A sphere of maximum volume is cut out from a solid hemisphere. What is the ratio of the volume of the sphere to that of the remaining solid?

Radius of the solid hemisphere = r
Radius of the sphere = r/2
Volume of the remaining solid = volume of the solid hemisphere - volume of the sphere
Volume of the remaining solid = $$\frac{2}{3} \pi r^3 - \frac{2}{3} \pi (\frac{r}{2})^3$$ =  $$\frac{2}{3} \pi r^3 - \frac{4}{3} \pi (\frac{r}{2})^3$$
=  $$\frac{2}{3} \pi r^3 - \frac{1}{6} \pi r^3$$ = $$ \frac{1}{2} \pi r^3$$
Ratio of the volume of the sphere to that of the remaining solid =  $$\frac{1}{6} \pi r^3 : \frac{1}{2} \pi r^3$$ = 1 : 3