Question

The height of a right circular cone is 20 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be `1/8` of the volume of the given cone, then at what height above the base is the section made?

Answer 1

We have,

Height of the given cone, H = 20 cm

Let the radius of the given cone be h and 

the height of the smaller cone be r.

Now, in ΔAQD and ΔAPC,

∠QAD = ∠PAC     (Common angle)

∠AQD = ∠APC = 90°

So, by AA criteria 

∠AQD _˜∠APC 

`rArr ("AQ")/"AP"=("QD")/("PC")`

`rArr "h"/"H" = "r"/"R"    .......(i)`

Volume of smaller cone `= 1/8xx "Volume of the given cone"`

`rArr "Volume of smaller cone"/"Volume of the given one" = 1/8`

`rArr ((1/3pi"r"^2"h"))/((1/3pi"R"^2"H"))=1/8`

`rArr ("r"/"R")^2xx("h"/"H") = 1/8`

`rArr ("h"/"H")^2xx("h"/"H")=1/8`       [Using (i)]

`rArr ("h"/"H")^3 = 1/8`

`rArr "h"/"H"=root(3)(1/8)`

`rArr "h"/"20" = 1/2`

`rArr  "h" = 20/2`

`rArr "h" = 10  "cm"`

∴ PQ = H - h = 20 - 10 cm

So, the section is made at the height of 10 cm above the base.