11.041. Determine the volume of a rectangular parallelepiped, the diagonal of which is equal to $l$ and forms an angle of $30^{\circ}$ with one face and $45^{\circ}$ with another.
## Solution.

$$
\begin{aligned}
& \quad V=a \cdot b \cdot c \cdot \mathrm{B} \triangle A_{1} B C \text { (Fig. 11.36) } \angle A_{1} B C=90^{\circ} ; \\
& a=B C=A_{1} C \cdot \sin 30^{\circ}=\frac{l}{2} ; A_{1} B=\frac{l \sqrt{3}}{2} \cdot \mathrm{B} \Delta A_{1} C_{1} C, \angle A_{1} C_{1} C=90^{\circ} ; \\
& C_{1} C=c=A_{1} C \cdot \sin 45^{\circ}=\frac{l \sqrt{2}}{2} ; A B=b=\sqrt{A_{1} B^{2}-c^{2}}=\sqrt{\frac{3 l^{2}}{4}-\frac{2 l^{2}}{4}}=\sqrt{\frac{l^{2}}{4}} \\
& V=a b c=\frac{l^{3} \sqrt{2}}{8} \\
& \quad \text { Answer: } \frac{l^{3} \sqrt{2}}{8}
\end{aligned}
$$