You are designing an athletic field in the shape of a rectangle x meters long capped at two ends by semicircular regions of radius r. The boundary of the field is to be a 400 meter track. What values of x and r will give the rectangle its greatest area?

x = 200 m

r = 100 m

Step-by-step explanation:

Let call "x" one of the sides of the rectangle (the one finishing in semicircular areas then as "r" is the radius of the semicircular areas

x + 2*r = 400 ⇒ 2*r = (400 - x)

Area of the rectangle is:

A (r) = x*y y = 2*r

Then the area of the rectangle as a function of x is:

A (x) = x * (400 - x) ⇒ A (x) = 400*x - x²

Taking derivatives on both sides of the equation we get:

A' (x) = 400 - 2*x

A' (x) = 0 ⇒ 400 - 2*x = 0

2*x = 400

x = 200 m

And r is equal to:

r = (400 - x) / 2

r = (400 - 200) / 2

r = 100 m