Question

A sphere and a cube have equal surface areas. Show that the ratio of the volume of sphere to that of the cube is
√6:√π​

Answer 1

Step-by-step explanation:

4πr² = 6a²

r² = 6a²/4π

r = √6*a/2√π

(4/3 πr³) / a³

(4πr³/3a³)

6a²r/3a³

2r/a

(2/a)*(√6a/2√π)

2√6a/2a√π

√6 / √π

ratio of their Volume is √6 : √π

Answer 2

Step-by-step explanation:

Let r and a be the radius of the sphere and edge of the cube respectively.

Given, Surface area of sphere = Surface area of cube

4πr2 = 6a2

(r/a)2 = 3 / 2π

r / a = √(3/2π)

Volume of sphere / Volume of cube = (4/3)πr3 / a3 = (4π/3)(r/a)3

= (4π/3)(√(3/2π))3

= (4π/3)(3/2π)(√(3/2π))

= 2√(3/2π)

= √(4x3/2π)

= √(6/π)

Thus, Volume of sphere : Volume of cube = √6 : √π.