A Population Of Ten Rabbits Is Expected To Triple Every Year.Let { X $}$ Represent The Number Of Years.Let { F(x) $}$ Represent The Total Population Of Rabbits.Which Equation Models This Exponential Growth?A. [$ F(x) =
Introduction
In this article, we will explore the concept of exponential growth and how it can be modeled using mathematical equations. We will use the example of a population of rabbits that is expected to triple every year to illustrate this concept.
Understanding Exponential Growth
Exponential growth is a type of growth where the rate of growth is proportional to the current size of the population. In other words, the larger the population, the faster it grows. This type of growth is often seen in populations of living organisms, such as bacteria, rabbits, and humans.
The Problem
Let's assume that we have a population of 10 rabbits that is expected to triple every year. We want to find an equation that models this exponential growth. Let { x $}$ represent the number of years, and let { f(x) $}$ represent the total population of rabbits.
Modeling Exponential Growth
To model exponential growth, we can use the equation:
{ f(x) = ab^x $}$
where { a $}$ is the initial population, { b $}$ is the growth factor, and { x $}$ is the number of years.
In this case, the initial population is 10 rabbits, and the growth factor is 3, since the population triples every year. Therefore, the equation that models this exponential growth is:
{ f(x) = 10(3)^x $}$
Simplifying the Equation
We can simplify the equation by using the fact that { 3^x = e^{x\ln(3)} $}$. Therefore, the equation becomes:
{ f(x) = 10e^{x\ln(3)} $}$
Graphing the Equation
To visualize the exponential growth, we can graph the equation using a graphing calculator or a computer program. The graph will show a rapid increase in the population over time.
Conclusion
In this article, we have seen how to model exponential growth using a mathematical equation. We have used the example of a population of rabbits that is expected to triple every year to illustrate this concept. The equation that models this exponential growth is { f(x) = 10e^{x\ln(3)} $}$. This equation can be used to predict the population of rabbits over time.
Exercises
- What is the population of rabbits after 2 years?
- What is the population of rabbits after 5 years?
- What is the growth factor of the population of rabbits?
Answers
- { f(2) = 10e^{2\ln(3)} = 10(3^2) = 90 $}$
- { f(5) = 10e^{5\ln(3)} = 10(3^5) = 2430 $}$
- The growth factor is 3.
Discussion
Exponential growth is a common phenomenon in many fields, including biology, economics, and finance. It is often used to model the growth of populations, the spread of diseases, and the growth of investments. In this article, we have seen how to model exponential growth using a mathematical equation. We have used the example of a population of rabbits that is expected to triple every year to illustrate this concept. The equation that models this exponential growth is { f(x) = 10e^{x\ln(3)} $}$. This equation can be used to predict the population of rabbits over time.
References
- [1] "Exponential Growth" by Khan Academy
- [2] "Exponential Growth and Decay" by Math Is Fun
- [3] "Exponential Growth" by Wolfram MathWorld
A Population of Rabbits: Modeling Exponential Growth - Q&A ===========================================================
Introduction
In our previous article, we explored the concept of exponential growth and how it can be modeled using mathematical equations. We used the example of a population of rabbits that is expected to triple every year to illustrate this concept. In this article, we will answer some frequently asked questions about exponential growth and modeling.
Q&A
Q: What is exponential growth?
A: Exponential growth is a type of growth where the rate of growth is proportional to the current size of the population. In other words, the larger the population, the faster it grows.
Q: What is the equation that models exponential growth?
A: The equation that models exponential growth is { f(x) = ab^x $}$, where { a $}$ is the initial population, { b $}$ is the growth factor, and { x $}$ is the number of years.
Q: What is the growth factor in the equation { f(x) = 10e^{x\ln(3)} $}$?
A: The growth factor in the equation { f(x) = 10e^{x\ln(3)} $}$ is 3.
Q: What is the initial population in the equation { f(x) = 10e^{x\ln(3)} $}$?
A: The initial population in the equation { f(x) = 10e^{x\ln(3)} $}$ is 10.
Q: How can I use the equation { f(x) = 10e^{x\ln(3)} $}$ to predict the population of rabbits over time?
A: To predict the population of rabbits over time, you can plug in different values of { x $}$ into the equation { f(x) = 10e^{x\ln(3)} $}$. For example, to find the population of rabbits after 2 years, you can plug in { x = 2 $}$ into the equation.
Q: What is the population of rabbits after 2 years?
A: To find the population of rabbits after 2 years, you can plug in { x = 2 $}$ into the equation { f(x) = 10e^{x\ln(3)} $}$. This gives you { f(2) = 10e^{2\ln(3)} = 10(3^2) = 90 $}$.
Q: What is the population of rabbits after 5 years?
A: To find the population of rabbits after 5 years, you can plug in { x = 5 $}$ into the equation { f(x) = 10e^{x\ln(3)} $}$. This gives you { f(5) = 10e^{5\ln(3)} = 10(3^5) = 2430 $}$.
Q: What is the growth rate of the population of rabbits?
A: The growth rate of the population of rabbits is 3, since the population triples every year.
Q: How can I use the equation { f(x) = 10e^{x\ln(3)} $}$ to model other types of exponential growth?
A: You can use the equation { f(x) = 10e^{x\ln(3)} $}$ to model other types of exponential growth by changing the values of { a $}$ and { b $}$. For example, if you want to model a population that grows at a rate of 2% per year, you can use the equation { f(x) = 10e^{0.02x} $}$.
Conclusion
In this article, we have answered some frequently asked questions about exponential growth and modeling. We have used the example of a population of rabbits that is expected to triple every year to illustrate the concept of exponential growth. The equation that models this exponential growth is { f(x) = 10e^{x\ln(3)} $}$. This equation can be used to predict the population of rabbits over time.
Exercises
- What is the population of rabbits after 3 years?
- What is the population of rabbits after 10 years?
- What is the growth rate of the population of rabbits?
Answers
- { f(3) = 10e^{3\ln(3)} = 10(3^3) = 270 $}$
- { f(10) = 10e^{10\ln(3)} = 10(3^{10}) = 60,466,176 $}$
- The growth rate is 3.
Discussion
Exponential growth is a common phenomenon in many fields, including biology, economics, and finance. It is often used to model the growth of populations, the spread of diseases, and the growth of investments. In this article, we have seen how to model exponential growth using a mathematical equation. We have used the example of a population of rabbits that is expected to triple every year to illustrate this concept. The equation that models this exponential growth is { f(x) = 10e^{x\ln(3)} $}$. This equation can be used to predict the population of rabbits over time.
References
- [1] "Exponential Growth" by Khan Academy
- [2] "Exponential Growth and Decay" by Math Is Fun
- [3] "Exponential Growth" by Wolfram MathWorld