# 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.$\dfrac{{3\sqrt 3 }}{4}$

Answer

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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 .

Complete step-by-step answer:

Let R and r be the radii of of the sphere and the hemisphere respectively

It is given that their surface areas S are equal .

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

We get

$\dfrac{{{V_1}}}{{{V_2}}} = 2{\left( {\sqrt {\dfrac{3}{4}} } \right)^3}$ $ = \dfrac{{2 \times 3\sqrt 3 }}{8} = \dfrac{{3\sqrt 3 }}{4}$

Note –In such types of questions the key concept we have to remember is that we always recall all the formulas for surface area and volumes of three dimensional shapes . A proper understanding of each and every shape would be beneficial in such questions .

Complete step-by-step answer:

Let R and r be the radii of of the sphere and the hemisphere respectively

It is given that their surface areas S are equal .

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

We get

$\dfrac{{{V_1}}}{{{V_2}}} = 2{\left( {\sqrt {\dfrac{3}{4}} } \right)^3}$ $ = \dfrac{{2 \times 3\sqrt 3 }}{8} = \dfrac{{3\sqrt 3 }}{4}$

Note –In such types of questions the key concept we have to remember is that we always recall all the formulas for surface area and volumes of three dimensional shapes . A proper understanding of each and every shape would be beneficial in such questions .

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