Multiple choice

A cylinder is inscribed in a sphere of radius r. What is the radius of the cylinder of maximum volume?

  1. $\displaystyle\frac{2r}{\sqrt{3}}$
  2. $\displaystyle\frac{\sqrt{2}r}{\sqrt{3}}$
  3. $r$
  4. $\sqrt{3}$r
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B Correct answer
Explanation

For a cylinder of radius x and height 2h inscribed in a sphere of radius r, x^2 + h^2 = r^2. Volume V = pi * x^2 * 2h = 2 * pi * (r^2 - h^2) * h = 2 * pi * (r^2*h - h^3). Setting dV/dh = 0 gives r^2 - 3h^2 = 0, so h = r/sqrt(3). Then x^2 = r^2 - r^2/3 = 2r^2/3, so x = r * sqrt(2/3).

AI explanation

Using the geometric property of an inscribed cylinder, r^2 + (h/2)^2 = r_sphere^2, we express the cylinder height as h = 2√(r^2 - r_c^2), where r_c is the cylinder radius. Substituting this into the volume formula yields V = 2πr_c^2√(r^2 - r_c^2). Setting dV/dr_c to zero results in the equation 2r^2 - 3r_c^2 = 0, so r_c^2 = 2r^2/3. The radius for maximum volume is √2r/√3.

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