A Home's Value Increases At An Average Rate Of $5.5 \%$ Each Year. The Current Value Is $\$ 120,000$. What Function Can Be Used To Find The Value Of The Home After $x$ Years? A. $f(x) = 120,000(1.055
Introduction
When it comes to calculating the value of a home over time, understanding the concept of exponential growth is crucial. In this scenario, we are given that a home's value increases at an average rate of 5.5% each year. The current value of the home is $120,000, and we need to find the value of the home after x years. To solve this problem, we can use a function that represents exponential growth.
Exponential Growth Function
The exponential growth function is given by the formula:
f(x) = P(1 + r)^x
where:
- P is the initial value (in this case, $120,000)
- r is the growth rate (in this case, 5.5% or 0.055)
- x is the number of years
Applying the Exponential Growth Function
Now, let's apply the exponential growth function to the given problem. We want to find the value of the home after x years, so we can plug in the values as follows:
f(x) = 120,000(1 + 0.055)^x f(x) = 120,000(1.055)^x
Discussion and Conclusion
In conclusion, the function that can be used to find the value of the home after x years is f(x) = 120,000(1.055)^x. This function represents the exponential growth of the home's value over time, taking into account the 5.5% annual growth rate.
Example Use Case
To find the value of the home after 5 years, we can plug in x = 5 into the function:
f(5) = 120,000(1.055)^5 f(5) = 120,000(1.302) f(5) = 156,240
Therefore, the value of the home after 5 years would be approximately $156,240.
Real-World Applications
The concept of exponential growth is not limited to the value of a home. It can be applied to various real-world scenarios, such as:
- Population growth: The population of a city or country can grow exponentially over time, leading to increased demand for resources and infrastructure.
- Investment growth: The value of an investment can grow exponentially over time, leading to significant returns on investment.
- Compound interest: The interest earned on an investment can grow exponentially over time, leading to significant returns on investment.
Conclusion
Introduction
In our previous article, we discussed how to find the value of a home after x years using the exponential growth function. In this article, we will answer some frequently asked questions related to the problem.
Q: What is the initial value of the home?
A: The initial value of the home is $120,000.
Q: What is the growth rate of the home's value?
A: The growth rate of the home's value is 5.5% per year.
Q: How can I find the value of the home after x years?
A: To find the value of the home after x years, you can use the exponential growth function:
f(x) = 120,000(1.055)^x
Q: What is the formula for exponential growth?
A: The formula for exponential growth is:
f(x) = P(1 + r)^x
where:
- P is the initial value
- r is the growth rate
- x is the number of years
Q: How can I apply the exponential growth function to the problem?
A: To apply the exponential growth function to the problem, you can plug in the values as follows:
f(x) = 120,000(1 + 0.055)^x f(x) = 120,000(1.055)^x
Q: What is the value of the home after 5 years?
A: To find the value of the home after 5 years, you can plug in x = 5 into the function:
f(5) = 120,000(1.055)^5 f(5) = 120,000(1.302) f(5) = 156,240
Therefore, the value of the home after 5 years would be approximately $156,240.
Q: Can I use the exponential growth function to model other real-world scenarios?
A: Yes, the exponential growth function can be used to model other real-world scenarios, such as:
- Population growth: The population of a city or country can grow exponentially over time, leading to increased demand for resources and infrastructure.
- Investment growth: The value of an investment can grow exponentially over time, leading to significant returns on investment.
- Compound interest: The interest earned on an investment can grow exponentially over time, leading to significant returns on investment.
Q: What are some common mistakes to avoid when using the exponential growth function?
A: Some common mistakes to avoid when using the exponential growth function include:
- Not considering the initial value and growth rate correctly
- Not using the correct formula for exponential growth
- Not plugging in the correct values into the function
- Not considering the impact of compounding on the growth rate
Conclusion
In conclusion, the exponential growth function is a powerful tool for modeling and predicting the growth of a quantity over time. By understanding the concept of exponential growth and applying it to real-world scenarios, we can make informed decisions and predictions about the future.