96. Inside a right circular cone, a cube is positioned such that one edge of the cube lies on the diameter of the cone's base, the vertices of the cube not belonging to this edge lie on the lateral surface of the cone, and the center of the cube lies on the height of the cone. Find the ratio of the volume of the cone to the volume of the cube.
96. Let the radius of the base of the cone be $R$, the height be $h$, and the edge of the cube be $a$. The section of the cone by a plane parallel to the base and passing through the center of the cube is a circle with radius $R \frac{2 h - a \sqrt{2}}{2 h}$, in which a rectangle (the section of the cube) with sides $a$ and $a \sqrt{2}$ is inscribed, i.e.,

$$
3 a^{2} = R^{2} \frac{(2 h - a \sqrt{2})^{2}}{h^{2}}
$$

The section of the cone, parallel to the base of the cone and passing through the edge of the cube opposite to the edge lying in the base, is a circle with radius $R \frac{h - a \sqrt{2}}{h}$. On the other hand, the diameter of this circle is $a$, i.e.,

$$
a = 2 R \frac{h - a \sqrt{2}}{h}
$$

From relations (1), (2) we obtain

$$
h = \frac{\sqrt{2}(5 + \sqrt{3})}{4} a, \quad R = \frac{2 \sqrt{3} - 1}{2} a
$$

Answer: $\frac{\pi(53 - 7 \sqrt{\overline{3}}) \sqrt{2}}{48}$.