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:

1

We know that,

Volume of cube = a³

Volume of sphere = (\(\frac{4}{3}\))π × a³

Total surface area of cube = 6a²

Total surface area of sphere = 4πr²

We have,

Ratio of volume of the cube to that of sphere = a

Ratio of surface area of the sphere to that of cube = b

Let side of cube = 3 m, then

Radius of sphere r = \(\frac{3}{2}\) m 

Volume of cube = 3³ = 27 m³ 

Total surface area of cube = 6 × 3² = 54 m²

Volume of sphere = \(\frac{4}{3}\) × π × (\(\frac{3}{2}\))³ = (\(\frac{4}{3}\)) × π × (\(\frac{27}{8}\)) = \(\frac{9π }{2}\) m³

Total surface area of sphere = 4π × (\(\frac{3}{2}\))² = 4π × (\(\frac{9}{4}\)) = 9π m²

According to the question

a = \(\frac{27}{\frac{9π }{2}}\)

= a = \(\frac{6}{ π }\)

b = \(\frac{9π }{ 54}\)

= b = \(\frac{π }{ 6}\)

ab = (\(\frac{6 }{ π }\)) × \(\frac{π }{ 6}\)

= ab = 1