4. (physics) From a piece of metal with mass $M$, a rectangular parallelepiped is cast. Then, a source of constant voltage is connected to different pairs of mutually opposite faces of the parallelepiped. It turned out that the ratio of the thermal powers dissipated in the parallelepiped when the source is connected to different pairs of faces is 1:2:8. Find the lengths of the edges of the parallelepiped. The density of the metal $\rho$ is known. The parallelepiped is connected to the source using wide bus bars that ensure the source contacts the entire area of the face, allowing the effects of current spreading to be neglected.
Solution. Let the edge lengths of the parallelepiped be $a$ (the smallest), $b$ (the middle), and $c$ (the largest). Then the resistance of the parallelepiped when electrical contacts are connected to the faces with sides $a$ and $b$ is

$$
R_{a b}=\frac{\rho_{0} a b}{c}
$$

where $\rho_{0}$ is the resistivity of the metal. Similarly, we find the resistance of the parallelepiped when electrical contacts are connected to other pairs of faces

$$
R_{a c}=\frac{\rho_{0} a c}{b}, \quad R_{b c}=\frac{\rho_{0} b c}{a}
$$

Obviously, the resistance values satisfy the following inequalities

$$
R_{a b}P_{a c}>P_{b c}
$$

where $P_{a b}, P_{a c}$, and $P_{b c}$ are the powers dissipated in the parallelepiped when electrical contacts are connected to the faces with sides $a$ and $b\left(P_{a b}\right)$, $a$ and $c\left(P_{a c}\right)$, and $b$ and $c\left(P_{b c}\right)$. Therefore, from the problem statement, we get

$$
\frac{P_{a c}}{P_{b c}}=\frac{R_{b c}}{R_{a c}}=\frac{b^{2}}{a^{2}}=2 \text { and } \frac{P_{a b}}{P_{a c}}=\frac{R_{a c}}{R_{a b}}=\frac{c^{2}}{b^{2}}=4
$$

or $b=\sqrt{2} a, c=2 b=2 \sqrt{2} a$. Since

$$
a b c=\frac{M}{\rho}
$$

we find

$$
a=\left(\frac{M}{4 \rho}\right)^{1 / 3}, b=\sqrt{2}\left(\frac{M}{4 \rho}\right)^{1 / 3}, c=2 \sqrt{2}\left(\frac{M}{4 \rho}\right)^{1 / 3}
$$