According to the problem,

$$
2 r \pi m: r \pi \sqrt{m^{2}+r^{2}}=8: 5
$$

from which

$$
9 m^{2}=16 r^{2}
$$

or, taking only the positive value, $r=\frac{3 m}{4}$. But $588=r m$ so $588=\frac{3 m^{2}}{4}$, from which

$$
m=28 \mathrm{~cm} \text { and } r=21 \mathrm{~cm}
$$

(Endre Sárközy, Keszthely.)

The problem was also solved by: L. Bánó, P. Füstös, Z. Harsányi, P. Heimlich, Gy. Jánosy, A. Paunz, S. Pichler, Gy. Schlesinger, Gy. Schuster, J. Szabó, B. Tóth.
The base and height of a right circular cylinder and a right circular cone are equal. The lateral surface areas of these solids are in the ratio $8: 5$. The area of the axial section of the cone is $588 \mathrm{~cm}^{2}$. What are the heights and the radii of the bases of these solids?