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Question

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


A
1:2
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B
1:4
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C
2:1
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D
2:1
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Solution

The correct option is D $$\sqrt 2 :1$$
Volume of right circular cylinder $$=\pi {r}^{2}h$$
Let height, radius and volume of first cylinder be $${h}_{1},{r}_{1},{v}_{1}$$ and of second cylinder be $${h}_{2},{r}_{2},{v}_{2}$$
$$\cfrac { { v }_{ 1 } }{ { v }_{ 2 } } =\cfrac { { \pi { r }_{ 1 }^{ 2 }h }_{ 1 } }{ { \pi { r }_{ 2 }^{ 2 }h }_{ 2 } } $$
$$\cfrac { { h }_{ 1 } }{ { h }_{ 2 } } =\cfrac { 1 }{ 2 } ;\cfrac { { v }_{ 1 } }{ { v }_{ 2 } } =1;\cfrac { { r }_{ 1 } }{ { r }_{ 2 } } =?$$
$$\cfrac { { v }_{ 1 } }{ { v }_{ 2 } } =\cfrac { { r }_{ 1 }^{ 2 } }{ { r }_{ 2 }^{ 2 } } .\cfrac { 1 }{ 2 } \Rightarrow 1={ \left( \cfrac { { r }_{ 1 } }{ { r }_{ 2 } }  \right)  }^{ 2 }=\times 2\Rightarrow { \left( \cfrac { { r }_{ 1 } }{ { r }_{ 2 } }  \right)  }^{ 2 }=2\Rightarrow \cfrac { { r }_{ 1 } }{ { r }_{ 2 } } =\sqrt { 2 } $$
$$\therefore { r }_{ 1 }:{ r }_{ 2 }=\sqrt { 2 } :1$$

Mathematics

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