12.090. A plane passing through a generatrix of a cylinder forms an acute angle $\alpha$ with the plane of the axial section containing the same generatrix. The diagonal of the rectangle obtained by the intersection of the cylinder with this plane is equal to $l$ and forms an angle $\beta$ with the plane of the base. Find the volume of the cylinder.
## Solution.

The volume of the cylinder is found as follows: $V=\pi R^{2} H$. The height of the cylinder $H=C A=l \sin \beta$ (from $\triangle C A B$) (Fig. 12.91). From $\triangle C A B: A B=l \cos \beta$. Consider $\triangle O F B\left(\angle O F B=90^{\circ}\right): F B=\frac{1}{2} A B=\frac{l}{2} \cos \beta$. The radius of the base of the cylinder is: $R=O B=\frac{F B}{\cos \alpha}=\frac{l \cos \beta}{2 \cos \alpha}$. Substituting, we finally get: $V=\pi \frac{l^{2} \cos ^{2} \beta}{4 \cos ^{2} \alpha} \cdot l \sin \beta=\frac{\pi l^{3} \sin 2 \beta \cos \beta}{8 \cos ^{2} \alpha}$.

Answer: $\frac{\pi l^{3} \sin 2 \beta \cos \beta}{8 \cos ^{2} \alpha}$.