







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. 





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. 





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. 





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. 





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. 





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. 





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. 





Minimize the cost 
20080426 

From A: A power line is to be constructed from the shore of a lake to an island that is 500 m away. The closest powerline ends 4km along the shore from the point on the shore closest to the island. If it costs 5 times as much to lay the powerline underwater as along the shore, how should the line be installed to minimize the cost? 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. 





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. 





Linear programming and optimization 
19990409 

From Shams: What is Linear programming and optimization? Answered by Jack LeSage and Penny Nom. 

