Improve Article Save Article Like Article Given a semicircle of radius r, we have to find the largest rectangle that can be inscribed in the semicircle, with base lying on the diameter. Let r be the radius of the semicircle, x one half of the base of the rectangle, and y the height of the rectangle. We want to maximize the area, A = 2xy. So from the diagram we have, y = √(r^2 – x^2) So, A = 2*x*(√(r^2 – x^2)), or dA/dx = 2*√(r^2 – x^2) -2*x^2/√(r^2 – x^2) Setting this derivative equal to 0 and solving for x, dA/dx = 0 or, 2*√(r^2 – x^2) – 2*x^2/√(r^2 – x^2) = 0 2r^2 – 4x^2 = 0 x = r/√2This is the maximum of the area as, dA/dx > 0 when x > r/√2 and, dA/dx < 0 when x > r/√2 Since y =√(r^2 – x^2) we then have y = r/√2 Thus, the base of the rectangle has length = r/√2 and its height has length √2*r/2. So, Area, A=r^2
OUTPUT : 25Time Complexity: O(1)
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