Select The Correct Answer.The Population Of A Community, P ( X P(x P ( X ], Is Modeled By This Exponential Function, Where X X X Represents The Number Of Years Since The Population Started Being Recorded: P ( X ) = 2 , 400 ( 1.025 ) X P(x) = 2,400(1.025)^x P ( X ) = 2 , 400 ( 1.025 ) X What Is The

Introduction

Exponential functions are a crucial concept in mathematics, particularly in modeling real-world phenomena such as population growth. In this article, we will delve into the world of exponential functions and explore how they can be used to model population growth. We will examine a specific exponential function, $p(x) = 2,400(1.025)^x$, where $x$ represents the number of years since the population started being recorded.

What is an Exponential Function?

An exponential function is a mathematical function of the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. The function $f(x)$ represents a quantity that grows or decays exponentially over time. In the case of the population function $p(x) = 2,400(1.025)^x$, the constant $a$ represents the initial population, and the constant $b$ represents the growth rate.

The Population Function

The population function $p(x) = 2,400(1.025)^x$ can be broken down into two parts: the initial population and the growth rate. The initial population is represented by the constant $2,400$, which is the population at $x = 0$. The growth rate is represented by the constant $1.025$, which is the factor by which the population increases each year.

Understanding the Growth Rate

The growth rate of $1.025$ represents a $2.5\%$ increase in the population each year. This means that if the population is $2,400$ at the start, it will increase by $2.5\%$ each year, resulting in a population of $2,400 \times 1.025 = 2,460$ at the end of the first year.

Calculating the Population at a Given Year

To calculate the population at a given year, we can plug in the value of $x$ into the population function. For example, to calculate the population at the end of $10$ years, we can plug in $x = 10$ into the function:

$p(10) = 2,400(1.025)^{10}$

Using a calculator, we can evaluate this expression to find the population at the end of $10$ years:

$p(10) = 2,400(1.025)^{10} \approx 3,509.19$

Solving for the Number of Years

To solve for the number of years it takes for the population to reach a certain value, we can set up an equation using the population function. For example, if we want to find the number of years it takes for the population to reach $5,000$, we can set up the equation:

$2,400(1.025)^x = 5,000$

To solve for $x$, we can use logarithms to isolate the variable. Taking the logarithm of both sides of the equation, we get:

$\log(2,400(1.025)^x) = \log(5,000)$

Using the property of logarithms that allows us to bring the exponent down, we can rewrite the equation as:

$x \log(1.025) = \log(5,000) - \log(2,400)$

Simplifying the equation, we get:

$x = \frac{\log(5,000) - \log(2,400)}{\log(1.025)}$

Using a calculator, we can evaluate this expression to find the number of years it takes for the population to reach $5,000$:

$x \approx \frac{\log(5,000) - \log(2,400)}{\log(1.025)} \approx 14.19$

Conclusion

In conclusion, the population function $p(x) = 2,400(1.025)^x$ is a powerful tool for modeling population growth. By understanding the initial population and the growth rate, we can use this function to calculate the population at a given year or solve for the number of years it takes for the population to reach a certain value. Whether you're a mathematician or a scientist, the exponential function is an essential tool for understanding and analyzing real-world phenomena.

References

• [1] "Exponential Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/exponential-and-logarithmic-functions/exponential-functions/v/exponential-functions.

Additional Resources

• [1] "Population Growth." World Bank, World Bank, www.worldbank.org/en/topic/population-growth.

Frequently Asked Questions

• Q: What is an exponential function? A: An exponential function is a mathematical function of the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable.
• Q: What is the population function $p(x) = 2,400(1.025)^x$? A: The population function $p(x) = 2,400(1.025)^x$ is a model of population growth, where $x$ represents the number of years since the population started being recorded.
• Q: How do I calculate the population at a given year? A: To calculate the population at a given year, you can plug in the value of $x$ into the population function $p(x) = 2,400(1.025)^x$.
• Q: How do I solve for the number of years it takes for the population to reach a certain value? A: To solve for the number of years it takes for the population to reach a certain value, you can set up an equation using the population function and use logarithms to isolate the variable.
  Q&A: Exponential Functions in Population Modeling =====================================================

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $f(x) = ab^x$, where $a$ and $b$ are constants, and $x$ is the variable. Exponential functions are used to model real-world phenomena that exhibit exponential growth or decay.

Q: What is the population function $p(x) = 2,400(1.025)^x$?

A: The population function $p(x) = 2,400(1.025)^x$ is a model of population growth, where $x$ represents the number of years since the population started being recorded. The function represents a population that grows at a rate of $2.5\%$ per year.

Q: How do I calculate the population at a given year?

A: To calculate the population at a given year, you can plug in the value of $x$ into the population function $p(x) = 2,400(1.025)^x$. For example, to calculate the population at the end of $10$ years, you would plug in $x = 10$ into the function.

Q: How do I solve for the number of years it takes for the population to reach a certain value?

A: To solve for the number of years it takes for the population to reach a certain value, you can set up an equation using the population function and use logarithms to isolate the variable. For example, if you want to find the number of years it takes for the population to reach $5,000$, you can set up the equation $2,400(1.025)^x = 5,000$ and solve for $x$.

Q: What is the significance of the growth rate in the population function?

A: The growth rate in the population function represents the rate at which the population increases each year. In the case of the population function $p(x) = 2,400(1.025)^x$, the growth rate is $2.5\%$ per year.

Q: How can I use the population function to make predictions about future population growth?

A: You can use the population function to make predictions about future population growth by plugging in different values of $x$ into the function. For example, you can use the function to predict the population at the end of $20$ years, $30$ years, or $50$ years.

Q: What are some real-world applications of exponential functions in population modeling?

A: Exponential functions are used in a variety of real-world applications, including:

• Modeling population growth in cities and countries
• Predicting the spread of diseases
• Analyzing the impact of environmental factors on population growth
• Developing policies to manage population growth and resource allocation

Q: How can I learn more about exponential functions and population modeling?

A: There are many resources available to learn more about exponential functions and population modeling, including:

• Online courses and tutorials
• Books and textbooks
• Research papers and articles
• Professional organizations and conferences

Q: What are some common mistakes to avoid when working with exponential functions in population modeling?

A: Some common mistakes to avoid when working with exponential functions in population modeling include:

• Failing to account for non-linear growth or decay
• Using incorrect or incomplete data
• Failing to consider the impact of external factors on population growth
• Not using logarithms to solve for the number of years it takes for the population to reach a certain value

Q: How can I apply exponential functions in population modeling to real-world problems?

A: You can apply exponential functions in population modeling to real-world problems by:

• Using the population function to make predictions about future population growth
• Analyzing the impact of external factors on population growth
• Developing policies to manage population growth and resource allocation
• Using logarithms to solve for the number of years it takes for the population to reach a certain value

Conclusion

In conclusion, exponential functions are a powerful tool for modeling population growth and making predictions about future population trends. By understanding the population function $p(x) = 2,400(1.025)^x$ and how to apply it to real-world problems, you can gain a deeper understanding of the complex relationships between population growth, resource allocation, and environmental factors.