Question

A running circuit is to be constructed in the shape of a rectangle with two semicircles attached at the east and west ends of the path (see figure). A picnic area will be set in the center of the circuit. If we want the total length of the circuit to be 2000 m (2km ), what will be the dimensions (r and L) that make the picnic area maximum? Give you answer in meters.

Answer 1

I will assume that we are just measuring the inner boundary of the path. Thus, the two lengths L and a circle of radius r has perimeter

p = 2L + 2πr
The inner picnic area is

a = L(2r) = (1000-πr)(2r)
= 2000r - 2πr^2

da/dr = 2000 - 4πr
da/dr=0 when r = 500/π

So, the picnic area is 500 by 1000/π

As usual, the length is divided equally between the lengths and the (curved) widths.