The ratio of volumes of two cones is 4:5 and the ratio of the radii of their bases is 2:3.Find the ratio of their vertical heights.

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Solution

Ratio in volumes of two cones =4:5

and ratio of radii=2:3

Let radius of the first cone $(r_{1})$ =2x

Then radius of second cone $(r_{2})$ =3x

Let $h_{1}, h_{2}$ be their heights respectively.

$∴ \frac{1}{3} π r_{1}^{2} h_{1} : \frac{1}{3} π r_{2}^{2} h_{2} = 4 : 5 \Rightarrow \frac{\frac{1}{3} π r_{1}^{2} h_{1}}{\frac{1}{3} π r_{2}^{2} h_{2}} = \frac{4}{5} \Rightarrow \frac{π r_{1}^{2} h_{1}}{π r_{2}^{2} h_{2}} = \frac{4}{5} \Rightarrow \frac{(2 x)^{2}}{(3 x)^{2}} \times \frac{h_{1}}{h_{2}} = \frac{4}{5} \Rightarrow \frac{4 x^{2}}{9 x^{2}} \times \frac{h_{1}}{h_{2}} = \frac{4}{5} \Rightarrow \frac{h_{1}}{h_{2}} = \frac{4}{5} \times \frac{9 x^{2}}{4 x^{2}} \frac{h_{1}}{h_{2}} = \frac{9}{5} ∴ R a t i o b e t w e e n t h e i r v e r t i c a l h e i g h t s = 9 : 5$

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