<p>A cone of height <em>H</em> and base radius <em>R</em> is given. A cylinder of height <em>h</em> and radius <em>r</em> is inscribed in the cone. Find the value of <em>H/h</em> when the volume of the cylinder is maximum.</p>
Step-by-Step Solution
Key Concept: Use the constraint that the cylinder is inscribed in the cone to express r in terms of h, then optimize volume V = πr²h by taking dV/dh = 0.
<p><strong>Step 1:</strong> Set up the constraint using similar triangles. The cone has height H and base radius R. At height h from the base, the radius of the cone is r_cone = R(H-h)/H. For the inscribed cylinder, r = R(H-h)/H.</p><p><strong>Step 2:</strong> Express volume of cylinder: V = πr²h = πR²(H-h)²h/H²</p><p><strong>Step 3:</strong> Differentiate with respect to h: dV/dh = πR²/H² [(H-h)²·1 + h·2(H-h)·(-1)] = πR²/H²[(H-h)² - 2h(H-h)] = πR²/H²(H-h)[(H-h) - 2h]</p><p><strong>Step 4:</strong> Set dV/dh = 0: (H-h)(H-3h) = 0. Since h ≠ H, we get H - 3h = 0, so h = H/3.</p><p><strong>Step 5:</strong> Therefore H/h = H/(H/3) = 3</p><p>∴ Answer: H/h = <strong>3</strong></p>
Correct Answer: B