The radius of the base of a straight cone is $r=7 \mathrm{~cm}$, and its height is $m=24 \mathrm{~cm}$; a sphere constructed with the cone's apex as the center divides the cone into two equal volumes. What is the volume of this sphere?
A sphere constructed from the apex of a cone cuts a spherical cap from the cone, whose volume is half the volume of the cone; thus

$$
\frac{2 R^{2} \pi m_{1}}{3}=\frac{r^{2} \pi m}{6}
$$

where $R$ is the radius of the sphere and $m_{1}$ is the height of the spherical cap; but

$$
m_{1}=R-R \cos \alpha
$$

where $\alpha$ is the angle between the slant height of the cone and its height; and since

$$
\cos \alpha=\frac{m}{\sqrt{m^{2}+r^{2}}}=\frac{24}{25}
$$

therefore

$$
m_{1}=\frac{R}{25}
$$

substituting this into (1):

$$
\frac{R^{3} \pi}{3.25}=\frac{r^{2} \pi m}{6}
$$

and thus the volume of the sphere is:

$$
\frac{4 R^{3} \pi}{3}=\frac{25 r^{2} \pi m}{3}=25 \cdot 49 \cdot 8 \cdot \pi=9800 \pi \mathrm{cm}^{3}
$$

(Zoltán Rehberger.)

The problem was also solved by: J. Kárf, A. Kiss, Gy. Krisztián, Gy. Lukhaub, G. Petrogalli.