A farmer has a rectangular property that needs to be fenced on three sides (a river surrounds the fourth. If he has 2400 feet of fencing material available, what maximum area will he be able to enclose?

1 Answer
May 8, 2017

The dimensions that give the maximum area are 600 feet by 1200 feet (that's to say, the side parallel to the river will measure 1200 feet).

Explanation:

Start by tracing a diagram.

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We now have:

2x + y = 2400

If we solve for y, we get:

y = 2400 - 2x

Now, we know that A = xy. Therefore,

A = (2400 - 2x)x

A = -2x^2 + 2400x

Now differentiate with respect to x.

(dA)/(dx) = -4x + 2400

Find critical numbers.

0 = -4x + 2400

x = 600

Since y = 2400 - 2x, we have that y = 2400 - 2(600) = 1200

Therefore, the dimensions the give the maximum area are 600 by 1200 feet. The maximum area is therefore 7,200,000.

We know that this is a maximum because the function A = -2x^2 + 2400x opens down and therefore has a maximum and no minimum.

Hopefully this helps!