Question

A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

Answer 1

Let the radius of the sphere be 'r₁'.

Let the radius of the hemisphere be 'r₂'

TSA of sphere = `4pi r_1^2`

TSA of hemisphere = `3pi r_2^2`

TSA of sphere = TSA of hemi-sphere

`4pir_1^2 = 3pir_2^2`

`=> r_2^2 = 4/3r_1^2`

`=> r_2 = 2/sqrt3r_1`

Volume of sphere, `v_1 = 4/3pir_1^3`

Volume of hemisphere, `v_2 = 2/3pir_2^3`

`v_2 = 2/3pir_2^3`

`=> v_2 = 2/3pi((r_1 2)/(3sqrt3))^3`

`=> v_2 = 2/3pi(r_2^3 8)/(3sqrt3)`

Dividing v₁ by v₂ 

`v_1/v_2 = (4/3pir_1^3)/(2/3pi 8/(3sqrt3)r_1^3`

`=> v_1/v_2 = (4/3)/(2/3 8/(3sqrt3)`

`=> v_1/v_2 = 4/3 xx (9sqrt3)/16`

`=> v_1/v_2 = (3sqrt3)/4`