A rectangle is to be inscribed in a semicircle of radius 5. What is the maximum possible area of such a rectangle?

Core Concept

Place the semicircle on a coordinate system with center at the origin. The equation of the semicircle is y = √(25 - x²). Let the rectangle have width 2x and height y. The area is A = 2xy = 2x√(25 - x²). To maximize A, we can maximize A² = 4x²(25 - x²) = 100x² - 4x⁴. Taking the derivative: d(A²)/dx = 200x - 16x³. Set to zero: 200x - 16x³ = 0, which gives x(200 - 16x²) = 0. Solutions are x = 0 (minimum) and x² = 200/16 = 25/2, so x = 5/√2. Then y = √(25 - 25/2) = √(25/2) = 5/√2. The maximum area is A = 2(5/√2)(5/√2) = 2(25/2) = 25.