The lateral surface area of a right circular cylinder is in the ratio of $5:3$ to its base area; the diagonal of the axial section is $39 \mathrm{~cm}$. What is the total surface area and volume of the cylinder?
Let $r$ be the radius of the base circle, and $m$ the height of the cylinder. Then

$$
2 r \pi m: r^{2} \pi=5: 3
$$

or

$$
6 m=5 r
$$

and

$$
(2 r)^{2}+m^{2}=39^{2}
$$

or

$$
4 r^{2}+\frac{25}{36} r^{2}=39^{2}
$$

from which the positive value of $r$ is:

$$
r=18 \mathrm{~cm} \text { and thus } m=15 \mathrm{~cm} .
$$

Therefore, the total surface area of the cylinder is

$$
F=2 r \pi m+2 r^{2} \pi=1188 \pi \mathrm{cm}^{2}
$$

and its volume is

$$
K=r^{2} \pi m=4860 \pi \mathrm{cm}^{3}
$$

(Leó Spitzer, Budapest.)

The problem was also solved by: Baján A., Basch R., Breuer P., Dénes M., Erdős V., Fried E., Koffer B., Lengyel K., Rosenberg E., Sichermann F., Szántó L., Szilárd V., Szóbel I., Ungar E., Vilcsek A., Viola R., Weinberger Gy.