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