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 r, in terms of y, of the semicircular ends of the track. r = Determine the distance d, in terms of y, around the inside edge of the two semicircular parts of the track. d = (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. y = (d) Use the result of part (c) to write the area A of the rectangular region as a function of x. A = (e) Use a graphing utility to graph the area function from part (d). Use the graph to approximate the dimensions that will produce a rectangle of maximum area. (Round your answers to two decimal places.) x = y =

Question

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 r, in terms of y, of the semicircular ends of the track.

r =

Determine the distance d, in terms of y, around the inside edge of the two semicircular parts of the track.

d =

(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.

y =

(d) Use the result of part (c) to write the area A of the rectangular region as a function of x.

A =

(e) Use a graphing utility to graph the area function from part (d). Use the graph to approximate the dimensions that will produce a rectangle of maximum area. (Round your answers to two decimal places.)