Question

Alton is making a sandbox for his kids in the backyard. He has 400 feet of board for the sides of the sandbox. A. Define the function A(w) to represent the area of the sandbox as a function of the width. Explain your reasoning. B. Determine the maximum area of the sandbox as well as the length and width that will result in the maximum area. Explain your reasoning.

Answer 1

The answer is below

Explanation:

a) Let the width be represented as w and the length be represented as l.

The perimeter of the sand box = 2(l + w)

Since there is 400 feet of material for the sides of the sandbox, hence:

400 = 2(l + w)

200 = l + w

l = 200 - w

The area (A) is:

A = length * width

A = (200 - w) * w

A(w) = 200w - w²

b) The maximum area is at A'(w) = 0. Hence:

A'(w) = 200 - 2w

200 - 2w = 0

2w = 200

w = 100 feet.

l = 200 - w = 200 - 100

l = 100 feet.

Area = length * width = 100 * 100 = 10000 feet²

The maximum area of the sandbox is 10000 feet², with a length of 100 feet and width of 100 feet.