An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter single-lane running track.
(a) Draw a diagram that illustrates the problem. Let $ x $ and $ y $ represent the length and width of the rectangular region, respectively.
(b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of $ y $ around the inside edge of each semicircular part of the track.
(c) Use the result of part (b) to write an equation, in terms of $ x $ and $ y $, for the distance traveled in one lap around the track. Solve for $ y $.
(d) Use the result of part (c) to write the area of $ A $ the rectangular region as a function of $ x $ What dimensions will produce a rectangle of maximum area?