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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.
Given:
The ratio of volumes of two cones is $4 : 5$ and the ratio of the radii of their bases is $2:3$.
To do:
We have to find the ratio of their vertical heights.
Solution:
Ratio of the volumes of two cones $= 4:5$
Ratio of the radii of the cones $= 2:3$
Let the radius of the first cone $(r_1)$ be $2x$ and the radius of the second cone $(r_2)$ be $3x$
Let $h_1$ and $h_2$ be the heights of the cones respectively.
Therefore,
$\frac{1}{3} \pi r_{1}{ }^{2} h_{1}: \frac{1}{3} \pi r_{2}{ }^{2} h_{2}=4: 5$
$\frac{\frac{1}{3} \pi r_{1}^{2} h_{1}}{\frac{1}{3} \pi r_{2}^{2} h_{2}}=\frac{4}{5}$
$\frac{\pi r_{1}^{2} h_{1}}{\pi r_{2}^{2} h_{2}}=\frac{4}{5}$
$\frac{(2 x)^{2}}{(3 x)^{2}} \times \frac{h_{1}}{h_{2}}=\frac{4}{5}$
$\frac{4 x^{2}}{9 x^{2}} \times \frac{h_{1}}{h_{2}}=\frac{4}{5}$
$\Rightarrow \frac{h_{1}}{h_{2}}=\frac{4}{5} \times \frac{9 x^{2}}{4 x^{2}}$
$\frac{h_{1}}{h_{2}}=\frac{9}{5}$
The ratio of the vertical heights of the two cones is $9: 5$.
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