Question

The surface area of a sphere and the total surface area of a cube are the same. Show that the ratio of the volume of the sphere to that of the cube is `sqrt6 : sqrtπ`

Answer 1

Given: Surface area of the sphere = Total surface area of the cube,

Formula to be used:

Surface area of a sphere (S_s) = 4πr²,

Total surface area of a cube (S_c) = 6a²,

Volume of a sphere (V_s) = `(4/3)`πr³,

Volume of a cube (V_c) = a³

Relating the dimensions using surface area:

⇒ 4π r² = 6 a²

Rearranging to find the ratio of `r/a`:

`r^2/a^2 = 6/(4π) = 3/(2π)`

∴ `r/a = sqrt(3/(2π))`    ....(Equation 1)

The ratio of the volume of the sphere to the volume of the cube is:

Ratio = `V_s/V_c = (4/3 π r^3)/a^3`

`V_s/V_c = 4/3 π (r/a)^3`

Substituting Equation 1 into the volume ratio:

`V_s/V_c = 4/3 π (sqrt(3/(2π)))^3`

`V_s/V_c = 4/3 π (3/(2π)) sqrt(3/(2π))`

`V_s/V_c = 4/2 sqrt(3/(2π))`

`V_s/V_c = 2 sqrt(3/(2π))`

`V_s/V_c = sqrt(4 xx 3/(2π))`

`V_s/V_c =sqrt(12/(2π))`

∴ `V_s/V_c =sqrt(6/π)`

Hence, the ratio of the volume of the sphere to the volume of the cube is `sqrt6 : sqrtπ`.