Multiple choice

The altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is

  1. $\dfrac { r }{ 2 } $
  2. $\dfrac { r }{ 3 } $
  3. $\dfrac { 3r }{ 4 } $
  4. $\dfrac { 4r }{ 3 } $
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D Correct answer
Explanation

For a cone of maximum volume inscribed in a sphere of radius r, the height h is 4r/3 and the base radius is sqrt(8/9) * r. This is a standard optimization problem result.

AI explanation

Let the sphere have center O and radius r, and the inscribed cone have height h and base radius x. The axial cross-section shows a right triangle where (h - r)^2 + x^2 = r^2. Solving for x^2 gives x^2 = 2rh - h^2. Substituting this into the cone volume formula V = (1/3)(pi)(x^2)h yields V = (pi/3)(2rh^2 - h^3). Taking the derivative and setting it to zero gives 4rh - 3h^2 = 0, so h = 4r/3.

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