Multiple choice

Height of the cylinder of maximum volume that can be inscribed in a sphere of radius 12 cm is

  1. $8\sqrt{3}$ cm
  2. $8$ cm
  3. $12\sqrt{3}$ cm
  4. $24$ cm
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A Correct answer
Explanation

For a cylinder of radius r and height h inscribed in a sphere of radius R, r^2 + (h/2)^2 = R^2. Volume V = pi * r^2 * h = pi * (R^2 - h^2/4) * h = pi * (R^2*h - h^3/4). Setting dV/dh = 0: R^2 - 3h^2/4 = 0, so h^2 = 4R^2/3, h = 2R/sqrt(3). With R=12, h = 24/sqrt(3) = 8*sqrt(3).

AI explanation

Using the volume of a cylinder formula and the Pythagorean relation that half the height squared plus the base radius squared equals the sphere's radius squared, we maximize the volume. Substituting the sphere radius of 12 into the derived maximum height formula, the height equals 12 divided by the square root of 3, which equals 8 times the square root of 3 cm.

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