Have vehicles gotten more fuel-efficient over the years? Between 1960 and 1999 the size and shape of automobiles in the United States has changed almost annually. The amount of fuel consumed by these vehicles has also changed. The following table describes the average fuel consumption per year per passenger car in gallons of gasoline.
| Year, $t$ | 1960 | 1970 | 1980 | 1990 | 1995 | 1997 | 1998 | 1999 |
|---|---|---|---|---|---|---|---|---|
| Gallons Consumed Per Passenger Car | 668 | 760 | 576 | 520 | 530 | 538 | 544 | 552 |
Determine the average rate of change, in gallons of gas per year, from 1960 and 1999 and then determine what the result means in this situation.
A. -0.33 gal/year; the average fuel consumption per year of a passenger car in the U.S. decreased by a third of a gallon per year.
B. 0.33 gal/year; the average fuel consumption per year of a passenger car in the U.S. increased by a third of a gallon per year.
C. 2.97 gal/year; the average fuel consumption per year of a passenger car in the U.S. increased by nearly 3 gal/year.
D. -2.97 gal/year; the average fuel consumption per year of a passenger car in the U.S. decreased by nearly 3 gal/year.
Answer (2)
Calculate the change in fuel consumption: 552 − 668 = − 116 gallons.
Calculate the number of years: 1999 − 1960 = 39 years.
Calculate the average rate of change: 39 − 116 ≈ − 2.97 gallons/year.
The average fuel consumption decreased by approximately 2.97 gallons per year: − 2.97 gal/year .
Explanation
Understanding the Problem We are given a table that shows the average fuel consumption per year per passenger car in the U.S. for various years. We need to determine the average rate of change in fuel consumption from 1960 to 1999 and interpret the result.
Calculating the Change in Fuel Consumption First, we need to calculate the change in fuel consumption between 1960 and 1999. In 1960, the average fuel consumption was 668 gallons, and in 1999, it was 552 gallons. So, the change in fuel consumption is: C han g e = F u e lC o n s u m pt i o n 1999 − F u e lC o n s u m pt i o n 1960 = 552 − 668 = − 116 This means that the fuel consumption decreased by 116 gallons.
Calculating the Number of Years Next, we need to calculate the number of years between 1960 and 1999: Y e a rs = 1999 − 1960 = 39 So, there are 39 years between 1960 and 1999.
Calculating the Average Rate of Change Now, we can calculate the average rate of change in gallons per year: A v er a g e R a t e O f C han g e = Y e a rs C han g e = 39 − 116 ≈ − 2.97 So, the average rate of change is approximately -2.97 gallons per year.
Interpreting the Result The negative sign indicates that the average fuel consumption per year decreased. Therefore, the average fuel consumption per year of a passenger car in the U.S. decreased by approximately 2.97 gallons per year.
Selecting the Correct Option Comparing our result with the given options, we find that option d matches our calculation and interpretation.
Final Answer Therefore, the average rate of change in fuel consumption from 1960 to 1999 is approximately -2.97 gallons per year, which means the average fuel consumption per year of a passenger car in the U.S. decreased by nearly 3 gallons per year.
Examples
Understanding the rate of change in fuel consumption can help in making informed decisions about energy policies and technological advancements in the automotive industry. For instance, if the rate of change is negative, it indicates that vehicles are becoming more fuel-efficient, which can lead to reduced greenhouse gas emissions and lower fuel costs for consumers. This information can be used to promote the development and adoption of more fuel-efficient vehicles and to set targets for reducing fuel consumption in the transportation sector. The calculation of average rate of change is a fundamental concept applicable in various fields, such as economics, environmental science, and engineering, to analyze trends and make predictions based on historical data. For example, if you are analyzing the stock market, you can calculate the average rate of change of a stock's price over a certain period to understand its performance and make investment decisions. Similarly, in environmental science, you can calculate the average rate of change of temperature or pollution levels to assess the impact of climate change or environmental policies.
The average rate of change in fuel consumption from 1960 to 1999 is approximately -2.97 gallons per year, indicating a decrease in fuel consumption. Therefore, the correct choice is option D. This implies that American passenger cars became more fuel-efficient during this period.
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