Problem 13. Determine the length and width of a rectangular plot of land that will maximize its area given a fixed perimeter.
## S o l u t i o n.

Let $x$ and $y$ be the sides of the rectangle. Then the perimeter will be $p=2x+2y$, and the area will be $S=xy$. By their nature, $x$ and $y$ are positive, as they represent the lengths of segments. According to the condition, the perimeter $p$ is a constant, meaning it reaches its maximum value when $x=y=\frac{p}{4}$. Therefore, the plot should be a square.