<p>From a given solid cone of height \(H\), another inverted cone is carved whose height is \(h\), such that its volume is maximum, then the ratio \(\dfrac{H}{h}\) is equal to:</p>
Step-by-Step Solution
Key Concept: Use similar triangles to relate the radius of the inverted cone to its height, then maximize volume as a function of h using calculus. The inverted cone's vertex is at the base of the original cone, so geometric constraints determine the relationship between dimensions.
<p><strong>Step 1:</strong> Set up the original cone with height H and base radius R. The inverted cone has height h (measured from the base of original cone upward) and radius r at its top.</p><p><strong>Step 2:</strong> By similar triangles, at height h from the base, the radius of the original cone is R(H-h)/H. Thus, r = R(H-h)/H for the inverted cone.</p><p><strong>Step 3:</strong> Volume of inverted cone: V = (1/3)πr²h = (1/3)π[R(H-h)/H]²h = (πR²/3H²)h(H-h)²</p><p><strong>Step 4:</strong> To maximize, differentiate with respect to h:</p><p>dV/dh = (πR²/3H²)[(H-h)² + h·2(H-h)(-1)] = (πR²/3H²)[(H-h)² - 2h(H-h)]</p><p>= (πR²/3H²)(H-h)[(H-h) - 2h] = (πR²/3H²)(H-h)(H-3h)</p><p><strong>Step 5:</strong> Setting dV/dh = 0: (H-h)(H-3h) = 0</p><p>Since h < H, we get H - 3h = 0, so h = H/3</p><p><strong>Step 6:</strong> Verify using second derivative test that this is a maximum (it is).</p><p><strong>Step 7:</strong> Therefore, H/h = H/(H/3) = 3</p><p>∴ Answer: B (which equals 3)</p>
Correct Answer: B