# Question #b9993

Jan 30, 2017

The total length of the partition should be $125$ feet, and the height of each partition should be $50$ feet (making each section $50$ feet x $31.25$ feet). This results in a maximum enclosed area of $6250 {\text{ feet}}^{2}$

#### Explanation:

Without loss of generality let us assume that the pen is divided as shown:

Let us set up the following variables:

$\left\{\begin{matrix}x & \text{Total height of the partition (feet)" \\ y & "Total length of the partition (feet)" \\ A & "Total Area enclosed by the partition (sq feet)}\end{matrix}\right.$

Our aim is to find $A \left(x\right)$, and to maximize the total area, A, wrt x (equally we could the same with $y$ and we would get the same result). ie we want a critical point of $\frac{\mathrm{dA}}{\mathrm{dx}}$.

Now, the total perimeter is given as $500$ (constant) and so:

$5 x + 2 y = 500$
$\therefore 2 y = 500 - 5 x$
$\therefore y = 250 - \frac{5}{2} x$ ..... [1]

And the total Area enclosed by the pen is given by:

$A = x y$

And substitution of the first result [1] gives us:

$A = x \left(250 - \frac{5}{2} x\right)$
$\setminus \setminus \setminus = 250 x - \frac{5}{2} {x}^{2}$

We no have the Area, A, as a function of a single variable, so Differentiating wrt $x$ we get:

$\frac{\mathrm{dA}}{\mathrm{dx}} = 250 - 5 x$ ..... [2]

At a critical point we have $\frac{\mathrm{dA}}{\mathrm{dx}} = 0 \implies$

$250 - 5 x = 0$
$\therefore \setminus \setminus \setminus \setminus \setminus 5 x = 250$
$\therefore \setminus \setminus \setminus \setminus \setminus \setminus \setminus x = 50$

And substituting $x = 50$ into [1] we get;

$y = 250 - \frac{5}{2} \left(50\right)$
$\setminus \setminus = 250 - 125$
$\setminus \setminus = 125$

We should check that $x = 50$ results in a maximum area. Differentiating [2] wrt x we get:

$\frac{{d}^{2} A}{\mathrm{dx}} ^ 2 = - 5 < 0$ when $x = 50$

Confirming that we have a maximum area, given by:

$A = \left(50\right) \left(125\right) = 6250 {\text{ feet}}^{2}$

We can visually verify that this corresponds to a maximum by looking at the graph of $y = A \left(x\right)$:
graph{250x - 5/2x^2 [-100, 200, -100, 7000]}