Question 51

Two cylinders of equal volume have heights in the ratio 1 : 9. The ratio of their radii is ___.

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Solution

3 : 1
Let $r_{1}$ , $r_{2}$ be the radii and $h_{1}$ , $h_{2}$ be the heights of two cylinders.
Given, $\frac{h_{1}}{h_{2}} = \frac{1}{9}$
Now, according to the question.,
$π r_{1}^{2} h_{1} = π r_{2}^{2} h_{2}$ $[∴ v o l u m e o f c y l i n d e r = π r^{2} h]$
$\Rightarrow \frac{r_{1}^{2}}{r_{2}^{2}} = \frac{h_{2}}{h_{1}} \Rightarrow {(\frac{r_{1}}{r_{2}})}^{2} = \frac{9}{1} \Rightarrow \frac{r_{1}}{r_{2}} = \frac{\sqrt{9}}{1} \Rightarrow \frac{r_{1}}{r_{2}} = \frac{3}{1}$
Hence, $r_{1} : r_{2} = 3 : 1$

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