A cylinder is inscribed in a sphere, the radius of which is $2 \mathrm{~cm}$ larger than the radius of the cylinder's base and $1 \mathrm{~cm}$ larger than the height of the cylinder. What is the volume of the sphere?

 untranslated portion:
Adott henger köré gömböt írunk, melynek sugara $2 \mathrm{~cm}$-rel nagyobb, mint a henger alapjának sugara és $1 \mathrm{~cm}$-rel nagyobb, mint a henger magassága. Mekkora a gömb köbtartalma? 

 translated portion:
A cylinder is inscribed in a sphere, the radius of which is $2 \mathrm{~cm}$ larger than the radius of the cylinder's base and $1 \mathrm{~cm}$ larger than the height of the cylinder. What is the volume of the sphere?
Let the radius of the base circle of the cylinder be denoted by $r$. Then the radius of the sphere is $r+2$, and the height of the cylinder is $r+1$, so

$$
(r+2)^{2}=r^{2}+\frac{(r+1)^{2}}{4}
$$

or

$$
r^{2}-14 r-15=0
$$

from which the positive value of $r$ is: $15 \mathrm{~cm}$. Therefore, the radius of the sphere is $17 \mathrm{~cm}$, and its volume is

$$
K=6550 \frac{2}{3} \pi \mathrm{cm}^{3}
$$

(Jakab Bauer, Zenta.)