Final answer:
The two positive numbers whose sum is 50 and whose product is the largest possible are both 25. By dividing 50 by 2, we obtain two equal numbers, which when multiplied give the maximum product of 625.
Step-by-step explanation:
To find two positive numbers whose sum is 50 and whose product is as large as possible, we can use the mathematical principle that for a given perimeter, a rectangle has the maximum area when it is a square. In this context, think of the sum of the two numbers as the 'perimeter' of the rectangle. Since the sum of the two numbers is fixed at 50, we consider them as sides of a rectangle and now we look for the 'square' condition, where both sides should be equal to have the maximum 'area' (which represents the product).
Therefore, the two numbers we are looking for need to be equal because a square has equal sides. To find the numbers, we divide the total sum by 2, yielding 25. So, both numbers are 25 and their product, 25 x 25, equals 625, which is the largest product possible with their sum being 50.