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(B) Let the radius of the sphere be $r_1$ and the radius of the hemisphere be $r_2$.The surface area of a sphere is $4\pi r_1^2$ and the curved surfac

Vedclass · Accepted Answer

$27:4$

Vedclass · Answer

(B) Let the radius of the sphere be $r_1$ and the radius of the hemisphere be $r_2$.The surface area of a sphere is $4\pi r_1^2$ and the curved surface area of a hemisphere is $2\pi r_2^2$.Given the ratio: $\frac{4\pi r_1^2}{2\pi r_2^2} = \frac{9}{2}$.Simplifying this,we get $\frac{2r_1^2}{r_2^2} = \frac{9}{2}$,which implies $\frac{r_1^2}{r_2^2} = \frac{9}{4}$.Taking the square root,we find $\frac{r_1}{r_2} = \frac{3}{2}$.The volume of a sphere is $V_1 = \frac{4}{3}\pi r_1^3$ and the volume of a hemisphere is $V_2 = \frac{2}{3}\pi r_2^3$.The ratio of their volumes is $\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{2}{3}\pi r_2^3} = 2 	imes \left(\frac{r_1}{r_2}ight)^3$.Substituting the ratio $\frac{r_1}{r_2} = \frac{3}{2}$,we get $\frac{V_1}{V_2} = 2 	imes \left(\frac{3}{2}ight)^3 = 2 	imes \frac{27}{8} = \frac{27}{4}$.Thus,the ratio of their volumes is $27:4$.

The ratio of the surface area of a sphere to the curved surface area of a hemisphere is $9:2$. Find the ratio of their volumes.

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