Suppose That The Dollar Value { V(t) $}$ Of A Certain House That Is { T $}$ Years Old Is Given By The Following Exponential Function:${ V(t) = 415,000(1.05)^t }$1. Find The Initial Value Of The House.2. Does The
Introduction
In this article, we will explore the concept of an exponential function and its application in determining the value of a house over time. The value of a house is often influenced by various factors such as its age, location, and condition. In this case, we are given an exponential function that represents the dollar value of a house that is t years old. Our goal is to find the initial value of the house and understand how its value changes over time.
The Exponential Function
The exponential function given is:
This function represents the dollar value of the house at time t, where t is the number of years the house has been standing. The function is in the form of an exponential growth function, where the base is 1.05 and the initial value is 415,000.
Finding the Initial Value
To find the initial value of the house, we need to find the value of v(t) when t = 0. This is because the initial value of the house is its value at the time of its construction, which is when t = 0.
Substituting t = 0 into the function, we get:
Since any number raised to the power of 0 is 1, we can simplify the expression as:
Therefore, the initial value of the house is $415,000.
Understanding the Growth Rate
The growth rate of the house's value is represented by the base of the exponential function, which is 1.05. This means that the value of the house increases by 5% every year. In other words, if the house is worth $100,000 today, it will be worth $105,000 next year.
Calculating the Value at a Given Time
To calculate the value of the house at a given time t, we can substitute the value of t into the function:
For example, if we want to find the value of the house after 10 years, we can substitute t = 10 into the function:
Using a calculator, we can evaluate the expression as:
Therefore, the value of the house after 10 years is approximately $675,511.50.
Conclusion
In this article, we have explored the concept of an exponential function and its application in determining the value of a house over time. We have found the initial value of the house to be $415,000 and have understood how its value changes over time. The growth rate of the house's value is 5% per year, and we have calculated the value of the house at a given time using the exponential function.
Applications of Exponential Functions
Exponential functions have many real-world applications, including:
- Population growth: Exponential functions can be used to model the growth of populations over time.
- Financial modeling: Exponential functions can be used to model the growth of investments over time.
- Science: Exponential functions can be used to model the growth of chemical reactions and other scientific phenomena.
Real-World Example
A real-world example of an exponential function is the growth of a population over time. Suppose a city has a population of 100,000 people and is growing at a rate of 5% per year. We can use an exponential function to model the growth of the population over time:
Using this function, we can calculate the population of the city at a given time t.
Conclusion
Q: What is an exponential function?
A: An exponential function is a 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 house value function, the value of the house (v(t)) is a constant power of the time (t).
Q: What is the base of an exponential function?
A: The base of an exponential function is the constant power to which the variable is raised. In the case of the house value function, the base is 1.05, which means that the value of the house increases by 5% every year.
Q: What is the initial value of the house?
A: The initial value of the house is the value of the house at time t = 0. In the case of the house value function, the initial value is $415,000.
Q: How does the value of the house change over time?
A: The value of the house increases exponentially over time, with a growth rate of 5% per year. This means that the value of the house will double approximately every 14.2 years.
Q: Can I use an exponential function to model other real-world phenomena?
A: Yes, exponential functions can be used to model a wide range of real-world phenomena, including population growth, financial investments, and scientific phenomena.
Q: How do I calculate the value of the house at a given time?
A: To calculate the value of the house at a given time, you can substitute the value of t into the function:
Q: What is the significance of the growth rate in an exponential function?
A: The growth rate in an exponential function represents the rate at which the value of the house increases over time. In the case of the house value function, the growth rate is 5% per year, which means that the value of the house will increase by 5% every year.
Q: Can I use an exponential function to model a decreasing value?
A: Yes, you can use an exponential function to model a decreasing value by using a base less than 1. For example, if the value of the house decreases by 5% every year, you can use the function:
Q: How do I determine the time it takes for the value of the house to reach a certain value?
A: To determine the time it takes for the value of the house to reach a certain value, you can set up an equation using the exponential function and solve for t. For example, if you want to find the time it takes for the value of the house to reach $675,511.50, you can set up the equation:
Solving for t, you get:
Therefore, it will take 10 years for the value of the house to reach $675,511.50.
Conclusion
In this article, we have answered some common questions about exponential functions and their application in determining the value of a house over time. We have explored the concept of an exponential function, its base, and its growth rate, and have provided examples of how to use an exponential function to model real-world phenomena.