11.184. A cylinder and a sphere are given. The radii of the base of the cylinder and the great circle of the sphere are equal. The total surface area of the cylinder is to the surface area of the sphere as \( m: n \). Find the ratio of their volumes.
Solution.

Let $R$ be the radius of the base of the cylinder and the radius of the sphere, $H$ be the height, $S_{1}$ be the total surface area, $V_{1}$ be the volume of the cylinder, $S_{2}$ be the surface area, and $V_{2}$ be the volume of the sphere. Then $\frac{S_{1}}{S_{2}}=\frac{2 \pi R^{2}+2 \pi R H}{4 \pi R^{2}}=\frac{R+H}{2 R}=\frac{m}{n}$. Let $R+H=m$. Then $2 R=n, R=\frac{n}{2}$ and $H=m-R=\frac{2 m-n}{2}$. Therefore, $\frac{V_{1}}{V_{2}}=\frac{\pi R^{2} H}{\frac{4}{3} \pi R^{3}}=\frac{3 H}{4 R}=\frac{3(2 m-n)}{2}: 2 n=\frac{6 m-3 n}{4 n}$.

Answer: $\frac{6 m-3 n}{4 n}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0883.jpg?height=767&width=705&top_left_y=73&top_left_x=334)

Fig. 11.74