







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 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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





A max min problem 
20120226 

From Christy: Hello, I have no idea where to start with this question.
Bob is at point B, 35 miles from A. Alice is in a boat in the sea at point C, 3 miles from the beach. Alice rows at 2 miles per hour and walks at 4.25 miles per hour, where along the beach should she land so that she may get to Bob in the least amount of time? Answered by Penny Nom. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





A maximum area problem 
20090113 

From Kylie: Help me please! I don't know how or where to start and how to finish.
The problem is: A window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 15 ft., find the dimensions that will allow the maximum amount of light to enter. 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. 





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. 





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. 





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. 





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. 





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. 





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. 





How much labor should the firm employ? 
20061028 

From Christy: A dressmaking firm has a production function of Q=LL(squared)/800. Q is the number of dresses per week and L is the number of labor hours per week. Additional cost of hiring an extra hour of labor is $20. The fixed selling price is P=$40. How much labor should the firm employ? What is the resulting output and profit? I am having a difficult time with this, HELP! Answered by Stephen La Rocque. 





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. 

