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:

Let the radius of the sphere = r

Since the sphere touches all the faces of the cube

The length of the each side of the cube = Diameter of the sphere = 2r

Given, 'a’ is the ratio of the volume of the cube to the volume of the sphere

$$=$$>  a = $$\frac{\left(2r\right)^3}{\frac{4}{3}\pi\ r^3}$$

$$=$$>  a = $$\frac{8r^3}{\frac{4}{3}\pi\ r^3}$$

$$=$$>  a = $$\frac{6}{\pi\ }$$

'b' is the ratio of the surface area of the sphere to the surface area of the cube

$$=$$>  b = $$\frac{4\pi\ r^2}{6\left(2r\right)^2}$$

$$=$$>  b = $$\frac{4\pi\ r^2}{24r^2}$$

$$=$$>  b = $$\frac{\pi\ }{6}$$

$$\therefore\ $$ab = $$\frac{6}{\pi\ }\times\frac{\pi\ }{6}$$ = 1

Hence, the correct answer is Option B