Question

A land owner is planning to build a fenced-in, rectangular patio behind his garage, using his garage as one of the "walls." He

wants to maximize the area using 80 feet of fencing. The quadratic function A(Z) = x(80 – 22) gives the area of the

patio, where x is the width of one side. Find the maximum area of the patio.

Answer 1

Maximum area = 800 square feet.

Explanation:

In the figure attached,

Rectangle is showing width = x ft and the side towards garage is not to be fenced.

Length of the fence has been given as 80 ft.

Therefore, length of the fence = Sum of all three sides of the rectangle to be fenced

80 = x + x + y

80 = 2x + y

y = (80 - 2x)

Now area of the rectangle A = xy

Or function that represents the area of the rectangle is,

A(x) = x(80 - 2x)

A(x) = 80x - 2x²

To find the maximum area we will take the derivative of the function with respect to x and equate it to zero.

`A'(x)=(d)/(dx)(80x-2x^(2))`

= 80 - 4x

A'(x) = 80 - 4x = 0

4x = 80

x =
`(80)/(4)`

x = 20

Therefore, for x = 20 ft area of the rectangular patio will be maximum.

A(20) = 80×(20) - 2×(20)²

= 1600 - 800

= 800 square feet

Maximum area of the patio is 800 square feet.