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Question

If the volumes of two cones are in the ratio $$1:4$$ and their diameters are in the ratio $$4:5$$, then the ratio of their heights is ___________.


A
1:5
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B
5:4
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C
5:16
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D
25:64
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Solution

The correct option is C $$25:64$$
Let $$V_1$$ and $$V_2$$ are the volumes of the cones.
 $$\dfrac{V_1}{V_2}=\dfrac{1/3\pi r_1^2 h_1}{1/3\pi r_2^2h_2}=\dfrac{r_1^2 h_1}{r_2^2h_2}\\ \Rightarrow \dfrac{1}{4}=\dfrac{4^2\times h_1}{5^2\times h_2}$$    ($$\because V_1:V_2=1:4$$ and $$r_1:r_2=4:5$$)
Thus, $$h_1:h_2=25:64$$

Mathematics

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