# 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

### 96-4y=0

### 96=4y

### y=24

### 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.