Exercise 38 Page 207

Let's define a variable for each dimension of the rectangle. Let $w$ be the width and $\ell$ be the length.

The perimeter of a rectangle is $P=2\ell +2w,$ but one side does not require fencing. This means the perimeter of the playground is $P=2\ell +w.$ Since there is $100\text{ ft}$ of fence, then $2\ell +w=100,$ which means that $w=100-2\ell .$

Now, we can write the area of the rectangular playground. This is a quadratic function that models the area of the playground. Let's rewrite it in standard form. Let's identify the coefficients of the above quadratic function. As we can see $a=\text{-}2,$ $b=100,$ and $c=0.$ Since $a$ is a negative number the parabola opens downward, which implies that its maximum point corresponds to the vertex. Let's find the $x\text{-}$coordinate of the vertex. Let's simplify above quotient to calculate $\ell .$ The length that maximizes the area of the playground is $\ell =25\text{ ft}.$ To know what is the greatest area, we must find $A(25).$ Consequently, the greatest area we can fence is $1250{\text{ ft}}^2.$