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 1

Let the radius of the small cone be r cm

Radius if the big cone = R cm

Given, height of the big cone = 20 cm

Let the height of section made = h cm

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

Now,

In △OAB and △OCD

∠AOB = ∠COD [common]

∠OAB = ∠OCD [each 90^o]

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 found that the section was made at a height of 16 cm above the base.