A right rectangle is constructed beneath the curve y=tanx in the first quadrant by choosing a point (x,tanx) on the curve with 0<x<3π. The rectangle’s lower left corner is fixed at the origin, its upper right corner is (x,tanx), and its sides are parallel to the coordinate axes.
(a) Express the area A(x) of the rectangle as a function of x. State the domain of this function in the context of the problem.
(b) Determine all critical values of A(x) on the open interval (0,3π) and show all supporting work.
(c) Use your result from part (b) to identify the value of x that yields the maximum rectangle area on the closed interval [0,3π]. Justify that your answer indeed gives the absolute maximum area over the specified domain.
(d) Compute the maximum area and provide the dimensions of the corresponding rectangle. Interpret each quantity in the geometric context.