11.072. 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.
Solution.

The required volume $V=S_{\text {base }} h$, where $S_{\text {base }}=A B \cdot A D$ (Fig. 11.66), $h$ is 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 $A D=\frac{n}{m} A B$. In $\triangle A B D$, we have $A B^{2}+A D^{2}=B D^{2}$, i.e., $A B^{2}+\frac{n^{2}}{\dot{m}^{2}} A B^{2}=Q \Rightarrow A B=\frac{m \sqrt{Q}}{\sqrt{m^{2}+n^{2}}}$, $A D=\frac{n \sqrt{Q}}{\sqrt{m^{2}+n^{2}}}$. Therefore, $V=A B \cdot A D \cdot h=\frac{m n Q \sqrt{Q}}{m^{2}+n^{2}}$.

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