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Question -

The height of a cone is 20 cm. A small cone is cut off from the top by a plane parallel to the base. If its volume be 1/125 of the volume of the original cone, determine at what height above the base the section is made.



Answer -


Let the radius of thesmall cone be r cm

Radius if the big cone= R cm

Given, height of thebig cone = 20 cm

Let the height ofsection made = h cm

Then, the height ofsmall cone will be = (20 – h) cm

Now,

In OAB and OCD

AOB = COD [common]

OAB = OCD [each 90o]

Then, OAB ~ OCD [by AA similarity]

So, by C.P.S.T we have

OA/ OC = AB/ CD

(20 – h)/ 20 = r/ R ……(i)

Also given,

Volume of small cone =1/125 x volume of big cone

1/3 π r2(20– h) = 1/125 x 1/3 πR2 x 20

r2/ R2 =1/125 x 20/ (20 – h) [From (i)] 

(20 – h)2/202 = 1/125 x 20/20 – h

(20 – h)3 =203/ 125

20 – h = 20/5

20 – h = 4

h = 20 – 4 = 16 cm

Therefore, it’s foundthat the section was made at a height of 16 cm above the base.

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