Question

We need to enclose a field with a fence. We have 500 feet of fencing material and a
building is on one side of the fencing, so we will not need any fencing along this side. Determine
the dimensions of the field that will enclose the largest area.

Answer 1

let the width be x and length be y

then the perimeter is

500 = 2x + y

so y = 500 - 2x

now the area is A = xy

or by substituting A = x( 500 - 2x)

A = 500x - 2x^2

next find the 1st and 2nd derivatives

\[\frac{dA}{dx} = 500 - 4x\]

and \[\frac{dA^2}{d^2x} = -4\]

the solution to the 1st derivative will give the maximum area as the 2nd derivative <0

so let 0 = 500 - 4x

solve for x to find the stationary points.


x = 125

so the maximum area occurs when the width is 125 feet and the length is 250 feet