The radii of two cylinders are in the ratio of 2 : 3 and their heights are in the ratio of 5 : 3. Find the ratio of their volumes.

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Solution

$Let the radii of the cylinders be r_{1} and r_{2}; and their heights be h_{1} and h_{2} . We have, r_{1} : r_{2} = 2 : 3 or \frac{r_{1}}{r_{2}} = \frac{2}{3} . . . . . (i) and h_{1} : h_{2} = 5 : 3 or \frac{h_{1}}{h_{2}} = \frac{5}{3} . . . . . (ii) Now, The ratio of the volumes of the cylinders = \frac{Volume of the first cylinder}{Volume of the second cylinder} = \frac{π {r_{1}}^{2} h_{1}}{π {r_{2}}^{2} h_{2}} = {(\frac{r_{1}}{r_{2}})}^{2} \times \frac{h_{1}}{h_{2}} = {(\frac{2}{3})}^{2} \times \frac{5}{3} [Using (i) and (ii)] = \frac{20}{27} = 20 : 27$

So, the ratio of the volumes of the given cylinders is 20 : 27.

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