A right circular cone has a volume of $V=30 \mathrm{~cm}^{3}$; its lateral surface area is 5 times the area of its base. What are the height $m$, slant height $l$, and radius $r$ of the base of the cone?
In the context of the task:

$$
r \pi l=n r^{2} \pi
$$

from which

$$
\begin{gathered}
l=n r \\
l^{2}=m^{2}+r^{2}
\end{gathered}
$$

by substituting (1), we get:

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

from which

$$
r^{2}=\frac{m^{2}}{n^{2}-1}
$$

Substituting (2) into the volume formula:

$$
v=\frac{m^{3} \pi}{3\left(n^{2}-1\right)}
$$

from which

$$
m=\sqrt[3]{\frac{3\left(n^{2}-1\right)}{\pi} v}
$$

Substituting the given values into (1), (2), and (3), we get:

$$
m=8.826 \mathrm{~cm}, \quad r=1.802 \mathrm{~cm}, \quad l=9.008 \mathrm{~cm}
$$

The task was solved by: Feuer Mór, Friedmann Bernát, Fröhlich Károly, Galter János, Geist Emil, Goldstein Zsigmond, Grünhut Béla, Hofbauer Ervin, Kántor Nándor, Reif Jenő, Schneider Béla, Szabó István.