Question

Choose the correct answer of the following question:

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

Options

• 1 : 2

• 1 : 4

• 1 : 6 

• 1 : 8

Answer 1

Let the radii of the smaller and given cones be r and R, respectively; and their heights b and H respectively 

We have, 

H = 2h   .........(i)

In ΔAQD and ΔAPC,

∠QAD = ∠PAC      (common angle)

∠AQD = ∠APC = 90°

So, by AA citeria

Δ AQD _˜ Δ APC

`=> "AQ"/"AP" = "QD"/"PC"`

`=> "h"/"H" = "r"/"R"`

`=> "h"/"2h" = "r"/"R"`

`=>1/2 = "r"/"R"`

⇒ R = 2r        ........(ii)

Now,

The ratio of the smaller cone to the whole cone`="Volume of the smaller cone"/"Volume of the whole cone"`

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

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

`=("r"/"2r")^2xx("h"/"2h")`          [Using (i) and (ii)]

`=(1/2)^2 xx (1/2)`

`=1/8`

= 1 : 8

Hence, the correct answer is option (d).