Q18 of 137 Page 19

A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area.


Let the length and breadth of rectangle ABCD be 2x and y respectively


Radius of semicircle = r (given)


In triangle OBA


r2 = x2 + y2 (Pythagoras theorem)


y2 = r2 - x2


…1


Let us say, area of rectangle = A =xy


A = x () (from equation 1)


Condition for maxima and minima is






r2 – x2 = x2


2x2 = r2


x =


Since, x cannot be negative


Hence,



For , < 0


A will be maximum for


From equation 1


y = =


Length of rectangle =


Breadth of rectangle =


Area of rectangle =


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