Multiple choice

The height of the cylinder of mas.Volume that can be inscribed in a sphere of radius $\alpha$ is

  1. $\dfrac{2\alpha}{\sqrt{3}}$
  2. $\dfrac{\alpha}{\sqrt{2}}$
  3. $\dfrac{5\alpha}{4}$
  4. $None\ of\ these$
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A Correct answer
Explanation

For a cylinder inscribed in a sphere of radius a, the volume is V = pi * r^2 * h. Since r^2 + (h/2)^2 = a^2, V = pi * (a^2 - h^2/4) * h = pi * (a^2 * h - h^3/4). Differentiating with respect to h and setting to zero gives a^2 - 3h^2/4 = 0, so h^2 = 4a^2/3, h = 2a / sqrt(3).

AI explanation

To maximize the cylinder volume inside a sphere, the cylinder height h and radius r satisfy the relation r^2 = alpha^2 - (h/2)^2. Substituting this into the cylinder volume formula V = pi * r^2 * h gives V = pi * (alpha^2 - h^2/4) * h, and differentiating this reveals the volume is maximum when h = 2 * alpha / sqrt(3).

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