Multiple choice

The radius of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius R is

  1. R
  2. $\displaystyle \frac{R}{2}$
  3. $\displaystyle \sqrt{\frac{2}{3}}R$
  4. $\displaystyle \sqrt{\frac{3}{2}}R$
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C Correct answer
Explanation

For a cylinder of radius r and height h inscribed in a sphere of radius R, the relationship is r^2 + (h/2)^2 = R^2. Maximizing the volume V = pi * r^2 * h leads to the optimal radius r = sqrt(2/3) * R.

AI explanation

Relating the cylinder's height h and radius r to the sphere gives r^2 + (h/2)^2 = R^2. Substituting h = 2√(R^2 - r^2) into the cylinder volume formula V = πr^2h gives V = 2πr^2√(R^2 - r^2). Setting the derivative dV/dr to zero requires 2(R^2 - r^2) - r^2 = 0, or 2R^2 = 3r^2. Solving for r gives the maximum volume radius as √(2/3)R.

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