12.130. The development of the lateral surface of a cylinder is a rectangle, in which the diagonal is equal to $a$ and forms an angle $\alpha$ with the base. Find the volume of the cylinder.
## Solution.

The volume of the cylinder $V=\pi R^{2} H$ (Fig. 12.130). The height of the cylinder from $\triangle A B C$ is $H=C B=a \sin \alpha$. The length of the circumference of the base $c=A B=a \cos \alpha$. On the other hand, $c=2 \pi R$. Then $R=\frac{a \cos \alpha}{2 \pi}$. Substituting, we get: $V=\pi \frac{a^{2} \cos ^{2} \alpha}{4 \pi^{2}} \cdot a \sin \alpha=\frac{a^{3} \cos ^{2} \alpha \sin \alpha}{4 \pi}$.

Answer: $\frac{a^{3} \cos ^{2} \alpha \sin \alpha}{4 \pi}$.

## Solutions to Chapter 13

## APPLICATION OF EQUATIONS TO PROBLEM SOLVING