Question

A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fencing along the river. What are the dimensions of the field that has the largest area?

Answer 1

600ft x 1200ft

Explanation:

Use derivative optimization to find the maximum area.

I'll call the two same sides "a", and the one different side "b"

The maximum perimeter (including 3 sides) is 2400 ft. so,

2400 = 2a + b

The area is length × width. so,

A = ab

Using substitution to combine the equations,

A = a × (2400 - 2a)

A = -2a² + 2400a

Find the maximum of A by finding the zeros of its derivative.

dA = -4a +2400

0 = -4a + 2400

The maximum occurs at a = 600

Substitute in the perimeter equation to find b.

2400 = 2(600) + b

b = 1200

600 x 1200