Optimization Problem

Question

A client has hired you to build a receptacle in the shape of a rectangle with two half-circles attached at either end. They would like to maximize the area of the rectangular region inside the receptacle. The client has 500 feet of material for you to build with, and they stipulate that the diameter of the two half circles on the ends must be at least 10 feet.

If you use the material only on the outer boundary of the receptacle (i.e., you do not build the two sides of the rectangle that are inside the receptacle), what is the maximum rectangular area you can enclose? What should the dimensions be?

Answer 1

One condition is the boundary is 500 feet long. Let d be the diameter of the semicircles and w be the width of the rectangle. Then we get the equation πd + 2w = 500 for our condition.

We're trying to maximize the area of the rectangle and the area of the rectangle = w*d.

Solve for w in the condition:

w = (500 - πd)/2

Then plug this into the function we're maximizing

Area = (500 - πd)/2 * d.

Find the absolute maximum value of this on the interval d ≥ 10

Do you think you'll be able to do this?