Match Each Function To The Correct Scenario. Not All Functions Will Be Used.1. The Price Of A Movie Ticket Is Currently $10 But Increases At A Rate Of 2.5% Per Year. Function: $f(x)=10(1.025)^x$2. There Are Currently 15 Students With

Feb 27, 2025 by ADMIN 238 views

Introduction

In this article, we will explore various mathematical functions and match each function to the correct scenario. We will examine two different functions and determine which one best fits each scenario. The functions we will be examining are exponential growth and decay functions, which are commonly used to model real-world phenomena.

Scenario 1: The Price of a Movie Ticket

The price of a movie ticket is currently $10 but increases at a rate of 2.5% per year. We can model this situation using an exponential growth function.

Exponential Growth Function

The exponential growth function is given by:

$f(x) = ab^x$

where $a$ is the initial value, $b$ is the growth rate, and $x$ is the number of years.

In this scenario, the initial value $a$ is $10, and the growth rate $b$ is $1.025$ (since the price increases at a rate of 2.5% per year). Therefore, the exponential growth function for this scenario is:

$f(x) = 10(1.025)^x$

Matching the Function to the Scenario

Based on the function $f(x) = 10(1.025)^x$ , we can conclude that this function best fits the scenario of the price of a movie ticket increasing at a rate of 2.5% per year.

Scenario 2: The Number of Students

There are currently 15 students with a 20% annual increase in the number of students. We can model this situation using an exponential growth function.

Exponential Growth Function

The exponential growth function is given by:

$f(x) = ab^x$

where $a$ is the initial value, $b$ is the growth rate, and $x$ is the number of years.

In this scenario, the initial value $a$ is 15, and the growth rate $b$ is $1.2$ (since the number of students increases at a rate of 20% per year). Therefore, the exponential growth function for this scenario is:

$f(x) = 15(1.2)^x$

Matching the Function to the Scenario

Based on the function $f(x) = 15(1.2)^x$ , we can conclude that this function best fits the scenario of the number of students increasing at a rate of 20% per year.

Scenario 3: The Population of a City

The population of a city is currently 100,000 people and is decreasing at a rate of 5% per year. We can model this situation using an exponential decay function.

Exponential Decay Function

The exponential decay function is given by:

$f(x) = ae^{-bx}$

where $a$ is the initial value, $b$ is the decay rate, and $x$ is the number of years.

In this scenario, the initial value $a$ is 100,000, and the decay rate $b$ is $0.05$ (since the population is decreasing at a rate of 5% per year). Therefore, the exponential decay function for this scenario is:

$f(x) = 100,000e^{-0.05x}$

Matching the Function to the Scenario

Based on the function $f(x) = 100,000e^{-0.05x}$ , we can conclude that this function best fits the scenario of the population of a city decreasing at a rate of 5% per year.

Conclusion

In this article, we have matched each function to the correct scenario. We have examined two different functions and determined which one best fits each scenario. The functions we have examined are exponential growth and decay functions, which are commonly used to model real-world phenomena. By understanding these functions and how to apply them to different scenarios, we can better model and analyze real-world data.

Exponential Growth and Decay Functions

Exponential growth and decay functions are commonly used to model real-world phenomena. These functions are characterized by a rapid increase or decrease in the value of the function as the input variable increases.

Exponential Growth Function

The exponential growth function is given by:

$f(x) = ab^x$

where $a$ is the initial value, $b$ is the growth rate, and $x$ is the number of years.

Exponential Decay Function

The exponential decay function is given by:

$f(x) = ae^{-bx}$

where $a$ is the initial value, $b$ is the decay rate, and $x$ is the number of years.

Real-World Applications

Exponential growth and decay functions have many real-world applications. Some examples include:

Modeling population growth and decline
Modeling the spread of diseases
Modeling the growth of investments
Modeling the decay of radioactive materials

Conclusion

Introduction

In our previous article, we explored exponential growth and decay functions and matched each function to the correct scenario. In this article, we will answer some frequently asked questions about exponential growth and decay functions.

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

A: Exponential growth occurs when a quantity increases at a rate proportional to its current value. Exponential decay occurs when a quantity decreases at a rate proportional to its current value.

Q: How do I determine whether a function is an exponential growth or decay function?

A: To determine whether a function is an exponential growth or decay function, look at the coefficient of the exponential term. If the coefficient is greater than 1, the function is an exponential growth function. If the coefficient is less than 1, the function is an exponential decay function.

Q: What is the formula for an exponential growth function?

A: The formula for an exponential growth function is:

$f(x) = ab^x$

where $a$ is the initial value, $b$ is the growth rate, and $x$ is the number of years.

Q: What is the formula for an exponential decay function?

A: The formula for an exponential decay function is:

$f(x) = ae^{-bx}$

where $a$ is the initial value, $b$ is the decay rate, and $x$ is the number of years.

Q: How do I calculate the growth rate or decay rate of an exponential function?

A: To calculate the growth rate or decay rate of an exponential function, you need to know the initial value and the value of the function at a later time. You can use the formula:

$b = \left(\frac{f(x)}{a}\right)^{\frac{1}{x}}$

where $b$ is the growth rate or decay rate, $f(x)$ is the value of the function at a later time, $a$ is the initial value, and $x$ is the number of years.

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

A: Exponential growth and decay functions have many real-world applications, including:

Modeling population growth and decline
Modeling the spread of diseases
Modeling the growth of investments
Modeling the decay of radioactive materials
Modeling the growth of bacteria and other microorganisms

Q: How do I graph an exponential growth or decay function?

A: To graph an exponential growth or decay function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph by hand.

Q: What are some common mistakes to avoid when working with exponential growth and decay functions?

A: Some common mistakes to avoid when working with exponential growth and decay functions include:

Confusing the growth rate and decay rate
Using the wrong formula for the function
Not checking the units of the variables
Not considering the initial value and the value of the function at a later time

Conclusion

In conclusion, exponential growth and decay functions are powerful tools for modeling real-world phenomena. By understanding these functions and how to apply them to different scenarios, we can better model and analyze real-world data. We hope this Q&A article has been helpful in answering some of your questions about exponential growth and decay functions.