A right triangle has legs of lengths 15 cm and 20 cm. A rectangle is to be inscribed in the triangle so that one side of the rectangle lies on the hypotenuse and the two adjacent vertices of the rectangle lie on the two legs, respectively.
(a) Introduce variables and express the area A of the rectangle in terms of a single geometric variable.
(b) Find the dimensions of the rectangle that maximize A. Include justification for why the critical point found yields the absolute maximum area within the feasible domain.
(c) Compute the maximum area to the nearest tenth of a square centimeter and compare it to the area of the entire triangle, providing the ratio as a simplified fraction.
(d) Explain, without additional calculation, how the optimal rectangle would change if the triangle’s legs were doubled in length while maintaining the same angle between them.