A cylinder is within the cube touching all the vertical faces. A cone is inside the cylinder. If their heights are same with the same base, find the ratio of their volumes.

Explanation

Let the length of each edge of the given cube be a units.
Then, volume of the cube `= a^(3)` cubic units `= V_(1)` (say)
It is given that the cylinder lies within the cube and touches all its vertical faces.

So, the radius of the base of the cylinder `= (a)/(2)` units and height of the cylinder = a units
Volume of the cylinder `= pi r^(2)h`
`= {(22)/(7) xx ((a)/(2))^(2) xx a}` cubic units
`= ((11a^(3))/(14))` cubic units `= V_(2)` (say)
A cone is drawn inside the cylinder such that both have the same base and same height.
`:.` radius of the base of the cone `= (a)/(2)` units
and height of the cone = a units
`:.` volume of the cone `= (1)/(3) pi r^(2) h`
`= ((1)/(3) xx (22)/(7) xx (a^(2))/(4) xx a)` cubic units
`= (11a^(3))/(42)` cubic units `= V_(3)` (say)
`:.` ratio of their volumes is given as
`V_(1) : V_(2) : V_(3) = a^(3) : (11a^(3))/(14) : (11a^(3))/(42) = 42 : 33 : 11`