A sphere is inscribed in an equilateral cone, whose volume is $100 \mathrm{~cm}^{3}$. What is the lateral surface area of the cone?
If the slant height of the cone is $l$, and the radius of the sphere is $r$, then

$$
v=\frac{4}{3} r^{3} \pi
$$

from which

$$
r=\sqrt[3]{\frac{3 v}{4 \pi}}
$$

and

$$
3 r=3 \sqrt[3]{\frac{3 v}{4 \pi}}=\frac{l}{2} \sqrt{3}
$$

from which

$$
l=\frac{6 \sqrt[3]{\frac{3 v}{4 \pi}}}{\sqrt{3}}
$$

and thus the lateral surface area

$$
P=\frac{l^{2} \pi}{2}=6 \pi \sqrt[3]{\left(\frac{3 v}{4 \pi}\right)^{2}}=6 \pi \sqrt[3]{\left(\frac{3 v}{4}\right) \pi}=6 \sqrt[3]{75^{2} \cdot \pi}=156.28 \mathrm{~cm}^{2}
$$

(Neumann Frida, Budapest.)