Q. 18

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, at what height above the base is the section made?

Answer :

Let the cutting plane be passing through points B and C as shown

Height of cone = AD = H = 20 cm

Height of small cone which we get after cutting = AB = hc

Let ‘r’ be the radius of small cone we have BC = r

‘R’ be radius of original cone which is to be cut we have DE = R

From figure consider ΔABC and ΔADE

ABC = ADE = 90˚

BAC = DAE …(common angle)

as two angles are equal by AA criteria we can say that


Let V1 be the volume of cone to be cut

Let V2 be the volume of small cone which we get after cutting

Volume of cone = (1/3)π(radius)2(height) cm3

V1 = (1/3) × π × R2 × hc

V2 = (1/3) × π × r2 × 20

Given is that the volume of small cone is (1/8) times the original cone

V2 = (1/8) V1

(1/3) × π × r2 × 20 = (1/8) × (1/3) × π × R2 × hc

Using equation (i) we get

hc3 = 203/8 cm

hc = 20/2 cm

hc = 10 cm

But we have to find the height from base i.e. we have to find BD from figure

20 = BD + hc

20 = BD + 10

BD = 10 cm

10 cm above base the section is made.

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RS Aggarwal - Mathematics