4. A right triangle has legs of lengths 3 and 4 . Find the volume of the solid formed by revolving the triangle about its hypotenuse.
Answer: $\frac{48 \pi}{5}$ (cubic units)
Solution: The solid consists of two conical solids with a common circular base of radius $\frac{3 \cdot 4}{5}=\frac{12}{5}$. If $h_{1}$ and $h_{2}$ are the heights of the two cones, then $h_{1}+h_{2}=5$. Hence, the volume of the solid is
$$
\frac{\pi}{3} \cdot\left(\frac{12}{5}\right)^{2} \cdot\left(h_{1}+h_{2}\right)=\frac{48 \pi}{5} .
$$