The largest possible sphere is carved out of a solid cylinder of radius 3 cm and height 10 cm. Find the volume of material wasted, in ${\text{cm}}^3$, in terms of π.

The sphere must fit across the cylinder as well as along it, and the cylinder is only 6 cm wide, so the sphere's radius is capped at 3 cm — the 10 cm height is more than enough. Cylinder = π$(3^2)(10)=90$π ${\text{cm}}^3$, sphere = (4/3)π(27) = 36π ${\text{cm}}^3$, waste = 90π − 36π = 54π ${\text{cm}}^3$. (36π and 90π are the sphere and the cylinder themselves; letting the sphere's diameter match the 10 cm height would give a radius of 5 cm, which cannot fit sideways; 72π subtracts a hemisphere, 18π, instead of the whole sphere.)