Multiple choice

A solid cylinder $8\ cm$ in diameter and $4\ cm$ in height is melted and $6$ spheres of equal sizes are made from the material. Then the diameter of each sphere is

  1. $4\ cm$
  2. $1\ cm$
  3. $1.2\ cm$
  4. $0.8\ cm$
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A Correct answer
Explanation

Volume of cylinder = pi * r^2 * h = pi * 4^2 * 4 = 64 * pi. Volume of 6 spheres = 6 * (4/3) * pi * R^3 = 8 * pi * R^3. Setting them equal: 64 * pi = 8 * pi * R^3 => R^3 = 8 => R = 2. The diameter is 2 * R = 4 cm.

AI explanation

The volume of the cylinder is found using the formula pi * r^2 * h, which gives (22/7) * 4^2 * 4 = 1408/7 cubic cm. This volume is melted to form 6 identical spheres, so each sphere has a volume of (1408/7) / 6 = 704/21 cubic cm. Using the sphere volume formula (4/3) * pi * r^3, we equate it to the volume of one sphere to get (4/3) * (22/7) * r^3 = 704/21. Solving this equation gives r^3 = 8, meaning the radius of each sphere is 2 cm and the diameter is 4 cm.

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