Question

A sphere of maximum volume is cut-out from a solid hemisphere of radius r, what is the ratio of the volume of the hemisphere to that of the cut-out sphere?

Answer 1

Since, a sphere of maximum volume is cut out from a solid hemisphere of radius.

i.e., radius of sphere

Therefore,

The volume of sphere

   `=4/3 pi (r/2)^3`

`v_1 = 1/6pir^3`…… (i)

The volume of hemisphere `v_2 = 2/3pir^3` …… (ii)

Divide (i) by (ii).

`v_1/v_2 = (1/6 pir^3)/(2/3 pir^3)`

     `=1/6 xx 3/2`

`v_1/v_2 = 1/4`

Hence , `v_2 :v_1 = 4:1`