# 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?

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.

We now have:

$2 x + y = 2400$

If we solve for $y$, we get:

$y = 2400 - 2 x$

Now, we know that $A = x y$. Therefore,

$A = \left(2400 - 2 x\right) x$

$A = - 2 {x}^{2} + 2400 x$

Now differentiate with respect to $x$.

$\frac{\mathrm{dA}}{\mathrm{dx}} = - 4 x + 2400$

Find critical numbers.

$0 = - 4 x + 2400$

$x = 600$

Since $y = 2400 - 2 x$, we have that $y = 2400 - 2 \left(600\right) = 1200$

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 $A = - 2 {x}^{2} + 2400 x$ opens down and therefore has a maximum and no minimum.

Hopefully this helps!