The height of cone is 30 cm. A small cone is cut off from the top portion by a plane parallel to the base. If the volume be () of the volume of the given cone, at what height above the base is the section made ?

The correct answer is: 20 cm

We have given that
height of the cone = 30 cm
Let the small cone is cut off at a height 'h' from the top.
Let the radius of big cone be be r₁​ and small cone be r₂​

v₁​=(1/3)​πr²h

Volume of the big cone.

v₁​= (1/3)​πr₁² x 30

= 10πr₁² ​cu.cm

Volume of the small cone,

v₂​= (1/3)πr²h

Given v₂​=(1/27)​.v1​

v₂ / ​v₁​​ = 1 / 27

(1/3)πr₂²h / 10πr₁² ​ = 1 / 27

r₂²h / 30πr₁² = 1 / 27

(r₂/​r₁​​)² × (h / 30​) = 1 / 27 ​...(1)
From the figure, △ACD∼△AOB (AA similarity criterion)

r₂ / r₁ ​​= h/ 30​
Hence equation (1), become's

(h / 30​)^2. h / 30) ​= 1 / 27​

(h / 30​)³=(1 / 3​)³

h / 30 ​= 1 / 3​

h = 10 cm
Thus at a height 20 cm above the base a small cone is cut.
Therefore, the correct option is c) 20 cm