Build a rectangular pen with three parallel partitions (meaning 4 total sections) using 500 feet of fencing. What dimensions will maximize the total area of the pen?

Dec 5, 2016

The dimensions that will maximize the area the total area of the pen will be $125 f t .$$b y$ $50 f t .$
Total area with these dimensions: $6 , 250 f t {.}^{2}$

Explanation:

Step 1: Set up a picture/diagram to help answer the question and write out the needed equations.

Diagram:

The total length will be $x$ and the height will be $y$.
Needed Equations:
Perimeter of this diagram $= 500 = 2 x + 5 y$
Total Area$= A = x y$

Step 2: Solve for $y$ using the equation $500 = 2 x + 5 y$
$5 y = 500 - 2 x$
$y = 100 - \frac{2}{5} x$

Step 3: Substitute the equation for $y$ into the function for area.
$A = x \left(100 - \frac{2}{5} x\right)$
$A = - \frac{2}{5} {x}^{2} + 100 x$

Step 4: Find the derivative of the equation for area.
$A ' = - \frac{4}{5} x + 100$

Step 5: Use the derivative equation in order to find the critical point(s) that maximize the area.
Critical points are when $A ' = 0$ and when $A '$ does not exist. It is also good to check the endpoints of an equation in order to check for a maximum or minimum.
Since $A '$ always exists, only find where $A ' = 0$ (there will be no endpoints to check since this is a pen).

$0 = - \frac{4}{5} x + 100$
$- 100 = - \frac{4}{5} x$
$100 = \frac{4}{5} x$
$x = 125$

$A '$ is positive when $x < 125$ and $A '$ is negative when $x > 125$, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.

Step 6: Find the height ($y$) when $x = 125$.
$500 = 2 \left(125\right) + 5 y$
$5 y = 250$
$y = 50$
The dimensions that will maximize the area the total area of the pen will be $125 f t .$$b y$ $50 f t .$

Step 7: Find the total area of the pen.
$A = x y$
$A = \left(125\right) \left(50\right)$
$A = 6 , 250 f t {.}^{2}$