You have \(100\) feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest possible area (in square feet)?

We have to maximize the area, \(A\). Since, \(100=2x+y\), it follows that

\(y=100-2x\).

We now express the area as the function of \(x\).

\(A(x)=\answer {x}\cdot (100-2x)\)

So, we have to find the global maximum of the function \(A\), given by

\(A(x)=100\cdot \answer {x}-2x^2\)

on its domain \([0,50]\).

We have to find all the critical points of \(A\) on \([0,50]\). Since

\(A'(x)=\answer {100}-4x\),

it follows that the function \(A\) has its only critical point at \(x=\answer {25}\).

We have to evaluate \(A\) at the end points and the critical point.

\(A(0)=\answer {0}\)

\(A(50)=\answer {0}\)

\(A(25)=\answer {1250}\)

Now we compare these three values, and determine the global maximum of \(A\).

\[ \answer {1250} ft^2 \]