492. The volumes obtained by rotating a rectangle about each of its sides are respectively $a$ cubic meters and $b$ cubic meters. Find the length of the diagonal of the rectangle.
492. If the measurements of a rectangle are $x$ and $y$, then the diagonal is

$$
\sqrt{x^{2}+y^{2}} .
$$

210

Since by condition $\pi x^{2} y=a$ and $\pi y^{2} x=b$, then

$$
\frac{y}{x}=\frac{b}{a}
$$

Multiplying the volumes gives $\pi^{2} x^{3} y^{3}=a b$, from which

$$
x y=\sqrt[3]{\frac{\overline{a b}}{\pi^{2}}}
$$

From $(1)$ :

$$
\frac{y^{2}}{x^{2}}=\frac{b^{2}}{a^{2}}
$$

and

$$
\frac{y^{2}+x^{2}}{y^{2}}=\frac{a^{2}+b^{2}}{b}
$$

or

$$
y^{2}+x^{2}=y^{2} \cdot \frac{a^{2}+b^{2}}{b^{2}}
$$

Multiplying (1) and (2) gives:

$$
y^{2}=\frac{b}{a} \cdot \sqrt[3]{\frac{a b}{\pi^{2}}}
$$

Substituting into (3) and solving.

$$
\sqrt{x^{2}+y^{2}}=\sqrt{\frac{a^{2}+b^{2}}{a b}} \cdot \sqrt[6]{\frac{a b}{\pi^{2}}}
$$