WeBWorK Standalone Renderer

You will build a rectangular sheep pen next to a river. There is no need to build a fence along the river, so you only need to build three sides. You have a total of $420$ feet of fence to use. Find the dimensions of the pen such that you can enclose the maximum area.

Let the width be $w$ ft, and the length be $l$ ft. Since there are only three sides of the fence, we have:

$\displaystyle{\begin{aligned}[t] l+2w &= 420 \\ l &= 420-2w \end{aligned} }$

If the pen’s width is $w$ feet, then its length is $420-2w$ feet. Now we can build a function for the area of the pen:

$\displaystyle{\begin{aligned}[t] A(w) &= (420-2w)w \\ A(w) &= -2w^2+420 w \end{aligned} }$

In this formula, $w$ represents the pen’s width in feet, and $A(w)$ represents the pen’s area in square feet.

A diagram of a rectangular pen. Along the top side is a river. The right side is labeled w ft, and the bottom side is labeled (420-2w) ft.

Answer the following questions:

To maximize the area of the pen, the length of the pen (parallel to the river) should be .

To maximize the area of the pen, the width of the pen (away from the river) should be .

The maximum area of the pen is .

(Use ft for feet, and ft^2 for square feet.)

You can earn partial credit on this problem.