The Dollar Value { V(t) $}$ Of A Certain Car Model That Is { T $}$ Years Old Is Given By The Function:${ V(t) = 27,500(0.78)^t }$Find The Initial Value Of The Car And The Value After 11 Years. Round Your Answers To The

Introduction

In this article, we will explore the concept of depreciation and how it affects the value of a car over time. We will use a mathematical function to model the value of a car as it ages. The function will be used to find the initial value of the car and its value after 11 years.

The Depreciation Function

The dollar value of a certain car model that is $t$ years old is given by the function:

$$v(t) = 27,500(0.78)^t$$

This function represents the value of the car at any given time $t$. The initial value of the car is the value when $t = 0$, and the value after 11 years is the value when $t = 11$.

Finding the Initial Value

To find the initial value of the car, we need to substitute $t = 0$ into the function:

$$v(0) = 27,500(0.78)^0$$

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

$$v(0) = 27,500(1)$$

$$v(0) = 27,500$$

So, the initial value of the car is $27,500.

Finding the Value After 11 Years

To find the value of the car after 11 years, we need to substitute $t = 11$ into the function:

$$v(11) = 27,500(0.78)^{11}$$

Using a calculator, we can evaluate the expression:

$$v(11) = 27,500(0.78)^{11} \approx 13,419.19$$

So, the value of the car after 11 years is approximately $13,419.19.

Discussion

The depreciation function $v(t) = 27,500(0.78)^t$ represents the value of the car at any given time $t$. The function shows that the value of the car decreases exponentially over time. The initial value of the car is $27,500, and the value after 11 years is approximately $13,419.19.

Conclusion

In this article, we used a mathematical function to model the value of a car over time. We found the initial value of the car and its value after 11 years. The function shows that the value of the car decreases exponentially over time, and the initial value is $27,500, while the value after 11 years is approximately $13,419.19.

Mathematical Background

The depreciation function $v(t) = 27,500(0.78)^t$ is an example of an exponential function. Exponential functions have the form $f(x) = ab^x$, where $a$ and $b$ are constants. In this case, $a = 27,500$ and $b = 0.78$. The function $v(t) = 27,500(0.78)^t$ represents the value of the car at any given time $t$.

Properties of Exponential Functions

Exponential functions have several properties that make them useful for modeling real-world phenomena. Some of these properties include:

• Exponential growth: Exponential functions can model exponential growth, where the value of the function increases exponentially over time.
• Exponential decay: Exponential functions can also model exponential decay, where the value of the function decreases exponentially over time.
• Constant rate of change: Exponential functions have a constant rate of change, which means that the value of the function changes at a constant rate over time.

Real-World Applications

Exponential functions have many real-world applications, including:

• Finance: Exponential functions can be used to model the value of investments over time.
• Biology: Exponential functions can be used to model the growth of populations over time.
• Physics: Exponential functions can be used to model the decay of radioactive materials over time.

Conclusion

Introduction

In our previous article, we explored the concept of depreciation and how it affects the value of a car over time. We used a mathematical function to model the value of a car as it ages. In this article, we will answer some frequently asked questions about the depreciation function and its applications.

Q: What is depreciation?

A: Depreciation is the decrease in value of an asset over time. In the case of a car, depreciation occurs as the car ages and its value decreases.

Q: What is the depreciation function?

A: The depreciation function is a mathematical function that models the value of a car over time. The function is given by:

$$v(t) = 27,500(0.78)^t$$

Q: What is the initial value of the car?

A: The initial value of the car is the value when $t = 0$. To find the initial value, we substitute $t = 0$ into the function:

$$v(0) = 27,500(0.78)^0$$

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

$$v(0) = 27,500(1)$$

$$v(0) = 27,500$$

So, the initial value of the car is $27,500.

Q: What is the value of the car after 11 years?

A: To find the value of the car after 11 years, we substitute $t = 11$ into the function:

$$v(11) = 27,500(0.78)^{11}$$

Using a calculator, we can evaluate the expression:

$$v(11) = 27,500(0.78)^{11} \approx 13,419.19$$

So, the value of the car after 11 years is approximately $13,419.19.

Q: How does the depreciation function model the value of a car over time?

A: The depreciation function models the value of a car over time by using an exponential function. The function shows that the value of the car decreases exponentially over time.

Q: What are some real-world applications of the depreciation function?

A: The depreciation function has many real-world applications, including:

• Finance: Exponential functions can be used to model the value of investments over time.
• Biology: Exponential functions can be used to model the growth of populations over time.
• Physics: Exponential functions can be used to model the decay of radioactive materials over time.

Q: How can I use the depreciation function in my own life?

A: You can use the depreciation function to model the value of your own assets over time. For example, if you own a car, you can use the function to model its value over time and make informed decisions about when to sell or trade it in.

Q: What are some common mistakes to avoid when using the depreciation function?

A: Some common mistakes to avoid when using the depreciation function include:

• Not accounting for external factors: The depreciation function assumes that the value of the asset decreases exponentially over time, but in reality, external factors such as market conditions and maintenance costs can affect the value of the asset.
• Not using the correct depreciation rate: The depreciation rate used in the function is an estimate and may not reflect the actual depreciation rate of the asset.
• Not considering the asset's condition: The depreciation function assumes that the asset is in good condition, but in reality, the asset's condition can affect its value.

Conclusion

In this article, we answered some frequently asked questions about the depreciation function and its applications. We hope that this article has provided you with a better understanding of the depreciation function and how it can be used in real-world scenarios.