Question

You have a 500-foot roll of chain link fencing and a large field. You want to fence in a rectangularplayground area. What are the dimensions of the largest such yard ? What is the largest yard? Field dimension Maximum area

Answer 1

The length of the fencing is 500 foot

That means the perimeter of the playground = 500

Assume that the dimensions of it are x and y

So 2x + 2y = 500 ---- perimeter of the rectangular yard

Divide both sides by 2

x + y = 250

Now subtract the two sides by x to find y in terms of x

x - x + y = 250 - x

y = 250 - x --------(1)

Since we need to find the maximum area, let us find the area in terms of x and y

Area of the yard is A

A = xy -------- area the rectangle

Now substitute y by (1)

A= x(250 - x)

A = 250x - x^2

For the maximum area, we will differentiate A with respect to x

$\frac{dA}{dx}=250-2x$

For maximum area equate dA/dx by zero

Since dA/dx = 0

250 - 2x = 0

Add 2x for both sides

250 - 2x + 2x = 0 + 2x

250 = 2x

Divide both sides by 2

125 = x