Mia Buys A New Tablet For $1,250. It Depreciates 36% Each Year. Which Exponential Function Models This Situation?
Introduction
Depreciation is a common phenomenon where the value of an asset decreases over time. In this scenario, Mia buys a new tablet for $1,250, and it depreciates by 36% each year. We need to find an exponential function that models this situation. In this article, we will explore how to create an exponential function that represents the depreciation of Mia's tablet.
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. The general form of an exponential function is:
f(x) = ab^x
where:
- a is the initial value (or the value when x = 0)
- b is the growth factor (or the rate of change)
- x is the independent variable (or the input)
Modeling Depreciation with Exponential Functions
To model the depreciation of Mia's tablet, we need to find the initial value (a) and the growth factor (b). The initial value is the value of the tablet when it is new, which is $1,250. The growth factor is the rate of depreciation, which is 36% per year.
Calculating the Growth Factor
The growth factor (b) is calculated as:
b = 1 - (rate of depreciation)
In this case, the rate of depreciation is 36%, so:
b = 1 - 0.36 b = 0.64
Creating the Exponential Function
Now that we have the initial value (a) and the growth factor (b), we can create the exponential function that models the depreciation of Mia's tablet:
f(x) = 1250(0.64)^x
where:
- f(x) is the value of the tablet after x years
- 1250 is the initial value (the value of the tablet when it is new)
- 0.64 is the growth factor (the rate of depreciation)
- x is the number of years
Interpreting the Exponential Function
The exponential function f(x) = 1250(0.64)^x represents the value of Mia's tablet after x years. For example, if x = 1, the value of the tablet is:
f(1) = 1250(0.64)^1 f(1) = 800
This means that after 1 year, the value of the tablet is $800. If x = 2, the value of the tablet is:
f(2) = 1250(0.64)^2 f(2) = 512
This means that after 2 years, the value of the tablet is $512.
Conclusion
In this article, we have created an exponential function that models the depreciation of Mia's tablet. The function is:
f(x) = 1250(0.64)^x
where:
- f(x) is the value of the tablet after x years
- 1250 is the initial value (the value of the tablet when it is new)
- 0.64 is the growth factor (the rate of depreciation)
- x is the number of years
This function can be used to calculate the value of the tablet after any number of years.
Example Use Cases
- Calculating the value of the tablet after 3 years:
f(3) = 1250(0.64)^3 f(3) = 256
The value of the tablet after 3 years is $256.
- Calculating the value of the tablet after 5 years:
f(5) = 1250(0.64)^5 f(5) = 128
The value of the tablet after 5 years is $128.
Real-World Applications
Exponential functions are used in many real-world applications, such as:
- Finance: Exponential functions are used to model the growth or decay of investments, such as stocks or bonds.
- Biology: Exponential functions are used to model the growth or decay of populations, such as bacteria or animals.
- Physics: Exponential functions are used to model the decay of radioactive materials.
Conclusion
Q: What is depreciation?
A: Depreciation is the decrease in value of an asset over time. It is a common phenomenon that affects many types of assets, including cars, electronics, and real estate.
Q: How do exponential functions model depreciation?
A: Exponential functions model depreciation by using a growth factor (b) that is less than 1. This growth factor represents the rate of depreciation, and it is used to calculate the value of the asset after a certain number of years.
Q: What is the formula for an exponential function that models depreciation?
A: The formula for an exponential function that models depreciation is:
f(x) = a(b)^x
where:
- f(x) is the value of the asset after x years
- a is the initial value (the value of the asset when it is new)
- b is the growth factor (the rate of depreciation)
- x is the number of years
Q: How do I calculate the growth factor (b)?
A: To calculate the growth factor (b), you need to subtract the rate of depreciation from 1. For example, if the rate of depreciation is 36%, the growth factor (b) would be:
b = 1 - 0.36 b = 0.64
Q: How do I use the exponential function to calculate the value of the asset after a certain number of years?
A: To use the exponential function to calculate the value of the asset after a certain number of years, you need to plug in the values of a, b, and x into the formula:
f(x) = a(b)^x
For example, if the initial value (a) is $1,250, the growth factor (b) is 0.64, and the number of years (x) is 2, the value of the asset after 2 years would be:
f(2) = 1250(0.64)^2 f(2) = 512
Q: What is the difference between depreciation and amortization?
A: Depreciation and amortization are both methods of accounting for the decrease in value of an asset over time. However, depreciation is used to account for the decrease in value of tangible assets, such as cars and electronics, while amortization is used to account for the decrease in value of intangible assets, such as patents and copyrights.
Q: Can I use exponential functions to model other types of growth or decay?
A: Yes, exponential functions can be used to model other types of growth or decay, such as population growth, radioactive decay, and chemical reactions.
Q: How do I choose the right exponential function to model a particular situation?
A: To choose the right exponential function to model a particular situation, you need to consider the following factors:
- Initial value: What is the initial value of the asset or quantity?
- Growth factor: What is the rate of growth or decay?
- Time period: What is the time period over which the growth or decay occurs?
By considering these factors, you can choose the right exponential function to model a particular situation.
Q: Can I use exponential functions to make predictions about future values?
A: Yes, exponential functions can be used to make predictions about future values. By plugging in the values of a, b, and x into the formula, you can calculate the value of the asset or quantity after a certain number of years.
Q: What are some common applications of exponential functions in real-world scenarios?
A: Exponential functions have many applications in real-world scenarios, including:
- Finance: Exponential functions are used to model the growth or decay of investments, such as stocks or bonds.
- Biology: Exponential functions are used to model the growth or decay of populations, such as bacteria or animals.
- Physics: Exponential functions are used to model the decay of radioactive materials.
- Economics: Exponential functions are used to model the growth or decay of economies, such as GDP or inflation rates.
Conclusion
In conclusion, exponential functions are a powerful tool for modeling real-world phenomena, such as depreciation. By understanding how to create and interpret exponential functions, you can better understand and analyze complex systems.