<p>A sector of a circle of radius 1 with angle α is bent to form a cone. The volume of the vessel (cone). If α = π:</p>
Step-by-Step Solution
Key Concept: The arc length of the sector becomes the circumference of the cone's base, and the sector's radius becomes the slant height. Use these relationships to find the base radius and height, then apply the cone volume formula.
<p><strong>Step 1:</strong> When the sector is bent into a cone, the sector's radius becomes the slant height: <strong>l = 1</strong></p><p><strong>Step 2:</strong> The arc length of the sector becomes the circumference of the cone's base: <strong>Arc length = α·r_sector = α·1 = α</strong></p><p>So: <strong>2πr = α</strong>, where r is the base radius of the cone</p><p>Therefore: <strong>r = α/(2π)</strong></p><p><strong>Step 3:</strong> For α = π: <strong>r = π/(2π) = 1/2</strong></p><p><strong>Step 4:</strong> Find height using Pythagorean theorem: <strong>h² + r² = l²</strong></p><p><strong>h² + (1/2)² = 1²</strong></p><p><strong>h² = 1 - 1/4 = 3/4</strong></p><p><strong>h = √3/2</strong></p><p><strong>Step 5:</strong> Volume of cone: <strong>V = (1/3)πr²h = (1/3)π(1/4)(√3/2)</strong></p><p><strong>V = (1/3)π · (√3/8) = π√3/24</strong></p><p>∴ <strong>Answer: C (or equivalent form)</strong></p>
Correct Answer: C