Answer

Verified

481.2k+ views

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 .

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

How do you graph the function fx 4x class 9 maths CBSE

Which are the Top 10 Largest Countries of the World?

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The largest tea producing country in the world is A class 10 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE