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

Question:

Two right circular cylinders of equal volumes have their heights in the ratio 1 : 2. What is the ratio of their radii?

Solution:

Let the radii of the two cylinders be $r$ and $R$; and the heights be $h$ and $H$.

We have,

$\frac{h}{H}=\frac{1}{2} \quad \ldots . .(\mathrm{i})$

Now,

Volume of the first cylinder $=$ Volume of the second sphere

$\Rightarrow \pi r^{2} h=\pi R^{2} H$

$\Rightarrow \frac{h}{H}=\frac{R^{2}}{r^{2}}$

$\Rightarrow \frac{1}{2}=\frac{R^{2}}{r^{2}}$

$\Rightarrow \frac{r^{2}}{R^{2}}=\frac{1}{2}$

$\Rightarrow\left(\frac{r}{R}\right)^{2}=\frac{2}{1}$

$\Rightarrow \frac{r}{R}=\frac{\sqrt{2}}{1}$

$\therefore r: R=\sqrt{2}: 1$

So, the ratio of their radii is $\sqrt{2}: 1$.

 

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