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:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4 : 1
The largest sphere that fits inside a hemisphere stands on the centre of the flat face and touches the top of the dome. Its diameter is then the height of the hemisphere, \(r\), so its radius is \(\frac{r}{2}\).
Volume of the hemisphere \(\displaystyle =\frac{2}{3}\pi r^3\)
Volume of the sphere \(\displaystyle =\frac{4}{3}\pi\left(\frac{r}{2}\right)^3=\frac{1}{6}\pi r^3\)
Ratio \(\displaystyle =\frac{2}{3}:\frac{1}{6}=4:1\)