Question

Write the answer as follows: area, dimension x dimension, y dimension I

Izzy the wonder dachshund wants her dog run to have the maximum area possible.
Amanda has 140 feet of fencing to build a rectangular dog run. To maximize the area
for Izzy, she is going to put the dog run along the side of the house so that the only
three sides of the run will need to be fenced. Find the maximum area possible and the
dimensions needed to achieve that area.
Your answer

Answer 1

1) The maximum area possible = 2450 ft²

2) Length, L = 70 ft Width , W = 35 ft

Explanation:

The length of the fencing available = 140 ft.

Let L, represent the length of the fence, and W, represent the Width of the fencing

We have;

L + 2·W = 140

∴ L = 140 - 2·W

The area = L × W = (140 - 2·W) × W

The area = (140 - 2·W) × W = 140·W - 2·W²

The area A = 140·W - 2·W²

The dimension of the width for the maximum area is obtained by finding the derivative and equating to zero as follows;

dA/dW = d(140·W - 2·W²)/dW = 140 - 4·W = 0

4·W = 140

W = 140/4 = 35 ft.

W = 35 ft.

The maximum area possible is obtained when W = 35

∴ The maximum area possible A₃₅, Where W = 140·35 - 2·35²

Where W = 35, A = 140 × 35 - 2×35² = 2450 ft.²

The maximum area possible = 2450 ft.²

Therefore, L = 140 - 2·W = 140 - 2 × 35 = 70 ft.

L = 70 ft.

The dimensions are;

Length, L = 70 ft. Width , W = 35 ft.