A track 400 m in periphery is to be laid out on the practice field. Each semicircular end must have a radius of at least 20 m, and each straight section must be at least 100 m. How should the track be laid out so that it encompasses the least area

r = 31.8

Explanation:

To start with, we make the following assumptions. Let

r be the radius of semicircular end

l be the length of straight section

a be the area inside track, then

p = 2πr + 2l = 400

a = πr² + 2rl

2πr + 2l = 400

l = 200 - πr, substituting for l in area,

a = πr² + 2r(200 - πr)

a = πr² + 400r - 2πr²

a = 400r - πr²

In order to find our minimum area, we differentiate with respect to r and then it equate to zero

da/dr = 400 - 2πr = 0

r = 200/π, substituting for r in equation for l,

l = 200 - πr

l = 0 m

And since this is not possible, l must be 100 m (the minimum), and

r = (200 - 100)/π

r = 100/π = 31.8 m