A cylinder's total surface area is in the ratio of $7: 4$ to the area of its base. What is the surface area and volume of the cylinder if the slant height of a cone inscribed in the cylinder is $30 \mathrm{~cm}$?
If the height of a cylinder is $m$, and the radius of its base circle is $r$, then according to the problem,

$$
\left(2 r^{2} \pi+2 r m \pi\right): 2 r m \pi=7: 4
$$

which means

$$
r=\frac{3}{4} m
$$

and

$$
r^{2}+m^{2}=\frac{9}{16} m^{2}+m^{2}=900
$$

from which

$$
m=24 \mathrm{~cm}, \quad \text { so } \quad r=18 \mathrm{~cm}
$$

thus

$$
\begin{gathered}
F=2 r^{2} \pi+2 r \pi m=1512 \pi \mathrm{cm}^{2} \\
K=r^{2} m \pi=7776 \pi \mathrm{cm}^{3}
\end{gathered}
$$