Question
A new car that costs $15,000 depreciates 21% each year. Write a function to model the value of the care after t amount of years. (2 points)
y=()
y=()
Answers
GPT-4o mini
To model the value of a car that depreciates by 21% each year, you can use the formula for exponential decay. The value of the car after \( t \) years can be expressed with the following function:
\[
y = C \cdot (1 - r)^t
\]
Where:
- \( y \) is the value of the car after \( t \) years,
- \( C \) is the initial cost of the car,
- \( r \) is the depreciation rate (in decimal form),
- \( t \) is the number of years.
For this scenario:
- The initial cost \( C \) is $15,000,
- The depreciation rate \( r \) is 21%, which is 0.21 in decimal.
Thus, the function becomes:
\[
y = 15000 \cdot (1 - 0.21)^t
\]
Simplifying the equation:
\[
y = 15000 \cdot (0.79)^t
\]
So, the function to model the value of the car after \( t \) years is:
\[
y(t) = 15000 \cdot (0.79)^t
\]
In shorthand notation, you might write the function as:
```plaintext
y(t) = 15000 * (0.79)^t
```
\[
y = C \cdot (1 - r)^t
\]
Where:
- \( y \) is the value of the car after \( t \) years,
- \( C \) is the initial cost of the car,
- \( r \) is the depreciation rate (in decimal form),
- \( t \) is the number of years.
For this scenario:
- The initial cost \( C \) is $15,000,
- The depreciation rate \( r \) is 21%, which is 0.21 in decimal.
Thus, the function becomes:
\[
y = 15000 \cdot (1 - 0.21)^t
\]
Simplifying the equation:
\[
y = 15000 \cdot (0.79)^t
\]
So, the function to model the value of the car after \( t \) years is:
\[
y(t) = 15000 \cdot (0.79)^t
\]
In shorthand notation, you might write the function as:
```plaintext
y(t) = 15000 * (0.79)^t
```