Problem 60
Question
The following table lists the annual average number of gallons of pure alcohol consumed by each person age 15 and older in the United States for selected years. $$ \begin{array}{rrrrr} \text { Year } & 1940 & 1960 & 1980 & 2000 \\ \hline \text { Alcohol } & 1.56 & 2.07 & 2.76 & 2.18 \end{array} $$ (a) Find the average rate of change during each \(20-\) year period. (b) Interpret the results.
Step-by-Step Solution
Verified Answer
Increasing rates: 1940-1960 (0.0255), 1960-1980 (0.0345); decreasing rate: 1980-2000 (-0.029).
1Step 1: Understand the Formula for Average Rate of Change
The average rate of change of a function over an interval is given by the formula: \( \frac{f(b) - f(a)}{b - a} \), where \( f(b) \) and \( f(a) \) are the values of the function at the endpoints \( b \) and \( a \) of the interval.
2Step 2: Calculate Each 20-Year Period Rate of Change
We need to calculate the rate of change for each period: 1940 to 1960, 1960 to 1980, and 1980 to 2000. For 1940 to 1960, use: \(\frac{2.07 - 1.56}{1960 - 1940} = \frac{0.51}{20} = 0.0255 \) gallons per year.For 1960 to 1980, use:\( \frac{2.76 - 2.07}{1980 - 1960} = \frac{0.69}{20} = 0.0345 \) gallons per year.For 1980 to 2000, use:\( \frac{2.18 - 2.76}{2000 - 1980} = \frac{-0.58}{20} = -0.029 \) gallons per year.
3Step 3: Interpret the Results
During 1940-1960, there was an increase in consumption at an average rate of 0.0255 gallons per year. The period 1960-1980 saw a larger increase with an average rate of 0.0345 gallons per year. Finally, 1980-2000 experienced a decrease in consumption at an average rate of -0.029 gallons per year.
Key Concepts
Alcohol ConsumptionFunction InterpretationRate Calculation
Alcohol Consumption
Alcohol consumption data provides insights into societal habits over time. In this exercise, the consumption is measured in gallons of pure alcohol per person aged 15 and older in the United States for four selected years: 1940, 1960, 1980, and 2000. By examining these figures, we can understand not just how much alcohol was consumed, but also how trends in drinking behavior have changed. This could be influenced by a variety of factors such as economic changes, legislation, or shifts in cultural attitudes towards alcohol.
Consider the data from 1940 where 1.56 gallons were consumed on average, compared to 2.76 gallons in 1980. This indicates a growing trend in alcohol consumption over these decades. However, by 2000, a decline to 2.18 gallons suggests a reversing trend, possibly hinting at health campaigns or other factors leading to reduced alcohol use. Understanding these figures helps policymakers assess the effectiveness of their alcohol control strategies.
Consider the data from 1940 where 1.56 gallons were consumed on average, compared to 2.76 gallons in 1980. This indicates a growing trend in alcohol consumption over these decades. However, by 2000, a decline to 2.18 gallons suggests a reversing trend, possibly hinting at health campaigns or other factors leading to reduced alcohol use. Understanding these figures helps policymakers assess the effectiveness of their alcohol control strategies.
Function Interpretation
Function interpretation involves understanding what numerical data and calculations tell us about real-world situations. In our case, we have a table mapping years to average gallons of alcohol consumed per person. This is essentially a function relating time (year) to alcohol consumption.
When we interpret this function, we look at how the output (alcohol consumed) changes as the input (year) changes. We can observe increasing numbers until 1980, followed by a decrease by 2000. This interpretation helps us determine trends—whether consumption is increasing, decreasing, or stable over specific periods. Additionally, it answers questions such as: Are people drinking more over time? Which decades saw the biggest change in behavior? Such function analysis is crucial as it turns mere numbers into narratives that explain how society evolves over time.
When we interpret this function, we look at how the output (alcohol consumed) changes as the input (year) changes. We can observe increasing numbers until 1980, followed by a decrease by 2000. This interpretation helps us determine trends—whether consumption is increasing, decreasing, or stable over specific periods. Additionally, it answers questions such as: Are people drinking more over time? Which decades saw the biggest change in behavior? Such function analysis is crucial as it turns mere numbers into narratives that explain how society evolves over time.
Rate Calculation
The rate of change is a crucial concept when analyzing trends over time. It tells us how quickly or slowly something is happening. The formula we use for this calculation is: \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \] This helps us find how much the alcohol consumption increased or decreased per year over given intervals. For each 20-year span in our dataset, we can calculate the average rate of change:
- For 1940 to 1960: \( \frac{2.07 - 1.56}{1960 - 1940} = 0.0255 \) gallons per year, indicating an increase in consumption.
- For 1960 to 1980: \( \frac{2.76 - 2.07}{1980 - 1960} = 0.0345 \) gallons per year, showing a larger increase.
- For 1980 to 2000: \( \frac{2.18 - 2.76}{2000 - 1980} = -0.029 \) gallons per year, reflecting a decrease in consumption.
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