A sphere has a radius $r=10 \mathrm{~cm}$; let us determine the volume of that spherical segment whose spherical surface area is in the ratio of $10: 7$ to the area of its base.
If the height of the spherical segment is $m$, and the radius of its base is $r_{1}$, then the surface area of the spherical segment is $2 r m \pi$ and the area of its base is

$$
t=r_{1}^{2} \pi=m(2 r-m) \pi
$$

According to the problem,

$$
2 r m \pi: m(2 r-m) \pi=10: 7
$$

from which

$$
m=\frac{6 r}{10} 6 \text{ cm}
$$

The sought volume is

$$
K=\frac{m^{2} \pi}{3}(3 r-m)=288 \pi \text{ cm}^{3}
$$

(Kürti lmre, Eger.)

The problem was also solved by: Bartók I., Deutsch E., Deutsch I., Eckstein J., Enyedi B., Haar A., Jánosy Gy., Kiss J., Liebner A., Losonczy J., Neidenbach E., Pám M., Pivnyik I., Preisich G., Riesz K., Riesz M., Rássy P., Schlesinger O., Schvemmer J., Szántó H., Szücs A.