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?

Answer: Option B

Solution

Answer: Option B

Solution:

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