Multiple choice

Solid spheres of radius 3 cm, 4 cm and 5 cm are melted and recast into a solid sphere. What will be the percentage decrease in the surface area?

  1. 12

  2. 14

  3. 16

  4. 28

  5. N/A
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Volumes: (4/3)pi(3^3 + 4^3 + 5^3) = (4/3)pi(27 + 64 + 125) = (4/3)pi(216). The new radius is 6. Original surface areas: 4pi(9 + 16 + 25) = 200pi. New surface area: 4pi(36) = 144pi. Decrease = (200-144)/200 = 56/200 = 28%.

AI explanation

The initial volume of the three spheres is the sum of their individual volumes, which is 4/3 times pi times the sum of their cubed radii (3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216). When recast into a single solid sphere, this volume is 4/3 times pi times the new radius cubed, meaning the new radius cubed is 216 and the new radius is 6 cm. The total initial surface area was 4 times pi times (9 + 16 + 25), which equals 200 times pi. The new surface area is 4 times pi times 6 squared, resulting in 144 times pi. The decrease in surface area is 56 times pi, which is a 28 percent decrease from the initial 200 times pi.

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