







The maximum area of a rectangle with a given perimeter 
20170602 

From Bob: How would I go about finding the maximum area of a rectangle given its perimeter (20m, for example)? 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. 





A waste oil tank 
20150613 

From Angela: a waste oil tank is 5 feet wide and 20 feet long, the empty tank on your truck holds 5000 gallons. If 7.48 gallons are in every cubic foot, what is the maximum depth the oil can be to completely fit into your truck? 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 cone of maximum volume 
20150316 

From Mary: I have to use a 8 1/2 inch by 11 inch piece of paper to make a cone that will hold the maximum amount of ice cream possible by only filling it to the top of the cone. I am then supposed to write a function for the volume of my cone and use my graphing calculator to determine the radius and height of the circle. I am so confused, and other than being able to cut the paper into the circle, I do not know where to start. Thank you for your help! Mary Answered by Robert Dawson. 





Largest cone in a sphere 
20150115 

From Alfredo: What is the altitude of the largest circular cone that may be cut out from a sphere of radius 6 cm? Answered by Penny Nom. 





Maximizing the ticket revenue 
20141007 

From Allen: An airplane whose capacity is 100 passengers is to be chartered for a flight to Europe. The fare is to be $150 per person, if 60 people buy tickets. However, the airline agrees to reduce the fare for every passenger by $1 for each additional ticket sold. How many tickets should be sold to maximize the ticket revenue for this flight? Answered by Chris Fisher. 





Two altitudes of a scalene triangle 
20120813 

From grace: Two of the altitudes of a scalene triangle ABC have length 4 and 12. If the length of the third altitude is also an integer, what is the biggest that it can be? Justify all of your conclusions. Answered by Chris Fisher. 





The maximum distance from the vertex of a triangle 
20120502 

From David: There are three towns A,B,and C, equidistant apart.
A car is 3 miles from town A, and 4 miles from town B.
(ie, somehwere outside of the triangle which the three towns form)
What is the maximum distance that the car can be from town A?
This was asked as quiz question in my local pub last Sunday.
The answer is 7. How do I prove it?
Best regards. David in Denton. Answered by Robert Dawson and Chris Fisher. 





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. 





Margie threw a ball 
20120216 

From mary: at 9:45 Margie threw a ball upwards while standing on a platform 35ft above the ground. The height after t seconds follows the equation:
h(t)= 0.6t^2 +72t+35
a) what will be the maximum height of the ball?
b)how long will it take the ball reach its maximum height?? Answered by Harley Weston. 





Maximum area of a rectangle 
20111004 

From Lyndsay: A rectangle is to be constructed having the greatest possible area and a perimeter of 50 cm.
(a) If one of the sides of the rectangle measures 'x' cm, find a formula for calculating the area of the rectangle as a function of 'x'.
(b) Determine the dimensions of the rectangle for which it has the greatest area possible. What is the maximum area? Answered by Penny Nom. 





A rectangle of largest possible area 
20110916 

From mary: Steven has 100 feet of fencing and wants to build a fence in a shape of a rectangle to enclose the largest possible area what should be the dimensions of the rectangle Answered by Penny Nom. 





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. 





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. 





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. 





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. 





Two max/min problems 
20100411 

From Amanda: 1) Find the area of the largest isosceles triangle that canbe inscribed in a circle of radius 4 inches.
2)a solid is formed by adjoining two hemispheres to the end of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. Answered by Tyler Wood. 





The maximum area of a rectangle 
20100103 

From Mohammad: determine the maximum area of a rectangle with each perimeter to one decimal place?
a)100 cm b)72 m c)169 km d)143 mm Answered by Penny Nom. 





Maximizing the area of a rectangle 
20091217 

From rachel: A rectangular field is to be enclosed by 400m of fence. What dimensions will give a maximum area? 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. 





The maximum number of right angles in a polygon 
20091005 

From Bruce: Is there way other than by trial and error drawing to determine the maximum number of right angles in a polygon? Secondary question would be maximum number of right angles in a CONVEX polygon. Is there a mathematical way to look at this for both convex and concave polygons? Or are we limited to trial and error drawing? Answered by Chris Fisher. 





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. 





Maximum Volume of a Cylinder Inscribed in a Sphere 
20090618 

From Jim: Hello I have a hard time finishing this question:
A right circular cylinder has to be designed to sit inside a sphere of radius 6 meters
so that each top and bottom of the cylinder touches the sphere along its complete
circular edge. What are the dimensions of the cylinder of max volume and what is the volume? Answered by Janice Cotcher. 





Application of Derivatives of Trig Functions 
20090521 

From Alannah: I have a word problem from my Calculus textbook that I can't figure out.
Triangle ABC is inscribed in a semicircle with diameter BC=10cm. Find the value of angle B that produces the triangle of maximum area.
I am supposed to set up an equation for the area of the triangle A=b x h/2 using Trig functions based on angle B to represent the base and height but I'm not sure how to do this when the side length given is not the hypotenuse. Answered by Janice Cotcher. 





The optimal retail price for a cake 
20090325 

From Shawn: Your neighbours operate a successful bake shop. One of their specialties is a cream covered cake. They buy them from a supplier for $6 a cake. Their store sells 200 a week for $10 each. They can raise the price, but for every 50cent increase, 7 less cakes are sold. The supplier is unhappy with the sales, so if less than 165 cakes are sold, the cost of the cakes increases to $7.50. What is the optimal retail price per cake, and what is the bakeshop's total weekly profit? Answered by Robert Dawson. 





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. 





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. 





Maximize revenue 
20081008 

From Donna: A university is trying to determine what price to charge for football tickets. At a price of 6.oo/ticket it averages 70000 people per game. For every 1.oo increase in price, it loses 10000 people from the average attendance. Each person on average spends 1.5o on concessions. What ticket price should be charged in order to maximize revenue.
price = 6+x, x is the number of increases.
ticket sales = 70000 10000x
concession revenue 1.5(70000  10000x)
I just do not know what to do with the concession part of this equation
(6+x) x (70000  10000x) I can understand but not the concession part please help. thx. Answered by Penny Nom. 





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. 





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. 





What is the greatest area she can have for her garden? 
20080512 

From angie: Mary has 12 wood boards, each board is 1 yard long. She wants her garden to be shaped like a rectangle. What is the greatest area she can have for her garden? Answered by Penny Nom. 





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. 





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. 





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 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. 





The maximum area of a rectangle 
20071123 

From Christy: Question from Christy, a student:
Show that among all rectangles with an 8m perimeter, the one with the largest area is a square.
I know this is simple but I'm not sure if I'm doing it correctly. Here is what I did.
1. A = xy
2. 8 = 2x+2y
3. y = 4x
4. A = x(4x) = 4xx^2
Not sure what to do from this point because I don't know if its right. Answered by Harley Weston. 





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. 





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. 





The range of a projectile 
20070918 

From Claudette: This is a maximum minimum problem that my textbook didn't even try to give an example of how to do it in the text itself. It just suddenly appears in the exercises.
Problem: The range of a projectile is R = v^2 Sin 2x/g, where v is its initial velocity, g is the acceleration due to gravity and is a constant, and x is the firing angle. Find the angle that maximizes the projectile's range.
The author gives no information other than the formula.
I thought to find the derivative of the formula setting that to zero, but once I had done that, I still had nothing that addressed the author's question.
Any help would be sincerely appreciated.
Claudette Answered by Stephen La Rocque. 





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. 





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. 





Maximizing the volume of a cone given the slant length 
20070514 

From Christina: A coffee filter for a new coffee maker is to be designed using a conical filter. The filter is to be made from a circle of radius 10cm with a sector cut from it such that the volume of coffee held in the filter is maximised. Determine the dimensions of the filter such that the volume is maximised. Answered by Stephen La Rocque and Kerstin Voigt. 





Maximum area 
20070429 

From fee: Given a perimeter of 24cm, calculate the maximum area using quadratics. Answered by Penny Nom. 





Find the maximum revenue 
20070405 

From Megan: The weekly revenue for a company is R= 3p+60p+1060, were p is the price of the company's product. Find the maximum revenue for this company. Answered by Stephen La Rocque. 





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. 





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. 





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. 





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. 





Maximizing revenue 
20050513 

From Jackie: 1.The manager of a 100unit apartment complex knows from experience that all units will be occupied if the rent is $400 per month. A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent. What rent should the manager charge to maximize revenue?
2.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. 





A trig problem 
20040802 

From A student: Given that the maximum value of [sin(3y2)]^2 [cos(3y2)]^2
is k. If y>7, Find the minimum value of y for which
[Sin(3y2)]^2  [cos(3y2)]^2 =k. Answered by Penny Nom. 





Percent difference 
20040410 

From A parent: For a school science project, my son Alex is taking measurements of plant growth at regular intervals. As part of the data, he must provide the maximum percent difference observed in the categories his team has identified.
So, for example he has six plants with four measurements each. (He has more, but I'll keep it simple) For the first plant he measured 2mm, 2.4mm, 2.9mm, and 3.2mm. For the 2nd, 3rd, and 4th plants, he has similar numbers. Is there a way to calculate the maximum percent difference between any two plants in his measurements during the project? Doing it for each combination would be tedious. Answered by Penny Nom. 





Which one has the most factors? 
20031031 

From Kristi: Of all the whole numbers less than or equal to 5000, which one has the most factors? Answered by 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. 





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. 





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. 





a+b=10 and ab=40 
20020427 

From April: What two numbers add to ten and multiply to forty? (a+b=10, a*b=40) I think the answer includes radicals and/or imaginary numbers. 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. 





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. 





Where is the fourth point? 
20011024 

From Mike: Four points are placed at random on a piece of paper. Connect the three points of the triangle of the largest area. What is the possibility that the fourth point is in the triangle? Answered by Penny Nom. 





Dividing a circle 
20011017 

From Ahmeen: I am having a hard time figuring out how a circle can be divided into 11 equal parts with only 4 cut allowed? My teacher gave this to us and I still can't cut my pie into eleven equal parts with only four cuts. Answered by Walter Whiteley. 





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. 





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. 





An emergency response station 
20010329 

From Tara: Three cities lying on a straight line want to jointly build an emergency response station. The distance between each town and the station should be as short as possible, so it cannot be built on the line itself, but somewhere east or west. Also, the larger the population of a city, the greater the need to place the station closer to that city. You are to minimize the overall sum of the products of the populations of each city and the square of the distance between that city and the facility. City A is 6 miles from the road's origin, City B is 19 miles away from the origin, and City C is 47 miles from the origin. The populations are 18,000 for City A, 13,000 for City B, and 11,000 for City C. Where should the station be located? Answered by Claude Tardif and Penny Nom. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





The isoperimetric theorem 
20000224 

From Raj Bobal: How can you prove Mathematically that the maximum area enclosed by a given length is a circle? Answered by Chris Fisher. 





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. 





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. 

