The lateral surface area of a cube is $256 \, m^2$. The volume of the cube is (in $m^3$)

Write True or False and justify your answer in each of the following: If a sphere is inscribed in a cube,then the ratio of the volume of the cube to the volume of the sphere will be $6: \pi$.

If $V$ is the volume of a cuboid of dimensions $a, b, c$ and $S$ is its total surface area,then prove that $\frac{1}{V} = \frac{2}{S} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$.

Write True or False and justify your answer in each of the following: $A$ cylinder and a right circular cone have the same base and the same height. The volume of the cylinder is three times the volume of the cone.

$A$ school provides milk to the students daily in cylindrical glasses of diameter $7 \, cm$. If the glass is filled with milk up to a height of $12 \, cm$, find how many litres of milk is needed to serve $1600$ students.

The surface area and volume of a cube with an edge length of $x \ cm$ are numerically equal. Find the value of $x$.