a rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of semicircle - Maths - Application of Derivatives

Question

a rectangle is inscribed in a semicircle of radius r with one of
its sides on the diameter of semicircle find the dimension of
rectangle so that its area is minimum find the area?

Garima Singhal · Accepted Answer

Let ABCD be the rectangle of length 2x and breadth y that is inscribed in a semicircle of radius r and centre O
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Let ∠BOC = θNow, x = r cos θ; y = r sin θArea of rectangle = length × breadthA = 2x × yA = r2 sin 2θdAdθ = 2r2 cos 2θFor maxima or minima,dAdθ = 0⇒2r2 cos 2θ = 0⇒cos 2θ = cos π2⇒θ = π4Now, d2Adθ2 = -4r2 sin 2θ⇒d2Adθ2θ=π/4  = -4r2 < 0So, area is maximum at θ = π4Now, length = 2x = 2r cos π4 = 2rbreadth = y = r sin θ = r sin π4 = r2 = 2r2

So, we do not get area of the rectangle minimum

a rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of semicircle . find the dimension of rectangle so that its area is minimum. find the area?