Final answer:
The pair of numbers whose difference is 58 and whose product is as small as possible is -29 and 29. By setting up an equation based on the conditions provided and using the identity involving squares, it is determined that equal numbers with opposite signs minimize the product.
Step-by-step explanation:
To solve the problem of finding the pair of numbers whose difference is 58 and whose product is as small as possible, we can set up a system of equations based on the given conditions. Let the two numbers be x and y, with x being the larger number. We are given that:
- The difference between the two numbers is 58, so x - y = 58.
- We want to minimize the product xy.
To minimize the product, we use the identity (x - y)2 = x2 - 2xy + y2. Since we know x - y = 58, this identity simplifies to 582 = x2 - 2xy + y2. This further simplifies to 2xy = x2 + y2 - 582. To minimize the product xy, we must minimize x2 + y2 because 582 is a constant. This occurs when x and y are as close to each other as possible. Since their difference is 58, the numbers that are closest to each other are -29 and 29 because their squares will give the minimum sum. Therefore, the pair that minimizes the product is -29 and 29.