Based on a spherical segment with a base radius $r=5 \mathrm{~cm}$ and height $m=2 \mathrm{~cm}$; inscribed in this segment and around this segment, we inscribe straight cones. How do the volumes of these three bodies relate to each other?
Let the volume of the segment be $v_{1}$, the inscribed cone be $v_{2}$, and the circumscribed cone be $v_{3}$, with its height being $h$; since

$$
h=\frac{2 m r^{2}}{r^{2}-m^{2}}
$$

therefore

$$
v_{3}=\frac{2 m \pi r^{4}}{3\left(r^{2}-m^{2}\right)}
$$

furthermore

$$
v_{1}=\frac{m \pi}{6}\left(3 r^{2}+m^{2}\right), v_{2}=\frac{r^{2} \pi m}{3}
$$

and thus

$$
v_{1}: v_{2}: v_{3}=\left(3 r^{2}+m^{2}\right): 2 r^{2}: \frac{4 r^{4}}{r^{2}-m^{2}}
$$

or substituting the given values:

$$
v_{1}: v_{2}: v_{3}=79: 50: \frac{2500}{21}=1659: 1050: 2500
$$

The problem was solved by: Barna D., Freibauer E., Krisztián Gy., Sasvári G., Weisz J.