Problem 61 - College Algebra

Question

Among all pairs of numbers whose sum is \(16,\) find a pair whose product is as large as possible. What is the maximum product?

Step-by-Step Solution

Answer

The pair of numbers whose sum is 16 and whose product is as large as possible is (8, 8), and their maximum product is 64.

Step 1: Set up a quadratic equation

Let's denote the two numbers as \(x\) and \(16 - x\). The sum of these two numbers is 16, which satisfies the conditions of the problem. Now, their product, which we want to maximize, is \(x(16-x)\), which can be rewritten as \(-x^2 + 16x\).

Step 2: Remind yourself of the vertex form of a parabola

A quadratic equation \(-x^2 + 16x = y\) can be rewritten as \(y = -(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. We can find \(h\) and \(k\) by completing the square.

Step 3: Find the x-coordinate of the vertex (h)

The x-coordinate of the vertex of the parabola \(y = ax^2 + bx + c\) is given by \(h = -b/(2a)\). Here, \(a = -1\) and \(b = 16\), so \(h = -16/(2*(-1)) = 8\). This is one of the two numbers.

Step 4: Find the other number

The other number is \(16 - h = 16 - 8 = 8\).

Step 5: Confirm that these numbers yield the largest possible product

We test our solution by adjusting the numbers slightly and confirming that the product decreases. If \(x\) exceeds 8 slightly, then \(16 - x\) will be slightly less than 8, and vice versa. Thus, 8 and 8 yield the largest possible product under the given conditions.

Step 6: Find the maximum product

The maximum product is \(8 * 8 = 64\).