# Calc2.50

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If a farmer wants to make a rectangular pen along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?

This problem is equivalent to asking us to find the absolute max of the area on an interval, though the interval is not given explicitly. We know that an absolute max can only occur at a critical point or at the endpoint of an interval. The information in this problem gives us one equation relating the length and width of this rectangle. The total length of fence must be 100 feet. Since we have three sides, two of which are equal, we have

$100=2x+y\,$

If the total fence to be used is 100 feet, it makes sense that $0\leq x\leq 50$. Thus, our interval for the problem is $[0,50]$. Now, the problem asks us to maximize the area. The derivative of a function can help us find the maximum, if one exists, so we need an equation for the area of this rectangle. That is simply

$A=xy\,$

This is all the information we need to solve this problem. Take the first equation and solve it for $y$. We can then plug this into our second equation so that the area is in terms of only one variable.

$A=x(100-2x)=100x-2x^{2}\,$

Remember, a critical point is a point where the original function is defined and the derivate is $0$ or undefined.

$A'=100-4x\,$

This derivative is always defined so no critical points will be found that way. The derivative is equal to $0$ when $x=25$ so we have one critical point. Thus are absolute maximum area will be found at $x=0$, $x=25$, or $x=50$.

$A(0)=0\,$

$A(25)=1250\,$

$A(50)=0\,$

Thus, the greatest area the fence can enclose in the prescribed manner is 1250 square feet when the side adjacent to the river is 50 feet long and the side perpendicular is 25 feet long.