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?

$$\eqalign{ & {\text{Sphere }}\left( {\text{V}} \right) = \frac{4}{3}\pi {\left( {\frac{R}{2}} \right)^3} = \frac{4}{3}\pi \times \frac{{{R^3}}}{8} \cr & {\text{Hemisphere }}\left( {\text{V}} \right) = \frac{2}{3}\pi {R^3} \cr & {\text{Sphere}}:{\text{Hemisphere}} = \frac{{4{R^3}}}{{3 \times 8}}:\frac{{2{R^3}}}{3} = 1:4 \cr & {\text{Sphere}}:{\text{Remaining Solid}} = 1:3 \cr} $$