2023 · Q65
Three solid lead spheres of diameters 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres?
Elementary MathematicsMensuration
- A25%
- B26%
- C27%
- D28%
Explanation
Diameters 6, 8 and 10 cm give radii 3, 4 and 5 cm. Volume is conserved when the spheres are melted together, so for the recast sphere R³ = 3³ + 4³ + 5³ = 27 + 64 + 125 = 216, giving R = 6 cm. The sum of the original surface areas is 4π(9 + 16 + 25) = 200π, while the new sphere's surface area is 4π·36 = 144π. The percentage diminution is (200π − 144π)/(200π) = 56/200 = 28%, option (D).