Sphere pressure

The longest diagonal of cube is equal to the diameter of the sphere.

Let the length of the side of cube be $x$ so the length of the longest diagonal is $x\sqrt{3}$ .

$\begin{aligned} \therefore x\sqrt{3} & =6×2 \\ x & =4\sqrt{3} \end{aligned} \\ \text{Volume}=x^3=(4\sqrt{3})^2=192\sqrt{3}$

From the figure, the diameter of the sphere is the space diagonal of the cube which is equal to $2(6)=12$ . If $a$ is the side length of the cube, the space diagonal is $a\sqrt{3}$ . So we have

$12=a\sqrt{3}$ $\implies$ $a=\dfrac{12}{\sqrt{3}}$

So the volume is $v=a^3=\left(\dfrac{12}{\sqrt{3}}\right)^3=\dfrac{1728}{3^{\frac{3}{2}}}=\dfrac{1728}{3^{\frac{1}{2}}\cdot 3}=\dfrac{576}{\sqrt{3}}=192\sqrt{3}$ .

The desired answer is $192$ .