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 is \(\frac{1}{{27}}\) of the volume of the cone, at what height, above the base, is the section made?

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 is \(\frac{1}{{27}}\) of the volume of the cone, at what height, above the base, is the section made? Correct Answer 20 cm

GIVEN :

⇒ Height of big cone = 30 cm

⇒ Radius of big cone = R

⇒ Height of small cone = h

⇒ Radius of small cone = r

FORMULA USED :

⇒ Volume of big cone = (1/3) × π × radius² × height

CALCULATION :

 ⇒ Volume of big cone = (πR² × 30) /3 = 10πR²

⇒ Volume of small cone = (1/3) × π × radius² × height

                                       = (πr²× h) /3

⇒ Volume of small cone = (1/27) × volume of big cone

⇒ (πr² × h) /3 = (1/27) × 10πR²

⇒ (R/r) ² = 9h/10

By using similarity of triangles,

⇒ R/r = 30/h

Solving equations,

⇒ 900/h² = 9h/10

⇒ h = 10 m

∴ Height at which small cone is cut = 30 – 10 = 20 m