The Problem

To make a pen for his new pony, Ted will use an existing fence as one side of the pen. If he has ninety-six meters of fencing, what are the dimensions of the largest rectangular pen he can make?
(Source: Mathematics Teaching in the Middle School, Nov-Dec1994).
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The Solution

To begin, lets look at a visual representation of this problem:


From this picture we can write the following formulas for the perimeter and the area of the pen Ted is building.

P= 96 +x

where x= the length of existing fence that Ted will use in addition to the new fence.

A= y (96-2y)

A= 96y -2y^2

Now we can take the derivative of the area formula to find the maximum value for y.


A'= 96-4y

Next we set the derivative equal to 0 to solve for the maximum value of y




If y= 24 then x = 96-2y, so x= 48

So the dimensions of Ted's new pen are 24x48, which produces an area of 1152 square feet for the pony.