The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be `1/(27)` of the volume of the given cone, at what height above the base is the section mode?

Explanation

Height of the givne cone = 30 cm
Let the radius of its base be R cm.
`"Volume of the given cone "=(1/3piR^(2)xx30)cm^(3)(10piR^(2))cm^(3)`.
Let the radius and height of the smaller cone be r cm and h cm respectively.
Then, `"volume of the smaller cone "=(1/3pir^(2)h)cm^(3)`.
`:." "1/3pir^(2)h=1/27(10piR^(2)r)" [given]"`

`rArr" "(R/r)^(2)=(9h)/10" ...(i)"`
Now, `DeltaOAB ~ Delta OCD`.
`:." "(AB)/(CD)=(OA)/(OC)rArrR/r=30/r" ...(ii)"`
From (i) and (ii), we get
`((30)/h)^(2)=(9h)/10rArr(30xx30)/(hxxh)=(9h)/10`
`rArr" "h^(3)=(30xx30xx10)/9=1000rArrh^(3)=(10)^(3)rArrh=10`.
`:." height of the smaller cone = 10 cm".`
Height of the section from the base = (30 - 10) cm = 20 cm.