# A rectangular playground is to be fenced off and divided

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?
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Janet Young
The five, fenced sides add up to 400. Subtract 2x from both sides
$2x+3y=400\to 3y=400-2x$
Divide 3 from both sides
$y=\frac{400}{3}-\frac{2}{3}x$
Area of a rectangle (in terms of x)
$Area=xy=x\left(\frac{400}{3}-\frac{2}{3}x\right)=-\frac{2}{3}{x}^{2}+\frac{400}{3}x=f\left(x\right)$
$x=\frac{-b}{2a}$
x-coordinate of vertex $=\frac{\frac{-400}{3}}{2\left(-\frac{2}{3}\right)}=100$
Plug in 100 for x into the y equation.
y-coordinate of vertex $=\frac{400}{3}-\frac{2}{3}\left(100\right)=66.6$
Plug in 100 for x and 66.6 for y into the area equation.
$Area=xy=100\left(66.6\right)=6666.6$