A rectangle is to be placed in a semicircle of radius 2. What can be the largest area of the rectangle and what are its dimensions. Solution Let the length of the rectangle be 2x. When the rectangle is placed in the semicircle we see that: length is 2x width is sqrt [4- x^2] Therefore the area is 2x*sqrt [4- x^2] Now we have a function for the area = f(x) = 2x*sqrt [4- x^2] To find the maximum value for area we need to find the derivative of f(x) and equate it to 0. f.