Question : The ratio of the volume of the two cones is 2 : 3, and the ratio of the radii of their bases is 1 : 2. The ratio of their heights is:
Option 1: 3 : 8
Option 2: 8 : 3
Option 3: 4 : 3
Option 4: 3 : 4
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Correct Answer: 8 : 3
Solution :
Let their radii be $x$ and $2x$ and their heights be $h$ and $H$ respectively.
According to the question,
$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2 h_1}{\frac{1}{3}\pi r_2^2 h_2}$
⇒ $\frac{V_1}{V_2}=\frac{r_1^2 h_1}{r_2^2 h_2}$
⇒ $\frac{2}{3}=\frac{x^2 h}{(2x)^2 H}$
⇒ $\frac{2}{3}=\frac{x^2 h}{4x^2 H}$
⇒ $\frac{h}{H}=\frac{8}{3}$
Hence, the correct answer is 8 : 3.
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