To model the value of the car after \( t \) years with an initial cost of $15,000 and an annual depreciation rate of 21%, you can use the following exponential decay function:
\[ y = P \cdot (1 - r)^t \]
Where:
- \( y \) is the value of the car after \( t \) years,
- \( P \) is the initial cost of the car ($15,000),
- \( r \) is the depreciation rate (0.21 for 21%),
- \( t \) is the number of years.
Substituting the values into the formula:
\[ y = 15000 \cdot (1 - 0.21)^t \]
This simplifies to:
\[ y = 15000 \cdot (0.79)^t \]
So, the function that models the value of the car after \( t \) years is:
\[ y = 15000 \cdot (0.79)^t \]