A cone and a cylinder have equal base radius. If they have equal volume, then find the ratio of height of the cone to that of the cylinder.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3:1
A cone holds one-third of the volume of a cylinder with the same base and height: \(V_{\text{cone}}=\frac{1}{3}\pi r^2h\), \(V_{\text{cylinder}}=\pi r^2h\). So for equal volumes on the same base, the cone must be three times as tall.
Equal volumes, with \(h_1\) the height of the cone and \(h_2\) that of the cylinder: \(\displaystyle \frac{1}{3}\pi r^2h_1=\pi r^2h_2\)
The common base area \(\pi r^2\) cancels: \(h_1=3h_2\)
Ratio of the heights: \(h_1:h_2=3:1\)