Let `V_(1)` be the volume of a given right circular cone with O as the centre of the base and A as its apex. Let `V_(2)` be the maximum volume of the right circular cone inscribed in the given cone whose apex is O and whose base is parallel to the base of the given cone. then the ratio `V_(2)//V_(1)` is -

Options

A. `3/25`
B. `4/9`
C. `4/27`
D. `8/27`

Answer

C

Explanation

`DeltaABC` and `DeltaAOP` are similar
` h/r = (H)/(R) rArr h = (rH)/(R)`
`V_(2) = 1/3 pir^(2) (H - h) = (pi)/(3)r^(2)H(1-r/R) = (piH)/(3) (r^(3) - (r^(3))/(R))`
`(dV^(2))/(dr) = 2r - (3r^(2))/(R) = 0 , r = (2R)/(3)`
`V_(2"max") = (4piR^(2)H)/(81)`
` V_(1) = (piR^(2)H)/(3)`
`:. (V_(2))/(V_(1)) = (4)/(27)`