A hemisphere, a cone and a cylinder stand on equal bases and their respective heights are in the ratio 1 : 2 : 3. Find the ratio of their volumes.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 : 2 : 9
“Equal bases” means the three solids have the same radius \(r\). The height of a hemisphere is its radius, so heights in the ratio 1 : 2 : 3 are \(r\), \(2r\) and \(3r\).
Hemisphere: \(\frac{2}{3}\pi r^3\)
Cone of height \(2r\): \(\frac{1}{3}\pi r^2(2r)=\frac{2}{3}\pi r^3\)
Cylinder of height \(3r\): \(\pi r^2(3r)=3\pi r^3\)
Ratio \(=\frac{2}{3}:\frac{2}{3}:3=2:2:9\) (multiplying each term by 3)