If the volumes of two cones are in the ratio 2 : 3 and the radii of their bases are in the ratio 1 : 2, then the ratio of their heights will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 8 : 3
The volume of a cone is \(\frac{1}{3}\pi r^2h\), so the height is proportional to the volume divided by the square of the radius; the \(\frac{1}{3}\pi\) cancels in the ratio.
Ratio of the volumes: \(\displaystyle \frac{V_1}{V_2}=\frac{r_1^2h_1}{r_2^2h_2}\), so \(\displaystyle \frac{h_1}{h_2}=\frac{V_1}{V_2}\times\frac{r_2^2}{r_1^2}\)
Putting in the volume ratio 2 : 3 and the radius ratio 1 : 2: \(\displaystyle \frac{h_1}{h_2}=\frac{2}{3}\times\frac{2^2}{1^2}=\frac{8}{3}\)