Question

Find the area of the largest rectangle that can be inscribed in a semicircle of radius 3 ft. (Hint: Let x be the angle formed by the lone from the center of the semicircle through one corner of the rectangle. Express the area of the inscribed rectangle as a function of x. Plot the graph of this function and determine its highest point)

Answer 1

From the figure, the area of the rectangle is length x width → A = 2xy. Remember that this is not the unit circle as the radius is 3. Therefore, sin α = y/3 and cos α = x/3. So A = 2(3sin α)(3cos α) or

A = 18(sin α)(cos α). Graph this and find the maximum using the trace.