Multiple choice

The altitude of a right circular cone of minimum volume circumscribed about a sphere of radius $r$ is

  1. $2r$
  2. $3r$
  3. $5r$
  4. $4r$
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D Correct answer
Explanation

For a cone of height h and base radius R circumscribed about a sphere of radius r, the volume is V = (1/3)pi R^2 h. Using similar triangles, R^2 = r^2 * h / (h - 2r). Minimizing V leads to h = 4r.

AI explanation

For a right circular cone circumscribed about a sphere of radius r, the altitude h, the sphere radius r, and the cone base radius x are geometrically related by the right triangle formed in the axial cross-section. The relation is r = (x^2 * h) / (x^2 + h^2). The cone volume is V = (1/3)(pi)(x^2)h. Substituting x^2 = (r * h^2) / (h - 2r) gives V = (pi/3) * (r * h^3) / (h - 2r). To minimize V, the derivative is set to zero, yielding h(h - 4r) = 0, so the altitude is 4r.

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