Question
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?
The correct answer is
25 : 6
Given:
Original sphere radius = 10 cm
It is melted into 125 identical small spheres
Step 1: Volume relation
Let radius of small sphere = r
Volume conservation:
4/3 π (10)^3 = 125 × 4/3 π r^3
Cancel 4/3 π:
1000 = 125 r^3
r^3 = 8 ⇒ r = 2 cm
Step 2: Surface area
Original sphere SA = 4π(10)^2 = 400π
One small sphere SA = 4π(2)^2 = 16π
Surface area of 6 small spheres = 6 × 16π = 96π
Step 3: Ratio
Original : 6 small spheres = 400π : 96π
Cancel π:
= 400 : 96 = 25 : 6
Final Answer: 25 : 6
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Important Questions from 3-D Mensuration
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