82. The vertex $A$ of a regular prism $A B C A_{1} B_{1} C_{1}$ coincides with the vertex of a cone, vertices $B$ and $C$ lie on the lateral surface of this cone, and vertices $B_{1}$ and $C_{1}$ lie on the circumference of its base. Find the ratio of the volumes of the cone and the prism, if $\left|A A_{1}\right|=2.4|A B|$.
82. Let $|A B|=a$, then $\left|A B_{1}\right|=\left|A C_{1}\right|=2.6 a$. Take points $K$ and $L$ on the lines $A B$ and $A C$ such that $|A K|=|A L|=\left|A B_{1}\right|=$ $=\left|A C_{1}\right|=2.6 a$. The isosceles trapezoid $K L C_{1} B_{1}$ is inscribed in the base circle of the cone. All sides of this trapezoid can be easily calculated, so the radius of the circumscribed circle around it can be found, which is $\frac{13}{20} \sqrt{7} a$.

Now we can find the volumes of the cone and the prism.

Answer: $\frac{15379 \pi}{4800 \sqrt{3}}$.