Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \(= \sqrt{3}\) is</p>
<p>\(\dfrac{4}{3}\sqrt{3}\,\pi\)</p>
<p>\(\dfrac{8}{3}\sqrt{3}\,\pi\)</p>
<p>\(4\pi\)</p>
<p>\(2\pi\)</p>
Step-by-Step Solution
Key Concept: Express cylinder volume in terms of one variable using the sphere constraint (x² + h² = 12 where x is radius, h is half-height), then optimize using calculus to find the maximum.
<p><strong>Step 1:</strong> Set up the constraint. For a cylinder inscribed in a sphere of radius R = √3, if the cylinder has radius r and height 2h, then: r² + h² = 3</p><p><strong>Step 2:</strong> Express volume in terms of one variable. V = πr²(2h) = 2πr²h. Substitute r² = 3 - h²:</p><p>V(h) = 2πh(3 - h²) = 2π(3h - h³)</p><p><strong>Step 3:</strong> Find critical points. dV/dh = 2π(3 - 3h²) = 0</p><p>⟹ h² = 1 ⟹ h = 1 (taking positive value)</p><p><strong>Step 4:</strong> Find corresponding radius. r² = 3 - 1 = 2 ⟹ r = √2</p><p><strong>Step 5:</strong> Verify maximum using second derivative. d²V/dh² = 2π(-6h) = -12π < 0 at h = 1 ✓</p><p><strong>Step 6:</strong> Calculate maximum volume. V = 2πr²h = 2π(2)(1) = <strong>4π</strong></p><p>∴ Answer: C</p>
Correct Answer: C