<p>Find the vertical angle of a right circular cone of minimum curved surface area that circumscribes a given sphere.</p>
<p>\(2\sin^{-1}(\sqrt{2}-1)\)</p>
<p>\(2\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)\)</p>
<p>\(2\sin^{-1}\left(\frac{1}{2}\right)\)</p>
<p>\(\sin^{-1}(\sqrt{2}-1)\)</p>
Step-by-Step Solution
Key Concept: For a cone circumscribing a sphere, the curved surface area S = πrl is minimized when we express r and l in terms of the sphere's radius R and the semi-vertical angle α, then optimize using calculus. The critical insight is that the sphere touches the slant surface, giving the constraint: R = r·sin(α)/(1 + sin(α)).
<p><strong>Step 1:</strong> Set up the constraint. For a cone with semi-vertical angle α, base radius r, and slant height l, when a sphere of radius R is inscribed (touching the base and lateral surface), the geometric relation is:</p><p>R = (r·sin α)/(1 + sin α)</p><p><strong>Step 2:</strong> Express cone dimensions in terms of α and R. From the constraint: r = R(1 + sin α)/sin α and l = R(1 + sin α)/sin α · 1/cos α</p><p><strong>Step 3:</strong> Write curved surface area S = πrl:</p><p>S = πR² · (1 + sin α)²/(sin² α · cos α)</p><p><strong>Step 4:</strong> Minimize S by taking dS/dα = 0. This leads to the equation:</p><p>sin² α - 4 sin α - 1 = 0</p><p><strong>Step 5:</strong> Solve for sin α: Using the quadratic formula: sin α = (4 ± √20)/2 = 2 ± √5. Since sin α ≤ 1, we have sin α = √5 - 2 ≈ 0.236</p><p><strong>Step 6:</strong> Find the vertical angle. The vertical angle = 2α, where sin α = √5 - 2, giving α ≈ 13.63°</p><p>∴ <strong>Vertical angle = 2 sin⁻¹(√5 - 2) ≈ 27.27° or π/6.6 radians</strong> (or equivalently, the semi-vertical angle α = sin⁻¹(√5 - 2))</p>
Correct Answer: A