Question

A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter
of the track is to be 1800 meters. What should the dimensions of the rectangle be so that the area of the rectangle is
maximized?

Answer 1

Answer:

Step-by-step explanation:

Let x be the track straights lengths

Let y be the track ends diameter and the other rectangle side lengths.

1800 = 2x + πy

y = (1800 - 2x) / π

A = xy

A = x((1800 - 2x) / π

A = (1/π)(1800x - 2x²)

dA/dx = (1/π)(1800 - 4x)

       0 = (1/π)(1800 - 4x)

       0 = 1800 - 4x

     4x = 1800

        x = 450 m

        y = (1800 - 2(450)) / π

        y = 900/π or approximately 286.5 m