The Dollar Value { V(t) $}$ Of A Certain Car Model That Is { T $}$ Years Old Is Given By The Following Exponential Function:${ V(t) = 25,900(0.76)^t }$Find The Initial Value Of The Car And The Value After 11 Years.
Introduction
In this article, we will explore the concept of exponential functions and their applications in real-world scenarios. Specifically, we will examine the dollar value of a certain car model that is t years old, given by the exponential function v(t) = 25,900(0.76)^t. We will first analyze the initial value of the car and then determine its value after 11 years.
Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, where one variable is a constant power of the other variable. In the case of the car model, the dollar value v(t) is a function of the number of years t, and the function is given by v(t) = 25,900(0.76)^t.
The Initial Value of the Car
To find the initial value of the car, we need to evaluate the function v(t) at t = 0. This means that we need to substitute t = 0 into the function and simplify.
v(0) = 25,900(0.76)^0 v(0) = 25,900(1) v(0) = 25,900
Therefore, the initial value of the car is $25,900.
The Value of the Car after 11 Years
To find the value of the car after 11 years, we need to evaluate the function v(t) at t = 11. This means that we need to substitute t = 11 into the function and simplify.
Q&A: Understanding the Exponential Function
In this article, we will continue to explore the concept of exponential functions and their applications in real-world scenarios. Specifically, we will examine the dollar value of a certain car model that is t years old, given by the exponential function v(t) = 25,900(0.76)^t. We will answer some frequently asked questions about the exponential function and its applications.
Q: What is an exponential function?
A: An exponential function is a type of mathematical function that describes a relationship between two variables, where one variable is a constant power of the other variable. In the case of the car model, the dollar value v(t) is a function of the number of years t, and the function is given by v(t) = 25,900(0.76)^t.
Q: What is the initial value of the car?
A: The initial value of the car is the value of the car when it is brand new, i.e., when t = 0. To find the initial value of the car, we need to evaluate the function v(t) at t = 0. This means that we need to substitute t = 0 into the function and simplify.
v(0) = 25,900(0.76)^0 v(0) = 25,900(1) v(0) = 25,900
Therefore, the initial value of the car is $25,900.
Q: What is the value of the car after 11 years?
A: To find the value of the car after 11 years, we need to evaluate the function v(t) at t = 11. This means that we need to substitute t = 11 into the function and simplify.
v(11) = 25,900(0.76)^11 v(11) = 25,900(0.076)^11 v(11) = 14,419.19
Therefore, the value of the car after 11 years is approximately $14,419.19.
Q: How does the exponential function change over time?
A: The exponential function changes over time in a predictable way. As the number of years t increases, the value of the function v(t) decreases. This is because the base of the exponential function, 0.76, is less than 1. As a result, the function v(t) approaches 0 as t approaches infinity.
Q: What are some real-world applications of exponential functions?
A: Exponential functions have many real-world applications, including:
- Population growth: Exponential functions can be used to model population growth, where the population increases at a constant rate.
- Compound interest: Exponential functions can be used to calculate compound interest, where the interest rate is applied to the principal amount at regular intervals.
- Radioactive decay: Exponential functions can be used to model radioactive decay, where the amount of radioactive material decreases at a constant rate.
Conclusion
In this article, we have explored the concept of exponential functions and their applications in real-world scenarios. We have examined the dollar value of a certain car model that is t years old, given by the exponential function v(t) = 25,900(0.76)^t. We have answered some frequently asked questions about the exponential function and its applications. We hope that this article has provided a useful introduction to the concept of exponential functions and their applications.
Additional Resources
For more information on exponential functions and their applications, please see the following resources:
- Khan Academy: Exponential Functions
- Mathway: Exponential Functions
- Wolfram Alpha: Exponential Functions
References
- "Exponential Functions" by Khan Academy
- "Exponential Functions" by Mathway
- "Exponential Functions" by Wolfram Alpha