Rectangle problem with shared wall split 4 ways (given perimeter)

Question

You plan to build four identical rectangular sheep pens in a row. Each adjacent pair of pens share a fence between them. You have a total of 304304 feet of fence to use. Find the dimension of each pen such that you can enclose the maximum area.

I have figured out the width of each pen (19 ft) but can't seem to get the length and area right

Do you really have 57+ miles of fencing? – DJohnM Mar 14 '17 at 05:16 — DJohnM, Mar 14 '17 at 05:16
no it should be 304 ft of fencing – janh callm Mar 14 '17 at 05:47 — janh callm, Mar 14 '17 at 05:47

score 0 · Answer 1 · answered Mar 14 '17 at 05:15

The largest rectangle will have dimensions $x$ and $y$, with area $A=xy$, such that $x\leq y$. The length of fence used is $L=5x+2y$ (It is evident that the longer edge should not be used as the length for the border between adjacent pens).

To maximize $A$, take the derivative of the area equation after substitution.

$A=x(\frac{L-5x}{2})=\frac{Lx}{2}-\frac{5x^2}{2}$

$\frac{dA}{dx}=L/2-5x$

Finding the maximum requires setting the derivative to zero, which is attained at $x=L/10$. Thus, the short side should be one tenth of the total fencing you have. Solving for the long side yields $y=L/4$.

Rectangle problem with shared wall split 4 ways (given perimeter)

1 Answers1