# Suppose you have 200 feet of fencing to enclose a rectangular plot. How do you determine dimensions of the plot to enclose the maximum area possible?

May 3, 2017

The length and width should each be $50$ feet for maximum area.

#### Explanation:

The maximum area for a rectangular figure (with a fixed perimeter) is achieved when the figure is a square. This implies that each of the 4 sides are the same length and (200" feet")/4=50" feet"
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Suppose we didn't know or didn't remember this fact:

If we let the length be $a$
and the width be $b$
then
$\textcolor{w h i t e}{\text{XXX}} 2 a + 2 b = 200$ (feet)

$\textcolor{w h i t e}{\text{XXX}} \rightarrow a + b = 100$
or
$\textcolor{w h i t e}{\text{XXX}} b = 100 - a$

Let $f \left(a\right)$ be a function for the area of the plot for a length of $a$
then
$\textcolor{w h i t e}{\text{XXX}} f \left(a\right) = a \times b = a \times \left(100 - a\right) = 100 a - {a}^{2}$

This is a simple quadratic with a maximum value at the point where it's derivative is equal to $0$

$\textcolor{w h i t e}{\text{XXX}} f ' \left(a\right) = 100 - 2 a$

and, therefore, at it maximum value,
$\textcolor{w h i t e}{\text{XXX}} 100 - 2 a = 0$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow a = 50$

and, since $b = 100 - a$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow b = 50$