Two cylinders of same volume have their radii in ratio 2:1 then the ratio of their height is

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Radius of the first cylinder is ${{r}_{1}}$, Radius of the second cylinder is ${{r}_{2}}$,Height of the first cylinder is ${{h}_{1}}$,Height of the second cylinder is ${{h}_{2}}$.$\therefore $ Volume of first cylinder is ${{V}_{1}}=\pi {{r}_{1}}^{2}{{h}_{1}}$.$\Rightarrow $ Volume of second cylinder is ${{V}_{2}}=\pi {{r}_{2}}^{2}{{h}_{2}}$.But we have the volumes of both the cylinders as same.$\begin{align} & \therefore {{V}_{1}}={{V}_{2}} \\ & \Rightarrow \pi {{r}_{1}}^{2}{{h}_{1}}=\pi {{r}_{2}}^{2}{{h}_{2}} \\ \end{align}$Cancelling the term $\pi $ which is on both sides and rearranging the above equation as radii of both cylinders at one place, then we will have$\begin{align} & \dfrac{r_{1}^{2}}{r_{2}^{2}}=\dfrac{{{h}_{2}}}{{{h}_{1}}} \\ & \Rightarrow \dfrac{{{r}_{1}}}{{{r}_{2}}}=\sqrt{\dfrac{{{h}_{2}}}{{{h}_{1}}}}...\left( \text{i} \right) \\ \end{align}$Given that, the heights of the cylinders are in the ratio $2:1$.$\begin{align} & \therefore {{h}_{1}}:{{h}_{2}}=2:1 \\ & \Rightarrow \dfrac{{{h}_{1}}}{{{h}_{2}}}=\dfrac{2}{1} \\ & \Rightarrow {{h}_{1}}=2{{h}_{2}} \\ \end{align}$Using the above value in equation $\left( \text{i} \right)$, then we will have$\begin{align} & \dfrac{{{r}_{1}}}{{{r}_{2}}}=\sqrt{\dfrac{{{h}_{2}}}{2{{h}_{2}}}} \\ & \Rightarrow \dfrac{{{r}_{1}}}{{{r}_{2}}}=\sqrt{\dfrac{1}{2}} \\ & \Rightarrow \dfrac{{{r}_{1}}}{{{r}_{2}}}=\dfrac{1}{\sqrt{2}} \\ \end{align}$

$\therefore $ Ratio of radii of the two cylinders is $1:\sqrt{2}$.

Note: Sometimes they may give one cylinder and one cone instead of two cylinders, then we will use the volume of cone formula as ${{V}_{c}}=\dfrac{1}{3}\pi {{r}^{2}}h$ and do the process according to the given conditions. So, we must be well versed with the formulas. Some students might make mistakes in taking the ratios, this might lead to the choice of the wrong option.

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Question 52 Two cylinders of same volume have their radii in the ratio 1: 6. Then, ratio of their heights is

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