The bank of the river is a straight line. Fence off from it a rectangular plot of total length $p$ with the largest possible area.

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Reflect the fence line relative to the shore.

## Solution

Reflect the fence line symmetrically from the shore line. We get a rectangle with a perimeter of $2 p$. This rectangle achieves the maximum area when it is a square with a side of $\frac{p}{2}$.

Indeed, if $x$ is the side of the rectangle, then the adjacent side is $p-x$. Then for the area $S$ of the rectangle, the inequality holds:

$$
S=x(p-x) \leqslant \frac{(x+(p-x))^{2}}{4}=\frac{p^{2}}{4}
$$

with equality only when $x=p-x$, i.e., when $x=\frac{p}{2}$. Therefore, the length of the plot is $\frac{p}{2}$, and the width is $\frac{p}{4}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8027e864227276d12944g-07.jpg?height=301&width=760&top_left_y=364&top_left_x=655)

## Answer

The length of the plot should be twice its width (half of a square).