Problem 59 - College Algebra with Modeling and Visualization

Question

Consumption The following table lists the number of cigarettes in billions consumed in the United States for selected years. $$ \begin{array}{|ccccc} \hline \text { Year } & 1900 & 1940 & 1980 & 2006 \\ \hline \text { Clgarettes } & 3 & 182 & 632 & 371 \\ \hline \end{array} $$ (a) Find the average rate of change during each time period. (b) Interpret the results.

Step-by-Step Solution

Answer

Consumption increased from 1900 to 1980 and decreased after 1980.

Step 1: Understanding the Average Rate of Change

The average rate of change of a function over an interval is a measure of how much the function's output varies per unit change in input. It is calculated as the difference in the values of the function at the two end points of the interval divided by the difference in the input values, i.e., \( \frac{f(b) - f(a)}{b - a} \). Here, we will calculate it for the number of cigarettes consumed, with respect to each time period.

Step 2: Calculate the Rate from 1900 to 1940

For the period from 1900 to 1940: Let \( a = 1900 \), \( b = 1940 \), \( f(a) = 3 \) billion cigarettes, and \( f(b) = 182 \) billion cigarettes.The average rate of change is:\[ \frac{182 - 3}{1940 - 1900} = \frac{179}{40} = 4.475 \]Thus, the average rate is 4.475 billion cigarettes per year.

Step 3: Calculate the Rate from 1940 to 1980

For the period from 1940 to 1980: Let \( a = 1940 \), \( b = 1980 \), \( f(a) = 182 \) billion cigarettes, and \( f(b) = 632 \) billion cigarettes.The average rate of change is:\[ \frac{632 - 182}{1980 - 1940} = \frac{450}{40} = 11.25 \]Thus, the average rate is 11.25 billion cigarettes per year.

Step 4: Calculate the Rate from 1980 to 2006

For the period from 1980 to 2006: Let \( a = 1980 \), \( b = 2006 \), \( f(a) = 632 \) billion cigarettes, and \( f(b) = 371 \) billion cigarettes.The average rate of change is:\[ \frac{371 - 632}{2006 - 1980} = \frac{-261}{26} = -10.038 \]Thus, the average rate is -10.038 billion cigarettes per year.

Step 5: Interpretation of Results

The average rate of change indicates the rate of increase or decrease in cigarette consumption during the specified periods. Between 1900 and 1940, cigarette consumption increased at an average rate of 4.475 billion per year. From 1940 to 1980, the consumption increased more sharply at 11.25 billion per year. However, from 1980 to 2006, consumption decreased at an average rate of 10.038 billion per year. This suggests a significant decline in smoking rates since 1980.

Key Concepts

Cigarette Consumption

Cigarette consumption is the total number of cigarettes smoked by people in a certain area over a specific period. In this exercise, we are looking at how cigarette consumption has changed in the United States over different time spans.
The table provided lists historical data for cigarette consumption over the years: 1900, 1940, 1980, and 2006. This allows us to see the trend of smoking habits over a century in the U.S.

• In 1900, cigarette consumption was low at only 3 billion.
• By 1940, there was a massive increase to 182 billion.
• The peak came in 1980 with 632 billion cigarettes consumed.
• In 2006, a noticeable decline was seen at 371 billion.

These numbers reflect the changing attitudes and regulations regarding smoking, including increased public awareness of health risks and stricter laws.

Function Over an Interval

The function over an interval is a mathematical way to represent how something changes over a set period. Think of it as looking at how one thing moves up or down over time.
When we talk about the 'function' in this context, we are referring to the number of cigarettes consumed. An interval is the years between two data points, such as 1900 to 1940 or 1980 to 2006.

• The interval helps us calculate the average rate of change by comparing the start and end points of two given years.
• Understanding this function over the intervals gives us insights into the trend and pace at which cigarette consumption progressed or regressed across different eras.

Smoking Rates

Smoking rates refer to the frequency or prevalence of smoking habits within a particular group or area, usually measured per year. Analyzing smoking rates over time provides valuable insights into public health trends and the effectiveness of anti-smoking campaigns.
In our example, the smoking rates reflect drastic changes between 1900 and 2006. Here are notable points:

• From 1900 to 1940, smoking rates surged, likely due to increased cigarette production and marketing.
• Between 1940 and 1980, smoking rates skyrocketed, reaching their highest recorded value indicating widespread cultural acceptance.
• However, between 1980 and 2006, the rates saw a downturn, which could hint at successful public awareness campaigns and stricter smoking laws.

By studying these rate changes, we can better understand how societal factors and policies have impacted smoking behaviors over time.

Mathematical Interpretation

Mathematical interpretation involves understanding data through mathematical calculations and conclusions. In this exercise, the focus is the average rate of change in cigarette consumption over specific periods.
Mathematically, this rate is determined by:\[\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}\]where:

• \(f(b)\) and \(f(a)\) are the cigarette consumption values at the beginning and end of an interval.
• \(b\) and \(a\) are the corresponding years.

The calculated average gives us a number, revealing the speed of increase or decrease over the years:

• A positive average rate means consumption increased over that interval.
• A negative average rate, on the other hand, means consumption decreased, reflecting a decline in smoking rates.

Each calculated rate for 1900-1940, 1940-1980, and 1980-2006 illustrates different trends in smoking culture and consumption. By interpreting these numbers, we gain insights into the impact of historical events, regulations, and social attitudes on smoking habits.