A rectangle has one of its diagonals $26 \mathrm{~cm}$. If we rotate the rectangle around one of its sides, a cylinder is formed. A plane perpendicular to the side of this cylinder is inscribed, and its total surface area is $2720 \mathrm{~cm}^{2}$. What is the height of the cylinder and the radius of its base?
Let the height of the cylinder be $m, \mathrm{~s}$ and the radius of its base be $r$. The surface area of the lateral surface

$$
8 r^{2}+8 r m=2720
$$

or

$$
r^{2}+r m=341
$$

and

$$
r^{2}+m^{2}=26^{2}=676
$$

If we solve for $m$ from (1) and substitute it into (2), we get the following equation:

$$
r^{4}-678 r^{2}+57800=0
$$

from which

$$
r_{1}^{2}=578 \text { and } r_{2}^{2}=100
$$

so the positive values of $r$ are:

$\sqrt{578} \mathrm{~cm}$ and $10 \mathrm{~cm}$,

and the corresponding values of $m$ are:

$\sqrt{98} \mathrm{~cm}$ and $24 \mathrm{~cm}$.