3.1. It is known that the area of the shaded region of the figure is $\frac{32}{\pi}$, and the radius of the smaller circle is 3 times smaller than the radius of the larger circle. What is the length of the smaller circle?

![](https://cdn.mathpix.com/cropped/2024_05_06_b6fc33c08418efc8d3bcg-1.jpg?height=485&width=485&top_left_y=1265&top_left_x=223)
# Answer: 4

Solution. Let the radius of the smaller circle be $R$, then the radius of the larger circle is $3R$. The area of the smaller circle is $S_{1}=\pi R^{2}$, and the area of the larger circle is $S_{2}=\pi(3R)^{2}=9\pi R^{2}$. Therefore, the area of the shaded part is $S_{2}-S_{1}=8\pi R^{2}$. We have $8\pi R^{2}=\frac{32}{\pi}$, from which $(\pi R)^{2}=4$ and $\pi R=2$. Thus, the length of the smaller circle is $2\pi R=4$.