Multiple choice

Three solid metallic spheres whose radii are 1 cm, x cm and 8 cm, are melted and recast into a single solid sphere of diameter 18 cm. The surface area (in cm2) of the sphere with radius x cm is:

  1. 144 π

  2. 72 π

  3. 64 π

  4. 100 π
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A Correct answer
Explanation

The volume of the original spheres is (4/3)pi(1^3 + x^3 + 8^3). The new sphere has a diameter of 18, so radius R = 9, and volume (4/3)pi(9^3). Equating volumes: 1 + x^3 + 512 = 729, so x^3 = 216, which means x = 6. The surface area of the sphere with radius 6 is 4 * pi * 6^2 = 144pi.

AI explanation

Because the three metallic spheres are melted and recast, the total volume of the three original spheres equals the volume of the new larger sphere. The new sphere has a diameter of 18 centimeters, meaning its radius is 9 centimeters and its volume is 4 thirds pi times 9 cubed, or 972 pi. Setting the sum of the initial volumes equal to this gives 1 cubed plus x cubed plus 8 cubed equals 972 pi divided by 4 thirds pi, which simplifies to 1 plus x cubed plus 512 equals 729. Solving this equation gives x cubed equals 216, so x is 6 centimeters. The surface area of this sphere is 4 pi times 6 squared, resulting in 144 pi.

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