Question

An ironsmith wants to melt 3 solid spheres in his furnace. If 3 solid metallic spheres of diameters 6 cm, 8 cm and 10 cm are melted and recast into a new solid sphere. The diameter of it is ________.

Answer 1

Sol:
Diameters of the spheres melted = 6 cm, 8 cm and 10 cm
Radii of the spheres = 3 cm, 4 cm and 5 cm
Volumes of the spheres:

V1 = 4 3 ×π ×r3
= 4 3 ×π ×(3)3
= 36 π cm3.

V2 = 4 3 ×π ×r3
= 4 3 ×π ×(4)3
= 256 3 π cm3.

V3 = 4 3 ×π ×r3
= 4 3 ×π ×(5)3
= 500 3 π cm3.

Sum of the volumes of the spheres
= 36 π + 256 3 π + 500 3 π
= 108 3 π + 256 3 π + 500 3 π
= 864 3 π
= 288 π

Let the radius of the new sphere formed be 'R'.

V = 4 3 ×π ×R3

⇒ 4 3 ×π ×R3 = 288 π

⇒ 4 3 ×R3 = 288

⇒ R3 = 288 × 3 4 

⇒ R3 = 216

⇒ R = 6

Radius of the new sphere formed = 6 cm

So, the diameter of the sphere formed = 12 cm.