## Task B-3.3.

Two spheres with radii of $3 \mathrm{~cm}$ and $5 \mathrm{~cm}$ are inscribed in two cones as shown in the figure. The cones have equal vertex angles and a common axis that is perpendicular to the bases of the cones and passes through their centers and the common vertex. The distance between the bases is $40 \mathrm{~cm}$. If each sphere touches the lateral surface and the base of its respective cone, calculate the total volume of the cones.

![](https://cdn.mathpix.com/cropped/2024_05_30_d18d2bbdee068dba95dfg-14.jpg?height=277&width=577&top_left_y=290&top_left_x=1296)
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_d18d2bbdee068dba95dfg-14.jpg?height=454&width=1288&top_left_y=744&top_left_x=384)

Note that the distance from the center of the sphere to the plane is $\left|S_{1} B\right|+\left|S_{2} B\right|=40-3-5=32 \mathrm{~cm}$.

Due to the similarity of triangles $S_{1} B D_{1}$ and $S_{2} B D_{2}$, we have $\left|S_{1} B\right|:\left|S_{2} B\right|=3: 5$.

It follows that $\left|S_{1} B\right|=12 \mathrm{~cm}$ and $\left|S_{2} B\right|=20 \mathrm{~cm}$.

$\left|B D_{1}\right|=\sqrt{12^{2}-3^{2}}=\sqrt{135}=3 \sqrt{15} \mathrm{~cm}$.

Due to the similarity of triangles $S_{1} B D_{1}$ and $E B A$, we have $\left|S_{1} D_{1}\right|:\left|B D_{1}\right|=|E A|:|A B|$.

It follows that $3: 3 \sqrt{15}=R_{1}: 15$, hence $R_{1}=\sqrt{15} \mathrm{~cm}$.

Similarly, $\left|B D_{2}\right|=\sqrt{20^{2}-5^{2}}=\sqrt{375}=5 \sqrt{15} \mathrm{~cm}$.

Due to the similarity of triangles $S_{2} B D_{2}$ and $F B C$, we have $\left|S_{2} D_{2}\right|:\left|B D_{2}\right|=|F C|:|C B|$.

It follows that $5: 5 \sqrt{15}=R_{2}: 25$, hence $R_{2}=\frac{5}{3} \sqrt{15} \mathrm{~cm}$.

The total volume of the two cones is

$$
\begin{aligned}
V & =V_{1}+V_{2}=\frac{1}{3} R_{1}^{2} \pi|A B|+\frac{1}{3} R_{2}^{2} \pi|B C|=\frac{1}{3} \pi\left(R_{1}^{2} \cdot|A B|+R_{2}^{2}|B C|\right) \\
& =\frac{1}{3} \pi\left(15 \cdot 15+\frac{125}{3} \cdot 25\right)=\frac{25}{9} \pi(27+125)=\frac{25}{9} \pi \cdot 152=\frac{3800}{9} \pi \mathrm{cm}^{3}
\end{aligned}
$$