Multiple choice

The internal and external radii of a metallic spherical shell are $4$ cm and $8$ cm, respectively. It is melted and recast into a solid right circular cylinder of height $9\displaystyle \frac{1}{3} $ cm. Find the diameter of the base of the cylinder.

  1. $16$ cm
  2. $18$ cm
  3. $12$ cm
  4. $14$ cm
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A Correct answer
Explanation

Volume of spherical shell = (4/3) * pi * (R^3 - r^3) = (4/3) * pi * (8^3 - 4^3) = (4/3) * pi * (512 - 64) = (4/3) * pi * 448. Cylinder volume = pi * r_c^2 * h = pi * r_c^2 * (28/3). Equating them: (4/3) * 448 = r_c^2 * (28/3). r_c^2 = (4 * 448) / 28 = 64. r_c = 8. Diameter = 2 * r_c = 16 cm.

AI explanation

The volume of the metallic shell is found using the spherical shell volume formula V = 4/3 * pi * (R^3 - r^3), which yields 4/3 * pi * (8^3 - 4^3) equaling 448 * pi. Equating this to the volume of the right circular cylinder, pi * r^2 * h, we get 448 * pi = pi * r^2 * 28/3. Canceling pi and solving for r^2 gives 48, meaning the base radius is 8 cm. Therefore, the diameter of the base of the cylinder is 2 * 8 = 16 cm.

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