Question

The surface areas of a solid sphere and a solid hemisphere are equal to S if their volumes are $V_1$ and $V_2$ respectively then $\frac{V_1}{V_2}$.

• A. $\sqrt{3}:2$
• B. $3\sqrt{3}:8$
• C. $3:4$
• D. $3\sqrt{3}:4$

Answer 1

The correct option is D $3\sqrt{3}:4$

Total surface are of a solid sphere $=4\pi R^2.$

Total surface area of a solid hemisphere $=3\pi r^2$

According to the question,

$\Rightarrow$ $4\pi R^2=3\pi r^2$

$\Rightarrow$ $\frac{R^2}{r^2}=\frac{3}{4}$

$\therefore$ $\frac{R}{r}=\frac{\sqrt{3}}{2}$ ------ ( 1 )

$\Rightarrow$ Volume of sphere $\left(V_1\right)=4\pi R^3$

$\Rightarrow$ Volume of hemisphere $\left(V_2\right)=2\pi r^3$

$\Rightarrow$ $\frac{V_1}{V_2}=\frac{4\pi R^3}{2\pi r^3}$

$\Rightarrow$ $\frac{V_1}{V_2}=2{\left(\frac{R}{r}\right)}^3$

$\Rightarrow$ $\frac{V_1}{V_2}=2{\left(\frac{\sqrt{3}}{2}\right)}^3$ [ From ( 1 ) ]

$\therefore$ $\frac{V_1}{V_2}=\frac{3\sqrt{3}}{4}$