Rectangles beneath a semi circle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 cm and with two vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

Solution Step 1: The equation of semicircle with radius 5 is 2 2 2 x + y = 5 x + y = 25 So y= 25 x 2 Let the length of rectangle be 2x And breadth of the rectangle be y Then area of the rectangle is given by A=2xy Step 2 2 The requirement of the rectangle lies on on the semicircle of y= 25 x That is A=2x( 25 x )2 ;0 x 5 The variable x is restricted between [0,5] at which both points yield a minimum value of 0 area