A metal sphere having a radius of 12 cm is melted down and molded into 8 identical smaller solid spheres. What is the ratio of the surface area of the original sphere to the total surface area of all 8 smaller spheres?
Show answer & explanation
Correct answer: A
Explanation
Given: Radius of original sphere, R = 12 cm
Melted into 8 identical smaller spheres of radius r.
Step 1: Find r (Volume is conserved during melting)
Volume of original sphere = 8 × Volume of one small sphere
(4/3)πR³ = 8 × (4/3)πr³
R³ = 8r³
r³ = R³/8
r = R/2 = 12/2 = 6 cm
Step 2: Surface area of original sphere
= 4πR²
= 4π(12)²
= 4π × 144
= 576π cm²
Step 3: Surface area of one small sphere
= 4πr²
= 4π(6)²
= 4π × 36
= 144π cm²
Step 4: Total surface area of 8 small spheres
= 8 × 144π
= 1152π cm²
Step 5: Required ratio
Surface area of original sphere : Total surface area of small spheres
= 576π : 1152π
= 1 : 2
Answer: 1 : 2