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# The surface areas of a solid sphere and a solid hemisphere are equal to S , if their volumes are are ${V_1}$ and ${V_2}$ respectively then $\dfrac{{{V_1}}}{{{V_2}}}$ A.$\dfrac{{\sqrt 3 }}{2}$ B.$\dfrac{{3\sqrt 3 }}{8}$ C.$\dfrac{3}{4}$ D.$\dfrac{{3\sqrt 3 }}{4}$

Last updated date: 15th Jul 2024
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Hint : In such kinds of questions we have to use the basic formulas for volume and surface area of sphere and hemisphere . Also the relation between hemisphere and sphere has to be used to find the ratio between their volumes .

We know that the surface areas of the sphere and hemisphere are $4\pi {R^2}$ and $3\pi {r^2}$ respectively .
$\Rightarrow 4\pi {R^2} = 3\pi {r^2}$
$\Rightarrow \dfrac{R}{r} = \sqrt {\dfrac{3}{4}}$ ( cancelling out similar terms )
Now , let ${V_1}$ and ${V_2}$ be the volumes of the sphere and hemisphere respectively .
Therefore , $\dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\dfrac{4}{3}\pi {R^3}}}{{\dfrac{2}{3}\pi {r^3}}}$
$= \dfrac{{2{R^3}}}{{{r^3}}}$ $= 2{\left( {\dfrac{R}{r}} \right)^3}$
Putting value of $\dfrac{R}{r}$ from above
$\dfrac{{{V_1}}}{{{V_2}}} = 2{\left( {\sqrt {\dfrac{3}{4}} } \right)^3}$ $= \dfrac{{2 \times 3\sqrt 3 }}{8} = \dfrac{{3\sqrt 3 }}{4}$