2.28. Find the ratio of the surface area and volume of a sphere to the surface area and volume of an inscribed cube.

## Group 5
2.28. Let the radius of the sphere be $R$, and the edge of the cube be $a$; then $R^{2}-\left(\frac{a}{2}\right)^{2}=\frac{a^{2}}{2}$, from which $a=\frac{2 R}{\sqrt{3}}$. Denote the volumes and surfaces of the sphere and the cube by $V_{1}, V_{2}$ and $S_{1}, S_{2}$, respectively. We have $V_{1}=\frac{4}{3} \pi R^{3}, V_{2}=a^{3}=\frac{8 R^{3}}{3 \sqrt{3}}, S_{1}=4 \pi R^{2}, S_{2}=6 a^{2}=8 R^{2}$, from which

$$
V_{1}: V_{2}=\pi \sqrt{3}: 2, S_{1}: S_{2}=\pi: 2
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-469.jpg?height=464&width=496&top_left_y=1418&top_left_x=42)

Answer: $\pi: 2 ; \pi \sqrt{3}: 2$.