Eight solid metal spheres with same radii are melted and recast into a sphere without losing any metal. What is the percentage change in the total surface area of the solid?
Explanation:
Let the radii of the eight spheres and the new hemisphere be r and R respectively.
Volume of the eight original spheres = Volume of the new sphere
∴ 8 × (4/3) × πr3 = (4/3) × πR3
∴ R = 2r
Original total surface area of the eight spheres = 8 × (4πr2) = 32πr2
New total surface area of the new sphere = 4πR2 = 4π(2r)2 = 16πr12
∴ Reduction in total surface area = 32πr2 – 16πr2 = 16πr2
∴ Percentage reduction = (16πr2/32πr2) × 100 = 50%
Hence, option (d).
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