3. Two spheres with radii 2 and 1 are placed inside a cone. The larger sphere touches the base and the lateral surface. Calculate the volume of the cone.
Solution. Let's consider the cross-section of the cone (see the drawing). Let $\overline{C O_{1}}=x$. From the similarity of triangles $\mathrm{CO}_{1} \mathrm{M}$ and $\mathrm{CO}_{2} \mathrm{~N}$, we get

$$
\overline{C O_{1}}: \overline{C O_{2}}=\overline{O_{1} M}: \overline{O_{2} N} \text {, i.e., } x:(x+3)=1: 2 \text {, }
$$

from which $x=3$. For the height $H$ of the cone, we have

$$
H=\overline{C D}=\overline{C O_{1}}+\overline{O_{1} O_{2}}+\overline{O_{2} D}=8
$$

From the similarity of triangles $C O_{1} M$ and $C D B$, we get

![](https://cdn.mathpix.com/cropped/2024_06_05_fb2f6c4e736d749cb214g-4.jpg?height=543&width=437&top_left_y=959&top_left_x=1029)

$$
\overline{O_{1} M}: \overline{C M}=r: \overline{C D}
$$

Since $\overline{O_{1} M}=1, \overline{C D}=8$ and $\overline{C M}^{2}={\overline{O_{1} C}}^{2}-{\overline{O_{1} M}}^{2}=8$, we get $r=2 \sqrt{2}$. Therefore,

$$
V=\frac{1}{3} r^{2} \pi H=\frac{64 \pi}{3}
$$