Volume and surface area

Question

A solid cuboid of size 10m*8m*6m is melted and recast into a cylinder of height 7 metre. How much minimum more molten material is required so that radius of the cylinder so formed is a natural number?

Answer 1

The volume of the cuboid is $\;10\cdot8\cdot6=480\,m^3\;$ , while the volume of the cylinder is $\;7\pi r^2 \,m^3\;$ , with $\;r=\;$ the cylinder's basis radius. If we compare, we get:

$$7\pi r^2=480\implies r=\sqrt\frac{480}{7\pi}\approx 4.67\,m$$

Thus, we need the radius to reach $\;5\;$ meters (I understand the problem as to get the radius to be the closest integer number...), so if we add $\;x\,m^3\;$ more molten material, we want:

$$7\pi\cdot25=480+x\implies x=175\pi-480\approx69.78\,m^3$$