A sphere is placed in a cube so that it touches all the faces of the cube. If 'a' is the ratio of the volume of the cube to the volume of the sphere, and 'b' is the ratio of the surface area of the sphere to the surface area of the cube, then the value of ab is:

Solution (By Examveda Team)

$$\eqalign{ & a = \frac{{{\text{Volume of cube}}}}{{{\text{Volume of the sphere}}}} \cr & a = \frac{{{1^3}}}{{\frac{4}{3}\pi \times \frac{{{1^3}}}{8}}} \cr & a = \frac{6}{\pi }{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( {\text{i}} \right) \cr & b = \frac{{{\text{Surface area of sphere}}}}{{{\text{Surface area of cube}}}} \cr & b = \frac{{4\pi \times \frac{{{1^2}}}{4}}}{{6 \times {1^2}}} \cr & b = \frac{\pi }{6}{\text{ }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}{\text{. }}\left( {{\text{ii}}} \right) \cr & a \times b = \frac{6}{\pi } \times \frac{\pi }{6} = 1{\text{ Answer}} \cr} $$