Optimization problem

Question

Boston College is building a new running track. It is to be the perimeter of a region obtained by

putting two semicircles on the ends of a rectangle. However, the administration decides they’d also like to have

an outdoor pool in the rectangular portion of the area surrounded by the track. Assume the track is to be 440

yards long.

a) Determine the necessary dimensions to build the track in order to maximize the area of the pool. (Assume

the edges of the pool are exactly along the straight line part of the track.)

State your conclusion clearly.

b) Could you use the Single Critical Point theorem to justify you’ve found the maximum? If so, do it. If not,

why not?

c) Could you use the Closed Interval Method? If so, do it. If not, why not?

Answer 1

Let the side of the rectangle where the semicircle track is built be 2x, and the other dimension of rectangle be y. What is the perimeter of the track in terms of x,y? The perimeter is the constraint as the track is 440 yards long. Use that to express y in terms of x.

Then, what is the area of the pool? This should be a function in x only. Make sure to write out the domain of this function. It will help in answering parts (b) and (c).

Proceed with differentiation to find critical point(s), and determine the maximum area.