## Task A-2.3. (4 points)

A wire of length $10 \mathrm{~km}$ is to be used to fence a rectangular plot of land that has a straight wall on one side (the wire must be used for the remaining three sides), so that the area of this plot is as large as possible. What is the area of the thus enclosed plot of land?
## Solution.

Let $x$ and $y$ be the lengths of the sides of the enclosed rectangle. We have $2x + y = 10$ (where $y$ is the length of the side parallel to the wall). The area of the land is

$$
P = xy = x(10 - 2x) = 10x - 2x^2
$$

We need to find the maximum value of the area as a function of $x$. Since $P$ is a quadratic function of $x$,

$$
P(x) = 10x - 2x^2 = \frac{25}{2} - 2\left(x - \frac{5}{2}\right)^2 \geqslant \frac{25}{2}
$$

which means the maximum area that can be enclosed is $12.5 \mathrm{~km}^2$.

Note. The area is maximized when $x = 2.5$ and $y = 5$.