10. (6 points) As shown in the figure, with the two vertices of one of the longer sides of a rectangle as the centers, and the width of the rectangle as the radius, draw $\frac{1}{4}$ circles. If the areas of the two shaded parts in the figure are equal, then the ratio of the length to the width of this rectangle is $\qquad$
【Solution】Solution: Let the length and width of the rectangle be $a$ and $b$, respectively, then $\frac{1}{4} \times \pi \times b^{2} \times 2=a b$
$$
\frac{1}{2} \pi b=a
$$

Therefore, $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\pi}{2}$.
Answer: The ratio of the length to the width of the rectangle is $\frac{\pi}{2}$. Hence the answer is: $\frac{\pi}{2}$.