A right circular cylinder is inscribed in a sphere of radius R. What is the maximum volume of such a cylinder?
Correct: A
Let the cylinder’s radius be r and height be h. From geometry: r² + (h/2)² = R². Volume = πr²h. Express r² = R² - h²/4. Substitute into volume: V = π(R²h - h³/4). Maximize V w.r.t h. Set dV/dh = 0 ⇒ π(R² - 3h²/4) = 0 ⇒ h = (2R√3)/3. Substituting, max volume = π(R²*(2R√3/3) - (2R√3/3)³/4) = 4πR³/3√3.