Question

A farmer is building a rectangular pen with three parallel partitions, shown in the diagram. He has 500 feet of

fencing available. What is the maximum area he can enclose?

Answer 1

10,416.67 ft²

Explanation:

If the length of the three parallel partitions is h, and the width of the two parallel walls is w, then the perimeter is:

500 = 3h + 2w

The area is:

A = hw

Solve for w in the first equation:

2w = 500 − 3h

w = 250 − 1.5h

Substitute into the second equation:

A = h (250 − 1.5h)

A = 250h − 1.5h²

We can find the maximum using vertex of a parabola:

h = -b / (2a)

h = -250 / (2 × -1.5)

h = 83.33

Or using calculus:

dA/dh = 250 − 3h

0 = 250 − 3h

h = 83.33

Therefore, the maximum area is:

A = 250h − 1.5h²

A = 10,416.67