An Inscribed Cube

Let the cube have side length equal to $a$ and let the radius of the sphere be $r$ . From the question we have $a^3 = 2 \times 96\sqrt{3}$ $\Rightarrow a = 4 \sqrt{3}$ . The main diagonal of the cube will have length $\sqrt{3} \times a = 12$ (this can be obtained by Pythagorean theorem). By symmetry, the center of the cube is equal to the center of the sphere, hence the main diagonal of the cube is the diameter of the sphere. Hence, the radius of the sphere is $\frac {12}{2} = 6$ .