Example 3. A combination of two inverted cones sharing the same base, the lateral development radius of one cone is 15, and the central angle is $288^{\circ}$, the lateral development radius of the other cone is 13, find the volume of its inscribed sphere.
Let the axial section of the composite body be SMVN (as shown in the figure), A be the area of the face of $v_{\mathrm{i}}$ SMVN, and C be the perimeter, then $2 \pi \cdot \mathrm{MO}$
$$
\begin{array}{l}
=2 \pi \cdot \mathrm{SM}_{360^{\circ}}^{288^{\circ}} \\
\quad \therefore \quad \mathrm{MO}=15 \times \frac{4}{5}=12 .
\end{array}
$$
$$
\begin{aligned}
\text { Also, } \mathrm{SO} & =\sqrt{\mathrm{SM}^{2}-\overline{\mathrm{MO}^{2}}}=9, \\
\mathrm{VO} & =\sqrt{\mathrm{VM}^{2}-\mathrm{MO}^{2}}=5, \\
\therefore \quad \mathrm{A} & =\frac{1}{2} \mathrm{MN} \cdot \mathrm{SV}=\frac{1}{2} \times 24 \times 14 \\
= & 168,
\end{aligned}
$$
$$
\begin{array}{l}
\mathrm{C}=2(\mathrm{SM}+\mathrm{MV})=2 \times 28=56 . \\
\therefore \quad \mathrm{r}=\frac{2 \mathrm{~A}}{\mathrm{C}}=\frac{2 \times 168}{56}=6 . \\
\therefore \quad V_{\text {sphere }}=\frac{4}{3} \pi \mathrm{r}^{3}=\frac{4}{3} \pi \cdot 6^{3}=288 \pi .
\end{array}
$$