Question

A sphere and a cube have equal surface areas. The ratio of volume of sphere to that of the cube is:

• A. $\sqrt{\pi }:\sqrt{6}$
• B. $\sqrt{2}:\sqrt{\pi }$
• C. $\sqrt{\pi }:\sqrt{3}$
• D. $\sqrt{\pi }:\sqrt{5}$

Answer 1

The correct option is A $\sqrt{\pi }:\sqrt{6}$

Given, Surface area of sphere = Surface area of cube

$\Rightarrow 4\pi r^2=6a^2$

$\Rightarrow \left(r/a\right)2=3/2\pi$

$\Rightarrow r/a=\sqrt{\left(3/2\pi \right)}$

Volume of sphere / Volume of cube $=\left(4/3\right)\pi r^3/a^3=\left(4\pi /3\right)(r/a{)}^3$

$=\left(4\pi /3\right)\left(\sqrt{\left(3/2\pi \right)}\right)3$

$=2\sqrt{\left(3/2\pi \right)}$

$=\sqrt{(4\times 3/2\pi )}$

$=\sqrt{\left(6/\pi \right)}$

Thus, Volume of sphere : Volume of cube $=\sqrt{6}:\sqrt{\pi }$