A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?

Given:

Radius of the original sphere = 10 cm

Number of smaller spheres = 125

Formula Used:

Volume of a sphere = (4/3)πr³

Surface area of a sphere = 4πr²

Ratio of surface areas = Surface area of original sphere / Total surface area of smaller spheres

Calculation:

Volume of the original sphere = (4/3)π(10)³

⇒ Volume = (4/3)π × 1000 = 4000π/3

Volume of one smaller sphere = Volume of original sphere / 125

⇒ Volume = (4000π/3) / 125 = 32π/3

Let the radius of each smaller sphere be r.

(4/3)πr³ = 32π/3

⇒ r³ = 32

⇒ r = 2 cm

Surface area of the original sphere = 4π(10)²

⇒ Surface area = 4π × 100 = 400π

Surface area of one smaller sphere = 4π(2)²

⇒ Surface area = 4π × 4 = 16π

Total surface area of 125 smaller spheres = 125 × 16π = 2000π

Ratio = Surface area of original sphere / Total surface area of 6 smaller spheres

⇒ Ratio = 400π / (6 × 16π)

⇒ Ratio = 400 / 96 = 25 : 6

The ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres is 25 : 6.