A sphere of maximum volume is cut out from a solid hemisphere of radius r. The ratio of the volume of the hemisphere to that of the cut out sphere is :

Solution (By Examveda Team)

Volume of hemisphere = $$\frac{2}{3}\pi {r^3}$$
Volume of biggest sphere :
= Volume of sphere with diameter r
$$\eqalign{ & = \frac{4}{3}\pi {\left( {\frac{r}{2}} \right)^3} \cr & = \frac{1}{6}\pi {r^3} \cr} $$
∴ Required ratio :
$$\eqalign{ & = \frac{{\frac{2}{3}\pi {r^3}}}{{\frac{1}{6}\pi {r^3}}} \cr & = \frac{4}{1}i.e.,4:1 \cr} $$