Two circular cylinders of equal volumes have their heights in the ratio $1 : 2$. Find the ratio of their radii.

 Given:

Two circular cylinders of equal volumes have their heights in the ratio $1 : 2$.

To do:

We have to find the ratio of their radii.

Solution:

Volumes of the two cylinders is equal.

Ratio in their heights $h_1 :h_2 = 1: 2$

$\frac{h_{1}}{h_{2}}=\frac{1}{2}$

Let $r_{1}$ and $r_{2}$ be the radii of the two cylinders.

Therefore,

$\frac{\pi r_{1}^{2} h_{1}}{\pi r_{2}^{2} h_{2}}=1$

$\frac{r_{1}^{2}}{r_{2}^{2}} \times \frac{h_{1}}{h_{2}}=1$

$\frac{r_{1}^{2}}{r_{2}^{2}} \times \frac{1}{2}=1$

$\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{1 \times 2}{1}$

$\frac{r_{1}^{2}}{r_{2}^{2}}=\frac{2}{1}$

$\Rightarrow \frac{r_{1}}{r_{2}}=\sqrt{\frac{2}{1}}$

$=\frac{\sqrt{2}}{1}$

Hence, the ratio between their radii is $\sqrt{2}: 1$.

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Updated on: 10-Oct-2022

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