Chapter 4.7, Problem 27E

Numerical Graphical and Analytic Analysis An exercise room consists of a rectangle with a semicircle on each end. A 200-meter running track runs around the outside of the room.

(a) Draw a figure to represent the problem. Let x and y represent the length and width of the rectangle, respectively.

(b) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maximum area of the rectangular region.

Length, x	Width, y	Area, xy
10	$\frac{2}{\pi }(100-10)$	$(10)\frac{2}{\pi }(100-10)\approx 573$
20	$\frac{2}{\pi }(100-20)$	$(20)\frac{2}{\pi }(100-20)\approx 1019$

(c) Write the area A of the rectangular region as a function of x.

(d) Use calculus to find the critical number of the function in part (c). Then find the maximum area and the dimensions that yield the maximum area.

(e) Use a graphing utility to graph the function in part (c) and verify the maximum area from the graph.