2.12. The sides of the base of a rectangular parallelepiped are in the ratio $m: n$, and the diagonal section is a square with an area equal to $Q$. Determine the volume of the parallelepiped.
2.12. The required volume $V=S_{\text {osn }} h$, where $S_{\text {osn }}=A B \cdot A D$ (Fig.2.12), $h$ - the height of the parallelepiped. According to the condition, $B B_{1} D_{1} D$ is a square, and thus, $h=\sqrt{Q}$. Let's find $A B$ and $A D$. Since $A B: A D=m: n$, then $A D=\frac{n}{m} A B . \mathrm{B} \triangle A B D$ we have $A B^{2}+A D^{2}=$ $=B D^{2}$, i.e., $A B^{2}+\frac{n^{2}}{m^{2}} A B^{2}=Q ;$ consequently, $A B=\frac{m \sqrt{Q}}{\sqrt{m^{2}+n^{2}}}, A D=\frac{n \sqrt{Q}}{\sqrt{m^{2}+n^{2}}}$. Therefore,

$$
V=A B \cdot A D \cdot h=\frac{m n Q \sqrt{Q}}{m^{2}+n^{2}}
$$

Answer: $\frac{m n Q \sqrt{Q}}{m^{2}+n^{2}}$.