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Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.

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Let ABCD be the rectangle inscribed in a semicircle of radius 1 unit such that the vertices A and B lie on the diameter. 

Let AB = DC = x and BC = AD = y. 

Let O be the centre of the semicircle. 

Join OC and OD. Then OC = OD = radius = 1. 

Also, AD = BC and m∠A = m∠B = 90°. 

∴ OA = OB

∴ OB = 1/2 AB = x/2

In right angled triangle OBC,

OB2 + BC2 = OC2 

For maximum value of f(x), f'(x) = 0

∴ by the second derivative test, f is maximum when x = 2

Hence, the area of the rectangle is maximum (i.e. rectangle has the largest size) when its length is √2 units and breadth is 1/√2 unit.

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