Problem 27 - College Algebra

Question

Table 4 gives the population of a town (in thousands) from 2000 to 2008 . What was the average rate of change of population (a) between 2002 and 2004 , and (b) between 2002 and 2006 ? $$\begin{array}{cccccccc}{\text { Year }} & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} & {2006} & {2007} & {2008} \\ {\text { Population (thousands) }} & {87} & {84} & {83} & {80} & {77} & {76} & {78} & {81} & {85}\end{array}$$

Step-by-Step Solution

Answer

(a) -3 thousand/year; (b) -1.25 thousand/year

Step 1: Understanding the Average Rate of Change

The average rate of change of a function over an interval [a, b] is given by \( \frac{f(b) - f(a)}{b - a} \), where \( f \) is the function's value. In this context, the population function represents the number of thousands of people in the town.

Step 2: Identify the Values for 2002 and 2004

From the table, the population in 2002 is 83 thousand and in 2004 is 77 thousand. We will use these values to calculate the average rate of change between 2002 and 2004.

Step 3: Compute Average Rate of Change from 2002 to 2004

Using the formula for the average rate of change: \[ \frac{77 - 83}{2004 - 2002} = \frac{-6}{2} = -3 \text{ thousand people per year}. \]

Step 4: Identify the Values for 2002 and 2006

From the table, the population in 2002 is 83 thousand and in 2006 is 78 thousand. We will use these values to calculate the average rate of change between 2002 and 2006.

Step 5: Compute Average Rate of Change from 2002 to 2006

Using the formula for the average rate of change: \[ \frac{78 - 83}{2006 - 2002} = \frac{-5}{4} = -1.25 \text{ thousand people per year}. \]

Key Concepts

Population Function

The population function in this context is a simple representation of how the population of a town changes over a period of time. Here, it's recorded as the number of thousands of people living in the town for each given year. The table shows the number of people at specific points in time, which you can use to understand how the population grows or declines over the years.

When dealing with a population function, it's important to record these values accurately as they will form the basis for any further calculations, like determining the rate at which the population is changing. In mathematical terms, this function provides us with discrete data points for each year. You can think of the population in each year as a snapshot that reflects social, economic, or environmental changes impacting the town's size.

• Population in 2000: 87 thousand
• Population in 2001: 84 thousand
• Population in 2002: 83 thousand
• Population in 2003: 80 thousand
• Population in 2004: 77 thousand
• Population in 2005: 76 thousand
• Population in 2006: 78 thousand
• Population in 2007: 81 thousand
• Population in 2008: 85 thousand

Interval Analysis

Interval analysis plays a crucial role when determining the average rate of change. It involves selecting specific time frames, or intervals, within which you assess how the population has changed. This allows you to zoom in and understand the nuances of population dynamics between specific years. In this exercise, we focus on two intervals: 2002 to 2004, and 2002 to 2006.

By isolating these intervals, one can see how the population trends differently over shorter versus longer periods. For these intervals:

• 2002 to 2004 shows a decrease in population by 6 thousand, over a period of 2 years.
• 2002 to 2006 also shows a decline, but the total decrease is 5 thousand over 4 years.

This highlights how different spans of time might provide insights into short-term versus long-term population changes. It's also a key technique in statistics and calculus, offering a practical way to understand and interpret changes in data.

Step-by-Step Solution

Breaking down the solution into simple steps makes it easier to follow and understand how you arrive at the answer. The process includes leveraging the formula for average rate of change, \[ \frac{f(b) - f(a)}{b - a} \], where the function \( f \) refers to the population size.

Step 1: Calculating for 2002-2004

• Identify populations: In 2002, it's 83 thousand; in 2004, it's 77 thousand.
• Apply the average change formula: \[ \frac{77 - 83}{2004 - 2002} = \frac{-6}{2} = -3 \]
• Result: The average rate of change is -3 thousand people per year.

Step 2: Calculating for 2002-2006

• Identify populations: In 2002, it's 83 thousand; in 2006, it's 78 thousand.
• Apply the average change formula: \[ \frac{78 - 83}{2006 - 2002} = \frac{-5}{4} = -1.25 \]
• Result: The average rate of change is -1.25 thousand people per year.

Remember, step-by-step solutions ensure that complex problems are simplified into manageable tasks, aiding understanding and clarity.