The Following Function, $f(x$\], Represents The Value Of A Car That Is Depreciating Over A Number Of Years:$f(x) = 15,000(0.85)^x$Consider The $y = Ab^x$ Form Of An Exponential Equation.Part A: What Is The $a$ Value

Introduction

In the world of finance, depreciation is a crucial concept that affects the value of assets over time. The value of a car is a classic example of depreciation, where its worth decreases as it ages. In this article, we will explore the exponential equation that represents the value of a car depreciating over a number of years. We will focus on the $y = ab^x$ form of an exponential equation and determine the value of the $a$ parameter.

The Exponential Equation

The given function, $f(x) = 15,000(0.85)^x$, represents the value of a car that is depreciating over a number of years. This equation is in the form of $y = ab^x$, where $a$ is the initial value, $b$ is the growth factor, and $x$ is the number of years.

Understanding the Parameters

In the equation $f(x) = 15,000(0.85)^x$, we can identify the parameters as follows:

• Initial Value ($a$): The initial value of the car, which is $15,000.
• Growth Factor ($b$): The growth factor, which is $0.85. This value represents the depreciation rate of the car.
• Number of Years ($x$): The number of years the car has been depreciating.

Determining the $a$ Value

To determine the $a$ value, we need to look at the given equation and identify the initial value. In this case, the initial value is $15,000.

The $a$ Value

The $a$ value is the initial value of the car, which is $15,000.

Conclusion

In conclusion, the $a$ value in the exponential equation $f(x) = 15,000(0.85)^x$ is $15,000. This value represents the initial value of the car, which is the starting point for the depreciation process.

Part B: What is the $b$ value

Introduction

In the previous section, we determined the $a$ value in the exponential equation $f(x) = 15,000(0.85)^x$. In this section, we will focus on determining the $b$ value, which represents the growth factor.

The Growth Factor ($b$)

The growth factor, $b$, is the value that represents the depreciation rate of the car. In the given equation, the growth factor is $0.85.

The $b$ Value

The $b$ value is the growth factor, which is $0.85. This value represents the depreciation rate of the car.

Conclusion

In conclusion, the $b$ value in the exponential equation $f(x) = 15,000(0.85)^x$ is $0.85. This value represents the depreciation rate of the car.

Part C: What is the value of the car after 5 years?

Introduction

In the previous sections, we determined the $a$ and $b$ values in the exponential equation $f(x) = 15,000(0.85)^x$. In this section, we will use these values to determine the value of the car after 5 years.

The Value of the Car after 5 Years

To determine the value of the car after 5 years, we need to substitute $x = 5$ into the equation $f(x) = 15,000(0.85)^x$.

Calculating the Value

$f(5) = 15,000(0.85)^5$

Using a calculator, we can evaluate the expression:

$f(5) = 15,000(0.85)^5$

$f(5) = 15,000(0.4437)$

$f(5) = 6,655.50$

Conclusion

In conclusion, the value of the car after 5 years is $6,655.50.

Conclusion

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Q&A: Frequently Asked Questions

Q: What is the purpose of the exponential equation in this article?

A: The exponential equation, $f(x) = 15,000(0.85)^x$, represents the value of a car that is depreciating over a number of years. It helps us understand how the value of the car changes over time.

Q: What is the initial value ($a$) in the exponential equation?

A: The initial value ($a$) in the exponential equation is $15,000. This represents the starting value of the car.

Q: What is the growth factor ($b$) in the exponential equation?

A: The growth factor ($b$) in the exponential equation is $0.85. This represents the depreciation rate of the car.

Q: How do I calculate the value of the car after a certain number of years?

A: To calculate the value of the car after a certain number of years, you need to substitute the number of years into the equation $f(x) = 15,000(0.85)^x$. For example, to find the value of the car after 5 years, you would substitute $x = 5$ into the equation.

Q: What is the value of the car after 5 years?

A: The value of the car after 5 years is $6,655.50.

Q: How does the exponential equation help me understand depreciation?

A: The exponential equation helps you understand how the value of the car changes over time. It shows that the value of the car decreases exponentially as it ages.

Q: Can I use this equation to calculate the value of any asset that depreciates over time?

A: Yes, you can use this equation to calculate the value of any asset that depreciates over time. However, you need to replace the initial value ($a$) and the growth factor ($b$) with the actual values for the asset you are calculating.

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

A: The exponential equation has many real-world applications, including:

• Calculating the value of assets that depreciate over time, such as cars, electronics, and furniture.
• Modeling population growth and decline.
• Understanding the spread of diseases.
• Predicting the behavior of financial markets.

Conclusion

In conclusion, the exponential equation $f(x) = 15,000(0.85)^x$ is a powerful tool for understanding depreciation. It helps us calculate the value of assets that depreciate over time and provides insights into the behavior of various systems. We hope this article has helped you understand the exponential equation and its applications.

Additional Resources

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

• Exponential Equation Wikipedia Page
• Depreciation Calculator
• Exponential Growth and Decay

We hope this article has been helpful in understanding the exponential equation and its applications. If you have any further questions or need additional assistance, please don't hesitate to contact us.