Question

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180ft of fencing on three sides of the yard. The quadratic equation A=−2x2+180x gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

Answer 1

a) length x = 45ft

b) maximum area = 4050 ft²

Explanation:

Given the quadratic equation A=−2x2+180x that gives the area A of the yard for the length x, to maximize the area of the yard then dA/dx must be equal to zero i.e dA/dx = 0

If A=−2x²+180x

dA/dx = -4x + 180 = 0

-4x + 180 = 0

Add 4x to both sides

-4x + 180 + 4x = 0 + 4x

180 = 4x

x = 180/4

x = 45

Hence the length of the building that should border the yard to maximize the area is 45 ft

To find the maximum area, we will substitute x = 45 into the modelled equation of the area i.e A=−2x²+180x

A = -2(45)²+180(45)

A = -2(2025)+8100

A = -4050 + 8100

A = 4050 ft²

Hence the maximum area of the yard is equal to 4050 ft²