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.