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Chapter 3: Problem 61

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

Short Answer

Expert verified
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 by step solution

01

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 \(-x^2 + 16x\).
02

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.
03

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.
04

Find the other number

The other number is \(16 - h = 16 - 8 = 8\).
05

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.
06

Find the maximum product

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

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