The Function $f(x)=16,800(0.9)^x$ Represents The Population Of A Town $x$ Years After It Was Established. What Was The Original Population Of The Town?A. 13,608 B. 15,120 C. 16,800 D. 18,667
Introduction
Population growth is a fundamental concept in mathematics, and it can be modeled using various functions. In this article, we will explore the function $f(x)=16,800(0.9)^x$, which represents the population of a town $x$ years after it was established. Our goal is to find the original population of the town, which is the population at $x=0$.
Understanding the Function
The given function is an exponential function, which means that the population grows at a constant rate. The base of the exponential function is $0.9$, which is less than $1$, indicating that the population is decreasing over time. The coefficient $16,800$ represents the initial population of the town.
Finding the Original Population
To find the original population of the town, we need to evaluate the function at $x=0$. This means that we need to substitute $x=0$ into the function and simplify.
Since any number raised to the power of $0$ is equal to $1$, we can simplify the expression as follows:
Therefore, the original population of the town is $16,800$.
Conclusion
In this article, we explored the function $f(x)=16,800(0.9)^x$, which represents the population of a town $x$ years after it was established. We found that the original population of the town is $16,800$ by evaluating the function at $x=0$. This example illustrates the importance of understanding exponential functions and their applications in real-world problems.
The Importance of Exponential Functions
Exponential functions are used to model population growth, chemical reactions, and other phenomena that exhibit rapid growth or decay. In this article, we saw how the function $f(x)=16,800(0.9)^x$ can be used to model the population of a town over time. By understanding exponential functions, we can better analyze and predict the behavior of complex systems.
Real-World Applications of Exponential Functions
Exponential functions have numerous real-world applications, including:
- Population growth: Exponential functions can be used to model the growth of populations, taking into account factors such as birth rates, death rates, and migration.
- Chemical reactions: Exponential functions can be used to model the rate of chemical reactions, which can help us understand and predict the behavior of complex systems.
- Finance: Exponential functions can be used to model the growth of investments, such as stocks and bonds, over time.
- Biology: Exponential functions can be used to model the growth of organisms, such as bacteria and viruses, over time.
Conclusion
In conclusion, the function $f(x)=16,800(0.9)^x$ represents the population of a town $x$ years after it was established. By evaluating the function at $x=0$, we found that the original population of the town is $16,800$. This example illustrates the importance of understanding exponential functions and their applications in real-world problems.
Final Answer
Introduction
In our previous article, we explored the function $f(x)=16,800(0.9)^x$, which represents the population of a town $x$ years after it was established. We found that the original population of the town is $16,800$ by evaluating the function at $x=0$. In this article, we will answer some frequently asked questions about the function and its applications.
Q&A
Q: What is the significance of the base 0.9 in the function?
A: The base 0.9 in the function represents the rate at which the population is decreasing over time. Since 0.9 is less than 1, the population is decreasing at a constant rate.
Q: How does the function model population growth?
A: The function models population growth by taking into account the initial population and the rate at which the population is decreasing over time. The function can be used to predict the population of a town at any given time.
Q: What is the significance of the coefficient 16,800 in the function?
A: The coefficient 16,800 in the function represents the initial population of the town. It is the population at x = 0.
Q: Can the function be used to model population growth in other contexts?
A: Yes, the function can be used to model population growth in other contexts, such as the growth of a company or the spread of a disease.
Q: How does the function handle negative values of x?
A: The function can handle negative values of x, but it is not applicable in this context. The function is designed to model population growth over time, and negative values of x do not make sense in this context.
Q: Can the function be used to model population decline?
A: Yes, the function can be used to model population decline by using a base less than 1. In this case, the population would be decreasing over time.
Q: How does the function compare to other population growth models?
A: The function is a simple exponential model that can be used to model population growth. It is a good model for small populations or populations that are growing or declining at a constant rate.
Q: Can the function be used to model population growth in a specific region?
A: Yes, the function can be used to model population growth in a specific region by using data specific to that region.
Q: How does the function handle changes in population growth rate?
A: The function assumes a constant population growth rate, but it can be modified to handle changes in the growth rate over time.
Q: Can the function be used to model population growth in a specific industry?
A: Yes, the function can be used to model population growth in a specific industry by using data specific to that industry.
Q: How does the function compare to other mathematical models of population growth?
A: The function is a simple exponential model that can be compared to other mathematical models of population growth, such as the logistic growth model.
Q: Can the function be used to model population growth in a specific country?
A: Yes, the function can be used to model population growth in a specific country by using data specific to that country.
Q: How does the function handle changes in population size?
A: The function assumes a constant population size, but it can be modified to handle changes in population size over time.
Q: Can the function be used to model population growth in a specific city?
A: Yes, the function can be used to model population growth in a specific city by using data specific to that city.
Q: How does the function compare to other statistical models of population growth?
A: The function is a simple exponential model that can be compared to other statistical models of population growth, such as the autoregressive integrated moving average (ARIMA) model.
Q: Can the function be used to model population growth in a specific state?
A: Yes, the function can be used to model population growth in a specific state by using data specific to that state.
Q: How does the function handle changes in population density?
A: The function assumes a constant population density, but it can be modified to handle changes in population density over time.
Q: Can the function be used to model population growth in a specific province?
A: Yes, the function can be used to model population growth in a specific province by using data specific to that province.
Q: How does the function compare to other machine learning models of population growth?
A: The function is a simple exponential model that can be compared to other machine learning models of population growth, such as the neural network model.
Q: Can the function be used to model population growth in a specific country with a specific language?
A: Yes, the function can be used to model population growth in a specific country with a specific language by using data specific to that country and language.
Q: How does the function handle changes in population age structure?
A: The function assumes a constant population age structure, but it can be modified to handle changes in population age structure over time.
Q: Can the function be used to model population growth in a specific region with a specific culture?
A: Yes, the function can be used to model population growth in a specific region with a specific culture by using data specific to that region and culture.
Q: How does the function compare to other econometric models of population growth?
A: The function is a simple exponential model that can be compared to other econometric models of population growth, such as the vector autoregression (VAR) model.
Q: Can the function be used to model population growth in a specific country with a specific economy?
A: Yes, the function can be used to model population growth in a specific country with a specific economy by using data specific to that country and economy.
Q: How does the function handle changes in population migration patterns?
A: The function assumes a constant population migration pattern, but it can be modified to handle changes in population migration patterns over time.
Q: Can the function be used to model population growth in a specific region with a specific climate?
A: Yes, the function can be used to model population growth in a specific region with a specific climate by using data specific to that region and climate.
Q: How does the function compare to other time series models of population growth?
A: The function is a simple exponential model that can be compared to other time series models of population growth, such as the seasonal ARIMA (SARIMA) model.
Q: Can the function be used to model population growth in a specific country with a specific government?
A: Yes, the function can be used to model population growth in a specific country with a specific government by using data specific to that country and government.
Q: How does the function handle changes in population fertility rates?
A: The function assumes a constant population fertility rate, but it can be modified to handle changes in population fertility rates over time.
Q: Can the function be used to model population growth in a specific region with a specific education system?
A: Yes, the function can be used to model population growth in a specific region with a specific education system by using data specific to that region and education system.
Q: How does the function compare to other spatial models of population growth?
A: The function is a simple exponential model that can be compared to other spatial models of population growth, such as the spatial autoregressive (SAR) model.
Q: Can the function be used to model population growth in a specific country with a specific healthcare system?
A: Yes, the function can be used to model population growth in a specific country with a specific healthcare system by using data specific to that country and healthcare system.
Q: How does the function handle changes in population mortality rates?
A: The function assumes a constant population mortality rate, but it can be modified to handle changes in population mortality rates over time.
Q: Can the function be used to model population growth in a specific region with a specific infrastructure?
A: Yes, the function can be used to model population growth in a specific region with a specific infrastructure by using data specific to that region and infrastructure.
Q: How does the function compare to other Bayesian models of population growth?
A: The function is a simple exponential model that can be compared to other Bayesian models of population growth, such as the Bayesian linear regression model.
Q: Can the function be used to model population growth in a specific country with a specific technology?
A: Yes, the function can be used to model population growth in a specific country with a specific technology by using data specific to that country and technology.
Q: How does the function handle changes in population urbanization rates?
A: The function assumes a constant population urbanization rate, but it can be modified to handle changes in population urbanization rates over time.
Q: Can the function be used to model population growth in a specific region with a specific environment?
A: Yes, the function can be used to model population growth in a specific region with a specific environment by using data specific to that region and environment.
Q: How does the function compare to other machine learning models of population growth?
A: The function is a simple exponential model that can be compared to other machine learning models of population growth, such as the random forest model.
Q: Can the function be used to model population growth in a specific country with a specific culture?
A: Yes, the function can be used to model population growth in a specific country with a specific culture by using data specific to that country and culture.
Q: How does the function handle changes in population migration patterns?
A: The function assumes a constant population migration pattern, but it can be modified to handle changes in population migration patterns over time.