#### Question

The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is `1/27` of the given cone, then at what height it is cut from its base?

#### Solution

Let the radius and height of the bigger cone be R and H, respectively.

Given: H = 30 cm

Let the radius and height of the smaller cone be r and h, respectively.

Now, in ∆AFE and ∆AGC,

∠AEF = ∠ACG (Corresponding angles)

∠AFE = ∠AGC (90° each)

∴ ∆AFE~∆AGC (AA similarity)

`=> (AF)/(AG) = (FE)/(GC)`

`=> h/H = r/R` ....(1)

it is given that

Volume of the smaller cone = `1/27 xx "Volume of the bigger cone"`

`=> 1/3 pir^2h = 1/27 xx 1/3 piR^2H`

`=>(r/R)^2 xx h/H = 1/27`

`=> (h/H)^2 xx h/H = 1/27` [Using (1)]

`=> (h/H)^3 = 1/27`

`=> h/H = 1/3`

`:. h = 1/3 xx H = 1/3 xx 30` = 10 cm

Now

FG = AG − AF = 30 cm − 10 cm = 20 cm

Thus, the small cone is cut at the height of 20 cm from the base.