Question

A farmer wants to build a rectangular pen with $80$ feet of fencing. The pen will be built against the wall of the barn, so one side of the rectangle won't need a fence. What dimensions will maximize the area of the pen $?$

Answer 1

Find the dimensions of the rectangular pen that maximize the area of the pen:

• Let the width of the rectangular pen be $\mathrm{x}$ feet and the length of the rectangular pen be $\mathrm{y}$ feet.
• Since no fencing is required along one side of the pen, the perimeter will be $\mathrm{y}+2\mathrm{x}\left[\mathrm{Perimeter}=2(\mathrm{length}+\mathrm{width})\right]$.
• The farmer has $80$ feet of fencing which will be equivalent to the perimeter of the pen:

$\begin{array}{rcl}\mathrm{y}+2\mathrm{x}& =& 80\\ \therefore \mathrm{y}& =& -2\mathrm{x}+80(\mathrm{first}\mathrm{equation})\end{array}$

• Now, find the area of the pen:

$\begin{array}{rcl}\mathrm{x}\times \mathrm{y}& =& \mathrm{x}\times (-2\mathrm{x}+80)\mathbf{[}\mathbf{Area}\mathbf{=}\mathbf{width}\mathbf{\times }\mathbf{length}\mathbf{]}\\ & =& -2{\mathrm{x}}^2+80\end{array}$

For the area to be maximum, $\frac{\mathrm{dA}}{\mathrm{dx}}=0$, which is equivalent to the following:

$\begin{array}{rcl}\frac{\mathrm{dA}}{\mathrm{dx}}& =& -4\mathrm{x}+80[\mathrm{for}\mathrm{a}\mathrm{maximum}\mathrm{area}]\\ & \Rightarrow & 0=-4\mathrm{x}+80\\ & \Rightarrow & \mathrm{x}=20\end{array}$

Expression is downward parabola so at $x=20$ area will be maximum.

Find the value of $\mathrm{y}$ by substituting the value of $\mathrm{x}$ into the first equation:

$\begin{array}{rcl}\mathrm{y}& =& (-2\times 20)+80\\ & =& -40+80\\ & =& 40\end{array}$

Hence, the width of the rectangular pen is $\mathbf{20}\mathbf{}\mathbf{feet}$ and the length of the rectangular pen is $\mathbf{40}\mathbf{}\mathbf{feet}$.