Question
A sphere and a cube have equal surface area. The ratio of the volume of the sphere to that of the cube is :
Answer: Option D
$$\eqalign{
& 4\pi {R^2} = 6{a^2} \cr
& \Rightarrow \frac{{{R^2}}}{{{a^2}}} = \frac{3}{{2\pi }} \cr
& \Rightarrow \frac{R}{a} = \frac{{\sqrt 3 }}{{\sqrt {2\pi } }} \cr} $$
$$\eqalign{
& \therefore \frac{{{\text{Volume of spere}}}}{{{\text{Volume of cube}}}} \cr
& = \frac{{\frac{4}{3}\pi {R^3}}}{{{a^3}}} \cr
& = \frac{4}{3}\pi {\left( {\frac{R}{a}} \right)^3} \cr
& = \frac{4}{3}\pi \frac{{3\sqrt 3 }}{{2\pi \sqrt {2\pi } }} \cr
& = \frac{{2\sqrt 3 }}{{\sqrt {2\pi } }} \cr
& = \frac{{\sqrt {12} }}{{\sqrt {2\pi } }} \cr
& = \frac{{\sqrt 6 }}{{\sqrt \pi }} \cr
& \text{or, }\sqrt 6 :\sqrt \pi \cr} $$
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$$\eqalign{
& 4\pi {R^2} = 6{a^2} \cr
& \Rightarrow \frac{{{R^2}}}{{{a^2}}} = \frac{3}{{2\pi }} \cr
& \Rightarrow \frac{R}{a} = \frac{{\sqrt 3 }}{{\sqrt {2\pi } }} \cr} $$
$$\eqalign{
& \therefore \frac{{{\text{Volume of spere}}}}{{{\text{Volume of cube}}}} \cr
& = \frac{{\frac{4}{3}\pi {R^3}}}{{{a^3}}} \cr
& = \frac{4}{3}\pi {\left( {\frac{R}{a}} \right)^3} \cr
& = \frac{4}{3}\pi \frac{{3\sqrt 3 }}{{2\pi \sqrt {2\pi } }} \cr
& = \frac{{2\sqrt 3 }}{{\sqrt {2\pi } }} \cr
& = \frac{{\sqrt {12} }}{{\sqrt {2\pi } }} \cr
& = \frac{{\sqrt 6 }}{{\sqrt \pi }} \cr
& \text{or, }\sqrt 6 :\sqrt \pi \cr} $$
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