5. The vertex of a right cone is at the center of one base of a cylinder. The other base of the cylinder and the base of the cone lie in the same plane and have a common center. The volumes of the cone and the cylinder are equal. The radius of the base of the cylinder is $r$, and the height of the cylinder is $h$.

a) Determine the radius of the base of the cone (expressed in terms of $r$).

b) What is the volume of the part of the cylinder that is inside the cone (expressed using $r$ and $h$).
Solution. Let the radius of the base of the cone be $R$. From $V_{k}=V_{c}$ we have

$$
\frac{R^{2} h}{3}=r^{2} h, \text { i.e. } R=r \sqrt{3}
$$

Now, from the similarity of triangles $B Q N$ and $O Q S$ we get

$$
\frac{B N}{O S}=\frac{R-r}{R}
$$

so

$$
\begin{aligned}
& B N=O S \frac{r \sqrt{3}-r}{r \sqrt{3}} \\
& h_{1}=h \frac{\sqrt{3}-1}{\sqrt{3}}=h \frac{3-\sqrt{3}}{3}
\end{aligned}
$$

Therefore, the volume of the part of the cylinder that is inside the cone is:

$$
V=\pi r^{2} h \frac{3-\sqrt{3}}{3}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_b58a8127523cd0cd5361g-5.jpg?height=730&width=697&top_left_y=296&top_left_x=803)