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

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This question is Based on Mensuration Chapter of M.L Aggarwal book for ICSE BOARD for class 10.
Here the height of a cone is given. A small cone is cut off at the top by a plane parallel to its base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section cut?
This is the Question Number 18, Exercise 17.2 of M.L Aggarwal.

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1. Given height of the cone, H = 30 cm

Let R be the radius of the given cone and r be radius of small cone.

Let h be the height of small cone.

Volume of the given cone = (1/3)R2H

Volume of the small cone = 1/27 th of the volume of the given cone.

(1/3)r2h = (1/27)Ã— (1/3)R2H

Substitute H = 30

(1/3)r2h = (1/27)Ã— (1/3)R2Ã—30

r2h/R2Â = 30/27

r2h/R2Â = 10/9 â€¦.(i)

From figure, r/R = h/H

r/R = h/30 â€¦.(ii)

Substitute (ii) in (i)

(h/30)2Ã—h = 10/9

h3/900 = 10/9

h3Â = 900Ã—10/9 = 1000

Taking cube root on both sides.

h = 10 cm

H-h = 30-10 = 20

The small cone is cut at a height of 20 cm above the base.

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