 inch thick.
The Contractor for the job wants to estimate the number of gallons of paint needed.
Use a differential to obtain an estimate (231 cubic inches/gallon) HINT: Approximate the change
in volume of hemisphere corresponding to increase of .01 inch in the radius. Answered by Robert Dawson. 





Graphing y=(4x^2)^5 
20091025 

From natalie: I want to graph the curve of y=(4x^2)^5 without using a graphing calculator. To do this, I'm suppose to find: domain, y and x intercepts, asymptotes, intervals of increase/decrease, local max/min, concavity and points of inflection. I got all the way to the step where I'm solving the concavity and I'm stuck. I found the f"(x) and it came out to be really large polynomial. I want to know how I can solve for the x of f"(x) without the use of a graphing calculator, when the polynomial has x^6 and x^8.
Thank you so much,
natalie Answered by Harley Weston. 





The hypotenuse of a right triangle 
20091018 

From steven: the perimeter of a right triangle is 20 cm. its area is 15 sq cm. find its hypotenuse. Answered by Penny Nom. 





A max/min problem 
20091012 

From avien: a rectangle has a line of fixed length Lreaching from the vertex to the midpoint of one of the far sides. what is the maximum possible area of such a rectangle? SHOW SOLUTION USING CALCULUS Answered by Penny Nom. 





A line tangent to a parabola 
20091001 

From kanchan: for what value of c a line y=mx+c touches a parabola y^2=4a(xa) Answered by Penny Nom. 





Sawdust 
20090929 

From joel: What is the density of saw dust Answered by Harley Weston. 





Probability 
20090927 

From Ed: My mother died 3 years to the day after her daughter died. what are the odds of that happening by chance? thanks Answered by Chris Fisher. 





Extraneous solutions 
20090920 

From iyana: what is an extraneous solution? what must you do to determine whether a extraneous solution? Answered by Stephen La Rocque and Harley Weston. 





solve integral of ( x^2+x+1)^5 
20090918 

From jaka: solve integral of ( x^2+x+1)^5 Answered by Robert Dawson. 





Sagitta 
20090910 

From Robert: Can you please tell me if there is a formula to figure out the Sagitta of an
arc when you know the radius, chord length, and arc length? Answered by Chris Fisher and Harley Weston. 





A circular border around a pool 
20090908 

From Calvin: A pool in the shape of a circle measures 10 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 3 inches, how wide will the border be? ( 1 cubic yard=27 cubic feet ) Answered by Stephen La Rocque. 





Simultaneous equations 
20090828 

From onias: solve 3/a  2/b = 1/2 , 5/a + 3/b = 29/12 Answered by Robert Dawson. 





The layout of an arch 
20090818 

From Steven: I am trying to layout a large radius between 2 points in a building and
need a formula to figure different senarios for example:
radius is 187'6"
distance between 2 points is 34'8"
need points 16" apart along the line between the 2 points to create the
radius
please help Answered by Stephen La Rocque. 





An antiderivative problem 
20090813 

From Indrajit: ∫4e^x + 6e^x/(9e^x + 4e^x)dx = Ax + Bloge(9e2x  4) + C
then A=?......B=?.....C=?
plz solve it...."^" stands for "to the power of".... Answered by Harley Weston. 





A rectangular pen 
20090813 

From Kari: A rectangular pen is to be built using a total of 800 ft of fencing. Part of this fencing will be used
to build a fence across the middle of the rectangle (the rectangle is 2 squares fused together so if you can
please picture it).
Find the length and width that will give a rectangle with maximum total area. Answered by Stephen La Rocque. 





Torricelli's trumpet 
20090729 

From Gary: I was reading about torricelli's trumpet which is described by the equation1/x which is then rotated around the x axis which results in a figure which looks like a trumpet. Now in order to find the volume the integral 1/x^2 dx is used which diverges when integrated so the volume is finite.However if you integrate 1/x dx which is the formula on the plane the answer diverges. Now if you took an infinite area then rotated it around the x axis shouldn't you get an infinite volume? Notice the area I am talking about is under the line 1/x not the surface area of the trumpet which is what the painters paradox is about What am I missing? Thanks Answered by Robert Dawson. 





Annular sector 
20090720 

From Ed: What is the name for the section of a flat ring (annulus)? Similar to a section on a radar screen, a quadrilateral but the top and bottom are curved. Answered by Robert Dawson. 





Simultaneous Equations 
20090706 

From Mukulu: Solve the equation simultaneously X/5=(Y+2)/2= (Z1)/4 ……………….eqt 1
3X+4Y+2Z25=0 ………………eqt 2 Answered by Janice Cotcher. 





The integral of x^x 
20090618 

From ANGIKAR: what would be the integration of (X^Xdx)?
give answer in details. Answered by Robert Dawson and Harley Weston. 





The radius of an arc 
20090612 

From Billy: I have tried using the formula 4h2(squared)plus L2(squared)divided by 8h
to find the radius of an arc, but I must be doing something wrong since I keep
getting the wrong answer. Can you tell me what I am doing wrong. The height
is 37.75 in. and the length is 18.875 in. Thank you for any help you can
give me. Answered by Stephen La Rocque. 





Extraneous solutions 
20090602 

From Ayana: solve and check for extraneous solutions.
3x+6/ x²4 = x+1/ x2
x can not = {2,2} Answered by Penny Nom. 





Two ships and a lighthouse 
20090527 

From Chelsey: I have a question in regards to how do I know when to use tangent or cosine when determining angles. The question is: Looking north from the observation deck of a lighthouse 60 m above the sea, a lighthouse keeper sees two ships. The angles of depression to the ships is 5 degrees and 10 degrees. How far apart are the ships?
I don't understand which one to use when solving the equation. Answered by Harley Weston. 





differentiate y sin[x^2]=x sin[y^2] 
20090511 

From mamiriri: derivate y sin[x^2]=x sin[y^2] Answered by Harley Weston. 





The integral of a to power x squared 
20090428 

From JIM: WHEN I ATTENDED U.OF T. (TORONTO ) MANY YEARS AGO
WE WERE TOLD THE FOLLOWING INTEGRAL COULD NOT BE
SOLVED : a to power x squared . is this still true ?
CURIOUS , JIM Answered by Robert Dawson. 





A maxmin problem 
20090420 

From Charlene: A fixed circle lies in the plane. A triangle is drawn
inside the circle with all three vertices on the circle and two of the vertices at the
ends of a diameter. Where should the third vertex lie to maximize the perimeter
of the triangle? Answered by Penny Nom. 





Sand falls from a conveyor belt 
20090401 

From Tracy: Sand falls from a conveyor belt at the rate of 10 cubic feet per minute onto a conical pile. The radius of the base is always equal to half the pile's height. How fast is the height growing when the pile is 5ft high? Answered by Stephen La Rocque. 





A spherical Tootsie Roll Pop 
20090401 

From Tracy: A spherical Tootsie Roll Pop you are sucking on is giving up volume at a steady rate of .8 ml/min. How fast will the radius be decreasing when the Tootsie Roll Pop is 20 mm across? Answered by Harley Weston. 





Uses of Pythagorean theory 
20090327 

From Britta: Please, give me some complex real life situation examples where the pythagorean theory is used. It must be a grade 8 or grade 9 level of thinking as that is what is my teacher's demands. Answered by Robert Dawson. 





An isosceles triangle 
20090326 

From sela: An isosceles triangle has two equal sides of length 10 cm. Theta is the
angle between two equal sides.
a) Express area of a triangle as a function of theta
b) If theta is increasing at a rate of 10 degrees/minute, how fast is area
changing at the instant theta=pi/3?
c) at what value of theta will the triangle have the maximum area?
Answered by Penny Nom. 





The rate of change of the volume of a sphere 
20090325 

From Kaylin: why the rate of change of volume of a sphere is not constant even though dr/dt is constant? Answered by Walter Whiteley. 





A maxmin problem 
20090324 

From Jay: Determine the area of the largest rectangle that can be inscribed between the xaxis and the curve defined by y = 26  x^2. Answered by Harley Weston. 





The diameter of a roll of plastic 
20090324 

From truong: hi. i have trouble to calculate the diameter of the plastic roll. the sheet is 765 m long and 0.8 mm to wrap around the core 400 mm in dia. please help me with the formula to calculate the dia of plastic roll, thanks in advance Answered by Harley Weston. 





The radius of a circle 
20090322 

From Justin: Find the radius of a circle with a circumference of 9.43cm. Answered by Penny Nom. 





The angles of a triangle 
20090311 

From Marissa: The angles in a triangle measure 7x1, 18x+2, and 5x+10. Determine whether the
triangle is acute, obtuse, or right. State your reasons clearly. Answered by Robert Dawson. 





A common tangent to two curves 
20090302 

From Jay: For what values of a and b will the parabola y = x^2 + ax + b be tangent to the curve y = x^3 at (1,1)? Answered by Penny Nom. 





Implicit differentiation 
20090301 

From Emily: determine the derivative y' at the point (1,0)
y= ln(x^2+y^2)
y'(1)= ?? Answered by Stephen La Rocque. 





Implicit differentiation 
20090218 

From Sunny: Find slope of the tangent line to the curve 2(x^2+y^2)2=25(x^2–y^2) at (3,1) Answered by Robert Dawson and Harley Weston. 





The area between the xaxis and a curve 
20090218 

From Lauren: This is from a homework question I can't figure out.
Let R be the region in the fourth quadrant enclosed by the xaxis
and the curve y= x^2  2kx, where k > 0. If the area of the region R is 36
then what is the value of k? Answered by Robert Dawson. 





The second derivative of h(x)=f(g(x)) 
20090216 

From Kristina: If h(x)=f(g(x)), and is differentiable, then find h"(x). Answered by Robert Dawson. 





A definite integral 
20090209 

From Mathata: Evaluate: integral from 0 to 1, x^2 e^x^3dx Answered by Harley Weston. 





A trig limit 
20090205 

From Samantha: lim x> 0 ( ( r*cos(wt +h) + r*cos(wt) )/ h )
Where r & w are constants. Answered by Harley Weston. 





A point on 8x^2+5xy+y^3=149 
20090204 

From Vivian: Consider the curve defined by 8x2+5xy+y3=149
a) find dy/dx
b) Write an equation for the line tangent to the curve at the point (4,1)
c) There is a number k so that the point (4.2,k) is on the curve. Using the tangent line found in part b), approximate the value of k.
d) write an equation that can be solved to find the actual value of k so that the point (4.2,k) is on the curve
e) Solve the equation found in part d) for the value of k Answered by Harley Weston. 





Simultaneous equations with an xy term 
20090201 

From angelee: xy+5x2y10=0
2x+y=1 Answered by Penny Nom. 





limit sinx/x 
20090130 

From Jackie: how to evaluate limit sinx/x as x tends to zero if x is in degrees Answered by Stephen La Rocque and Harley Weston. 





An integral from 1 to infinity 
20090124 

From Ray: Determine the area bounded by the xaxis and the curve y=1/(x^2) from x=1 to x=infinity.
A. 1.00
B. infinity
C. indeterminate
D. 2.00 Answered by Harley Weston. 





Archimedes' formula for parabolic arches 
20090123 

From La: Use calculus to verify Archimedes' formula for y=9x^2. Prove Archimedes' formula for a general parabolic arch. Answered by Harley Weston. 





In the shadow of a flagpole 
20090122 

From La: How fast is the length of the shadow of an 18 foot flagpole growing when the angle of elevation of the sun is 45 degrees and is decreasing at a rate of 10 degrees per hour? Answered by Harley Weston. 





The parabola with vertex (7,2) and directrix y = 3 
20090121 

From Deann: Find an equation of the parabola with vetrex (7,2) and directrix y =(3) Answered by Penny Nom. 





Partial derivatives 
20090117 

From Meghan: I have a question I've been working at for a while with maxima/minima of partial derivatives.
"Postal rules require that the length + girth of a package (dimensions x, y, l) cannot exceed 84 inches in order to be mailed.
Find the dimensions of the rectangular package of greatest volume that can be mailed.
(84 = length + girth = l + 2x + 2y)" Answered by Harley Weston. 





Negative rate of change 
20090112 

From hemanshu: when i have to find rate of change of decrease in any value my ans comes in negative why?????????? Answered by Penny Nom. 





What is the maximum revenue? 
20090109 

From Kristy: A skating rink manager finds that revenue R based on an hourly fee x for
skating is represented by the function R(x) = 200x^2 + 1500x
What is the maximum revenue and what hourly fee will produce
maximum revenues? Answered by Harley Weston. 





A max/min problem 
20090109 

From Angelica: have 400 feet of fence. Want to make a rectangular play area. What dimensions should I use to enclose the maximum possible area? Answered by Robert Dawson. 





The area of a region bounded by two curves 
20090107 

From Rogerson: Find the area, S, enclosed by the given curve(s) and the given line.
y = x^2  x  1, y = x+2 Answered by Harley Weston. 





A kennel with 3 individual pens 
20090106 

From Jean: An animal clinic wants to construct a kennel with 3 individual pens, each with a gate 4 feet wide and an area of 90 square feet. The fencing does not include the gates.
Write a function to express the fencing as a function of x.
Find the dimensions for each pen, to the nearest tenth of a foot that would produce the required area of 90 square feet but would use the least fencing. What is the minimum fencing to the nearest tenth? Answered by Harley Weston. 





The area enclosed by a curve and the xaxis 
20090104 

From Rogerson: Find the area, S, enclosed by the curve y = x^2 + 6x  5 and the xaxis in the interval 0≤x≤4. Answered by Harley Weston. 





Determine y'' by implicitly differentiating twice 
20090104 

From Walter: Given x^3  3xy + y^3 = 1 , determine y'' by implicitly differentiating
twice. I cannot solve this. Would you be kind enough to perform the
mathematics and show the steps involved in obtaining the solution? Answered by Harley Weston. 





The area of a region in the plane 
20090103 

From Rogerson: Find the area, S, of the shaded region enclosed by the given cureve, the given line and the xaxis.
y = x^2 + 1
line x = 2 Answered by Harley Weston. 





The radius of a cone 
20090102 

From kalpaj: A conical funnel holds 100ml. If the height of the funnel is 10 cm, determine
its radius, to the nearest tenth of a centimeter. Answered by Penny Nom. 





The volume of a pipe elbow 
20081231 

From mhd: how i can find volume of the pipe elbow of 4inches? Answered by Stephen La Rocque. 





Pouring angles for a crucible 
20081220 

From Richard: I am trying to work at pouring angles and volume left in during pouring a crucible, The crucible is cylindrical and flat bottomed.
I know the diameter, radius and volume of the crucibles. and the volume of liquid going into it.
So lets say the crucible is only half full firstly I need to work out the angle just before its going to pour. ( I can work this out as long as there is a certain volume of liquid if its not enough I cant do it)
Now the problem I also need to work out how much I should tilt the crucible to allow a certain amount out and be able to do this untill the volume reaches 0 at 90' turn. This is where I am stuck.
The reason for needing to be able to work this out is so i can develop a constant flow for example 10Kg of metal per second.
Thank you very much for you time Answered by Harley Weston. 





Integral of cos^2 X between pi/2 and 0 
20081218 

From Wanda: Integral or Area of cos^2 X between pi/2 and 0.
The answer that I got is pi/4. Is this correct? If not, how did you come up with your answer? Answered by Robert Dawson. 





Solve for x 
20081216 

From Melissa: I have a test tomorrow and I'm hoping you can help me before then. I can never seem to solve the "RESOLVE X" problems, or in french resous pour x/
They look like this. 3x+2\6=2x5\3. I only understand NOTHING from that.
Another equation is 2(x+1)=3(x+2). Answered by Robert Dawson and Penny Nom. 





The middle term of an arithmetic sequence 
20081215 

From Leigh: Find the sum of the first fifteen terms of an arithmetic series if the middle term is 92 Answered by Penny Nom. 





A sphere in a can of water 
20081212 

From Meghan: A cylindrical can open at the top has (inside) base radius equal to 1.
The height of the can is greater than 2.
Imagine placing a steel sphere of radius less than 1 into the can, then pouring water into the can until the top of the sphere is just covered.
What should be the radius of the sphere so the volume of water used is as large as possible? Answered by Harley Weston. 





The third vertex of a right triangle 
20081210 

From prashant: how to calculate coordinates of third vertex in a right angle triangle given the coordinates of hypotenuse as (1,3) and (4,1) Answered by Robert Dawson. 





What is so important about quadratics? 
20081129 

From zoe: what is so important about quadratics? Answered by Harley Weston. 





How fast is the distance between the airplanes decreasing? 
20081110 

From Crystal: At a certain instant, airplane A is flying a level course at 500 mph. At the same time, airplane B is straight above airplane A and flying at the rate of 700 mph. On a course that intercepts A's course at a point C that is 4 miles from B and 2 miles from A. At the instant in question, how fast is the distance between the airplanes decreasing? Answered by Harley Weston. 





A trig limit 
20081104 

From Teri: Although I have this problem completely worked out in front of me I still cannot understand
how it was done. The problem is:
Find the limit.
lim x>0 sin2x/tan7x. Answered by Harley Weston. 





Separating variables 
20081104 

From Terry: by separating variables solve the initial value problem
(x+1)y' + y = 0 y(0) = 1 Answered by Harley Weston. 





Taxes in Taxylvania 
20081022 

From April: Taxylvania has a tax code that rewards charitable giving. If a person gives p% of his income to charity, that person pays (351.8p)% tax on the remaining money. For example, if a person gives 10% of his income to charity, he pays 17 % tax on the remaining money. If a person gives 19.44% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 19.44% of his income to charity. Count Taxula earns $27,000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving? Answered by Harley Weston. 





Antiderivative of 1/(x(1  x)) 
20081022 

From Matt: derivative of dx/(x(1x))
From what I've seen I should break apart the equation as such
derivative of dx/x  dx/(1x)
and then get the 2 corresponding log functions.
If that is correct why does this factoring work, if that is incorrect what is the proper way to find the derivative. Answered by Harley Weston. 





The slope of a tangent line 
20081018 

From Amanda: If f(x)=square root of (x+4), and the slope of the tangent line at x=21 was 1/n for some integer n, then what would you expect n to be? Answered by Stephen La Rocque. 





Two equations in two unknowns 
20081017 

From Dushayne: Please help me in solving this problem:
a. 3x4y=32
5x+2y=10
b. 2x+3y=11
4x+3y=10 Answered by Penny Nom. 





Two modular equations 
20081008 

From Mhiko: please solve this Chinese remainder problem..and give me a solution or rule in order to solve this problem/
x=2mod15
x=1mod25 Answered by Stephen La Rocque. 





Finding the Distance Between Two Latitudes 
20081002 

From Samua: Assuming that the Earth is a sphere of radius 4000 miles and that the cities
are on the same longitude (one city is due north to the other). Find the distance between
the cities with the latitudes of 37 degrees 47'36'' and another city with 47 degrees 37'18''. Heeeeeeeeeeelp! Answered by Janice Cotcher. 





Extraneous solutions 
20080925 

From crystal: /6x+7/=5x+2 Answered by Penny Nom. 





The region between two circles 
20080924 

From Carol: Good day!
Here is a picture of the problem that we need to solve. (I send the picture through email.)
A small circle is inside a larger circle, the only given in the problem is the chord of the larger circle tangent to the smaller circle which measures 16cm. The question is, what is the area of the shaded region?
Can you answer this question? Thanks! :) Answered by Harley Weston. 





The biggest right circular cone that can be inscribed in a sphere 
20080908 

From astrogirl: find the volume of the biggest right circular cone that can be inscribed in a sphere of radius a=3 Answered by Harley Weston. 





An exclusion zone around a triangle 
20080907 

From Awrongo: A long time ago Mr Gibson found an island shaped as a triangle with three straight shores of length 3 km,4 km and 5 km. He declared an 'exclusion zone' around his island and forbade anyone to come within 1 km of his shore. What was the area of his exclusion zone? Answered by Stephen La Rocque and Penny Nom. 





A string around the world 
20080818 

From Terry: I heard this on T.V. and was wondering if it was true. Place a string around
the world, you would need 25,000 miles of string. If you raised the string
off of the ground by 1 foot all the way around. How much more string
would you need? Answered by Penny Nom. 





[f(x)f(1)]/(x1) 
20080814 

From katie: Evaluate (if possible) the function of the given value of the independent
variable:
f(x)=(x^3)x:
[f(x)f(1)]/(x1) Answered by Penny Nom. 





Arclength and sectorangle 
20080806 

From Benson: If chord length, radius are given, How to find the sector angle and arclength Answered by Janice Cotcher. 





Integral of X^2 
20080728 

From Hemanshu: Integral of X^2 Answered by Janice Cotcher. 





Simultaneous equations 
20080723 

From Franco: Solve
3 D + E  F = 10
2 D  F = 4
3 D  4 E  F = 25
Franco Answered by Penny Nom. 





The maximum range of a projectile 
20080722 

From kwame: the range R of projectile fired with an initial velocity Vo ,at an angle of elevation (@ )theta from the horizontal is given by the equation R = (Vo(squared) sin2theta)/g. where g is the accelation due to gravity . Find the angle theta such that the projectile has maximum range . Answered by Harley Weston. 





A square and a circle 
20080720 

From kobina: 4 ft of a wire is to be used to form a square and a circle. how much of the wire is to be used for the square and how much should be used for the square in order to enclose the maximum total area Answered by Harley Weston. 





Chords and arcs 
20080711 

From Ronnie: We are trying to build a semi life size ark decoration , and we are trying to cut the sides out . The curved sides and we can't figure our radius , all we know is that our chord length is 24ft. any suggestions on how to find the radius or maybe even the arc length or circumference or diameter? Answered by Harley Weston. 





A difference quotient 
20080710 

From Rita: Find the difference quotient of f, that is, find [f (x + h)  f (x)]/h, where
h does not = 0 for the given function. Be sure to simplify.
f(x) = 1/(x + 3) Answered by Janice Cotcher. 





A dog tied to a round building 
20080708 

From maitham: i have this question which i don't know how to solve it :
One dog was linked to the outer wall of a building round of 20 meters in diameter. If the length of chain linking the dog sufficient turnover of half the distance around the building,
What area can guard dog?
they said that we can solve it by integral .. can you solve it for me? Answered by Harley Weston. 





CIRCLES 
20080707 

From daryl: Find the equation of the smaller circle that is tangent to the axes and the circle x(squared)+y(squared)=2x+2y1? Answered by Penny. 





If the arc is 75mm, what is the radius? 
20080612 

From malcolm: If the are is 75mm, what is the radius? Answered by Janice Cotcher and Harley Weston. 





Two rhombi 
20080612 

From Malik: ("rhombus" diagram is attached)
ABCD and EJCD are rectangles.
EFHG and HIJK are similar rhombus.
HIJK is 4times bigger than EFHG.
In each rhombus the larger diagonal is double of the smaller one.
If AB = 30cm, what is the lenght of the diagonals of the smaller rhombus ? Answered by Penny Nom. 





The rate of change in the depth of the water 
20080612 

From Liz: A rectangular pool 50ft long and 30ft. wide has a depth of 8 ft. for the first 20 ft. for its length and a depth of 3 ft. on the last 20ft. of its length and tapers linearly for the 10 ft in the middle of its length. the pool is being filled with water at the rate of 3ftcubed/ min
at what rate is the depth of the water in the pool increasing after 15 hours? Answered by Harley Weston. 





The radius of a sphere if you know the volume 
20080611 

From Cey: how to i find the radius of a sphere with a volume of 1000cm cubed using the formula v=4/3 pi r squared???????/ Answered by Penny Nom. 





Two circles 
20080610 

From cey: the diameter of the larger circle is 20cm, and the smaller 10cm. what is the shaded area?? Answered by Janice Cotcher. 





The length of a shadow 
20080527 

From Simon: A figure skater is directly beneath a spotlight 10 m above the ice. IF she skates away from the light at a rate of 6m/s and the spot follows her, how fast is her shadow's head moving when she is 8m from her starting point? The skater is (almost) 1.6m tall with her skates on. Answered by Stephen La Rocque and Harley Weston. 





The weight of a concrete column 
20080511 

From russell: a cylindrical form is filled with a slow curing concrete. The base of the form
is 10 ft in radius, and height is 25 ft. while the concrete hardens, gravity
causes the density to vary from a density of 90 lbs/ft^3 at the bottom to a
density of 50 lb/ft^3 at the top. Assume that the density varies linearly
from the top to the bottom, and compute the total weight of the resulting
concrete column Answered by Harley Weston. 





A lidless box with square ends 
20080428 

From Chris: A lidless box with square ends is to be made from a thin sheet of metal. Determine the least area of the metal for which the volume of the box is 3.5m^3.
I did this question and my answer is 11.08m^2 is this correct? If no can you show how you got the correct answer. Answered by Stephen La Rocque and Harley Weston. 





At what value of t is the maximum acceleration? 
20080425 

From Mary: Velocity of a function (which is the first derivative of its position) is defined over the interval 0 to 12 using the following piecewise function: v(t)=1 from 0 to 4, v(t)=x5 from (4 to 8 and v(t)=x+11 from (8 to 12. At what value of t is the maximum acceleration? Answered by Stephen La Rocque. 





The radius of a circle 
20080425 

From kathy: How do you find the radius of a circle if the area is 803.84 and using 3.14 for pi. Answered by Penny Nom. 





A volume of revolution 
20080424 

From Sabahat: Hi, i have a region enclosed by both axes, the line x=2 and the curve y=1/8 x2 + 2 is rotated about the yaxis to form a solid . How can i find the volume of this solid?. (Please note that y equation is read as y =1 over 8 times x square plus 2.) I will be really grateful if you answer this question. :) Answered by Harley Weston. 





An open box 
20080423 

From Le: Metal Fabrication; If an open box is made from a tin sheet 8 in square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. Answered by Harley Weston. 





What is the integral of 13sin^3(x)*cos^7(x)dx? 
20080422 

From Cathrine: I am having trouble integrating this problem. It says to evaluate the integral but I don't know what to do or how to do it.
It is the integral of
13sin^3(x)*cos^7(x)d Answered by Harley Weston. 





f(x)=sin^3(3x^2) find f ' (x) 
20080421 

From Michael: f(x)=sin^3(3x^2) find f ' (x) Answered by Harley Weston. 





The perimeter of a rhombus 
20080418 

From susana: how do you find the perimeter of a rhombus? Answered by Penny Nom. 





The area bounded by 3 curves 
20080413 

From Sabahat: Hi, I have enclosed a diagram.
The diagram shows the curve y=(2x5)4. The point P has coordinates (4,81) and the tangent to the curve at P meets the xaxis at Q.
Find the area of the region (shaded in the diagram) enclosed between the curve, PQ and the xaxis . (Please note that the equation y is read as y=2x 5 whole raise to power 4.) Answered by Stephen La Rocque. 





f(x) =ax^blnx 
20080413 

From charles: supposef(x) =ax^blnx is a real valued function. Determine exact values(not decimal approximations) fro nonzero constants a and b so that the function f has a critical point at x=e^3 and a maximum value of 1/2e Answered by Harley Weston. 





A volume of revolution 
20080404 

From ted: Consider the region bounded by y=x^2 + 1, y=53x and y=5. Sketch and
shade the given region; then set up but dont evaluate teh integrals to find
the following:
a) The volume of the solid generated by rotating the region about the line
y=5
b) the volume of the solid generated by rotating the region about the yaxis Answered by Penny Nom. 





lim as x approaches infinite of 5x + 2/x1 
20080404 

From Jordan: how to solve this.
lim as x approaches infinite of 5x + 2/x1 Answered by Stephen La Rocque and Harley Weston. 





Finding the radius when only given chord length 
20080403 

From Lorraine: There are two chords in a circle, an 8 inch chord and a 10 inch chord. The 8 inch chord
is twice the distance from the center as the 10 inch chord. What is the radius? Answered by Stephen La Rocque. 





The integral of dx / (4x^2  25)^3/2 
20080401 

From Meghan: I have a question from the trigonometric substitution of my calculus course.
integral of dx / (4x^2  25)^3/2 Answered by Harley Weston. 





A maxmin problem 
20080327 

From LSL: show that of all rectangle with a given area, the square has the smallest perimeter. Answered by Penny Nom. 





The radius of a circle 
20080322 

From danny: waht is the radius of a circle, if the circumference is 800? Answered by Penny Nom. 





A train and a boat 
20080315 

From Sabrina: A railroad bridge is 20m above, and at right angles to, a river. A person in a train travelling at 60 km/h passes over the centre of the bridge at the same instant that a person in a motorboat travelling at 20km/h passes under the centre of the bridge. How fast are the two people separating 10s later? Answered by Harley Weston. 





The centre and radius of a circle 
20080312 

From Ryan: hello and thank you for such a wonderful service.
This problem I think needs to be checked could you take a gander at it and tell me if i get it correct thanks
find the center and the radius of this circle x^2+y^2=8x2y+15=0
I cam up with center 2, 1/2 and a radius of 11 3/4 Answered by Harley Weston. 





What point on the graph y = e^x is closest to the origin? 
20080303 

From elvina: What point on the graph y = e^x is closest to the origin? Justify your answer. Answered by Stephen La Rocque. 





Simultaneous equations 
20080229 

From CONOR: I was wondering if you could help me with this problem
7x  5y = 1
3y = 4x Answered by Penny Nom. 





The radius of a circle 
20080228 

From SteVonee: Estimate the radius of a circle with the given circumference that is 192ft Answered by Penny Nom. 





I cut the cylinder at a 45 degree angle 
20080226 

From Shannon: I have a cylinder with a radius of 2' 1 5/8".
How do I calculate the radius increase when I cut the cylinder at a 45
degree angle? Answered by Harley Weston. 





A Norman window 
20080225 

From Jason: If the perimeter of a Norman window is 20 feet, what is the maximum area of the window? Answered by Stephen La Rocque. 





A ball bearing is placed on an inclined plane 
20080215 

From Leah: A ball bearing is placed on an inclined plane and begins to roll.
The angle of elevation of the plane is x.
The distance (in meters) that the ball bearing rolls in t seconds is s(t) = 4.9(sin x)t^2.
What is the speed of the ball bearing,
and what value of x will produce the maximum speed at a particular time? Answered by Penny Nom. 





Two regions with equal area 
20080213 

From James: There is a line through the origin that divides the region bounded by the parabola y=3x5x^2 and the xaxis into two regions with equal area. What is the slope of that line? Answered by Harley Weston. 





The circumference and radius of a circle 
20080210 

From Ray: How do you find the circumference or radius of an area presuming it is a circle. Or in other words how do you find the c or r given only the area is 50 sq metres Answered by Penny Nom. 





Classifying a triangle 
20080207 

From kevin: scalene triangle 8 ft base right side 9.5 left side 12 ft what is the angles Answered by Penny Nom. 





Integration by parts 
20080130 

From seth: hi i really dont understand integr
ation by parts. for example, the integral(t^2sintdt. i have u=t^2 and v'=sint also u'=t^/3 v=cost
for the formula i have uvintegralvu' dx this is all well and good but i cant get it right. Answered by Harley Weston. 





Belledout pier 
20080128 

From Gina: I need to know how to find the total yards needed to fill a concrete pier that is 54"/ 108" and 26' deep.
That is...54" @ the top of the pier belled to 108" @ the bottom...26' deep. Answered by Stephen La Rocque. 





Inflection points 
20080125 

From Armando: Hi, Im trying to write a program that takes an equation ( f(x) = 0 ) and returns a list of the inflexion points in a given interval.
there must be (I think) a mathematical method or algorithm to do this, probably involving the (second) derivate of the function.
However I have not found such a method yet. Any help on this will be much appreciated. Answered by Stephen La Rocque and Harley Weston. 





Finding the area of an isosceles triangle given one angle and the inradius 
20080124 

From Saurabh: Given an isosceles Triangle, whose one angle is 120 and inradius is √3. So area of triangle is? Answered by Stephen La Rocque. 





A parallelogram and a rhombus 
20080122 

From miguel: i have a problem proving a parallelogram a rhombus.. if a diagonal of a parallelogram bisects an angle
of the parallelogram , then its a rhombus
prove Answered by Stephen La Rocque and Walter Whiteley. 





Maximize income 
20080118 

From Chris: Lemon Motors have been selling an average of 60 new cars per month at
$800 over the factory price. They are considering an increase in this
markup. A marketing survey indicates that for every $20 increase, they
will sell 1 less car per month. What should their new markup be in order
to maximize income? Answered by Stephen La Rocque and Harley Weston. 





The radius of a planet 
20080106 

From Ben: Two people who are both h feet tall are standing on a spherical planet. One person walks a distance d in feet away from the the other person. At this point, the person walking turns around and can no longer see the top of the other persons head. What is the radius of the planet? Answered by Stephen La Rocque and Harley Weston. 





The integral of 1/ (x(x+1)^0.5) 
20071229 

From Nooruddin: Integral of
dx / x(x+1)^0.5
(boundaries are 5 and 3) Answered by Harley Weston. 





Differentiate 
20071228 

From taiwo: i am finding it difficult to use first principle to differentiate this question: y=xcos2x. can u help me. Answered by Penny Nom. 





lim sinx/(x +tanx) 
20071216 

From shimelis: i have problem how do you solve this equation
lim sinx/(x +tanx) Answered by Harley Weston. 





A 454590 triangle 
20071213 

From Aaditya: explain to me please how to do the 454590 theorem when one of the legs(not the
hypotenuse) is 3. How do you find the remaining two sides? please help me out. Answered by Leeanne Boehm. 





A right triangle 
20071206 

From Shubhomoy: The coordinates of a hypotenuse are (1,3) and (4,1). Find the equations of the perpendicular sides. Answered by Harley Weston. 





System of equations 
20071206 

From Jenn: change the equation,xy=4 to form y=mx+b
the solution to the system of equations y=2x and y=x+3 is Answered by Stephen La Rocque. 





Chicken and goat feet 
20071205 

From Kim: Old McDonald raises goats and chickens. The animals have a total of
100 heads adn 360 feet. How many goats and how many chickens does Mr.
McDonald have? Answered by Stephen La Rocque and Penny Nom. 





A radius and a tangent to a circle 
20071204 

From elizabeth: show that the radius of a circle meets a tangent line to the circle in a 90 degree angle.
hint: start by assuming they are not perpendicular and at a contradiction. Answered by Penny Nom. 





Proving a quadrilateral is a rhombus 
20071203 

From Jeanie: How do you prove that a quadrilateral is a rhombus because the diagonals
of the quadrilateral are perpendicular and bisect each other using the 2column
proof method? Answered by Stephen La Rocque. 





Maximize the product 
20071125 

From David: Hi i have this site call calcchat.com, but i dont understand how they explained this can you take a look? The question is:
Direction: Find two positive numbers that satisfy the given requirements.
The sum is S and the product is a maximum
this is what they did
1) Let x and y be two positive numbers such that x + y = S
2)P = xy
3) = x (S  x)
4) =Sx  x^2
5)...etc. the thing i dont get is how did they go from step 2 to step 3
and also i know this sound dumb but how did they get step 2? =) Answered by Harley Weston. 





A rectangular plot of farmland 
20071125 

From Christy: A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a singlestrand electric fence. With 800m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? Answered by Harley Weston. 





A curve sketch 
20071122 

From Ahson: Find critical points, determine the monotonicity and concavity and sketch
a graph of f(x) with any local maximum, local minimum and inflection
points labeled:
1. f(x) = x^4  x^3  3x^2 + 1 Answered by Harley Weston. 





Elimination of mayan prisoners 
20071119 

From Jim: An evil Mayan emperor decides to make an example of 1,000 prisoners. He stands them in a circle with numbers one to a thousand marked on their shirts. He then starts counting: "one in, two, in three out" The third man is immediately executed. This continues round and round and round the circle. While the number in the circle shrinks, every third prisoner is pushed out and executed. And it continues even when there are only two prisoners left alive. What number is on the last prisoner's shirt? Answered by Victoria West. 





A rectangle in an ellipse 
20071118 

From David: I need to find the max area of a rectangle inscribed in an ellipse with the equation
x^2+4y^2=4.. What I have so far is f(x,y)=4xy
g(x,y)=x^2+4y^24=0,
y=sqrtx^24/4
f'(x)=2x^2/sqrt4x^2+2(sqrt4+x^2).
What I need to know is how to finish the problem and find the actual mas area of the rectangle.
David Answered by Penny Nom. 





Find the radius of a circle given the center and a point on the circle 
20071118 

From Raymund: Find the radius if the center is at (0, 5) and one point on the circle is (2,3) Answered by Stephen La Rocque. 





lim [x + squareroot(x^2 + 3)] as x>inf 
20071116 

From David: Find the limit. (Hint: treat the expression as a fraction whose denominator is 1, and rationalize the numerator.)
lim [x + squareroot(x^2 + 3)] as x>inf
i got to
lim 3/(x  squareroot(x^2 + 3)) as x>inf
but i'm having trouble understanding why the answer is 0 plz explain thx Answered by Harley Weston. 





Local maxima, minima and inflection points 
20071113 

From Russell: let f(x) = x^3  3a^2^ x +2a^4 with a parameter a > 1.
Find the coordinates of local minimum and local maximum
Find the coordinates of the inflection points Answered by Harley Weston. 





Maximize his profit 
20071112 

From apoorva: During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.
a. Find the demand function, assuming it is linear.
b. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit? Answered by Penny Nom. 





Family of functions 
20071112 

From Russell: Consider the family of functions
f(t)= Asin3t + Acos3t +Bsin8t + Bcos8t
find exact values of parameters A and B so that f(0) = 2 and f ' (0) = 1 Answered by Stephen La Rocque. 





The radius of an arch 
20071110 

From Mark: How do you determine the raduis or diameter of a circle based on the folowing information:
1. The distance along the circle between two points is 35'2". This creates an arch.
2. The (chord) distance between the two points is 30'8".
3. The distance from the center of the chord (on a 90 degree) to the arch is 6'10 3/4". Answered by Harley Weston. 





Two integrals 
20071109 

From Akilan: how to integrate these (tan x)^6(sec x)^4 and sinh(x)(cosh(x))^2.
Please send me how to do this question. Having exams on Monday. Please help. Answered by Harley Weston. 





Increasing and decreasing for functions 
20071109 

From David: Direction: Identify the open intervals on which the function is increasing or decreasing.
f(x)=1/(x^2)
f'(x)= 2/(x^3)
i understand how to get up until there, and the undf. is x=0, but now i'm having problem setting up the number table chart. i cant remember how, and where to place the increase and decrease +  the
chart, for example <0> where would the increase and the decrease be place? Answered by Harley Weston. 





f(x+y) = f(x) + f(y) + 2xy 
20071101 

From Marcia: For all real numbers x and y, let f be a function such that f(x+y) = f(x) + f(y) + 2xy and such that the limit as h > 0 of f(h) / h = 7, find: f(0), use the definition of the derivative to find f'(x), and find f(x). Answered by Penny Nom. 





How to solve related rates problems 
20071027 

From David: Can you plz explain how and where you come up with an equation to solve this?
Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2 centimeters per second. Answered by Stephen La Rocque. 





lim x>1 (root x  x^2)/{1  root x) 
20071016 

From Meghan: Hi! I have a question from my Calculus textbook that I've been picking at for a while and I'm stuck.
lim x>1 (root x  x^2)/{1  root x). Answered by Stephen La Rocque and Penny Nom. 





How do you find the radius of a circle if you only know its area 
20071015 

From s: how do you find the radius of a circle if you only know the area of the circle. Do you somehow
reverse the Pi formula. Answered by Penny Nom. 





Four triangles in a square 
20071015 

From Kristina: A square with side lengths of 6 cm is divided into 3 right triangles and a larger isosceles triangle. If the three right triangles have equal area, find the exact area of the isosceles triangle. Answered by Stephen La Rocque. 





13 year and 17 year locusts 
20071012 

From stefan: how many years pass between the years when both 13 year and 17 year locusts are out at the same time? Answered by Penny Nom. 





The average rate of change of a function 
20071011 

From vern: Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(X)=sinX for the inverval [0,pi/6]? Answered by Harley Weston. 





Substitution method 
20071011 

From Kevin: 3xx+2y=36y=11 Answered by Stephen La Rocque. 





Given the arc length and chord length, what is the radius? 
20071010 

From Wayne: I have the actual length of an arc plus the length of the cord. How do I determine the radius of the arc. Answered by Harley Weston. 





Parabolic suspension bridge 
20071009 

From Jessica: A suspesion bridge with weight uniformly distributed along its length has twin towers
that extend 75 meters abouve the road surfce and are 400 meters apart.The cables are
parabolic in shape and are suspended from the tops of the towers. The cables touch the road
surface at the center of the bridge. Find the height of the cables at a point
100 meters from the center. (Assume that the road is level.) Answered by Stephen La Rocque. 





Coin jar 
20071007 

From a student: Sally empties his jar of coins. It contains $3.75 in nickels, dimes, and
quarters. The number of dimes is twice the number of nickels and the number
of quarters is three less than the number of nickels. Determine how many nickels,
dimes, and quarters were in the in the jar. Answered by Stephen La Rocque. 





Solving four simultaneous equations (system of four linear equations) 
20071007 

From Johan: I need some help in solving this question
x + 2y  3z + 4w = 12
2x + 2y  2z + 3w = 10
0 + y + z + 0 = 1
x  y + z  2w = 4 Answered by Stephen La Rocque. 





Arc lengths, central angles and radii 
20071004 

From Ashutosh: Jose can remember that the length of an arc is 440cm, but he cannot remember the radius of the arc or the angle at the center. He does know that the angle was a whole number of degrees and the radius was less than 100cm. Find three possible angles and write down the size of each of the possible radii. Answered by Stephen La Rocque. 





How many ten thousands makes one million? 
20071002 

From Payton: how many ten thousands makes one million? Answered by Penny Nom. 





Finding equations, intersection point of two lines at right angles 
20070922 

From Yaz: Find the equation of the line joining A(1,9) to B(6,120). Another line passes through C(7,5) and meets AB at rigth angle of D. Find the euation of CD and calculate the coordinates of D. Answered by Stephen La Rocque. 





The hypotenuse 
20070920 

From Kasey: What is the hypotenuse of 96 squared and 108 squared? Answered by Penny Nom. 





How many two digit numbers contain at least one 7? 
20070906 

From Janet: How many two digit numbers contain at least one number seven? Answered by Penny Nom. 





The area of a circle knowing only the length of a chord 
20070905 

From James: I need some help in the right directions with a problem. I was presented with a problem where I need to find the area of a circle knowing only the length of a chord.
the is a circle in the center of a larger circle (which the size of either could change) the only thing that matter is that the chord is 100 ft long and rests on top of the smaller circle. Answered by Stephen la Rocque and Brennan Yaremko. 





The tangent to y = x^3 at x = 0 
20070904 

From Amit: consider the equation = x^3. The equation of tangent to this curve (which is smmetrical in Ist and IVth quadrant) at (0,0) is y=0, which is xaxis.
but graphically one can visulize that xaxis intersects the curve, so how can it be the tangent to the curve. Please help. Answered by Harley Weston. 





A frustum of a right pyramid 
20070824 

From Andrew: Find the volume of a frustum of a right pyramid whose lower base is a square with a side 5 in., whose upper base is a square with a side 3in., and whose altitude is 12 in. Round your answer to the nearest whole number.
A. 47cu in. C. 226 cu in.
B. 196 cu in. D. 1036 cu in. Answered by Stephen la Rocque. 





Where do you use trigonometry? 
20070821 

From jenny: where do you use trigonometry besides architecture and engineering? Answered by Stephen La Rocque. 





A geometry problem 
20070820 

From samhita: ABC is a triangle. Let D be a point on side BC produced beyond B such that BD=BA. Let M be the midpoint of AC. The bisector of angle ABC meets DM at P. Prove that angle BAP=angle ACB. Answered by Chris Fisher. 





A right triangle 
20070811 

From Peter: Ok, is it possible to find the height and base of a right triangle when all the information you are given is the length of the hypotenuse?
I also know that the angle between side b and the hypotenuse is 45 degrees. please help. Answered by Walter Whiteley. 





The swaying of a building in the wind 
20070811 

From San: During a strong wind, a tall builing, such as the CN Tower, can sway
back and forth as much as 100cm, with a period of 10 seconds.
Please help me to determine the equation for this function, in the form
y=asinkx Answered by Stephen La Rocque. 





Diameter of an octagon 
20070807 

From Bree: I am trying to find the diameter of a octagon with 20' sides . What formula do I use? Answered by Stephen La Rocque. 





A complex number in polar form 
20070723 

From roland: write the given complex number z in polar form lzl(p+qi) where lp + qil=1 for 3  4i. Answered by Harley Weston. 





f(x) = (x^4)  4x^3 
20070722 

From Michael: I'm a student who needs your help. I hope you'll be able to answer my question.
Here it is: Given the function f(x)=(x^4)4x^3, determine the intervals over which the function is increasing, decreasing or constant. Find all zeros of f(x) and indicate any relative minimum and maximum values of the function.
Any help would be appreciated. Thank you for your time. Answered by Harley Weston. 





A normal to a curve 
20070716 

From Samantha: The function f is defined by f:x > 0.5x^2 + 2x + 2.5
Let N be the normal to the curve at the point where the graph intercepts the yaxis. Show that the equation of N may be written as y = 0.5x + 2.5.
Let g:x> 0.5x + 2.5
(i) find the solutions of f(x) = g(x)
(ii) hence find the coordinates of the other point of intersection of the normal and the curve Answered by Penny Nom. 





The isosceles triangle of largest area with perimeter 12cm 
20070716 

From sharul: find the dimension of isosceles triangle of largest area with perimeter 12cm Answered by Harley Weston. 





Implicit Derivatives 
20070713 

From Charles: I need help computing y' by implicit differentiation the question is:
y^2 + x/y + 4x^2  3 Answered by Stephen La Rocque. 





Derivative of a Function 
20070709 

From Bob: What is the derivative of the function a sub n = [n/(n+1)]^n ? Answered by Stephen La Rocque. 





Finding the radius of an inscribed circle 
20070705 

From Maria: I need to find the radius of a circle which is inscribed inside an obtuse triangle ABC. I know all the angles and all the lengths of the triangle. Answered by Stephen La Rocque and Chris Fisher. 





Using calculus to prove the formula for the area of a triangle 
20070704 

From Apratim: Using calculus how can one show that the area of any triangle is 1/2 times its base times its height? Answered by Stephen La Rocque. 





A rhombus with all right angles 
20070629 

From Rachel: Can a rhombus with all right angles still be tagged as a rhombus, or is it then said to be a square? Answered by Stephen La Rocque, Harley Weston and Walter Whiteley. 





A question on continuity 
20070628 

From Mac: f(x) = (1/x) + (1/(2x)) be the function and [0,2] be the interval.
1) It is continuous at the end points ?
2) is f(0) equal to f(2) ? Answered by Harley Weston. 





Log base 2 of log base 2 of x 
20070627 

From alex: y = log base 2 of lag base 2 of x
The slope of the tangent to the given curve at its xintercept is..? Answered by Harley Weston. 





sinx and cosx 
20070625 

From Mac: Can anyone tell me whether sinx and cosx is differentiable at x=0 ?
As far as i know, cos(x) and sin(x) is differentiable at all x. Answered by Penny Nom and Stephen La Rocque. 





Limits as x approaches a constant 
20070625 

From Mac: can you please tell me what is the reason they say "denominator is a negative quantity"
in the solution 11 and "denominator is a positive quantity" solution 10 ??
If i guess correctly, for solution 10, its because of x^2 in the denominator. Answered by Penny Nom. 





Simultaneous equations : the Elimination method 
20070621 

From Patricia: I need to find the value of X and Y using the Elimination method.
5/x + 3/y=4
25/x2/y=3 Answered by Stephen La Rocque. 





Simultaneous inequalities 
20070618 

From Freddy: Watson Electric has production facilities in Valley Mills, Marlin,and Hillsboro.
Each one produces radios,stereos,and TV sets.
There production capacities are
Valley Mills: 10 radios, 12 stereos, and 6 TV sets per hour
Marlin: 7 radios, 10 stereos, and 8 TV sets per hour
Hillsboro: 5 Radios, 4 Stereos, amid 13 TV sets per hour
QUESTION
How many hours should each plant be scheduled to fill an order of 1095 radios, 1230 stereos,
and 1490 TV sets? Answered by Stephen La Rocque. 





Area of a circular garden 
20070618 

From Cynthia: Hi is this the correct formula for this problem?
What is the approximate area of a circular garden that is 20 feet in
diameters? Answered by Stephen La Rocque. 





Using the Pythagorean Theorem 
20070618 

From cynthia: Hi,
If I have a question with a right triangle and it asks....
If ABC is say 400 miles. How much shorter will the miles be if I travel
from BC?
I don't exactly remember the question but, I would I solve a problem
similiar to this one? Answered by Stephen La Rocque. 





Angles of depression 
20070613 

From Phonda: The pilot of a small private plane can look forward and see the control tower for a small airstrip. Beyond that is a large factory that is 3 milies from the airstrip. The angles of depression are 12.5 degrees and 4.8 degrees respectively.
Find the airplane's altitude, to the nearest ten feet. Answered by Stephen La Rocque. 





Two tangent lines to y=x^3 
20070607 

From stephanie: find the equations of two tangent lines to the y=x^3 function through the point (2,8) Answered by Penny Nom. 





The limit of a rational function 
20070528 

From Imad: 3 _______ 3 _______
lim \/ 1 + x  \/ 1  x
x>0  
x Answered by Penny Nom. 





A circular blob of molasses 
20070528 

From Julie: A circular blob of molasses of uniform thickness has a volume of 1 m^3.
The thickness of the molasses is decreasing at a rate of 0.1 cm/hour.
At what rate is the radius of the molasses increasing when the radius is 8
m?
Thanks,
Julia Answered by Penny Nom. 





More on quadrilateral shape names 
20070526 

From Don: If North Americans call a quadrilateral with no parallel sides a trapezium, is a kite merely a special type of trapezium? Can a rhombus be a kite? Answered by Walter Whiteley and Penny Nom. 





System of equations 
20070524 

From Chris: Find all real solutions (x,y,z,w) of the system of equations:
2y= x + x/17, 2z= y + y/17, 2w = z + z/17, 2x= w + w/17 Answered by Penny Nom. 





Set up two simultaneous equations 
20070521 

From Admire: The cost of producing windscreen wipers blades at a factory ais partly fixed (due to operating overheads) and partly dependent on the number of blades produced. It costs $300 to produce 1000, and $600 to produce 5000 blades. How much would it cost to produce 24000 blades? Answered by Penny Nom. 





Finding the hypotenuse without Pythagorus 
20070511 

From Shelbie: How do i find the hypotenuse of a right traingle not using the pythagorean thereom if i have the measurements of the legs? Answered by Stephen La Rocque. 





Pattern for a truncated cone 
20070511 

From Mike: I have been trying to get this cone flat so I can build this column. Can you please help me so I can figure this out? Thanks for your help. Answered by Stephen La Rocque. 





A ton of sawdust 
20070510 

From David: I am trying to do a conversion. A tractor trailer is loaded with 165 cubic
yards of sawdust (I do not know the actual weight). I knw the wholesale
cost as $1,000 for this amount, but would like to convert this to find out
what the equivalent cost for a metric ton would be. Answered by Stephen La Rocque. 





Area of region between circle and inscribed octagon 
20070507 

From amy: I have to find the area of the shaded region where there is an octagon inscribed in a circle
The radius is 4 inches. The shaded region is everything besides the octagon inside the circle.
How can I find the area of the shaded region?
Thank you! Answered by Stephen La Rocque. 





Optimization  carrying a pipe 
20070505 

From A student: A steel pipe is taken to a 9ft wide corridor. At the end of the corridor there is a 90° turn, to a 6ft wide corridor. How long is the longest pipe than can be turned in this corner? Answered by Stephen La Rocque. 





Hypotenuse 
20070503 

From ashley: how do you find the hypotenuse Answered by Penny Nom. 





Edging surrounding a round pool 
20070503 

From Carol: Hello,
I am new at this and very rusty on my math. I am getting a 24 Ft. round pool and would like to put 2ft width stone (small) edging around it. How much would I need to buy. I have to buy it by the yard (cubic yard) I am not looking for an exact, just an approx. even would be great.
Thanks
Carol Answered by Stephen La Rocque. 





Continuity of y = x 
20070502 

From moulipriya: Is the curve y =  x  continuous everywhere? Answered by Penny Nom. 





Two concentric circles form an annulus 
20070502 

From A student: In the diagram below, two concentric circles form an annulus. The
vertical line is tangent to the inner circle, and forms the diameter of
a third circle.
Explain why the areas of the annulus and third circle are the same. Answered by Penny Nom. 





A thousand, is it M or K? 
20070501 

From Larry: I have heard that Million is annotated as MM. But Ihave heard two answers for Thousands (K, M). Which is correct? Answered by Penny Nom. 





A tugboat's speed 
20070430 

From Amanda: a tugboat must travel 24 miles against a 4 mile per hour current on the Potomac River and return. At what
constant speed must the tugboat travel to make the trip in 12 hours. Round answer to the nearest tenth mph. Answered by Stephen La Rocque. 





The area of a pyramid 
20070428 

From Alexander: Total area of the plate required to fabricate a vessel(pyramid) the base is 0.6mx0.6m and height of 1.0m. Answered by Stephen La Rocque. 





Maximize the volume of a cone 
20070427 

From ashley: hello,
I've been stumped for hours on this problem and can't quite figure it out.
The question is: A tepee is a coneshaped shelter with no bottom. Suppose you have 200
square feet of canvas (shaped however you like) to make a tepee. Use
calculus to find the height and radius of such a tepee that encloses the
biggest volume.
Can you help?? Answered by Stephen La Rocque and Penny Nom. 





A cylinder inside a sphere 
20070425 

From Louise: i need to find the maximum volume of a cylinder that can fit inside a sphere of diamter 16cm Answered by Penny Nom. 





Liquid is being poured into the top of a funnel 
20070419 

From neroshan: Liquid is being poured into the top of a funnel at a steady rate of 200cm^3/s.
The funnel is in the shape of an inverted right circular cone with a radius
equal to its height. It has a small hole at the bottom where the liquid is
flowing out at a rate of 20 cm^3/s. How fast is the height of the liquid
changing when the liquid in the funnel is 15 cm deep?
At the instant when the height of the liquid is 25cm, the funnel becomes clogged
at the bottom and no more liquid flows out. How fast does the height of the
liquid change just after this occurs? Answered by Penny Nom. 





Minimum cost for a fixed volume 
20070418 

From James: My question goes: A silo is to be constructed and surmounted by a hemisphere. The material of the hemisphere cost twice as much as the walls of the silo. Determine the dimensions to be used of cost is to be kept to a minimum and the volume is fixed. Answered by Penny Nom. 





Simultaneous equations 
20070416 

From kyrie: simultaneous equation
4x + 3y = 21
2x * y = 8 Answered by Penny Nom. 





The second derivative 
20070414 

From Gerry: In mathematical context,what do you understand by the term "Second Derivative" Answered by Penny Nom. 





An arc shaped groove into a peice of metal 
20070412 

From daniel: hello i work at an engineering workshop the other night i was asked to machine an arc shaped groove into a piece of metal the cord length was 6 mm and the height from the middle of the cord to the arc was 1mm i was hoping to find the diameter of the cutter needed to do the job and also the formula to work out how to find the diameter. i believe it is 10mm dia thankyou for your time and knowledge Answered by Stephen La Rocque. 





Pythagoras was right 
20070411 

From Vineet: in a right angle triangle, hypotenuse side is less than the sum of other two
sides, how the square of hypotenuse is equal to the sum of squares of other two sides? Answered by Stephen La Rocque. 





What is the limit of 3.x^(3/x) as x approaches +infinity? 
20070411 

From Teodora: What is the limit of 3.x^3/x as x approaches +infinity ? Answered by Haley Ess. 





Find the volume of the solid 
20070407 

From tricia: a solid is constructed so that it has a circular base of radius r centimeters
and every plane section perpendicular to a certain diameter of the base is
a square, with a side of the square being a chord of the circle.
find the volume of the solid
at first i thought the length of a side of the square would be r, but that
isn't awlays be true only when the chord is in the center.
so how can i solve this without any values? i dont understand the relationship
between the chord and radius, except that the radius intercepts
the chord at the midpoint.
i know i hav to take the integral to get the volume,
but how do i even find the area of one of the squares?
please help,
thanks,
tricia Answered by Penny Nom. 





What is the hypotenuse of a right traingle 
20070404 

From debbie: what is the hypotenuse of a right triangle with sides of 38 meters and 24.2 meters. Answered by Stephen La Rocque. 





A set of points in space 
20070404 

From Lenny: What is a set of points in space the same given distance from its center point called? Answered by Stephen La Rocque. 





A beam on a lighthouse 
20070328 

From Lisa: A beam on a lighthouse 2000 metres away from the nearest point P on a straight shoreline revolves at the rate of 10 pi radians per minute. How fast is the beam of the light moving along the shoreline when it is 500 metres from P? Answered by Stephen La Rocque. 





The foci of an ellipse 
20070327 

From Brad: I am trying to figure out how to find the foci of an ellipse x^2/7 + y^2/16 = 1.
Since 16 is the largest denominator I know the major axis is going to be the y axis.
Do I now take 7c^2=16. c^2=167, c^2=9, c=3. So is my foci (0,+3). Answered by Penny Nom. 





y = sin(2x) 
20070322 

From bader: sin(2x)
find dx/dy Answered by Penny Nom. 





A rhombus 
20070304 

From Sally: As a kindergarten teacher, I am trying to introduce the term rhombus to my class. What would be the best mathematical, but simple language to use? The diamond shape, which I am trying to label as rhombus, is still "a dimaond" to my kindergarten students. Help! Answered by Steve La Rocque and Diane Hanson. 





Mutually exclusive events 
20070301 

From kalyssa: will you me an example of two events that are mutually exclusive and could you explain to me what mutually exclusive means? Answered by Steve La Rocque, Pam Fowler and Penny Nom. 





Simultaneous equations with fractions 
20070228 

From Alyca: Hello Math Central, I am a grade 10 student taking Academic math. Our unit right now is method of substitution and elimination. I'm stuck on this one question that I've been doing forever. Please help =)
*For this equation I have to do method of elimination, but it's so much harder with fractions...could some one please explain to me how to do it step by step?* x y 2    =   3 6 3 x y 1    = 1 12 4 2
Answered by Steve La Rocque and Ashley Mang. 





At what rate is the area of the triangle changing? 
20070224 

From mac: two sticks 3.5 feet long are hinged together and are stood up to form an isosceles triangle with the floor. The sticks slide apart, and at the moment when the triangle is equilateral, the angle is increasing at the rate of 1/3 radian/sec. At what rate is the area of the triangle increasing or decreasing at that moment? Mac Answered by Penny Nom. 





Find the area of the triangle 
20070220 

From Christina: Graph the function f(X)= x+1/x1 and graph the tangent line to the function at the points A:(2,3) and B:(1,0). The point of intersection of the two tangent lines is C. Find the area of the triangle ABC. Answered by Stephen La Rocque. 





Optical illusions 
20070218 

From Jami: Hi, I'm Jami and I'm in 10th grade.I'm doing a geometry research project on optical illusions and need to know how math is involved.I have an idea already of how our eyes percept 2 dimensional images and construct them into 3D images but, that isn't my question.There are many books that contain optical illusion pictures that have secret messages in them or have objects popping out. Is there a mathematical way in that optical illusion pictures are constructed? Answered by Walter Whiteley and Harley Weston. 





An augmented matrix 
20070213 

From Mary: I've been trying for quite some time now to figure this out. I have to solve this by using the GaussJordan Method: 3x  y = 15 2x + 3y = 10 Can anyone help me? Answered by Penny Nom and Gabriel Potter. 





Exponential form of complex numbers 
20070212 

From Austin: When dealing with imaginary numbers in engineering, I am having trouble getting things into the exponential form. The equation is 1+i now I do know that re^(theta)i = r*cos(theta) + r*i*sin(theta). Just not quite understanding the order of operations. Thanks Answered by Penny Nom. 





Volume of an inner tube 
20070210 

From Bubba: For a science project, I'm collecting methane gas in an inner tube. In addition to measuring psi of the inner tube, I'd like to calculate the volume of gas collected. What is the formula for or how would I calculate the volume of an inner tube? I appreciate any help you can give me.
Thanks so much. Bubba Answered by Penny Nom. 





Simultaneous equations with envelopes 
20070208 

From Mick: There were 17 envelopes bought, some were brown, some were white. The brown envelopes cost one cent more per envelope than the white ones. The total cost was 80 cents. How much of each type of envelope was bought? Many thanks! Answered by Stephen La Rocque. 





The substitution method 
20070131 

From Victoria: how do i solve this problem using the substitution method? 2x5= 14 7x+14y= 5 Answered by Stephen La Rocque. 





The elimination method 
20070131 

From Addrianna: x2y=2 3x5y=7 Answered by Stephen La Rocque. 





The centre and radius of a circle 
20070127 

From A student: x^2+y^2=121 is the equation of the Circle C
(1) Write down the center and the radius of C. Answered by Stephen La Rocque. 





How many locations for the lampposts are possible? 
20070121 

From Madeeha: Maria's backyard has two trees that are 40 feet apart, as shown in the accompanying diagram. She wants to place lampposts so that the posts are 30 feet from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lampposts are possible? Answered by Penny Nom. 





The volume of a frustum of a pyramid 
20070117 

From Sam: Find the volume of a frustum of a pyramid with square base of side b, square top of side a, and height h. Answered by Penny Nom. 





An arc, a cord and the radius of a circle 
20070114 

From Kevin: I have the length of the cord and the distance from the cord to the arc, is it possible to find the radius with just these parameters? Answered by Penny Nom. 





An octagonal bird house 
20070113 

From Soren: I'm in the process of building a birdhouse that is an octagon (based on previous questions, looks like that's a familiar tune). The essential elements are known, but I get stuck when trying to determine the angle for the cuts that would be made to the thickness of the wood so that they all fit together when assembled. Each octagonal section is 7 inches in width and the peak of the roof will be 2 inches higher than the sides. My sense is that the angle cuts that need to be made to the 'height' of each piece of wood. By height I mean the thinnest part of the wood that is neither the length nor the width to use colloquial terms. While it's clear that a slight angle is needed, it would seem that the angle would necessarily change as the distance from the top of any one side to the peak changes. Please advise if more clarification is needed. The 2 inches is random and can be changed if more convenient. Whew! Answered by Harley Weston. 





Integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 
20070109 

From James: How do you integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 Answered by Penny Nom. 





What are the dimensions of the most economical container? 
20070104 

From Ashely: A cylindrical container costs $2.00 per square foot for the sides and $3.00 a square foot for the top and bottom. The container must hold 100 cubic feet of material. What are the dimensions of the most economical container. Answered by Stephen La Rocque. 





An octagonal birdhouse 
20061230 

From Verner: I am building a octagon birdhouse,what degree would I cut each side of each piece of wood to assemble the birdhouse? Answered by Penny Nom. 





A rhombus 
20061226 

From Jose: show mathematically that a quadrilateral whose vertices are A(2,1),B(6,2) C(10,1),and D(6,4) is a rhombus Answered by Penny Nom. 





Rolle's Theorem 
20061207 

From Erika: If f(x) = (x^2)(square root of [3x]) on the interval [0,3] is given, Does Rolle's Theorem apply? If yes, find any values of c such that f '(c)=0 Answered by Penny Nom. 





A Norman window 
20061130 

From Joe: a norman window is a rectangle with a semicircle on top. If a norman window has a perimeter of 28, what must the dimensions be to find the maximum possible area the window can have? Answered by Stephen La Rocque. 





The radius of a hemisphere 
20061129 

From Emma: how do you calculate the radius of a hemisphere when you are given the volume? Answered by Stephen La Rocque. 





The radius of an arch 
20061115 

From Kelly: I am trying to achieve an arc height of .375 on the length of 17.375. Answered by Penny Nom. 





Tangent lines 
20061109 

From Melissa: let f be a function with f(1)=4 such that for all points (x,y) on the graph of f the slope is given by (3x^(2)+1)/(2y)
a.)Find the slope of the graph of f at the point where x=1. b.)Write an equation for the line tangent to the graph of f at x=1 and use it to approximate f(1.2) c.) Find whether f is concave up or concave down when x=1. Is your answer in part b an overestimate or an underestimate? Answered by Stephen La Rocque. 





Simultaneous equations 
20061106 

From An other: e^2yx+2=0 ln(x+3)2y1=0 Answered by Penny Nom. 





Water is being pumped into the pool 
20061024 

From Jon: A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deeps at the deep end. Water is being pumped into the pool at 1/4 cubic meters per minute, an there is 1 meter of water at the deep end.
a) what percent of the pool is filled?
b) at what rate is the water level rising? Answered by Stephen La Rocque. 





An approximation 
20061022 

From Ellen: consider the curve 8x^2 +5xy+y^3 +149 =0 Write an equation for the line tangent to the curve at (4, 1) use this equation to approximate the value of K at the point (4.2, K) Answered by Penny Nom. 





How fast is the water level rising when the water is 1 meter deep? 
20061019 

From Don: The cross section of a 5meter trough is an isosceles trapezoid with a 2meter lower base, a 3meter upper base and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter per minute. How fast is the water level rising when the water is 1 meter deep? Answered by Stephen La Rocque. 





The hypotenuse 
20061002 

From Ashley: How do you find the hypotenuse of a right triangle? I don't understand how to find c. Answered by Stephen La Rocque. 





The focus of a parabola 
20061001 

From Lily: I have a mathematical assignment which includes applications of parabolas, hyperbolas and ellipses in the real world. I have been searching the internet and now I am ware that most of the applications of parabolas have a connection with what people call "the focus". However, I do not think I clearly understand what "the focus" of a parabola is. Would you please explain it to me? Answered by Penny Nom. 





The area of a rhombus 
20060910 

From Lillian: In a rhombus, each side is 14 in. long. Two of the sides form a 60 degree angle. Find the area of the rhombus. Round your answer to the nearest square inch. Answered by Stephen La Rocque. 





The radius of a cone 
20060908 

From Hermanson: I know the cone is 20 degrees at the top and 80 degrees at the bottom. What is the formula for finding the radius? Answered by Stephen La Rocque. 





The velocity of a pendulum, part II 
20060907 

From Erin: We saw the question in your database about the velocity of a pendulum swinging.....It is the same exact question....but there is another question......it says....
"estimate the instantaneous rate of change of d with respect to t when t = 1.5. At this time, is the pendulum moving toward or away from the wall? Explain." Answered by Harley Weston. 





How many thousands are in ten million. 
20060824 

From Echoe: How many thousands are in ten million. Answered by Stephen La Rocque. 





Differentiate Y= sin3x + cos7x 
20060822 

From james: Differentiate the function of x using the basic rules.
Y= sin3x + cos7x Answered by Stephen La Rocque. 





How fast is the water level rising 
20060812 

From Erin: Water runs into a conical tank at the rate of 9ft^{3}/min. The tank stands point down and has a height of 10 ft. and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep? (V=1/3 pi r^{2} h). Answered by Penny Nom. 





An Integral 
20060730 

From Aniket:
I am Aniket studing in 12 th standard At Mumbai
I have following integration problem please give me a solution
integral of 1/under root of (5x^{2}  2x) dx
Answered by Penny Nom. 





Find the radius knowing the chord length and... 
20060728 

From Jim: If I know the length of a chord and its distance from the diameter, how do I calculate the radius of the circle?
Answered by Stephen La Rocque. 





Minimizing a cost 
20060725 

From Edward: The cost of running a car at an average speed of V km/h is given by c= 100 + (V2 / 75) cents per hour. Find the average speed (to the nearest km/h) at which the cost of a 1000 km trip is a minimum. Answered by Stephen La Rocque. 





The area of a house 
20060628 

From Michael: I would like to know how to measure the area of a house? Answered by Penny Nom. 





Fahrenheit and Celsius 
20060612 

From Doris: I just know how to do problems with fahrenheit or celcius. Could you show me step by step how to do these? Then I can do it if you can show me each step of the way. Answered by Stephen La Rocque and Penny Nom. 





What is the sum of the first 100 whole numbers? 
20060531 

From Jo: what is the sum of the first 100 whole numbers? Answered by Natasha Glydon, Paul Betts and Penny Nom. 





Simultaneous Equations 
20060524 

From Angie:
Question: solve the equations
2x3yz=0
3x2y+z=5
x+3y2z=14
for x,y,z
Answered by Stephen La Rocque. 





differentiate the volume of a cylinder with V respect to h 
20060524 

From A student: differentiate the volume of a cylinder with V respect to h Answered by Stephen La Rocque. 





integral of tan^4 x 
20060514 

From Aqil: integral of tan^{4} x Answered by Penny Nom. 





How many thousands make 1million? 
20060510 

From Raj: How many thousands make 1million? Answered by Penny Nom. 





Rate of ladder falling 
20060430 

From Harsh: A ladder 4 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 30 cm/s, how quickly is the top of the ladder sliding down the wall when the bottom of the ladder is 2 m from the wall? Answered by Stephen La Rocque. 





School bus reliability  a probability question 
20060427 

From Peggy: The school bus arrives at Janet's stop on time on 75% of school mornings. What is the probability it will arrive on time each day in a 5day week? Answered by Stephen La Rocque. 





Three circles inside a larger circle 
20060416 

From Meghan: Given three congruent circles tangent to one another (radii = 1), what is the radius of a circle circumscribed around them?
Answered by Stephen La Rocque. 





2x+5y=3 And x+3y=7 
20060403 

From Lloyd: simplify 2x+5y=3 And x+3y=7
Answered by Penny Nom. 





The centre and radius of a circle 
20060402 

From Kaye: I need to calculate Dimension E and F. I am given A, B, C, (or over all A+B+C), D, G. The radius is one continuous unknown radius.
Example: A = 23.50
B = 35.50
C = 0.50
D = 11.50
G = 23.50
I have calculated this for angles but my mind is drawing a blank for the radius calculation. I can draw it but I need to put into Excel spreadsheet.
Answered by Harley Weston. 





Find the point of inflexion for the curve y = e^x/(x^21) 
20060331 

From Sam: Hi, i am trying to find the point of inflexion for the curve y = e^{x}/(x^{2}1) and i got a really complex expression for y". I can't seem to solve x^{4}4x^{3}+4x^{2}+4x+3=0 so does that mean there is no point of inflexion? Answered by Penny Nom. 





A fence around a pen 
20060330 

From Daryl: I hope you can help me out with the attached problem, It has been driving me crazy. Answered by Stephen La Rocque and Penny Nom. 





Can an equilateral triangle have an obtuse angle? 
20060326 

From Chris: Can an equilateral triangle have an obtuse angle?
I'm thinking not, because all sides must be equal, but
does that also imply that all angles are equal?
Answered by Stephen La Rocque. 





2x+3y=0 and 3xy=0 
20060314 

From Lisa: my name is lisa I am doing math for work and i have a math problem to solve and i need help with it here is the question 2x+3y=0 and 3xy=0 this is one question can you help me please Answered by Penny Nom. 





A nine digit number 
20060306 

From Ryan: What is the total number of possible combinations of a nine digit number (ie., social security number) including repeating numbers? Answered by Stephen La Rocque and Penny Nom. 





The path of a submarine 
20060226 

From Meadow: Suppose that a submarine has been ordered to follow a path that keeps it equidistant from a circular island of radius r and a straight line shoreline that is 2 units from the edge of the island. Derive an equation of the submarine path, assuming that the shoreline has equation x = p and that the center of the island is on the xaxis. Answered by Penny Nom. 





A locus problem 
20060208 

From Jason: If the distance from p(1,5) is 3/4 that of the distance of a(4,3)? find the equation of a locus. Answered by Penny Nom. 





The box of maximum volume 
20060201 

From Elizabeth: A box factory has a large stack of unused rectangular cardboard sheets with the dimensions of 26 cm length and 20 cm width.
The question was to figure what size squares to remove from each corner to create the box with the largest volume.
I began by using a piece of graph paper and taking squares out. I knew that the formula L X W X H would give me volume. After trial and error of trying different sizes I found that a 4cm X 4cm square was the largest amount you can take out to get the largest volume. My question for you is two parts
First: Why does L X H X W work? And second, is their a formula that one could use, knowing the length and width of a piece of any material to find out what the largest possible volume it can hold is without just trying a bunch of different numbers until you get it. If there is, can you explain how and why it works. Answered by Penny Nom. 





how can i find the height of a triangle if i have the base and the hypotenuse 
20060127 

From Kelsey: how can i find the height of a triangle if i have the base and the hypotenuse Answered by Penny Nom. 





The diameter of a pipe 
20060127 

From An other:
I know the base length of my arc (10 inches)  I also know the height at the center to the arc (2 inches). I don't think the end is at the midpoint tho. How do I figure out how long the arc length is?
My question involves being able to cut a round pipe into an arc that is 10 inches wide and 2 inches tall. I need to know the smallest diameter pipe to buy in order to fulfill these requirements.
Answered by Penny Nom. 





Measuring an octagon 
20060126 

From Travis: I am looking to do a project for work where I must find the radius of an octagon but I cannot directly measure it. I've found that on a regular hexagon I can find the radius by using the distance between the bolts to find the radius to the line connecting the bolts but also to the outside of a circle to cut it out. I do not understand however how this works for an octagon. What do I do to find the radius of an octagon with only the ability to measure the distance of the bolts? The center has a cutout in it and is mounted currently and I cannot get accurate measurements.
Answered by Penny Nom. 





One boundary of a pond is parabolic in shape. 
20060120 

From Glenn: One boundary of a pond is parabolic in shape. The boundary passes through the points A(20,45), B(40,40) and E(30,35). The equation of the parabola is of the form y=ax2+bx+c. Find the equation of the parabola and the coordinates of the vertex of the parabola. Any assistance you could provide would be greatly appreciated. Answered by Penny Nom. 





Differentiation, powers and logs 
20060106 

From Claudia:
Question: how do I find the derivative of
x* ln(x+(e^2))^2
x^lnx
x^(e^(x^2))
Answered by Penny Nom. 





Extraneous solutions 
20060101 

From Liz:
Question: solve and check for extraneous solution
3(w + 1)^{1/2} = 6
Answered by Penny Nom. 





Two related rates problems 
20051229 

From Shimaera:
#1. A manufacturer determines that the cost of producing x of an item is C(x)=0.015x^{2}+12x+1000 and the price function is p(x)=250+2x/10. Find the actual and marginal profits when 500 items are produced.
#2. At 9 a.m a car is 10km directly east of Marytown and is traveling north at 100 km/h. At the same time, a truck leaves Marytown traveling east at 70 km/h. At 10 a.m, how is the distance between the car and the truck changing?
Answered by Penny Nom. 





The Mean Value Theorem 
20051222 

From Candace: Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Find all numbers "c" that satisfy the Mean Value Theorem. 11. f(x)=3x^{2} + 2x +5 [1, 1] Answered by Penny Nom. 





Simultaneous Equations 
20051221 

From Matt: I have these two equations,
336 = 60a + 10b
and
432 = 84a + 6b
Am I right in saying both a and b are 4.8? Answered by Penny Nom. 





A maxmin problem 
20051216 

From Julie: A car travels west at 24 km/h. at the instant it passes a tree, a horse and buggy heading north at 7 km/h is 25 km south of the tree. Calculate the positions of the vessels when there is a minimum distance between them. Answered by Penny Nom. 





Mrs. Faria lives on an island 
20051215 

From Julie: Mrs. Faria lives on an island 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. the nearest drug store is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time? Answered by Penny Nom. 





Inclusive definitions 
20051214 

From Layla:
recently the solvable quandary of 5+5+5=550 came up (the question says that you have to put 1 straight line somewhere in the equation to make it true with out turning the "=" into a "not=" sign).
So two answers were put forward:
545+5=550 (the use of a line converting a + into a 4)
AND
5+5+5(less than or equal to)550
There is currently an argument about the second solution. The disagreement is about whether this sign can be used. One person is arguing that the "less than or equal to" sign defines that the number on the left is in the range 550 and below. The other is saying that since the number (which is clearly defined with no variables) can never equal 550, then the "less than or equal to" sign cannot be used in this case.
Which one is the correct definition?
Answered by Walter Whiteley. 





A point is twice the distance from y = 5 + 2x as it is from y = 5  2x 
20051209 

From Hazel: A point moves so that its distance from the line y=5+2x is twice its distance from the line y=52x. Find the general form of the equation of its locus. Answered by Penny Nom. 





Four tangent circles 
20051206 

From Ananth:
I have one bigger circle A with radius 15.
Inside this bigger circle i have another circle B with radius 3 which touch this bigger circle. Have another circle C with radius 4 which touches A and B. I would like to draw a biggest circle which touches A,B and C.
Answered by Chris Fisher. 





Notation for the second derivative 
20051108 

From Mussawar: my question is ^{d}/_{dx}( ^{dy}/_{dx}) = ^{d2y}/_{dx2}. why it is not equal to ^{d2y}/_{d2x}. Answered by Penny Nom. 





Percent or percentage 
20051103 

From Kenneth:
Which word should be used in the following?
Change a (percent or percentage) to a decimal.
Should the word percent be used only when a number precedes it as in 45 percent?
Answered by Harley Weston and Chris Fisher. 





Velocity and acceleration 
20051027 

From Candace: When taking the integral of the position function, you get the velocity function, and the same for velocity to acceleration. So when you do each of these, you get a function. But when you integrate on a graph, you get an area under a curve. The area is un units squared where do the units go when you make it an equation? How can a function be an area? Answered by Harley Weston. 





Can we take the derivative of independent variable 
20051018 

From Mussawar: why we take derivative of dependent variable with respect to independent variable .can we take the derivative of independent with respect to dependent.if not why. Answered by Walter Whiteley. 





How would I find the length of the radius? 
20051015 

From Stace: If given the length of a chord (121") and the distance from the midpoint of the arc to the midpoint of a chord (12"), how would I find the length of the radius?
Answered by Penny Nom. 





Simultaneous equations 
20051013 

From Daniel: 5x + 3y = 22
4x  7y = 20 Answered by Penny Nom. 





Prove that a rhombus' diagonals are perpendicular 
20051002 

From Tania: How do you prove that a rhombus' diagonals are perpendicular using the 2 column proof method? Answered by Walter Whiteley. 





U'(X)  U(X) = 0; U(0) = 2 
20050923 

From David: Out of interest could you please answer the following questions?
U'(X)  U(X) = 0; U(0) = 2
and
U''(X)  U'(X) = 0; U'(0) = U(0) = 2
Answered by Harley Weston. 





A point is moving on the graph of x^3 + y^2 = 1 in such a way that 
20050917 

From Gina: A point is moving on the graph of x^{3} + y^{2} = 1 in such a way that its y coordinate is always increasing at a rate of 2 units per second. At which point(s) is the x coordinate increasing at a rate of 1 unit per second. Answered by Penny Nom. 





How do you differentiate y=(x)^(x^x)? 
20050914 

From Calebius: How do you differentiate y=(x)^{(xx)}? Answered by Penny Nom. 





At what rate is the circumference of the circle increasing? 
20050808 

From John:
A mathematics professor is knitting a sweater. The main part of the sweater is knit in a large spiral, ending up with a diameter of 30 inches. She knits at a constant rate of 6/7 square inches per minute.
1. At what rate is the circumference of the circle increasing when the diameter is 2 inches?
2. How long will it take her to finish this piece of the sweater?
Answered by Penny Nom. 





What is the radius of this planet 
20050805 

From Kelly: Assuming that a NorthSouth line has been established, you set up two camps that are 67 miles apart. You now set up poles at each camp perpendicular to the ground. On a certain day at "noon" the pole at the South camp casts no shadow, while at the North camp a shadow is cast. The shadow makes an angle of 89 degrees with the horizontal. What is the radius, diameter, and circumference of this planet? Answered by Chris Fisher. 





The equation of an ellipse 
20050717 

From Allan: I working on a problem that asks me to give the equation of an ellipse when only the location of the directrix and the length of the latus rectum are given. No other points on the ellipse are given. Again, the only "givens" are:
Length of latus rectum = 12
Location of directrix is x = 16
If I could determine the eccentricity, I could proceed from there by taking the ratio of the distance from a focus to the latus rectum point to the distance of the point from the directrix, but I lack the x coordinate of c. I've searched the text, and feel I've "missed something" somewhere! I note that the latus rectum segment is unique in one respect in that it is parallel to the directrix, where any other line segment on the ellipse to the focus would not be. Please indicate where I'm going wrong. Answered by Chris Fisher. 





A lighthouse is located on a small island,... 
20050714 

From Brittnee: A lighthouse is located on a small island, 3 km away from the nearest point P on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? Answered by Penny Nom. 





The volume of a hopper 
20050528 

From Brian: I would like to know the volume of this rectangular hopper. can you help Answered by Penny Nom. 





Logarithmic differentiation 
20050523 

From Richard: I need to convince myself that I understand the process of
differentiating y=x^{x}.
The specific question is that if I have to take the logarithm of both sides
of the equation how can differentiate the following?
y= {(x+2)^{(x+2)}}/{(x+1)^{(x+1)}}  {(x+1)^{(x+1)}}/(x^{x}),
I have an idea that the differential of this fairly complex function
is itself ... am I right or wrong. Answered by Penny Nom. 





L'hopital's rule 
20050515 

From Abraham: Find the limit of [(1/(x+4))(1/4)]/x as x approaches zero.
How do you use l"hopital's rule to find this limit. I know how to do it with multiplying everything by 4(x+4), and getting the answer, 1/16.But how do you apply derivatives with l'hopitals rule to this type of problem? Answered by Penny. 





A Taylor series for ln(x) 
20050416 

From Anood: i have to represent ln(x) as a power series about 2
i`m not getting the final answer which is ln 2+ sigma (((1)^{(n+1)}/
(n*2^{n}))*(x2)^{n}). i don`t get the ln 2 part
i show you my trial
f(x)= ln x.
f(x)=(1/x) .
f(x)= (1/x^{2})*1/2!
f(x)= (2/x^{3})*1/3!
f(x)= (6/x^{4})* 1/4!
so the pattern shows me that f(n)= ((1)^{(n+1)})/x^{n} *n)
so f(2)= sigma ((1)^{(n+1)})/2^{n} *n) *(x2)^{n}
so as you see i don`t get ln 2
Answered by Penny Nom. 





A torus and a sphere 
20050327 

From Tony: Is it possible to shrink a torus into a sphere? Answered by Andrei Volodin and Penny Nom. 





Dimensions of a roof 
20050318 

From A roofer?: A right triangle (roof of a house) has a base of 7 feet and a 22 degree angle. What is the height of the roof and what is the hypothenus of the triangle. Answered by Penny Nom. 





The square root of 2 
20050312 

From Madhumita: From Pythagoras theorem we can draw square root 2 as a finite distance but it is irrational number which is endless. Explain how we can equate these two. Answered by Harley Weston. 





Each interior angle of a particular polygon is an obtuse angle... 
20050222 

From Victoria: Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have? Answered by Walter Whiteley. 





Extraneous solutions 
20050204 

From Heather: My teacher wants to know why there are extraneous solutions in logarithms? Answered by Penny Nom. 





Solve for x 
20050202 

From Christie: Solve for x
.387 = (.40  .265x)/(sqrt(1x2)) Answered by Penny Nom. 





Differentiating F(x,y) = 0 
20050123 

From Jacob: In calculus, we often mention to the students that if F(x,y) = 0, then we can differentiate both sides and still get an equality. The problem is that we can't perform the same operation on F(x) = 0, say x = 0, otherwise 1 = 0, which is absurd. What is the reason? Answered by Walyer Whileley and Harley Weston. 





The radius of a circle 
20050118 

From A student: find the radius of a circle whose area is 1256sq cm.Use pi as an appoximation for pi. Answered by Penny Nom. 





A line from the center of the patch to the periphery 
20050101 

From Sandrine: I am currently researching a patch disease of grasses. These patches are roughly circular. I need a term for a line from the center of the patch to the periphery. Since the patches are not perfectly circular, my supervisors tell me I cannot use the word 'radius'. What else could I use? Answered by Denis Hanson and Harley Weston. 





Three calculus problems 
20041209 

From Ashley: Hi, I am having a lot of trouble with three calculus questions and was wondering if you could help Answered by Penny Nom. 





Implicit differentiation 
20041024 

From Emily: If x^3+3xy+2y^3=17, then in terms of x and y, dy/dx = Answered by Penny Nom. 





A 40% increase in garage space 
20041020 

From Dianna: A bus company recently expanded and no longer has enough room in its garage for all of its buses. Twelve of the buses have to be stored outside. If the company decides to increase their garage space by 40%, this will give them enough room for all of their current buses, plus enough room to store another twelve in the future. How many buses does the company own? Answered by Penny Nom. 





The hypotenuse of a right triangle 
20040920 

From Shannon: I am trying to find the hypotenuse of a right triangle with only the length of the opposite side. What is the formula as I don't have the length of the adjacent side? Can I compute it without knowing what the other two angles are? Answered by Penny Nom. 





The length of a cut 
20040917 

From Florita: My daughhter, who is a 9th grader is attempting to cut a piece of wood after determining the length of the cut for the hypotenuse. These are the measures:
a=4squared, b=6squared.
She determined that c should equal 52. But when she measured the actual piece to be cut, c measured 39.5 inches! Can you offer any insight as to what she is doing wrong? I have suggested that she may be working with an Acute rather than a Right angle . But she insists that it is a Right angle after using a "framing square". Answered by Claude Tardif. 





1+3+5+...+(2n+1) 
20040910 

From Emma: Prove that 1+3+5+...+(2n+1)= (n+1)^{2} Answered by Penny Nom. 





The radius of a circle 
20040824 

From Peter: If you slice any circle with a line, and call the distance of the line between intersections the "y" length and the perpendicular length to the shorter side of the curve the "x" length, what is the resulting equation for the radius? Answered by Penny Nom. 





A division symbol 
20040818 

From William: i was wondering what the mathematical name for this division sign (÷). Answered by Penny Nom. 





The integrating factor method 
20040805 

From A student: Whilst using the integrating factor method, I am required to integrate a function multipled by another function.
say f(t) = exp(kt) and some other function g(t); where exp = exponential and k is some constant.
Integral f(t)*g(t) dt or
Integral exp(kt)*g(t) dt
What would the result of this integral be? I have never met an integral like this before. Would it simply be exp(kt)*g(t)/k?
More specifically, the problem and my attempted answer is in PDF format:
In my attempted solution, I am unsure about the last two lines I have written out, as it relates to integrating a function multipled by another function. Answered by Harley Weston. 





Integrating e^sin(x) 
20040804 

From A student: I need to know that how to solve the integral " e^sin x", Answered by Harley Weston. 





Differentiation 
20040804 

From A parent: I am a parent trying to understand higher level of maths and would be very grateful if you could help with differentiating the following functions, identifying general rules of calculus:
a) f(x)=e^2^xIn(cos(8x))
b) f(x)=secx/SQRTx^4+1 Answered by Penny Nom. 





Extraneous solutions 
20040728 

From Nicole: When I have a problem like (2x + 3)/5 = (x + 1)/6 and the question asks to check for extraneous solutions, how do you solve that? Answered by Penny Nom. 





An Octagonal playhouse 
20040713 

From Levi: I'm building an octagon playhouse for my son that is 8 feet wide.
what would be the measurements of each of the eight sides. Answered by Harley Weston. 





The circle through three points 
20040706 

From Jim: I am a student trying to solve math problem. I'd like to calculate the radius of the circle that exactly fits any three points. If the points are (X1,Y1), (X2,Y2), and (X3,Y3), what is the radius of the circle that contains those three points? Answered by Penny Nom. 





The sum of some positive integers 
20040607 

From A student: Find the sum of all positive integers not greater than 10000 that are divisible by neither 3 nor 7. Answered by Penny Nom. 





Maximizing the angle to the goal mouth 
20040515 

From Yogendra: You are running down the boundary line dribbling the ball in soccer or hockey. Investigate where in your run the angle the goal mouth makes with your position is at a maximum. Answered by Penny Nom. 





Three dice 
20040510 

From A student: If one has 3, 6 sided dice what is the probability of the numbers that are rolled to total 4 through 10 inclusively?
Subsequent to this, what is the probability to do this consecutively...say 3 times? Answered by Peny Nom. 





Related rates and baseball 
20040426 

From Bethany: A baseball diamond is the shape of a square with sides 90 feet long. A player running from second to third base at a speed of 28 feet/ second is 30 feet from second base. At what rate is the player's distance from home plate changing? Answered by Penny Nom. 





The problem of Apollonius 
20040425 

From Mitja: There are given 2 circles lying one out of another and one point out of both circles. How to construct a circle passing through a given point and internally tangent to one and externally tangent to the other cirlce? Answered by Chris Fisher. 





A changing rectangle 
20040403 

From A student: The width x of a rectangle is decreasing at 3 cm/s,
and its length y is increasing at 5 cm/s. At what rate
is its area A changing when x=10 and y=15? Answered by Penny Nom. 





Some calculus problems 
20040401 

From Weisu:
I have questions about three word problems and one
regular problem, all dealing with derivatives.
 Find all points on xy=e^{xy} where the tangent line
is horizontal.
 The width x of a rectangle is decreasing at 3 cm/s,
and its length y is increasing at 5 cm/s. At what rate
is its area A changing when x=10 and y=15?
 A car and a truck leave the same intersection, the
truck heading north at 60 mph and the car heading west
at 55 mph. At what rate is the distance between the
car and the truck changing when the car and the truck
are 30 miles and 40 miles from the intersection,
respectively?
 The production P of a company satisfies the
equation P=x^{2} + 0.1xy + y^{2}, where x and y are
the inputs. At a certain period x=10 units and y=8
units. Estimate the change in y that should be made to
set up a decrease of 0.5 in the input x so that the
production remains the same.
If you could just give me some hints on these
questions, I'd really appreciate it. Thanks! Answered by Penny Nom. 





A partial derivative 
20040319 

From Penny Nom: Is it possible to differentiate the following equation, if so could
you please explain.
S=SQRT(T(5/X^2))
I would like the derivative of S with respect to X. Answered by Harley Weston. 





The radius of a circle 
20040306 

From A student: what is the radius of a circle with the circumference of 12 inches? Answered by Penny Nom. 





Billions and more! and even more! 
20040301 

From Steph: What comes after undecillion? Answered by Penny Nom. 





Rearranging an expression 
20040224 

From Corey:
use the following formula to answer the question
F=9c divided by 5 plus 32
rearange the formula to solve for c
how many degrees fahrenheit would it be it it was:
0 degrees celcius
10 degrees celcius
46 degrees celcius
Answered by Penny Nom. 





The derivative of x to the x 
20040214 

From Cher: what about the derivative of x to the power x? Answered by Penny Nom. 





A pyramidshaped tank 
20040213 

From Annette: The base of a pyramidshaped tank is a square with sides of length 9 feet, and the vertex of the pyramid is 12 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at a rate of 3 cubic feet per second. Find the rate of change of the depth of water in the tank. (Hint: the volume of a pyramid is V = 1/3 B h , where B is the base area and h is the height of the pyramid.) Answered by Harley Weston. 





Some trig problems 
20040118 

From Weisu:
I have some questions about precalculus.
(1) (2(cos(x))^2)+3sin(x)1=0
(2) sin(x)cos(x)=(1/2)
(3) 3sin(x)=1+cos(2x)
(4) tan(x)*csc(x)=csc(x)+1
(5) sin(arccsc(8/5))
(6) tan(arcsin(24/25))
(7) arccos(cos(11pi/6))
the last problem uses radian measure.
Answered by Penny Nom. 





Unusual occurances 
20040108 

From Martin: My wife and I have a question about the probability of something that happened to us a few years ago. So far, no one has been able to give me even an approximate answer. On my 32nd birthday, my wife and I went out to eat at local Japanese hibachi style restaurant. At the restaurant, couples/families are sat together around the hibachi where the cook performs a show. There was a fifteen minute or so wait, so my wife and I sat in the lounge waiting for our name to be called. When they called our names for the reservation, this is what happened. The first group called was the Martin family. Then they called the Francis family. We were the next family to be called, the Ashton family. My full name is Martin Francis Ashton! I think the odds of that happening to someone are very unlikely, but it did, and there is more. Next, we were all sat at the same table in that order, "Martin" family, "Francis" family, then us, the "Ashton" family. Again, it formed my full name! Answered by Penny Nom. 





Business trip 
20031219 

From Ameer: A businnessman drives from Washington, D.C., to Boston, a distance
of 442 miles, and then makes the return trip. On the way to Boston,
he drives 65 miles per hour, taking an 1hour rest stop during the
drive. After finishing his business in Boston, he make the return
trip driving at 60 miles per hour and takes a 45minute rest stop
halfway through the trip. Which leg of the journey, Washington, D.C.
to Boston, or Boston to Washington, D.C., takes the longer time? Answered by Penny Nom. 





A locus 
20031202 

From Tash:
Question:
a)Find the equation of the locus of the point P which moves so that its distance from A(1,2) is always three times its distance from B(5,6)
b) Show that this locus is a circle and states the coordinates of its centre and the length of its radius
Answered by Penny Nom. 





A riddle 
20031119 

From Sarah: Ok, our teacher gave us this riddle, and I cannot for the life of me figure it out. He said that there are three problems with the following proof: Answered by Penny Nom. 





Symmetries of a rhombus 
20031102 

From Tonia: why cant an equal sided rhombus have 3 lines of symmetry? you have one line of symmetry on each of the diagonals, and there should be one vertically on an angle. can you please explain the rules of symmetry to me? Answered by Walter Whiteley. 





The sketch of a graph 
20031007 

From A student: I was wondering how do you figure out if a graph has a horizontal tangent line. One of my homework problem was to sketch the graph of the following function; (4/3)x^{3}2x^{2}+x. I set f''(x) ( the second derivative) of the function equal to zero and got the inflection point:(1/2,1/6). Also i am having trouble finding the concavity for x>1/2 and x<1/2, i am getting a different answer from the back of the book, the graph i draw looks completely different from the correct answer. Answered by Penny Nom. 





Indeterminate forms 
20031006 

From A teacher: Is it possible for me to find any geometrical interpretation without using calculus to explain indeterminate forms? Answered by Chris Fisher. 





Functions, graphs and derivatives 
20031005 

From Jathiyah: I wanted to know how would you tell (on a graph diplaying two funtions), which funtion is the derivative of the other? Answered by Walter Whiteley. 





The slope of a tangent 
20031001 

From A student:
find the slope of the tangent to each curve at the given point f(x)=square root 16x, where y=5 Answered by Penny Nom. 





The mean house price 
20030910 

From Carol:
Question: I have to find the mean from the following example: Price Range £000  No of Houses  55 and under 60  3  60 and under 65  6  65 and under 70  13  70 and under 80  21  80 and under 100  15  100 and under 130  7  130 upwards  1  I know when calculating the mean you use the mid points of the classes, but how does this work for the 130 upwards class? Also, does this still work given the difference in the classes (ie. 1st class is 5, 5th class is 20, etc). Any help to get me started would be greatly appreciated. Answered by Penny Nom. 





A helicopter rises vertically 
20030902 

From Kate: A helicopter rises vertically and t seconds after leaving hte ground its velocity is given in feet per second by v(t) = 8t + 40 / (t+2)^{2} How far above the ground will the helicopter be after 3 seconds? Answered by Penny Nom. 





Two precalculus problems 
20030804 

From Kate:
Please help me verify the identity: cos2x(sec2x1)=sin2x Also I am having trouble withdetermining whether f(x) is odd, even, or neither f(x)=x3x Answered by Penny Nom. 





Natural logarithms 
20030722 

From Amanda: I'm going into my senior year of high school. I will be taking AP calculus, and my teacher gave us some homework over the summer. However, there are two things that I do not understand how to do. The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles. I have used the unit circle in trigonometry, however I was never taught how to draw it. Secondly, I need to know how to do natural logarithms without a calculator. I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator. Answered by Claude Tardif. 





Odd powers of sine and cosine 
20030625 

From Antonio: Can you please tell me how to integrate a trig function involving sine and cosine? I know if the powers of both the sine and cosine are even and nonnegative, then I can make repeated use of the powerreducing formulas. But for the question I have on my hand, the powers of both sine and cosine are odd: ( sin3x + cos7x ) dx. Answered by Harley Weston. 





Circumference 
20030509 

From A parent: Find the circumference use 3 1/7 for pi
1. r= 28 ft. 2. D=98 cm Answered by Penny Nom. 





Integrating e^x sin(x) 
20030503 

From Lech: I am having trouble integrating the following expression by parts: ex sin(x) I used the integrator at http://www.integrals.com/ to find the solution, ? 1/2 ex cos(x) + 1/2 ex sin(x). This is easy to confirm by differentiation, however I am confounded as how to arrive at the answer. Answered by Penny Nom and Claude Tardif. 





The volume of air flowing in windpipes 
20030502 

From James: The volume of air flowing in windpipes is given by V=kpR^{4}, where k is a constant, p is the pressure difference at each end, R is the radius. The radius will decrease with increased pressure, according to the formula: R_{o}  R = cp, where R_{o} is the windpipe radius when p=0 & c is a positive constant. R is restricted such that: 0 < 0.5*R_{o} < R < R_{o}, find the factor by which the radius of the windpipe contracts to give maximum flow? Answered by Penny Nom. 





The square of my age was the same as the year 
20030414 

From Pat: Augustus de Morgan wrote in 1864, "At some point in my life, the square of my age was the same as the year." When was he born? Answered by Penny Nom. 





Uses of conic sections 
20030401 

From William: My name is William and I am doing a research paper on conic sections for my 12th grade math class. Part of the project is to find two conic sections in our world today and explain what there purpose is. I really need help in this area because I've been searching the internet for where conic sections are used in our world today and I really can't find anything. If you can tell me specific building or a pyramid that contains conic sections that would be great. Or even something in the universe would be helpful. Answered by Leeanne Boehm. 





A royal flush 
20030324 

From Vikki:
A poker hand consists of 5 cards selected randomly from an ordinary deck of cards: find the probability of a ROYAL FLUSH : the 10 , jack, queen,king and ace of the same suit. I was thinking somewhere along the lines of: *the number of ways to get the suit is 4C1 *the number of ways to get a 10 out of the 13 cards etc.... ...but Im not sure I am going about this the right way, could you help? Answered by Andrei Volodin. 





Surface area of a sphere 
20030311 

From Kim: a sphere has a surface area of 128 pi sq. units. What is its exact radius? formula is 4 pi r^{2} I believe but how do I get radius Answered by Penny Nom. 





Can a square be a rhombus? 
20030304 

From Beth: Can a square be a rhombus? Some sources say yes, some say no. Some sources define a rhombus as a quadrilateral and parallelogram with equal sides, but without right angles. Some sources say a square is a special case of a rhombus. Clarity, please! Answered by Walter Whiteley. 





Hundreds, thousands, millions,... 
20030219 

From Karissa and Jasmeen: hundreds, thousands, millions,  can you help us with the rest of this sequence  we are trying to find the largest number Answered by Penny Nom. 





Wrap a rope around the equator 
20030212 

From Ali: If you take a rope and wrap it around the equator and then overlap it 6ft. and cut off the extra then you loosen it up so the ends meet how far would it stand off the ground? Answered by Penny Nom. 





Extraneous solutions 
20030124 

From Paul:
What is an extraneous solution and in what cases do you get one? How do you know it is extraneous? Answered by Penny Nom. 





Integration of 1/(2+cos(x)) 
20030107 

From A student: integral from pi to 0 of dx/(2+cos x) i used the substitution t=tan(x/2) and i ended up with integral from +infinity to 0 of 2dt/(t^{2}+3) which looks like an inverse tan function , and i ended up with sqr(27)/2 pi , which is not the same as my calculator's answer , so i suspct i am doing some thing wrong. can some one tell me where i am going wrong please. Answered by Penny Nom. 





Constructions of polygons 
20030103 

From Garrett: Our teacher just finished the constructions unit, and he mentioned briefly about odd sided figures such as pentagons and septagons, only that they're very hard. My question is, how do you draw, with a compass and a straight edge, a pentagon and septagon? Answered by Chris Fisher. 





How many billions equal one trillion? 
20021207 

From Ryan and Aylah: How many billions equal one trillion? We believe that the answer is one thousand times one billion equals one trillion. Please help us with the answer to this question. Answered by Penny Nom. 





The length of an arc 
20021127 

From Nancy: If all I have is the length between 2 ends of an arc (72"), how do I find the length of the arc at its apex and the radius? Answered by Penny Nom. 





Differentiating inverses 
20021120 

From Amy: f(x)= x^{3}+x+1, a=1 find g'(a) (g = f^{ 1}). I am having trouble finding g(a). Answered by Penny Nom. 





Round to hundredths 
20021119 

From Brittany: Can u tell me how to do a problem like this:
35 divded by 4.8= Round to hundreths. Answered by Penny Nom. 





A bus is 60% occupied 
20021109 

From Joe: A bus with a seating capacity of 60 people is 60% occupied. At the next stop onethird of the people get off the bus and 3 people get on the bus. The bus is now ___% occupied. Answered by Penny Nom. 





Mathematics and Music 
20021101 

From Hannah: I am looking for a science fair project to compare math and music and how they relate. If you have any project ideas for me, they would be greatly appreciated. Answered by Walter Whiteley. 





How would you find the length of the chord? 
20021031 

From A draftsperson: If given the length of an arc and the distance from the midpoint of the arc to the midpoint of a chord, how would you find the length of the chord and the radius of the arc. The chords endpoints are the same as the the arcs endpoints. Answered by Penny Nom. 





Nixon, Jefferson, and Madison 
20021008 

From Lisa: The longestlived US presidents are John Adams(age90), Herbert Hoover (also90), and Harry Truman (88). Behind them are James Madison, Thomas Jefferson, and Richard Nixon. The lattter three men lived a total of 249 years, and their ages at the time of death form consecutive odd integers. For how long did Nixon, Jefferson, and Madison live? Answered by Penny Nom. 





A max/min problem 
20020921 

From Evelina: A window is the shape of a rectangle with an equilateral triangle on top. The perimeter of the window is 300 cm. Find the width that will let the maximum light to enter. Answered by Penny Nom. 





The entire earths' population would fit in the state of Texas 
20020918 

From Roz: I have been told that the entire earths' population would fit in the state of Texas and each person would have 1/2 acre. Is this true. Answered by Chris Fisher. 





A Circle is evenly divided into six equal triangles 
20020916 

From Marilynn: A Circle is evenly divided into six equal triangles leaving an area between the outside of the circle and the one side of the triangle. This area is measured as 3.14. What is the length of the radius, one line on the triangle? Answered by Paul Betts. 





Sums of evens 
20020914 

From Rosa: How do I find a geometric way to easily compute sums of consecutive even numbers 2 + 4 + 6 + .... Answered by Leeanne Boehm and Harley Weston. 





The sum of the first one hundred even numbers 
20020910 

From Arthur: What is the sum of the first one hundred even numbers? Answered by Penny Nom. 





Two equations 
20020726 

From Derek: 1. 3x + 2y = 4 2. 7x + 2y = 24 finding x and y. Answered by Penny Nom. 





Musical Scales 
20020724 

From Terence: Given that there are 12 notes in a musical octave, what is the maximum number of musical scales possible within that octave, if each scale has a minimum of 5 notes and a maximum of 9 and we start all the scales from the same note? In case you don't know anything about music, a scale is a progression of notes where you start on a specific note and end on that same note an octave higher. There are twelve different notes between these two similar notes. Which notes you choose to play determine the sound of the scale. Anything less than five notes would not make for a very interesting scale. Anything more than nine and you would be playing almost 'every' note in the scale, not leaving much room for distinction in how you organize these notes. I assume you first have to figure out the maximum number of variations possible in a 5note scale (with 12 notes at your disposal). Then do the same for a 6note scale, then a 7note, then an 8note, and so on. Then add up the results. How to find this maximum number of variations for each scale size though is what I don't know. Answered by Leeanne Boehm. 





Integrating x^x 
20020618 

From Jeremy: I am a student at the University of Kansas and I am wondering if there is a general antiderivative for x^{ x} (i.e. the integral of x^{ x} dx)? I've looked in a bunch of Table of Integrals and have found nothing (can you guys find it?), so I'm sort of wondering if this may be a research type question. Answered by Claude Tardif. 





A good rule of thumb when driving 
20020613 

From Lisa: A good rule of thumb when driving is that you should be about one car length away from the car in front of you for every 10 miles per hour that you are travelling. Suppose you follow this rule perfectly (so you are exactly the correct distance away). You are waiting at a stop light with your front bumper just touching the car in front of you. The light turns green and the car in front accelerates at a constant rate "r". Calculate how you should accelerate in order to follow the rule. Answered by Penny Nom. 





Overlapping circles 
20020529 

From Naman: There are two circles, big circle with radius R and small one with radius r. They intersect and overlap in such a way that the common area formed is 1/2 pi r^{ 2} (half the area of the small circle) If r=1, find the Radius of the big circle (R)? Answered by Harley Weston. 





A spotlight shines on a wall 
20020525 

From Barb: A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight toward the bldg at a speed of 1.6 m/s, how fast is his shadow on the bldg decreasing when he is 4m from the bldg? Answered by Penny Nom. 





What is Calculus About? 
20020513 

From A student: I am a high school senior and have to write an essay answering the question "What is Calculus?" I need some ideas. Thanks Answered by Walter Whiteley. 





The law of cosines and obtuse angles 
20020509 

From Bryant: The question that I am pondering is that I need to derive the law of cosines for a case in which angle C is an obtuse angle. Answered by Penny Nom. 





A rectangular marquee 
20020507 

From Alyaa: a marquee with rectangular sides on a square base with a flat roof is to be constructed from 250 meters square of canvas. find the maximum volume of the marquee. i find this topic so hard Answered by Harley Weston. 





How will I use calculus in my career? 
20020506 

From Meridith: How will I, hopefully a future secondary mathematics teacher, use calculus in my career if I'm not teaching calculus? Answered by Walter Whiteley. 





Arithmetic progressions 
20020424 

From David: I have been searching everywhere for the formula to mathamatical progression. Answered by Penny Nom. 





Arc length 
20020417 

From Vix: Find the point on the curve r(t)=(12sint)i(12cost)j+5tk at a distance 13pi units along the curve from the point (0,12,0) when t=0 in the direction opposite to the direction of increasing arc length. Answered by Harley Weston. 





Related rates 
20020417 

From Molly: A tanker spilled 30 ft cubed of chemicals into a river, causing a circular slick whose area is expanding while its thickness is decreasing. If the radius of the slick expands at the rate of 1 foot per hour, how fast is them thickness of the slick decreasing when the area is 100 feet squared? Answered by Penny Nom. 





Pairs of equations 
20020404 

From A student:
high school level student is asking
y=4x x=4y
x+y=5 3x+2y=20
y=x1 3xy=4
x+y=3 2x3y=9
x+5y=4
3x+15y=1
. . .
Answered by Penny Nom. 





Some 5 card hands 
20020328 

From A student: From a standard deck of cards how many 5 card hands are possible consisting of a. exactly 4 hearts
b. two cards of one kind and three of another(like a full house). Answered by Penny Nom. 





The slope of a tangent line 
20020304 

From Ridley: Suppose a function f(x) has the line 3x+4y=2 as its tangent line at x=5. Find f'(5). Answered by Harley Weston. 





The substitution method 
20020224 

From Joe: whats the answer to this question? 3x+y=11 x+2y=3 its substitution method i am having alot of trouble figuring it out. send the answer as soon as possible. thank you Answered by Penny Nom. 





Alfredos house number 
20020221 

From Aunt Patty: Alfredos house number is between 20 and 35. The sum of the digits is less than 5. If you subtract 1 from it you would get a multiple of 3. If you add three, you get a multiple of 5. What is Alfredos house number. Answered by Penny Nom. 





Diameter of a pipe 
20020216 

From Landry: I am trying to calculate the dia. of a pipe 60 inches long that will hold a gallon of water. What is the formula? Answered by Penny Nom. 





Getting to B in the shortest time 
20011219 

From Nancy: A motorist in a desert 5 mi. from point A, which is the nearest point on a long, straight road, wishes to get to point B on the road. If the car can travel 15 mi/hr on the desert and 39 mi/hr on the road to get to B, in the shortest possible time if...... A.) B is 5 mi. from A B.) B is 10 mi. from A C.) B is 1 mi. from A Answered by Penny Nom. 





Simultaneous equations 
20011217 

From Matthew: 4x + y = 4 2x  3y = 5 what is x and y Answered by Penny Nom. 





Two equations in two unknowns 
20011204 

From Courtney: y = 3x + 2 y = 4x  5
solve for x Answered by Penny Nom. 





3, 6, 10, 15, 21 
20011129 

From Patrick: we are trying to find the expression to solve for the nth term in the pattern 3, 6, 10, 15, 21 Answered by Denis Hanson. 





A lighthouse and related rates 
20011129 

From Melissa: A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline, and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? Answered by Penny Nom. 





A tangent line 
20011121 

From A student: write an equation of the line tangent to the graph of
e^{y} + ln(xy) = 1 + e at (e,1) Answered by Harley Weston. 





Gallons 
20011119 

From Shawn: Do we use British or American gallons in Canada? Answered by Chris Fisher. 





Asymptotes 
20011109 

From Frank:
given the function: f(x) = (x^{2}) / (x1) the correct answer to the limit of f(x) as x approaches infinity is: y = x+1 all math references point to this answer and the method they all use is long division of x1 into x^{2} however if one were to multiply both the numerator and denominator by 1/x and then take the limit, one gets: y=x how can the descrepency between the two answers be explained? Answered by Chris Fisher and Penny Nom. 





A lighthouse problem 
20011102 

From A student: A lighthouse at apoint P is 3 miles offshore from the nearest point O of a straight beach. A store is located 5 miles down the beach from O. The lighthouse keeper can row at 4 mph and walk at 3.25 mph.
a)How far doen the beach from O should the lighthouse keeper land in order to minimize the time from the lighthouse to the store?
b)What is the minimum rowing speed the makes it faster to row all the way? Answered by Harley Weston. 





Three problems 
20011028 

From Brenda:
 Joseph is planting bushes around the perimeter of his lawn. If the bushes must be planted 4 feet apart and Joseph's lawn is 64 feet wide and 124 feet long, how many bushes will Joseph need to purchase?
 The cost of a long distance phone call is $1.50 for the first two minutes and $0.60 for each additional minute. How much will Maria pay for a 24 minute phone call?
 Find the next three numbers in the pattern. 1,3,7,15,31,___,____,___.
Answered by Penny Nom. 





Concavity of f(g) 
20011025 

From Troy: Suppose f & g are both concave upward on (infinity,infinity). Under what condition on f will the composite function h(x)= f(g(x)) be concave upward? Answered by Walter Whiteley. 





Tenths, hundredths, and thousandths 
20011017 

From Cheri: I would like the French terms for the decimal place values of tenths, hundredths, and thousandths. (I would be interested in the Spanish terms also if you had them.) Answered by Maria Torres and Claude Tardif. 





Odd plus even is odd 
20011014 

From James: Why is the sum of an odd number and an even number always odd? Answered by Peny Nom. 





Maximize the area 
20011013 

From Mike:
I have no clue how to do this problem. Here is what the professor gave to us: A=LW
C=E(2L+2W) + I(PL) Where P = # of partitions E= cost of exterior of fence I = cost of interior of fence C = total cost of fence . . . Answered by Harley Weston. 





4 sinx cosy = 1 
20011010 

From A student: How would i differentiate the following example in terms of t (x and y are functions of t) 4 sinx cosy = 1 Answered by Claude Tardif. 





Locusts 
20011008 

From A parent: JOHN CONJECTURED THAT BOTH 13YR.&17YR. LOCUSTS CAME OUT THE SAME YEAR. ASSUME IT IS CORRECT. HOW MANY YEARS PASS BETWEEN THE YRS. WHEN BOTH 13 YR. & 17YR. LOCUSTS ARE OUT AT THE SAME TIME? EXPLAIN. NEXT, SUPPOSE THERE WERE 12YR.,14 YR.,& 16YR. LO CUSTS,& THEY ALL COME OUT THIS YR. HOW MANY YRS. WILL IT BE BEFORE THEY ALL COME OUT TOGETHER AGAIN? Answered by Leeanne Boehm. 





The height of the lamppost 
20011002 

From Werner: I am working on question 51,section 3.7 ,page 191 of Stewart's Single Variable Calculus. The question involves a lamp post which is casting a shadow around the eliipse whose formula is x^{2} + 4*y^{2} = 5. I have found the derivative of the elllipse both explicitly: x/4(((5x^{2})/4)^{0.5}) and implicitly : y' =  x/(4*y). Answered by Harley Weston. 





(x^25x6)/(x6) 
20011002 

From Bill: given f(x) = (x^{2}5x6)/(x6) find f'(6). Answered by Harley Weston. 





1+2+3+...+1000 
20011001 

From Louise: Find a quick way to add all Intergers (whole Numbers) between 1 and 1000? Answered by Walter Whiteley. 





Sharing a donut 
20010906 

From Amanda: You have invited 11 people over to your house one day, and your friends are hungry. You go into the kitchen and find out you only have 1 donut (with a hole in the middle). In order to feed 12 people (including you), you must cut the donut into 12 pieces with only using a straight knife and cutting 3 times. This is NOT a trick question. Answered by Claude Tardif. 





The radius of a planet 
20010730 

From Jessica: A satellite is orbiting the earth at an altitude of 100 miles. If the angle of depression from the satellite to the horizon is 50 degrees, what is the radius (to the nearest mile) of the planet? Answered by Harley Weston. 





The Mean Value Theorem 
20010723 

From Corrie: I need to find if the mean value theorem exists. and if so, find all values c guaranteed by the theorem. f(x) = x^{2}25 on the interval [10,0] Answered by Harley Weston. 





Rhombus 
20010716 

From William: Calculate the internal angles of a rhombus given measurments of all four sides only. Answered by Walter Whiteley. 





Area between curves 
20010613 

From Phil:
question 1 find the area bound by the curves y = x^{2} + 2x + 3 and y = 2x + 4 question 2 Find the volume generated by rotating the curve x^{2} + y^{2} = 9 about the xaxis Answered by Harley Weston. 





Mutually exclusive 
20010605 

From Marje: What does the mathmatical term "mutually exclusive" mean. Pleas diagram if possible. Answered by Penny Nom. 





Common solution 
20010602 

From Samantha:
 Solve for common solution: x+y=6 2x3y=2
 Solve for y in terms of x: 3xy=4
Answered by Penny Nom. 





National consumption function 
20010509 

From Brian: If consumption is $11 billion when disposable income is 0 and the marginal propensity to consume is dC/dy = 1/(2y+4)1/2+0.3(in billions of dollars), find the national consumption function. Answered by Harley Weston. 





Maximize profit 
20010509 

From Brian: The marginal cost for a certain product is given by MC = 6x+60 and the fixed costs are $100. The marginal revenue is given by MR = 1802x. Find the level of production that will maximize profit and find the profit or loss at that level. Answered by Harley Weston. 





The average value of a continuous function 
20010508 

From Esther: The average value of a continuous function y = f(x) on the interval [a,b] is given by ________________? Answered by Harley Weston. 





A Taylor series 
20010427 

From Karan: Given the following information of the function  f''(x) = 2f(x) for every value of x
 f(0) = 1
 f(0) = 0
what is the complete Taylor series for f(x) at a = 0 Answered by Harley Weston. 





Oil revenue 
20010421 

From Brian: Suppose that t months from now an oil well will be producing crude oil at the rate of r(t), not a constant, barrels per month and that the price of crude oil will be p(t), not a constant, dollars per barrel. Assume that the oil is sold as soon as it is extracted from the ground.  Find an expression for the total revenue from the oil well, R(t).
 A certain oil well that currently yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $18 per barrel and is expected to rise at a constant rate of 3 cents per barrel per month. What will be the total revenue from this well? {Hint: Model the degraded production rate with the equation:
r(t) = (ABt)e^{0.04t}} Answered by Harley Weston. 





Differentiation 
20010417 

From Esther: Could you please tell me what the first derivative is of the following: y = 2/(2x+e^{2x}) Is it (1+xe^{2x})/(2x+e^{2x})^{2} or perhaps 4(1+e^{2x})/(2x+e^{2x})^{2} ? I am a little confused between the two! Answered by Harley Weston. 





Integration by parts 
20010409 

From A student: how do you integrate x tan^{1}x dx, i know it can be done by integration by parts maybe, but i'm not sure.... Answered by Claude Tardif and Harley Weston. 





The domain of a function 
20010408 

From Mina: Let f(x) = (2x^{2}+3x17)/(x+5) What is the domain of f? What are the values of x for which f'(x) does not = 0? Answered by Harley Weston. 





The normal to a curve 
20010408 

From Varenne: I am having SO much trouble tackling this question and don't know what the right answer is... can you help me out? The question is
Find the equation of the normal to the curve y=(x2)^{2}/(1x)^{2} that is parallel to the line x+4y+7=0 Answered by Harley Weston. 





Common tangents 
20010408 

From Anne: I have been working on this problem for a while but I'm not sure I'm getting the right answer: Find the common tangents of 2y=x^{2} and 2y=x^{2}16 Thanks for the help. :) Answered by Harley Weston`. 





Where do the lines y=2x4 and y=x1 intesect? 
20010406 

From Bryce: solve the following problem by setting them equal to each other. Solve for x and y. Where do the lines y=2x4 and y=x1 intesect? Answered by Penny Nom. 





12 RTV's 
20010327 

From Christine:
 In order to promote the 2000 Census and encourage participation, 12 Road Tour Vehicles(RTV's) set out from 12 locations across the US for a 10 week promotion tour last February. The RTV's reportedly traveled a total combined distance of 97,911 miles. What was the aberage number of miles traveled in a day by each RTV?
 2. The 12 RTV's mentioned above traveled a total combined distance equivalent to traveling 12.35 times around the earth at the equator. To the closest mile, what is the radius of the earth? Use 3.14 as your approximation for pi.
Answered by Leeanne Boehm. 





Airflow in windpipes 
20010325 

From Ena: The volume of air flowing in windpipes is given by V=kpR^{4}, where k is a constant, p is the pressure difference at each end, R is the radius. The radius will decrease with increased pressure, according to the formula: Ro  R = cp, where Ro is the windpipe radius when p=0 & c is a positive constant. R is restricted such that: 0 < 0.5*Ro < R < Ro, find the factor by which the radius of the windpipe contracts to give maximum flow? Answered by Harley Weston. 





A suspension bridge 
20010324 

From Janna: The cables of a suspension bridge hang in a curve which approximates a parabola. The road bed passed through the vertex. If the supporting towers are 720m apart and 60m high, find: a) an equation of the parabola (it's y = ^{1}/_{2160}x^{2}) b)the height of the cables at a point 30m from the vertex. I substituted 30 in for the x value and got 0.42 and the answer is 42. What did I do wrong? Answered by Denis Hanson and Claude Tardif. 





The repetend in repeating decimals 
20010321 

From Sharon: What is the name for the bar over the repetend in repeating decimals? Also, what is the name of the long division "house"? Answered by Penny Nom. 





Systems of equations 
20010316 

From joy: How do u solve problems using systems of equations? ~ finding x and y~ ex:
26 = 3x  2y 42 = 4x + y Answered by Penny Nom. 





A jogger 
20010312 

From Bill: At time t=0 a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly propotional to time t. This brings the jogger to a stop in 10 minutes. a) write an expression for the velocity of the jogger at time t. b) what is the total distance traveled by the jogger in that 10minute interval. Answered by Harley Weston. 





Two locus problems 
20010308 

From Janna: A point P moves such that it is always equidistant from the point G(2,5) and the line defined by y=3. Find the equation of the locus. I got as far as the equation: 3y^{2} 4y = x^{2} + 4x  16 and didn't know what to do from there. Of, course that whole equation could be wrong. Question 2: P is always twice as far from A(8,0) as it is from B(2,0). Find the equation of the locus. Once again, I got as far as y^{2} = x^{2} 8x 56, and got stuck. Answered by Harley Weston. 





The substitution method 
20010305 

From A student: Solve each system of equations by the substitution method. Show your work.  y = 8
7x = 1  y
 y = x  1
4x  y = 19 Answered by Penny Nom. 





The domain of the derivative 
20010222 

From Wayne: I know that the domain of f'(x) is a subset of the domain of f(x). Is it necessarily true that the subset will have at most one less element than the domain of the original function? Answered by Harley Wesston. 





Differentiation of y = x^{ n} 
20010217 

From Jashan: i am studying differentation at the moment i have drawn some graphs such as y=x^{ 2}. i have found the formula for the gradient of this curve, this being 2x obtained by using differentation, but i need to know the general case for the formula where y=x^{n } in order for me to understand this topic more throughly, i would also like to know how u derived this general formula Answered by Harley Weston. 





A quartic equation 
20010215 

From George: Let P(x) = x^{4} + ax^{3} + bx^{2} + cx + d. The graph of y = P(x) is symmetric with respect to the yaxis, has a relative max. at (0,1) and has an absolute min. at (q, 3) a) determine the values for a, b c, and d using these values, write an equation for P(x) b) find all possible values for q. Answered by Harley Weston. 





Find an exprression for f(x) 
20010207 

From A 12th grade AP Calc student: Let f be the function defined for all x > 5 and having the following properties. Find an expression for f(x). i) f^{ ''}(x) = 1/ (x+5)^{1/3} for all x in the domain of f ii) the line tangent to the graph of f at (4,2) has an angle of inclination of 45 degress. Answered by Harley Weston. 





1 + 1 = 1 
20010123 

From Stephanie: My friend has this as a bonus question the other day and I want to figure it out. I don't know how 1+1 in any form could equal 1. Please let me know how you come about geting that. Answered by Claude Tardif. 





The hypotenuse of a right triangle 
20010122 

From Phillipe: How do you find the hypotenuse of a right triangle? Answered by Penny Nom. 





Polynomials and exponents 
20010115 

From A student: I am duing a project in math on polynomials and exponents. I need a real life usage of polynomials and exponents for my project. Answered by Penny Nom. 





Height of the lamp 
20001231 

From Joey: The figure shows a lamp located three units to the right of the yaxis and a shadow created by the elliptical region x^{2} + 4y^{2} < 5. If the point (5,0) is on the edge of the shadow, how far above the xaxis is the lamp located? Answered by Harley Weston. 





Bush fractals 
20001230 

From Anita Wisecup: My son has a report due on fractals. He needs information on bush fractals, but we cannot seem to find anything out about them. Answered by Penny Nom. 





How do you integrate secant(theta)? 
20001222 

From Robert Williamson: How do you integrate secant(theta)? I know the answer is ln [sec(theta) + tan(theta)] but how do you get there? Answered by Claude tardif. 





A limit using l'hopital's rule 
20001213 

From Wassim: I need to know how to solve the: limits of (x ( to the power lamda) 1 )/LAMDA when lamda tends to zero ( the answer is that the functional form is ln x) and I still don't know how using hopital rule leads to this answer. Answered by Harley Weston. 





Optical illusions 
20001206 

From Jessica: Hi, my name is Jessica, 7th grade, and Im doing a Math Fair project on optical illusions. As one of the required factors, we need a "mathmatical significance" paragraph. Unfortunatley, I can only think of one way that optical illusions have to do with math, and thats time because some optical illusions tell you to look at the picture for a certain amount of time. Answered by Penny Nom. 





A nonintegrable function 
20001203 

From Mark Spilker: I have a proof that I cannot do here it goes. Let F(x)= 1 if x is a rational number 0 if x is an irrational number Prove the function is not intregrable on the interval (0,1). Hint: Show that no matter how small the norm of the partition, the Riemann Sum for the SigmaNotation F(w_{i}) Delta_{i}X is not unique. Answered by Harley Weston. 





Comparing an integral and a sum 
20001121 

From Douglas Norberg: A fellow teacher asked me about a problem she wanted to give to her students. It involved whether to take a million dollars or a penny doubled a number of times. I was able to determine the number must have been .01 * 2^{30} which is about $10 million and a lot more than $1 million. To check that I was right I used a spreadsheet and did a Riemann sum. When I finished I reasoned that I had done the task in several steps and I could have done it in 1 step. Thus I integrated .01 * 2^{x} from 0 through 30 but the number I got was $15,490,820.0324. Why the difference? Answered by Harley Weston. 





Bridges and parabolas 
20001118 

From Lauren: My name is Lauren, and Im a secondary school student in Ontario. For my gr11 advanced math class I have to find out how and why parabolics are used in arch bridges and write 3 paragraphs on it. People who cohse satelites and whatnot are lucky  I've found a ton of info, but for arch bridges there seems to be nothing. Answered by Harley Weston. 





Inscribing a circle in a rhombus 
20001116 

From Jacky: A rhombus ABCD is drawn in which the diagonals are 12 and 20 units long. A circle is inscribed in the quadrilateral with the centre of the circle right on the intersection point of the 2 diagonals. The circumference of the circle touches all 4 sides of the rhombus. Is it possible to find the radius of the inscribed circle? If so, how and what is it? Answered by Chris Fisher. 





Rhombuses 
20001107 

From Melissa: What in real life is the shape of a rhombus? Answered by Chris Fisher and Walter Whiteley. 





Concavity 
20001022 

From Alex: the question is: on what interval is f(x)=(x^{2})(e^{x})? ive found the 2nd derivative which is e^{x}(x^{2}+4x+2) and i did the quadratic to get 22^{0.5} and 2+2^{0.5}, but i dont know what the interval is. Answered by Harley Weston. 





A chord length 
20001017 

From Al Paas: How to determine the length of a chord given the diameter of the circle and the maximum distance from the chord to The circle? Answered by Chris Fisher. 





Pillows and Cushions 
20000927 

From Fiona:
The following problem was given to grade eleven algebra students as a homework assignment. To manufacture cushions and pillows, a firm uses two machines A and B. The time required on each machine is shown. Machine A is available for one full shift of 9.6 hours. Machine B is available for parts of two shifts for a total of 10.5 hours each day. Answered by Harley Weston. 





Dividing a region in half 
20000921 

From Kerry: There is a line through the origin that divides the region bounded by the parabola y=xx^{2} and the xaxis into two regions with equal area. What is the slope of the line? Answered by Penny Nom. 





A cycloid in Cartesian form 
20000920 

From Billy: The parametric equation of cycloid is given: x=r(tsint) y=r(1cost) How to eliminate t? Answered by Harley Weston. 





A proof that 1=2 
20000919 

From sporky: Why does the proof for 1=2 not work? x = 1 x^{2} = 1 x = x^{2} 1 = 2x (derivitive) 1 = 2(1) 1 = 2 ??? please tell me where the false logic is. Answered by Walter Whiteley. 





Two linear equations 
20000914 

From David Dean: 2a + 1b = 3.39 3a + 3b = 6.59 What formula do I use to find what a = ? Answered by Harley Weston. 





1 + 2 + 3 + ... + 50 
20000914 

From Vicki Charron: How can you calculate the total of the numbers one through fifty, without adding up the individual numbers? Answered by Penny Nom. 





Derivatives, there must be an easier way 
20000906 

From Brad Goorman: The direction read: Take the derivative of each expression. y = {1+[x+(x^{2} +x^{3})^{4}]^{5}}^{6}
Answered by Harley Weston. 





Velocity of a pendulum 
20000828 

From Mekca: A pendulum hangs from the ceiling. as the pendulum swings, its distance,d cm, form one wall of the room depends on the number of seconds,t, since it was set in motion. assume that the equation for d as a function of t is: d=80+30cos3.14/3t, t>0. estimate the instantaneous rate of change of d at t=5 by finding the average rates for t=5 to 5.1, t=5 to 5.01, and t=5 to 5.001. Answered by Harley Weston. 





Some trigonometry 
20000811 

From Angela: I have some PreCal questions. I am a student at the secondary level. I would be very grateful for your help. Solve the equation for theta (0 <= theta < 2pi). tan^{2}(theta) = 3 I know sec^{2}(theta) 1 = tan^{2}(theta) . . . Answered by Harley Weston. 





L'Hospital's Rule 
20000719 

From Dan Krymkowski: The limit of the following as x goes to infinity is 2*y. Y is a constant. lim 2*x*log(x/(xy)) = 2*y Answered by Harley Weston. 





Divisors of 2000 
20000606 

From Amanda Semi:
 find the product of all the divisors of 2000
 dog trainer time has 100m of fencing to enclose a rectangular exercise yard. One side of the yard can include all or part of one side of his building. iff the side of his building is 30 m, determine the maximum area he can enclose
Answered by Claude Tardif. 





A derivative problem 
20000604 

From Jeff Ellis: If F(x)=(4+x)(3+2x^{2})^{2}(2+3x^{3})^{3}, find F'(0) Answered by Harley Weston. 





Calculus Research Questions 
20000522 

From William Wright: I am a Calculus Teacher, and me and my class ran into these two problems without solutions in my manual, we got answers, but are unable to check them. If anyone gets this email and can respond to this with the solutions it be greatly appreciated. . . . Answered by Harley Weston. 





Radioactive decay 
20000518 

From Catherine Sullivan: Please help me with the following: The radioactive isotope carbon14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to carbon12 at a rate proportional to the amount of C14 present, with a half life of 5730 years. Suppose C(t) is the amount of C14 at time t.  Find the value of the constant k in the differential equation: C'=kC
 In 1988 3 teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of C14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?
Answered by Harley Weston. 





Related Rates 
20000507 

From Derek: How can you show that if the volume of a balloon is decreasing at a rate proportional to its surface area, the radius of the balloon is shrinking at a constant rate. Answered by Harley Weston. 





An improper integral 
20000504 

From A high school senior: Hi, I am a high school senior and I need help stugying for a final. I am stuck on one of the questions on my review sheet. Does the improper integral from 5 to infinity of (38/97)^{x} converge or diverge? If it converges I also need to know how to find the approximate value accurate to .01 of its actual value. Answered by Harley Weston. 





Thearcius Functionius 
20000503 

From Kevin Palmer: With the Olympics fast approaching the networks are focusing in ona new and exciting runner from Greece. Thearcius Functionius has astounded the world with his speed. He has already established new world records in the 100 meter dash and looks to improve on those times at the 2000 Summer Olympics. Thearcius Functionius stands a full 2 meters tall and the networks plan on placing a camera on the ground at some location after the finish line(in his lane) to film the history making run. The camera is set to film him from his knees(0.5 meters up from the ground) to 0.5 meters above his head at the instant he finishes the race. This is a total distance of two meters(the distance shown by the camera's lens). Answered by Harley Weston. 





An indefinite integral 
20000503 

From Bonnie Null: I am to find the indefinite integral of: (e^{x}  e^{x})^{2} dx Answered by Claude Tardif. 





Minimizing the metal in a can 
20000502 

From May Thin Zar Han: A can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. Answered by Harley Weston. 





Two calculus problems 
20000501 

From Kaushal Shah: How Do WE Integrate the following Functions,  Integral xtanx dx
 How was natural base "e" discovered and why e=2.7.......
Answered by Claude Tardif. 





The area of a triangle using calculus 
20000415 

From Todd Bowie: Hi, I am not a student but am reviewing calculus for an upcoming interview. I would like to know how to derive the area of a triangle using calculus. Thanks! Answered by Patrick Maidorn. 





y = x^x^x^x... 
20000405 

From Michael Hackman: Find the derivative of: y = x^x^x^x... on to infinity. Answered by Claude Tardif. 





Riemann sums 
20000330 

From Joshua D. Parham: If n is a positive integer, then
lim (1/n)[1/(1+1/n) + 1/(1+(2/n) + ... + 1/(1+n/n)]
n>infinity
can be expressed as the integral from 1 to 2 of 1/x dx Answered by Penny Nom. 





Compounding continuously 
20000321 

From Gina: You deposit $1500 in an account that pays 6.5% annual interest, compounded continuously. Find the balance after 10 years. I'm not sure what to do with the "compounded continuously" part. Answered by Penny Nom. 





Functions that satisfy f' = f 
20000316 

From Kevin Palmer: Recently my calculus teacher asked his students to try and find any functions whose derivatives where the exact same as the original function. The only function then I have determined that statement to be accurate in is all the natural exponential functions. Ex. f(x) = e^{x}, f'(x) = e^{x} If possible could you please email me all the functions that you can find in which the original function and its derivative is identical. Answered by Claude Tardif. 





Maximize 
20000312 

From Tara Doucet: My question is Maximize Q=xy^2 (y is to the exponent 2) where x and y are positive integers such that x + y^2 ( y is to the exponent 2)=4 Answered by Harley Weston. 





Simultaneous equations 
20000311 

From Laura Molck: My name is Laura Molck and I am in Year 11 in Australia. Please help me with the following. I know that they are all simultaneous equations which I can do but I have trouble with the formulae to work the equations. Can you please help!! 1. A tent manufacturer produces 2 models, Outback and Bushwalker. From earlier sales records it is known that 20% more of the Outback model is sold than the Bushwalker. A profit of $200 is made on each Outback sold, but $350 is made on each Bushwalker. If during the next year a profit of $177,000 is planned how many of each model must be sold? Answered by Penny Nom. 





Systems of linear equations 
20000310 

From Ann Marie Devereux: hi there!!, I guess I have a problem!!! 3x+4y=10 (over) 4x+y=9 2x=5y+3 (over) x=3y+1 Answered by Penny Nom. 





A mixture problem 
20000306 

From Rebecca Edwards: A tank in which cholocate milk is being mixed contains a mixture of 460 liters of milk and 40 liters of chocolate syrup initially. Syrup and milk are then added to the tank at the rate of 2 liters per minute of syrup and 8 liters of milk per minute. Simultaneously the mixture is withdrawn at the rate of 10 liters per minute. Find the function giving the amount of syrup in the tank at time t. Answered by Harley Weston. 





Two calculus problems 
20000303 

From Tara Doucet:
The height of a cylinder with a radius of 4 cm is increasing at rate of 2 cm per minute. Find the rate of change of the volume of the cylinder with respect to time when the height is 10 cm. A 24 cm piece of string is cut in two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so the sum of the areas is a maximum? Answered by Harley Weston. 





Slant height of a cone 
20000224 

From Jocelyn Wozney: I need help with this problem for my high school calculus class. Any help you can give me will be greatly appreciatedI am pretty stumped. "Express the volume of a cone in terms of the slant height 'e' and the semivertical angle 'x' and find the value of 'x' for which the volume is a maximum if 'e' is constant. Answered by Harley Weston. 





Some integration problems 
20000223 

From Tim Valentine: I am having a great deal of difficulty with the following integrals, can you help? I think they need the use of trig substitution or integration by parts but I cannot figure out how to begin. Thanks! The integral of 1/(2+3x^{2}) dx. and The integral of x * square root of (4x+5) dx. Answered by Harley Weston. 





A moving point on the graph of y=sinx 
20000222 

From Veronica Patterson: Find the rate of change of the distance between the origin and a moving point on the graph of y=sinx if dx/dt=2 centimeters per second. Answered by Harley Weston. 





The quotient rule 
20000221 

From Charlene Anderson: Question: I came across a question in our book that states: Let Q(x) = N(x) / D(x) Then rewrite Q(x) in a form that can utilize the Power and Product Rules. Use this rearranged form to derive the Quotient Rule. The Quotient Rule can be derived from the Power Rule and the Product Rule. One must also use the chain rule too, right? Answered by Harley Weston. 





Filbert Family Circus 
20000204 

From Sarah: As Clyde moves his broom around the circus ring, he thinks that he has finally found a job where he can make a clean sweep of things. Clyde is sweeping the ring where the lions perform in the Filbert Family Circus. The ring is 76 feet across and Clyde is using a broom 3 feet wide. He starts at the outside edge and works his way to the middle, making circles around the ring. After sweeping 3/4 of the ring, Clyde sees the lions coming with their trainer and scurries out of the ring. How many trips around the ring did he make? Answered by Penny Nom. 





Play ball 
20000203 

From Jessie: Here's a calc question that is probably a lot easier than I am making it. If you have a legendary "baseball problem" for the related rates section of Calc I, and you are given that the runner is running from 2nd to 3rd base at a given rate, and the umpire is standing at home plate, and you are given the distance between the bases on the field, how do you find the rate of change of the angle between the third base line (from the point of the umpire) and the runner? Here is a sample prob: Runner is moving from 2nd to 3rd base at a rate of 24 feet per second. Distance between the bases is 90 feet. What is the rate of change for the angle (theta, as described previously) when the runner is 30 feet from 3rd base? Answered by Harley Weston. 





A problem with a radius. 
20000201 

From Howard B Davis: We start a Line that goes up 1 unit, then it goes to the Right for 5 units long, and then goes down 1 unit which is the end point. If we draw a circle that is tangent to both ends as well as the midpoint of the horizontal line: How do we find the radius of the arc; in Mathematics, with only this information? Answered by Chris Fisher. 





Functions 
20000123 

From Tara: Hi my name is Tara, I have two math problems that I need help with in my calculus math class.  If f(x)= x  2 show that (x+3)f(x)(x+2)f(x+1)+4=0
 Graph this function and use the graph to determine the range y=2x^{2}  8x  3
Answered by Harley Weston. 





The limit of f(x)/x 
20000122 

From Laurent Jullien: I would appreciate help to prove that a twice continuously differentiable convex function from R+ to R has the property that f(x)/x has a limit when x tends to infinity. Answered by Claude Tardif. 





Why study calculus? 
20000105 

From Trlpal: I am a high school senior enrolled in a precalculus class. Could you tell me what the benefits of taking calculus are and why it would be important to take the class. Answered by Walter Whiteley and Harley Weston. 





zero 
20000101 

From Jason: What civilization first used zero? Answered by Penny Nom. 





A decreasing ellipsoid 
19991215 

From A student instructor: The volume of an ellipsoid whose semiaxes are of the lengths a,b,and c is 4/3 *pi*abc. Suppose semiaxes a is changing at a rate of A cm/s , the semiaxes b is changing at B cm/s and the semiaxes c is changing at C cm/s . If the volume of the ellipsoid is decreasing when a=b=c what can you say about A,B,C? Justify. Answered by Harley Weston. 





Two calculus problems 
19991213 

From Alan: I have 2 questions that are very new to me, they were included on a quiz and the material was never covered. Our teacher never explained the purpose and detailed explanation of how to solve the problem. Could you help? Thanks. Question 1: A ball is falling 30 feet from a light that is 50 feet high. After 1 sec. How fast is the shadow of the ball moving towards the light post. Note that a ball moves according to the formula S=16t^2 Question 2: How many trapezoids must one use in order for the error to be less than 10^8 if we want to find the area under the curve Y=1/X from 1 to 2. Find the exact area, Graph the function and use the trap rule for the "N" that you found. Answered by Harley Weston.






A calculus problem 
19991208 

From JT Wilkins: These are the questions:  Show that there exists a unique function that meets the following requirements:
a) f is differentiable everywhere b) f(0)= f'(0)= 0 c) f(x+y)= f(x)+ f(y), for all real values of x,y  Consider the function F: R>R (All Reals)
F(x) = 0, for x irrational & 1/q, x=p/q gcd(p,q)=1 q > 0 a)determine the values x where f is continuous, respectively discontinuous. b)determine the values x when f is differentiable and for each of these values compute f'(x). Answered by Penny Nom. 





Advanced Calculus 
19991207 

From Kay: Hi, my name is Kay. Please helpthese problems are driving me crazzzzy!!!! Your help would be greatly appreciated!  Let a,b be contained in R, a
 .
. . Answered by Claude Tardif. 





Systems of equations 
19991206 

From Roger Hornbaker: I am having problem figuring out x and y solutions.  5x + y = 4
3x  y = 4  3x + 2y = 6
 3x + y = 0 Answered by Penny Nom. 





The chain rule 
19991203 

From Jennifer Stanley: This problem is making me dizzy. I would greatly appreciate a little help! Express the derivative dy/dx in terms of x. y=u^2(uu^4)^3 and u=1/x^2 Answered by Harley Weston. 





Two calculus problems 
19991201 

From O'Sullivan: Question #1 Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? It's under the chain rule section of differentiation if that any help. I've set up a ratio and tried to find the constant but am stuck. Question #2 The figure shows a lamp located three units to the right of the yaxis and a shadow created by the elliptical region x^2 + 4y^2 < or= 5. If the point (5,0) is on the edge of the shadow, how far above the x axis is the lamp located? The picture shows an x and y axis with only the points 5 and 3 written on the x axis. the lamp is on the upper right quadrant shining down diagonally to the left. There's an ellipse around the origin creating the shadow. It's formula is given as x^2 + 4y^2=5. Answered by Harley Weston. 





Two derivatives 
19991116 

From Gina Renicker: The derivative of: y=e^{(xlnx)} and y=x^{2arctan(x1/2)} Answered by Harley Weston. 





Parabolic mirrors 
19991107 

From Andy White: I am working on a project concerning parabolic mirrors. I need to create a mirror to focus sunlight on a focal point, but I don't know how to do it. Is there some equation that tells where a focal point will be in relation to a parabola? What is a directrix? Answered by Penny Nom. 





Area of a circle and an inequality 
19991030 

From Adam Anderson: I have two problems. The first: prove that the area of a cirlce is pi times radius squared without using calculus. The second: show that ln(x) < x  1 for all x > 0. Answered by Harley Weston.






Clockwise or Counterclockwise? 
19991027 

From Tim: A particle moves around the circle x^{2} + y^{2} = 1 with an xvelocity component dx/dt = y  Find dy/dt
 Does the particle travel clockwise or counterclockwise around the circle? Why?
Answered by Harley Weston. 





Derivatives with logs 
19991026 

From Kate: What is the derivative of 5 to the 5x2 at x equals 0.8? Answered by Harley Weston. 





log(a) 
19991022 

From Brenda Miskimmin: I need to know the mercury concentration in mg/L or ng/L for the following: log M (Hg) = 8.5 where mw of Hg=200.59 (it's the negative sign in front of log that confuses me). Answered by Harley Weston. 





l'Hospital's Rule 
19991018 

From Yannick Gigandet: How can I solve these two limits :  lim when n approches 1 of n[a^{1/n} 1]
 lim when x approches 0 of (e^{ax}  e^{bx}) / x
Thanks for the answer! Answered by Harley Weston. 





A famous mathematician 
19991012 

From Yvette Perez: Another way to write 3/15. Remove 0 add a line, unscramble, you have the name of a famous mathematician. Answered by Claude Tardif. 





Length of a line 
19991010 

From Dagmara Sarudi: My question has to do with the length of a diagonal. This problem came up when I thought about the shortest distance between two points, for example walking from one point to another in my neighborhood. I can choose a zig zag route and assuming the blocks I walk are exactly the same length, it shouldn't matter what route I took, the distance I travel should still be the same when I reached my goal. If, on the other hand I could travel in a diagonal line, the distance would be shorter. But what if, in my zig zag motion, the sections get so small the route approaches a diagonal. Shouldn't it be that each separate section added together equals the value of the two original sides? Or would it suddenly equal the value of the diagonal (which, of course was shorter than the two sides added together)? What gives? Answered by Chris Fisher and Harley Weston.






A trig limit 
19991006 

From Yannick Gigandet: What is the limit, as x approaches pi/3, of (12cosx) / sin(x(pi/3)) ? Answered by Penny Nom. 





The circumference of a circle 
19991005 

From Mara Frost: what is the formula to find the circumference of a circle, or if there is no formula, how do you find the circumference of a circle? Answered by Penny Nom. 





Two limits 
19991002 

From Jennifer: How do I find lim (1cosx)/(x^2) as x> 0 and lim (tan3x)/x as x>0 Answered by Harley Weston. 





Temperatures 
19990927 

From Eula: How do you cahnge farenheit degrees to celsius degrees? Answered by Penny Nom. 





Numbers with the digit 2 in 1...1000 
19990920 

From Jessica: Is there a trick to finding out how many numbers containing the digit two is there from 1 to 1000? Answered by Walter Whiteley. 





Distance between the windows 
19990919 

From Lawrence: An observer on level ground is at distance d from a building. The angles of elevation to the bottom of the windows on the second and third floors are a and b respectively. Find the distance h between the bottoms of the windows in terms of a b and d Answered by Harley Weston. 





2 to the x and x squared 
19990917 

From John: For what values of x is 2 to the exponent x greater than x squared? Answered by Harely Weston. 





Y2K? 
19990903 

From Mike Putzakulish: In Y2K, the "K" stands for thousand, but where did the "K" come from? I know it's not a Roman Numeral, but what is it?? Answered by Chris Fisher and Harley Weston. 





A double negative 
19990901 

From Dennis: If b = 2 what does b = ? As in (a + 8.5)  [(b) + c] a = 1.5, c = 1.7 Answered by Penny Nom. 





Parametric Equations 
19990806 

From Nicholas Lawton: Show that an equation of the normal to the curve with parametric equations x=ct y=c/t t not equal to 0, at the point (cp, c/p) is : yc/p=xp^2cp^3 Answered by Harley Weston. 





A calculus problem 
19990722 

From Nicholas Lawton: The curve y= e^x(px^2+qx+r) is such that the tangents at x=1 and x=3 are parallel to the xaxis. the point (0,9) is on the curve. Find the values of p,q and r. Answered by Harley Weston. 





The shortest ladder 
19990626 

From Nicholas: A vertical wall, 2.7m high, runs parallel to the wall of a house and is at a horizontal distance of 6.4m from the house. An extending ladder is placed to rest on the top B of the wall with one end C against the house and the other end, A, resting on horizontal ground. The points A, B, and C are in a vertical plane at right angles to the wall and the ladder makes an angle@, where 0<@ Answered by Harley Weston. 





Even and Odd Function 
19990617 

From Kent: There is one function with the domain of all real numbers that is both even and odd. Please give me the answer to this question before I go insane. Answered by Penny Nom. 





A circle in a square 
19990526 

From Jose V Peris: A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle expands, the square expands to maintain the condition of tangency. find the rate at which the perimeter of the square is increasing. find the rate of increase in the area enclosed between the circle and the square at the instant when the area of the circle is 25(pi) square inches. Answered by Harley Weston. 





Related rates 
19990513 

From Tammy: The sides of a rectangle increase in such a way that dz/dt=1 and dx/dt=3*dy/dt. At the instant when x=4 and y=3, what is the value of dx/dt? (there is a picture of a rectangle with sides x and y, and they are connected by z, which cuts the rectangle in half) Answered by Harley Weston. 





A Polar Plot 
19990506 

From Irene: Consider the polar equation r=23Cos(theta/2) In the interval [o, 4Pi], how would you find the area of one of the leaves and also the length of one of the edges of a leaf? Answered by Harley Weston. 





Radius of an arc 
19990422 

From Rusty Riddleberger: I need to find the equation for finding the radius of an arc; I know the length of the arc (i.e the distance of the line connecting the two ends of the arc) and the height; (i.e the rise of the arc at its apex,) I had the formula years ago but it has lost me; this would be invaluable for work in new homes i.e. where we need to build an "arch" with a rise of 21" between two columns 11 feet apart Answered by Chris Fisher. 





Radius of convergence 
19990421 

From Nowl Stave: Why is the radius of convergence of the first 6 terms of the power series expansion of x^(1/2) centered at 4 less than 6? Answered by Harley Weston. 





Circles 
19990421 

From Alex Elkins: How do you find the circumference of a circle if you only know the radius and the square feet or inches of the circle if the radius is 18 inches, If done in inches do you multiply by 12 to get the square feet? Answered by Jack Lesage and Harley Weston. 





The average rate of change of a function 
19990420 

From Tammy: Suppose that the average rate of change of a function f over the interval from x=3 to x=3+h is given by 5e^h4cos(2h). what is f'(3)? I would appreciate any help with this question. Answered by Harley Weston. 





A Frustum 
19990329 

From Monica Armour: What do you call a square pyramid that has had the top chopped off? Answered by Chris Fisher and Jack LeSage. 





Rhomboid 
19990325 

From Monica Armour: I need to see a net of a rhomboid. Where can I find one on the net? Is it like a square paramid with the top chopped off? Help! This has me stumped. Answered by Jack LeSage. 





Graphing the Derivative 
19990118 

From Milena Ghebre: This question has been nagging me for sometime now. Is there a way of finding out the derivative of a function, just by looking at the graph of it? Answered by Walter Whiteley. 





Calculus 
19990116 

From Kaylea Rankin: Differentiate the following. y = 1 /(2+3/x) Answered by Jack LeSage and Penny Nom. 





Absolute value of i 
19990106 

From Wayne Bagley: I would like to know what is the absolute value of i. I need an answer suitable for the secondary level. Answered by Harley Weston. 





The area and the circumference of a circle. 
19980827 

From Jason Wright: I was looking at the relationship of the area of a circle and the circumference when I realized that 2*pi*r is the derivative of pi*r^2. I was wondering if there is any connective deep dark meaning as to why this appears to be related. Thanks for any help you can give me! Answered by Walter Whiteley. 





Parabolas 
19980724 

From Danica: how do you find the focus, vertex, and directrix of 4xy^22y33=0 Answered by Penny Nom. 





Volumes of Revolution 
19980724 

From Lorraine Wall: I'm on the section fpr The Computation of Volumes of Solids of Revolution and the following question is giving me problems: Consider the region in the first quadrant bounded by the xand yaxes, the vertical line x=3, and the curve y=1/(xsquared + 3) I can determine the volume of the solid by rotating the region about the yaxis using the shell method but I can't seem to be able to get started with the volume when rotated about the xaxis. Answered by Harley Weston. 





Calculus problems 
19980713 

From Lorraine: I'm stuck again. Can you help? This involves integration using the method of partial fractions the integral of: 7x(to the 5th)  2x(cubed) + 3 dx  x(to the fourth)  81 Do I have to do long division to reduce the numerator to the fourth power? the integral of: 4 16x +21x(squared) + 6x(cubed)  3x(fourth) dx  x(cubed)(x  2)(squared) Lorraine Answered by Harley Weston. 





A Calculus Problem 
19980628 

From Lorraine: I'm a postsecondary student taking calculus by correspondence. I'm stuck on the following question (and similar ones) Can you help? Evaluate the following indefinite integral: d(theta)  1 + sin (theta) (It says to multiply both numerator and denominator by: 1  sin(theta) Thanks Lorraine Answered by Harley Weston. 





A Logic Problem 
19980607 

From Anthony Bacigalupo: My name is Anthony Bacigalupo and I take Sequential ][ Math and am taking a practice regents. When doing a logic problem, I encountered the following statements, where I am trying to prove P ( I left out steps unrelated to the question).... Answered by Chris Fisher. 





A trig limit 
19980528 

From Ann: This problem is a calculus 1 limit problemhigh school level. I'm teaching myself calc over the summer and I'm already stumped. find the limit lim sec^(2)[(sqrt2)(p)]1 p>0  1sec^(2)[(sqrt3)(p)] I'm Ann. Answered by Harley Weston. 





A Tightrope Walker. 
19980219 

From Amy Zitron: A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A.... Answered by Harley Weston. 





Pi 
19971031 

From Ryan McKinnon: What Is Pi? Answered by Chris Fisher. 





Some Calculus Problems. 
19971030 

From Roger Hung:
 What real number exceeds its square by the greatest possible amount?
 The sum of two numbers is k. show that the sum of their squares is at least 1/2 k^2.
 .
. . Answered by Penny Nom. 





A Trigonometric Limit 
19970918 

From Brian Ray: What is the limit, as x approaches 0, or tan^23x/x^2? (read, tan squared 3x over...)? Answered by Harley Weston. 





A Limit Problem 
19970916 

From Robert Reny: what is the limit, as x approaches 0, of 3x/2x[x]? [] means absolute value. Answered by Harley Weston. 





The Division Bracket. 
19970409 

From Judy Riley: A fellow teacher recently asked if I remembered the exact word for a division bracket (not the symbol with dots, the horizontal line in a fraction, or a solidus). I couldn't. Can you help? Answered by Walter Whiteley and Harley Weston. 





Mathematical Induction and the Derivative 
19970318 

From Shuling Chong: "Obtain a formula for the nth derivative of the product of two functions, and prove the formula by induction on n." Any educated tries are appreciated. Answered by Penny Nom. 





Parabolic Mirrors 
19970128 

From Megan Wennberg: Consider a ray of light that passes through a chord of a parabola (the chord is above the focus and parallel to the directrix), hits the parabola at a point (x,y) and is reflected through the focus. If d1 is the distance from the chord to the point of incidence (x,y) and d2 is the distance from (x,y) to the focus, can you prove that the sum of the distances d1+d2 is constant, independent of the particular point of incidence. Answered by Penny Nom. 





Foci of an Ellipse 
19970122 

From David Gilliam: How do I find the focii of the following equation? 4x^2 + 9y^2 = 36 Answered by Harley Weston. 





A tangent to a circle is perpendicular to the radius at the point of contact. 
19961022 

From Rita Leung: I wonder if there is any proof for this theorem  A tangent to a circle is perpendicular to the radius at the point of contact. If there is any proof for that, can you tell me please? Answered by Chris Fisher and Harley Weston. 





Could you tell me the name for the bar in a division problem? 
19961021 

From Linda: Could you tell me the name for the bar in a division problem. Not the line with dots on either side but the line that divides the two numbers? My name is Linda. I am asking for my niece who is in 8th grade. Answered by Chris Fisher. 





Thousands, millions, billions and trillions 
19960822 

From Blaine: What are the names of the periods in groups of numbers like 123,456,789. I need to know the names of them for school. I already know the first ones: units thousands millions billions trillions. Answered by Penny Nom and Diane Hanson. 





What are fractals and are they of any practical use? 
19960626 

From Ron Lewis: What are fractals and are they of any practical use? Answered by Chris Fisher. 





Show that this construction yields a rhombus. 
19960616 

From LennyB: Hello, I have a problem. I doubt you can help me. If you have an isosceles trapezoid, and you connect the midpoints of the four sides of the isosceles trapezoid forming a quadrilateral, how do you prove that it's a rhombus in a 2 column proof?? Answered by Penny Nom. 





Area of an annulus 
19960404 

From Ryan Levering: What is the area of this annulus? Answered by Penny Nom. 





(3)x(2) 
19951025 

From Azmat: Why is (3)x(2) = 6? Answered by Herley Weston and Ed Giesbrecht. 





une écriture plus simple des expressions 
20110113 

From jessie:
trouver une écriture plus simple des expressions suivantes sachant que : x + y = 3 et x  y =2
A=x1+y+2 B=x1+y2
C=x+1+y+2 C=x+1+y2 Answered by Claude Tardif. 





soustraction 
20100114 

From LUCAS: comment faire l'operation suivant car je ne me souvient plus tres bien
D'avance merci
4h44mn53s2h52mn45s Answered by PierreLouis Gagnon et Claude Tardif. 





Le plus grand commun diviseur 
20091111 

From Katie: Question: Pendant mon cours de MAT 1300, mon enseignant m'a posé la question suivante:
(a, b) = 12
(a², b) = 24
(b, 72) = ?
Pouvezvous m'aider s'ilvousplaît? Answered by Claude Tardif. 





Un système de plusieurs équations et plusieurs inconnues 
20090122 

From Cédrick: J'ai un problème écrit à résoudre estce que vous pouvez m'aider ?
Le premier est le tiers de la somme des deux autres.
La somme du premier et du deuxième est 13.
Le produit du deuxième et du troisième est 56.
Réponse: Le premier nombre est___________________
Le deuxième nombre est _________________
Le troisième nombre est _________________
Answered by PierreLouis Gagnon, Antoine Letarte at Claude Tardif. 





calcul des hh,mn,sec 
20080119 

From gillot: bonjour, mon fils à des additions et soustractions d'horaire et je n'arrive pas à trouver
la règle pour lui espliquer comment il faut faire
il y a t il une formule pour les additions et soustractions
merci par avance pour vos réponses Answered by Claude Tardif. 





Illusion géométrique 
20070404 

From Dominique: Je n'arrive pas expliquer la modofocation de surface. Answered by Claude Tardif. 





soustractions avec les heures 
20060301 

From Bossedi: Je m'appelle Bossedi et malheureusement j'ai un petit souci en ce qui concerne le calcul des heures! Je ne me rappelle plus comment s'effectue une addition ou un soustraction avec les heures comme par exemples 22h307h00. Merci infiniment de bien vouloir m'aider.
P.S.: Je me souviens qu'une lectrice du surnom Jennifer avait deja poser ce genre de probleme, mais avec la seule difference que dans son cas, c'etait i j'ai bonne memoire, 09h45 moins 07h52! Quelques choses de ce genre. Doisje +tot ercire 11h3007h00? Answered by Claude Tardif. 





quel sont les plus grands diviseurs communs? 
20050907 

From Sylvain: quel sont les plus grands diviseurs communs? Answered by Claude Tardif. 





Soustraction avec les heures 
20030408 

From Jennifer: Je m'appelle Jennifer et malheureusement j'ai un petit souci. Pour le calcul des heures c'ets à dire je ne me rappelle plus comment s'effectue une addition ou un soustraction avec les heures comme par exemples 9h457h52. Si vous pouviez m'aider j'ens erais trés contente. Answered by Diane Hanson and Claude Tardif. 





Comment fait on pour justifier que... 
20020930 

From Lucie: Comment fait on pour justifier que p au carré est égal à 2 fois q au carré? Answered by Claude Tardif. 





isomorphisme 
20000810 

From Romain Kroes: Pour les beoins d'un ouvrage d'économie que je suis en train de terminer, pouvezvous me dire qui est (sont) l'inventeur de l' "isomorphisme" en mathématiques (calcul tensoriel)? Answered by Claude Tardif. 





Derivées partielle 
19991019 

From Arnaud Flandin: Quel est la definition des derivées partielle Answered by Claude Tardif. 

