Question

The manager at a daycare facility wants to put a fence around a rectangular play yard just outside of the building. She will use a wall of the building as one side of this rectangle, so that fencing is only needed on the other three sides. If there are 96 feet of fencing available, and x represents the length of one of the sides of the rectangle that is perpendicular to the building's wall, then find the value of x that will maximize the area of this play yard. (see the picture below)

Answer 1

24

Explanation:

If x is the length of the perpendicular sides, then the length of the parallel side must by 96−2x.

The total area is:

A = x (96 − 2x)

A = 96x − 2x²

This is a parabola, so we know the maximum is at the vertex (at x = -b/(2a)).

x = -96 / (2 · -2)

x = 24

Or, we can optimize using calculus. Find dA/dx and set to 0:

dA/dx = 96 − 4x

0 = 96 − 4x

x = 24

The value of x that maximizes the area is x = 24 ft.