Question

The surface areas of a sphere and a cube are equal. Find the ratio of their volumes.

Answer 1

Surface area of the sphere = 4πr² 
Surface area of the cube= 6a² 
Therefore,

4πr² = 6a²

⇒ 2πr² = 3a² 

`=>2pi"r"^2 = 3"a"^2`

`=> r^2 = (3"a"^2)/(2pi)`

`=>"r" = sqrt(3/(2pi)a)`

Ratio of their Volumes `= (4//3 pi"r"^3)/a^3 = (4pi"r"^3)/3"a"^3`

`= (4pi)/"3a"^3xx(3sqrt(3"a"^3))/(2pisqrt(2pi)) ["since" "r" = sqrt(3/2pi)]`

`=(2sqrt(3))/sqrt(2pi)`

`=(2sqrt(3))/(sqrt(2)xxsqrt(22)/sqrt(7)`

`= (2xxsqrt(3)xxsqrt(7))/(sqrt(2)xxsqrt(2)xxsqrt(11))`

`=sqrt(21)/sqrt(11)`

Thus, the ratio of their volumes is `sqrt(21):sqrt(11)`