Question Detail

Let the two positive numbers be x and y.Given that their sum is 105: x + y = 105, so y = 105 - x.Their product P is given by P = xy = x(105 - x) = 105x - x².To find the maximum product, we find the first derivative of P with respect to x and set it to zero:dP/dx = 105 - 2x.Set dP/dx = 0: 105 - 2x = 0 => 2x = 105 => x = 52.5.Then y = 105 - 52.5 = 52.5.The numbers are 52.5 and 52.5, and their product is 52.5 * 52.5 = 2756.25.Among the given integer options, the pair that sums to 105 and has the largest product will be the one closest to (52.5, 52.5):A. 60 * 45 = 2700B. 30 * 75 = 2250C. 50 * 55 = 2750D. 80 * 25 = 2000Option C, (50, 55), gives the maximum product among the choices and is the closest integer pair to the exact solution.