A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1 : 2 : 3
Equal bases mean the same radius r for all three. A hemisphere of radius r is r high, so “the same height” means h = r for the cone and the cylinder as well. Each volume is then a multiple of \(\pi r^3\).
Volume of the cone: \(\displaystyle\frac{1}{3}\pi r^2h=\frac{1}{3}\pi r^3\)
Volume of the hemisphere: \(\displaystyle\frac{2}{3}\pi r^3\)
Volume of the cylinder: \(\pi r^2h=\pi r^3\)
Ratio, after dividing by \(\pi r^3\) and multiplying by 3: \(\displaystyle\frac{1}{3}:\frac{2}{3}:1=1:2:3\)