Question

A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.

Answer 1

•If solid of one shape is converted into solid of another shape, then Total volume of the solid to be converted = Total volume of the solid into which the given solid is to be converted.

SOLUTION :
Given : Internal diameter of hollow sphere(d)= 4 cm.
Internal radius of hollow sphere (r) = 4/2= 2 cm
external diameter of hollow sphere (D) = 8 cm.
external radius of hollow sphere( R )= 8/2= 4 cm.

Volume of the Hollow sphere = 4/3π(R³ - r³)
Volume of the Hollow sphere = 4/3π(4³ - 2³)
Volume of the Hollow sphere = 4/3π(64 - 8)
Volume of the Hollow sphere = 4/3π(56) cm³

Diameter of the cone(d1) = 8 cm
radius of the cone( r1)= 8/2 = 4 cm

Let the height of the cone be h cm.
Volume of the cone = ⅓ πr1²h
= ⅓ π × 4² × h = 16πh/3

Volume of the cone = Volume of the hollow sphere
16πh/3 = 4/3π(56)
16h = 4 ×56
h = (4 × 56)/16
h = 56/4 = 14 cm

Hence, the height of the cone is 14 cm.