A right circular cylinder is inscribed in a right circular cone of base radius R and height H. What height h of the cylinder maximizes its volume?

Answer and explanation

Correct answer: D. H / 3

  1. Step 1: By similar triangles, the radius r of the cylinder at height h satisfies r / R = (H - h) / H, so r = R*(1 - h/H).
  2. Step 2: Volume of cylinder: V(h) = pi r^2 h = pi R^2 h * (1 - h/H)^2.
  3. Step 3: Expand: V(h) = pi R^2 (h - 2*h^2/H + h^3/H^2).
  4. Step 4: Differentiate with respect to h: V'(h) = pi R^2 (1 - 4h/H + 3h^2/H^2) = 0.
  5. Step 5: Factor: (1 - h/H)(1 - 3h/H) = 0. For maximum volume (0 < h < H), 1 - 3h/H = 0, so h = H / 3.
Answer
H / 3.
Why not the others: H / 2 maximizes the curved surface area; 2*H / 3 gives the radius ratio, not the height; H / 4 is a sub-optimal height.

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