The heights of a cone,cylinder,and hemisphere | vedclass

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(C) Let the common height be $h$. For a hemisphere,the height is equal to its radius,so $h = r_3$. Given that the heights of the cone,cylinder,and hem

Vedclass · Accepted Answer

$4: 27: 2$

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(C) Let the common height be $h$. For a hemisphere,the height is equal to its radius,so $h = r_3$. Given that the heights of the cone,cylinder,and hemisphere are equal,we have $h = h_{cone} = h_{cylinder} = r_{hemisphere}$.Let the radii be $r_1 = 2x$,$r_2 = 3x$,and $r_3 = 1x = x$. Since the height of the hemisphere is its radius,$h = x$.Volume of cone $V_1 = \frac{1}{3} \pi r_1^2 h = \frac{1}{3} \pi (2x)^2 (x) = \frac{4}{3} \pi x^3$.Volume of cylinder $V_2 = \pi r_2^2 h = \pi (3x)^2 (x) = 9 \pi x^3$.Volume of hemisphere $V_3 = \frac{2}{3} \pi r_3^3 = \frac{2}{3} \pi (x)^3 = \frac{2}{3} \pi x^3$.The ratio $V_1 : V_2 : V_3 = \frac{4}{3} \pi x^3 : 9 \pi x^3 : \frac{2}{3} \pi x^3$.Multiplying by $\frac{3}{\pi x^3}$,we get $4 : 27 : 2$.

The heights of a cone,cylinder,and hemisphere are equal. If their radii are in the ratio $2: 3: 1$,then the ratio of their volumes is

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