# You have 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?

Refer to explanation

#### Explanation:

Let $x , y$ the sides of a rectangle hence the perimeter is

$P = 2 \cdot \left(x + y\right) \implies 500 = 2 \cdot \left(x + y\right) \implies x + y = 250$

The area is

$A = x \cdot y = x \cdot \left(250 - x\right) = 250 x - {x}^{2}$

finding the first derivative we get

$\frac{\mathrm{dA}}{\mathrm{dx}} = 250 - 2 x$ hence the root of the derivative gives us the maximum value hence

$\frac{\mathrm{dA}}{\mathrm{dx}} = 0 \implies x = 125$

and we have $y = 125$

Hence the largest area is $x \cdot y = {125}^{2} = 15 , 625$ ft^2

Obviously the area is a square.