Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(xy\).

Step 2: Convert the Problem to a Single Variable Problem

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

Step 3: Complete the Square

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

Step 4: Find the minimal value

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

Step 5: Find the corresponding x-value

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

Step 6: Calculate the Product

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