# Calc2.51

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The fence and pens in question. x is the entire length of the top border, and y is the height of each line.

If a fence is to be made with three pens, the three connected side-by-side, find the dimensions which give the largest total area if 400 feet of fence are to be used.

This problem describes a large rectangle with length x and height y. However, this rectangle is split into three parts with two extra lengths of fence of length y. We know the total fence to be used is 400 feet, so

$400=2x+4y\Rightarrow 200=x+2y\,$

From this equation we can see that $0\leq y\leq 100$ which gives us our interval. Now, to maximize the area, we need a function for the area. Since this is a rectangle, the area is simply

$A=xy\,$

Solving the first equation for x and plugging it into the second equation for area gives

$A=(200-2y)y=200y-2y^{2}\,$

To find the absolute maximum area, we need to find the critical points of this area function.

$A'=200-4y\,$

Thus, when $y=50$ we have a critical point and this is our only critical point. Our absolute maximum area, then, must come when $y=0$, $y=50$, or $y=100$.

$A(0)=0\,$

$A(50)=5000\,$

$A(100)=0\,$

Thus, our absolute maximum area of 5000 square feet comes when the length is 100 feet and the height is 50 feet.