At the time, the required materials are as follows.



Preserving the source text's line breaks and format, the translation result is output directly as above.
$S=x: \sqrt{r^{2}+h^{2}}$. For the given condition,
$$
\begin{aligned}
S^{2} & =\pi^{2} r^{2}\left(r^{2}+h^{2}\right)=x^{2} r^{4}+\frac{1}{2} \pi^{2} r^{2} h^{2}+\frac{1}{2} x^{2} r^{2} h^{2} \\
& >3 \sqrt[3]{\left(x^{2} r^{1}\right)\left(\frac{1}{2} x^{2} r^{2} h^{2}\right)\left(\frac{1}{2} \pi^{2} r^{2} h^{2}\right)} \\
& =3 \sqrt[3]{\frac{1}{4} \pi^{3} r s h^{4}}=3 \sqrt[8]{\frac{81}{4} x^{2} V_{0}^{4}}=\text { constant. }
\end{aligned}
$$

When $x^{2} y^{4}=\frac{1}{2} x^{2} r^{2} h^{2}=\frac{1}{2} x^{2} r^{2} h^{2}$, $S^{2}$ has a minimum value. From this equation, we get $h=r \sqrt{2}$.