Praveen has a solid hemisphere of metal, he cut out a sphere of maximum volume. Then find the ratio of the volume of the sphere to that of the remaining solid ?

For a hemisphere of radius R, the maximum sphere that can be cut has radius r = R/2 (inscribed in hemisphere). Volume of sphere = (4/3)π(R/2)³ = (4/3)πR³/8 = (1/6)πR³. Volume of hemisphere = (2/3)πR³. Remaining volume = (2/3)πR³ - (1/6)πR³ = (1/2)πR³. Ratio of sphere to remaining = (1/6)πR³ : (1/2)πR³ = 1:3. This matches option C.