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. 18,667 B. 15,120 C. 13,608 D. 16,800

Introduction

The function $f(x) = 16,800(0.9)^x$ represents the population of a town $x$ years after it was established. This function is an example of exponential growth, where the population increases at a constant rate over time. In this article, we will explore the concept of exponential growth and use the given function to find the original population of the town.

Understanding Exponential Growth

Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the growth rate is constant, but the actual growth is not. This type of growth is often seen in populations, where the number of individuals increases at a constant rate over time.

The function $f(x) = 16,800(0.9)^x$ represents the population of the town $x$ years after it was established. The base of the exponential function is 0.9, which means that the population decreases by 10% each year. The initial population is 16,800, which is the value of the function when $x = 0$.

Finding the Original Population

To find the original population of the town, we need to find the value of the function when $x = 0$. This is because the original population is the population at the time when the town was established, which is $x = 0$.

We can find the original population by plugging in $x = 0$ into the function:

$f(0) = 16,800(0.9)^0$

Since any number raised to the power of 0 is 1, we can simplify the expression:

$f(0) = 16,800(1)$

$f(0) = 16,800$

Therefore, the original population of the town is 16,800.

Conclusion

In this article, we explored the concept of exponential growth and used the function $f(x) = 16,800(0.9)^x$ to find the original population of a town. We learned that the function represents the population of the town $x$ years after it was established and that the original population is the value of the function when $x = 0$. By plugging in $x = 0$ into the function, we found that the original population of the town is 16,800.

Answer

The correct answer is D. 16,800.

Additional Information

Exponential growth is a common phenomenon in many fields, including biology, economics, and physics. It is often used to model population growth, chemical reactions, and financial investments. The function $f(x) = 16,800(0.9)^x$ is a simple example of exponential growth, but it can be used to model more complex phenomena.

Real-World Applications

Exponential growth has many real-world applications, including:

• Population growth: Exponential growth can be used to model population growth in cities, countries, and the world.
• Chemical reactions: Exponential growth can be used to model chemical reactions, such as the growth of bacteria in a petri dish.
• Financial investments: Exponential growth can be used to model the growth of investments, such as stocks and bonds.
• Epidemiology: Exponential growth can be used to model the spread of diseases, such as the growth of a virus in a population.

Conclusion

Introduction

In our previous article, we explored the concept of exponential growth and used the function $f(x) = 16,800(0.9)^x$ to find the original population of a town. In this article, we will answer some common questions related to the function and its applications.

Q&A

Q: What is the meaning of the base 0.9 in the function $f(x) = 16,800(0.9)^x$?

A: The base 0.9 represents the rate of growth or decay of the population. In this case, the population decreases by 10% each year, which means that the base is 0.9.

Q: What is the significance of the initial population 16,800 in the function $f(x) = 16,800(0.9)^x$?

A: The initial population 16,800 represents the population of the town at the time when it was established, which is $x = 0$. This is the starting point for the population growth or decay.

Q: How can we use the function $f(x) = 16,800(0.9)^x$ to model population growth in real-world scenarios?

A: The function can be used to model population growth in cities, countries, and the world. For example, if we want to model the population growth of a city over a period of 10 years, we can plug in $x = 10$ into the function to get the population at that time.

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

A: Exponential growth has many real-world applications, including:

• Population growth: Exponential growth can be used to model population growth in cities, countries, and the world.
• Chemical reactions: Exponential growth can be used to model chemical reactions, such as the growth of bacteria in a petri dish.
• Financial investments: Exponential growth can be used to model the growth of investments, such as stocks and bonds.
• Epidemiology: Exponential growth can be used to model the spread of diseases, such as the growth of a virus in a population.

Q: How can we use the function $f(x) = 16,800(0.9)^x$ to find the population of a town after a certain number of years?

A: To find the population of a town after a certain number of years, we can plug in the value of $x$ into the function. For example, if we want to find the population of a town after 5 years, we can plug in $x = 5$ into the function to get the population at that time.

Q: What is the difference between exponential growth and linear growth?

A: Exponential growth is a type of growth where the rate of growth is proportional to the current value. Linear growth, on the other hand, is a type of growth where the rate of growth is constant. Exponential growth is often seen in populations, where the number of individuals increases at a constant rate over time.

Q: Can we use the function $f(x) = 16,800(0.9)^x$ to model population decline?

A: Yes, we can use the function to model population decline. If the base is less than 1, the population will decrease over time. In this case, the base is 0.9, which means that the population will decrease by 10% each year.

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 answering some common questions related to the function and its applications, we have gained a deeper understanding of exponential growth and its significance in real-world scenarios.

Additional Resources

For more information on exponential growth and its applications, please refer to the following resources:

• Mathematics textbooks: Exponential growth is a common topic in mathematics textbooks, particularly in the chapters on functions and modeling.
• Online resources: There are many online resources available that provide information on exponential growth, including videos, articles, and interactive simulations.
• Real-world examples: Exponential growth can be seen in many real-world scenarios, including population growth, chemical reactions, and financial investments.