30. A hemisphere is placed on a sphere of radius $100 \mathrm{~cm}$. The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in $\mathrm{cm}$ ).
30. Answer: 300 .
Let the radius of the $i$ th hemisphere be $r_{i}$ metre $\left(r_{0}=1\right)$. Set $h_{i}=\sqrt{r_{i-1}^{2}-r_{i}^{2}}$.

By Cauchy inequality,
$$
\begin{aligned}
\left(\sqrt{r_{0}^{2}-r_{1}^{2}}+\sqrt{r_{1}^{2}-r_{2}^{2}}+\sqrt{r_{2}^{2}-r_{3}^{2}}+r_{3}\right)^{2} & \leq 4\left(r_{0}^{2}-r_{1}^{2}+r_{1}^{2}-r_{2}^{2}+r_{2}^{2}-r_{3}^{2}+r_{3}^{2}\right) \\
& =4 r_{0}^{2}=4 .
\end{aligned}
$$

The total height $h \leq r_{0}+\sqrt{r_{0}^{2}-r_{1}^{2}}+\sqrt{r_{1}^{2}-r_{2}^{2}}+\sqrt{r_{2}^{2}-r_{3}^{2}}+r_{3} \leq 1+\sqrt{4}=3 \mathrm{~m}=300 \mathrm{~cm}$. The equality holds if $r_{i}=\frac{\sqrt{4-i}}{2}, i=0,1,2,3$.