13.201. A sports field has the shape of a rectangle, the length of which is $b$ m longer than its width. The field is surrounded by a path of uniform width of $a$ m. What are the dimensions of the sports field if its area is equal to the area of the surrounding path?
## Solution.

Let $x$ m be the width of the plot, $b+x$ m - its length (Fig. 13.12). The area of the plot is $x(x+b)$ m². The area of the path is $2 a x + 2(b + x + 2 a) a$, from which we get $x = \frac{\sqrt{b^{2} + 32 a^{2}} - b + 4 a}{2} \text{ m};$

$$
b + x = \frac{4 a + b + \sqrt{b^{2} + 32 a^{2}}}{2} \text{ m}
$$

Answer: $\left(\sqrt{b^{2} + 32 a^{2}} - b + 4 a\right) / 2$ and $\left(\sqrt{b^{2} + 32 a^{2}} + b + 4 a\right) / 2$ m.