Answer:
x = 24 and y = 24
Explanation:
Let's use algebra to solve this optimization problem.
Let x be the first number, and y be the second number. Then we have the following two equations based on the problem statement:
x + y = 48 (sum of the two numbers is 48)
xy = ? (product of the two numbers, which we want to maximize)
To solve for x and y in terms of each other, we can use the fact that:
(x + y)^2 = x^2 + 2xy + y^2
Expanding the left side of the equation gives:
x^2 + 2xy + y^2 = 2304
And substituting xy for its value in terms of x and y gives:
x^2 + 2xy + y^2 = x^2 + 2(48 - x)y + y^2 = 2304
Simplifying this equation gives:
2y^2 - 96y + x^2 - 2304 = 0
To maximize the product xy, we need to maximize the value of xy = x(48 - x) = 48x - x^2. This function is a quadratic that opens downwards, and therefore, its maximum value occurs at the vertex of the parabola, which is located at x = -b/2a = -48/(2*-1) = 24.
Thus, the two positive real numbers that sum up to 48 and their product is a maximum are x = 24 and y = 24.