Challenge: You have a 500-foot roll of chain link fencing and a large field. You want to fence in a rectangular

playground area. What are the dimensions of the largest such playground area you can enclose? What is the area

of the playground?

Dimensions of the rectangular area:

x = 125 ft

y = 125 ft

A (max) = 15625 ft²

Step-by-step explanation:

Lets call x and y the wide and the height of the rectangular area. And P the perimeter

Then

A = x*y

And as we have 500 ft of fencing material

P = 2*x + 2*y ⇒ 500 = 2*x + 2*y ⇒ y = (500 - 2x) / 2

y = 250 - x

Then rectangular area is:

A = x * y ⇒ A (x) = x * (250 - x) ⇒ A (x) = 250*x - x²

Taking derivatives on both sides of the equation we get:

A' (x) = 250 - 2*x

A' (x) = 0 250 - 2*x = 0 ⇒ 2x = 250 ⇒ x = 125 ft

And y = 250 - x ⇒ y = 125 ft

We really got a square of side 125 ft

The area

A (max) = 125*125 = 15625 ft²