Question

Optimization

a) An outdoor physical fitness complex consists of a rectangular region (football field) with a semicircle on each end. The perimeter of the complex is to be a 400-meter running track. What dimensions will create the maximum area of the rectangular section of the fitness complex?​

Answer 1

length = 100 m

width = 63.66 m

Explanation:

Let's call

x: lenght of rectangle

y: width of rectangle, and also diameter of the semicircles

Perimeter: 2*x + π*y = 400

or:

x = 200 - π/2*y

Area of the rectangle = x*y

Replacing:

A = (200 - π/2*y)*y

A = 200*y - π/2*y²

At the maximum:

dA/dy = 200 - π*y = 0

y = 200/π

y = 63.66 m

and

x = 200 - π/2*63.66

x = 100 m