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Chapter 3: Problem 63

Among all pairs of numbers whose difference is \(16,\) find a pair whose product is as small as possible. What is the minimum product?

Short Answer

Expert verified
The minimum product is -64 when one number is 8 and the other is -8.

Step by step solution

01

Formulate the Problem Using Variables

Let the two numbers be \(x\) and \(y\). So, \(x - y = 16\). We need to find the minimum value of \(xy\).
02

Convert the Problem to a Single Variable Problem

From the first step, we know that \(x = y + 16\). Substitute \(x\) in \(xy\) as \(y (y + 16)\).
03

Complete the Square

Expression \(y^2 + 16y\) can be written as \((y + 8)^2 - 64\), by completing the square.
04

Find the minimal value

To get the minimum value, set \(y + 8 = 0\), then \(y = -8\).
05

Find the corresponding x-value

The corresponding x value is then \(x = y+16 = -8+16 = 8\).
06

Calculate the Product

Plug the x and y values into the product xy equation to get the minimal product, which is \(xy = 8 * -8 = -64\).

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Most popular questions from this chapter

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Use $$ \frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2 $$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely

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I graphed \(f(x)=(x+2)^{3}(x-4)^{2},\) and the graph touched the \(x\)-axis and turned around at \(-2\)

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