The radii of two cones are in the ratio 2:1, | vedclass

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(B) Let the radii of the two cones be $r_1$ and $r_2$,and their heights be $h_1$ and $h_2$ respectively.Given that the ratio of the radii is $\frac{r_

Vedclass · Accepted Answer

$1:4$

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(B) Let the radii of the two cones be $r_1$ and $r_2$,and their heights be $h_1$ and $h_2$ respectively.Given that the ratio of the radii is $\frac{r_1}{r_2} = \frac{2}{1}$.Since the volumes of the two cones are equal,we have $V_1 = V_2$.The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.Therefore,$\frac{1}{3} \pi r_1^2 h_1 = \frac{1}{3} \pi r_2^2 h_2$.Canceling $\frac{1}{3} \pi$ from both sides,we get $r_1^2 h_1 = r_2^2 h_2$.Rearranging to find the ratio of heights,we get $\frac{h_1}{h_2} = \frac{r_2^2}{r_1^2} = \left(\frac{r_2}{r_1}\right)^2$.Substituting the given ratio $\frac{r_1}{r_2} = \frac{2}{1}$,we have $\frac{r_2}{r_1} = \frac{1}{2}$.Thus,$\frac{h_1}{h_2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.So,the ratio of their heights is $1:4$.

The radii of two cones are in the ratio $2:1,$ and their volumes are equal. Find the ratio of their heights.

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