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The ratio of the respective heights and the respective radii of two cylinders are 1 : 2 and 2 : 1 respectively. Then their volumes are in the ratio
A. 4 : 1
B. 1 : 4
C. 2 : 1
D. 1 : 2

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Last updated date: 13th Jun 2024
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Answer
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Hint: We are given the ratio of the heights of the cylinders to be 1 : 2 and ratio of the radii of the cylinders to be 2 : 1 and we known that the volume of a cylinder having radius r and height h is given as $\pi {r^2}h{\text{ }}cu.units$. Using this and the given values above we can find the ratio of their volumes.

Complete step by step solution:
We are given that the ratio of heights of two cylinders is 1 : 2
Let the heights of the cylinders be ${h_1}$and ${h_2}$respectively
Since there are in the ratio 1 : 2
It can be written as
$ \Rightarrow \dfrac{{{h_1}}}{{{h_2}}} = \dfrac{1}{2}$ …………(1)
We are given that the ratio of radii of two cylinders is 2 : 1
Let the radii of the cylinders be ${r_1}$and ${r_2}$respectively
Since there are in the ratio 2 : 1
It can be written as
$ \Rightarrow \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{2}{1}$ ……………(2)
Now we need to find the ratio of their volumes
At first , the volume of a cylinder having radius r and height h is given as $\pi {r^2}h{\text{ }}cu.units$
Same way the volume of a cylinder having radius ${r_1}$ and height ${h_1}$ is given as $\pi {r_1}^2{h_1}{\text{ }}cu.units$
And the volume of a cylinder having radius ${r_2}$ and height ${h_2}$ is given as $\pi {r_2}^2{h_2}{\text{ }}cu.units$
Hence their ratios are given by
$
   \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\pi r_1^2{h_1}}}{{\pi r_2^2{h_2}}} \\
   \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{r_1^2{h_1}}}{{r_2^2{h_2}}} \\
 $
Now using the values from (1) and (2)
$
   \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = {\left( {\dfrac{2}{1}} \right)^2}\times \dfrac{1}{2} \\
   \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{4}{1} \times \dfrac{1}{2} \\
   \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{4}{2} = \dfrac{2}{1} \\
 $
From this we get that the ratio of their volumes is 2 : 1

Therefore the correct answer is option C.

Note :
1) A ratio between two or more quantities is a way of measuring their sizes compared to each other.
2) The ratio of two quantities expressed in terms of the same unit is the fraction that has the first quantity as numerator and the second as denominator.