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Chapter 3: Problem 64

Among all pairs of numbers whose difference is \(24,\) find a pair whose product is as small as possible. What is the minimum product?

Short Answer

Expert verified
The minimum product of pairs of numbers whose difference is 24 is -144, which corresponds to the pair (12, -12)

Step by step solution

01

Understand the Problem

To solve the problem, you need to find two numbers x and y such that their difference is 24, i.e., \( |x - y| = 24 \). Then, you need to find the minimum possible value of their product.
02

Set up the Equation

Because the absolute difference between the two numbers is 24, we can write the equation \( y = x - 24 \) or \( y = x + 24 \) depending on whether x is larger or y larger. Since the problem does not specify the magnitude of x and y, we can assume x to be larger, hence the equation \( y = x - 24 \). Then, we're looking to minimize \( P = xy \)
03

Substitute y in the Product

Replace y in the expression for the product, so \( P = x(x - 24) = x^2 - 24x \)
04

Minimize the Product

The expression \( x^2 - 24x \) is a quadratic function which opens upwards, its minimum point is at \( x = -(-24)/2 = 12 \)
05

Find the Corresponding Value of y

Substitute x = 12 in the equation \( y = x - 24 \), we get \( y = -12 \)
06

Find the Minimum Product

Substitute x = 12 and y = -12 into the equation \( P = xy \), we get minimum product \( P = 12*(-12) = -144 \)

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