In Exercises, use the steps shown to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.

x + y = 105 and xy² is maximized.

(a) Solve x + y = 180 for y.
(b) Substitute the result from part (a) into P = xy, the equation for the variable that is to be maximized.

(c) Find the domain of the function P found in part (b).
(d) Find dP/dx. Solve the equation dP/dx = 0.
(e) Evaluate P at any solutions found in part (d), as well as at the endpoints of the domain found in part (c).
(f) Give the maximum value of P, as well as the two numbers x and y whose product is that value.