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Chapter 11: Problem 60

Among all pairs of numbers whose sum is \(20,\) 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 20 and whose product is as large as possible are both 10. The maximum product is 100.

Step by step solution

01

Understanding the Problem

We are given two numbers whose sum is fixed at 20. The task is to maximize the product of these two numbers. We can let one number be \(x\) and the other number will be \(20 - x\), because the sum of the two numbers is 20.
02

Setting up the Equation

The product of the two numbers is \(x\) times \((20 - x)\), or \(20x - x^2\). This is the equation that needs to be maximized.
03

Finding the Maximum

The maximum value of a quadratic equation \(ax^2 + bx + c\) happens at \(x = -b/(2a)\). Applying this to our problem, \(a = -1\), and \(b = 20\). Plugging these into the formula gives \(x = -(-20) / (2*(-1)) = 20/2 = 10\).
04

Verifying the Solution

When \(x = 10\), the other number is also \(20 - 10 = 10\). The product of 10 and 10 is \(10 * 10 = 100\), which would indeed be the maximum product. If \(x\) were to be smaller or larger than 10, the product would decrease as one number increases while the other decreases, which gives a smaller product overall.

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