# Question c8b0c

May 9, 2017

$\frac{500}{3} \setminus \textrm{m b y} 125 \setminus \textrm{m}$

#### Explanation:

The are of one of the rectangular areas will be
$A = x y$

The length of the fenced area will be
$1000 = 3 x + 4 y$

We have two parts of the fence on the outside and one in the middle connecting them. Between the outside parts and connecting fence there are two parts of fence. So we have four of those.

We can make the area function a function of $x$ by solving for the second equation.
$4 y = 1000 - 3 x$
$y = 250 - \frac{3 x}{4}$

Then we plug this into the area function
$A = x \left(250 - \frac{3 x}{4}\right)$
$A = 250 x - \frac{3 {x}^{2}}{4}$

This is a parabola opening downwards, so the maximum will be where the slope is zero. To do this we find the derivative of the area function.
A`=250-{6x}/4=250-{3x}/2#

We then solve for when it is equal to zero.
$0 = 250 - \frac{3 x}{2}$
$\frac{3 x}{2} = 250$
$3 x = 500$
$x = \frac{500}{3}$

We can then plug this into our total length function
$y = 250 - \frac{3 \left(\frac{500}{3}\right)}{4}$
$y = 250 - \frac{500}{4}$
$y = 250 - 125$
$y = 125$

This is the graph of the area function and the max is at $x = \frac{500}{3} = 166.66$
graph{250x-(3x^2)/4 [-1, 340, -1, 22000]}