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Question

(A) Let the radius of the solid sphere be $r_1$ and the radius of the solid hemisphere be $r_2$.Total Surface Area $(TSA)$ of a sphere $= 4 \pi r_1^2$

Vedclass · Accepted Answer

$3 \sqrt{3} : 4$

Vedclass · Answer

(A) Let the radius of the solid sphere be $r_1$ and the radius of the solid hemisphere be $r_2$.Total Surface Area $(TSA)$ of a sphere $= 4 \pi r_1^2$.Total Surface Area $(TSA)$ of a hemisphere $= 3 \pi r_2^2$.Given that their total surface areas are equal:$4 \pi r_1^2 = 3 \pi r_2^2$$\frac{r_1^2}{r_2^2} = \frac{3}{4}$$\frac{r_1}{r_2} = \frac{\sqrt{3}}{2}$.Now,the ratio of their volumes is:$\frac{V_{	ext{sphere}}}{V_{	ext{hemisphere}}} = \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 $\frac{r_1}{r_2} = \frac{\sqrt{3}}{2}$:Ratio $= 2 	imes \left(\frac{\sqrt{3}}{2}ight)^3 = 2 	imes \frac{3 \sqrt{3}}{8} = \frac{3 \sqrt{3}}{4}$.

$A$ solid sphere and a solid hemisphere have the same total surface area. The ratio of their volumes is

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