Question
A car was purchased for \( \$ 16,500 \). Each year since, the resale value has decreased by \( 25 \% \). Let \( t \) be the number of years since the purchase. Let \( y \) be the resale value of the car, in dollars. Write an exponential function showing the relationship between \( y \) and \( t \).
Ask by Kelly Williams. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The resale value \( y \) after \( t \) years is given by the function:
\[
y = 16500 \cdot (0.75)^t
\]
Solution
1. The original purchase price is \( P = 16500 \).
2. Each year, the car retains \( 75\% \) of its value, so it decreases by \( 25\% \). This means the multiplication factor for the resale value each year is \( 0.75 \).
3. After \( t \) years, the exponential function describing the resale value is:
\[
y = 16500 \cdot (0.75)^t
\]
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Simplify this solution The Deep Dive
The exponential function that describes the resale value of the car over time can be expressed as: \[ y = 16500 \times (0.75)^t \] Here, \( 0.75 \) represents the remaining value after a \( 25\% \) depreciation each year, meaning that each year the car retains \( 75\% \) of its value. Now, if you take a deeper look into depreciation models, similar functions are used in various fields besides cars, such as electronics and real estate. This type of exponential decay can help you understand how assets lose value over time, assisting in budgeting and investment strategies.
