Question

If the radii of the bases of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3, find the ratio of their volumes.

Answer 1

Given: Ratio of radii of cylinder to cone:

\[\dfrac{r_{1}}{r_{2}} = \dfrac{3}{4}\] 

Ratio of heights of cylinder to cone:

\[\dfrac{h_{1}}{h_{2}} = \dfrac{2}{3}\]

To find: Ratio of their volumes = `(V_{1} : V_{2})`.

Formula: Volume of cylinder \[V_{1} = \pi r_{1}^{2} h_{1}\] 

Volume of cone \[V_{2} = \dfrac{1}{3}\pi r_{2}^{2} h_{2}\] 

Ratio of volumes: \[\dfrac{V_{1}}{V_{2}} = \dfrac{\pi r_{1}^{2} h_{1}}{\dfrac{1}{3}\pi r_{2}^{2} h_{2}} = 3 \times \left(\dfrac{r_{1}}{r_{2}}\right)^{2} \times \left(\dfrac{h_{1}}{h_{2}}\right)\]

Substitution & Calculation: \[\dfrac{V_{1}}{V_{2}} = 3 \times \left(\dfrac{3}{4}\right)^{2} \times \left(\dfrac{2}{3}\right)\] 

\[\dfrac{V_{1}}{V_{2}} = 3 \times \dfrac{9}{16} \times \dfrac{2}{3}\] 

\[\dfrac{V_{1}}{V_{2}} = \dfrac{18}{16} = \dfrac{9}{8}\]

Answer: \[\text{Ratio of their volumes} = 9 : 8\]