Problem 660

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

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