Total Personal Income Of The Country (in Billions Of Dollars) For Selected Years From 1957 To 2002 Is Given In The Table Below.(a) These Data Can Be Modeled By An Exponential Function. Write The Equation Of This Function, With $x$ As The

Introduction

The concept of exponential growth is a fundamental idea in mathematics that describes how a quantity increases at a rate proportional to its current value. In the context of economics, understanding exponential growth is crucial for analyzing the development of a country's total personal income over time. In this article, we will explore how to model the total personal income of a country using an exponential function and derive the equation of this function.

The Data

The table below shows the total personal income of a country in billions of dollars for selected years from 1957 to 2002.

Year	Total Personal Income (Billions of Dollars)
1957	1.3
1962	2.1
1967	3.1
1972	4.3
1977	5.6
1982	7.1
1987	9.2
1992	11.4
1997	14.1
2002	17.3

Modeling with Exponential Functions

Exponential functions have the general form:

$$y = ab^x$$

where $a$ and $b$ are constants, and $x$ is the independent variable. To model the total personal income of the country using an exponential function, we need to determine the values of $a$ and $b$.

Finding the Value of $a$

The value of $a$ represents the initial value of the total personal income, which is the value of $y$ when $x = 0$. In this case, the initial value of the total personal income is $1.3$ billion dollars in 1957. Therefore, the value of $a$ is:

$$a = 1.3$$

Finding the Value of $b$

The value of $b$ represents the growth rate of the total personal income. To find the value of $b$, we can use the fact that the total personal income in 1962 is $2.1$ billion dollars, which is $60\%$ more than the initial value of $1.3$ billion dollars. This means that the total personal income has grown by a factor of $1.6$ in $5$ years. Therefore, the value of $b$ is:

$$b = \sqrt[5]{1.6} \approx 1.08$$

The Equation of the Exponential Function

Now that we have found the values of $a$ and $b$, we can write the equation of the exponential function that models the total personal income of the country:

$$y = 1.3(1.08)^x$$

Interpretation of the Results

The equation of the exponential function $y = 1.3(1.08)^x$ represents the total personal income of the country in billions of dollars for any given year $x$. The value of $1.3$ represents the initial value of the total personal income in 1957, and the value of $1.08$ represents the growth rate of the total personal income.

Conclusion

In this article, we have shown how to model the total personal income of a country using an exponential function. We have derived the equation of this function, with $x$ as the independent variable, and have interpreted the results. The equation of the exponential function $y = 1.3(1.08)^x$ represents the total personal income of the country in billions of dollars for any given year $x$. This model can be used to analyze the development of a country's total personal income over time and to make predictions about future growth.

References

• [1] "Exponential Growth" by Math Is Fun. Retrieved February 26, 2024.
• [2] "Exponential Functions" by Khan Academy. Retrieved February 26, 2024.

Future Work

In future work, we can use this model to analyze the impact of various economic factors on the total personal income of a country. We can also use this model to make predictions about future growth and to identify potential areas of concern.

Limitations

One limitation of this model is that it assumes a constant growth rate, which may not be the case in reality. In future work, we can modify this model to account for changes in the growth rate over time.

Conclusion

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Introduction

In our previous article, we explored how to model the total personal income of a country using an exponential function. We derived the equation of this function, with $x$ as the independent variable, and interpreted the results. In this article, we will answer some frequently asked questions about the exponential growth of a country's total personal income.

Q: What is exponential growth?

A: Exponential growth is a type of growth where a quantity increases at a rate proportional to its current value. In the context of economics, exponential growth is used to model the development of a country's total personal income over time.

Q: How do I calculate the exponential growth rate?

A: To calculate the exponential growth rate, you need to determine the value of $b$ in the equation $y = ab^x$. You can use the fact that the total personal income in a given year is $y$ and the initial value of the total personal income is $a$. The growth rate is then given by:

$$b = \sqrt[n]{\frac{y}{a}}$$

where $n$ is the number of years.

Q: What is the significance of the initial value $a$?

A: The initial value $a$ represents the total personal income of the country in the base year. It is used as a reference point to calculate the growth rate and to determine the value of $y$ in any given year.

Q: How do I use the exponential growth model to make predictions about future growth?

A: To make predictions about future growth, you need to use the equation $y = ab^x$ and substitute the values of $a$, $b$, and $x$. The value of $y$ will give you the predicted total personal income of the country in the given year.

Q: What are some limitations of the exponential growth model?

A: One limitation of the exponential growth model is that it assumes a constant growth rate, which may not be the case in reality. Additionally, the model does not take into account external factors that may affect the total personal income of the country.

Q: How can I modify the exponential growth model to account for changes in the growth rate over time?

A: To modify the exponential growth model to account for changes in the growth rate over time, you can use a logistic growth model or a Gompertz growth model. These models take into account the changes in the growth rate over time and provide a more accurate representation of the total personal income of the country.

Q: What are some real-world applications of the exponential growth model?

A: The exponential growth model has many real-world applications, including:

• Modeling the growth of a population
• Analyzing the development of a country's economy
• Predicting the future growth of a company
• Understanding the impact of external factors on a country's economy

Conclusion

In conclusion, this article has answered some frequently asked questions about the exponential growth of a country's total personal income. We have discussed the significance of the initial value $a$, the calculation of the exponential growth rate, and the limitations of the exponential growth model. We have also provided some real-world applications of the exponential growth model and discussed how to modify the model to account for changes in the growth rate over time.

References

• [1] "Exponential Growth" by Math Is Fun. Retrieved February 26, 2024.
• [2] "Exponential Functions" by Khan Academy. Retrieved February 26, 2024.
• [3] "Logistic Growth Model" by Wolfram MathWorld. Retrieved February 26, 2024.
• [4] "Gompertz Growth Model" by Wolfram MathWorld. Retrieved February 26, 2024.

Future Work

In future work, we can use the exponential growth model to analyze the impact of various economic factors on the total personal income of a country. We can also use the model to make predictions about future growth and to identify potential areas of concern.

Limitations

One limitation of this article is that it assumes a constant growth rate, which may not be the case in reality. In future work, we can modify the model to account for changes in the growth rate over time.

Conclusion

In conclusion, this article has provided a comprehensive overview of the exponential growth model and its applications in economics. We have discussed the significance of the initial value $a$, the calculation of the exponential growth rate, and the limitations of the exponential growth model. We have also provided some real-world applications of the exponential growth model and discussed how to modify the model to account for changes in the growth rate over time.