Certainly! To determine the value of the car after 3 years, given that it depreciates by 5% each year, we can follow these steps:
### Step-by-Step Solution
1. Understand Depreciation: Depreciation means the car loses value each year. A 5% depreciation rate implies the car retains 95% (100% - 5%) of its value each year.
2. Write the Exponential Function:
- In general, the formula for depreciation can be written as:
\(\[ V(t) = P \times (1 - r)^t \]\)
where:
- \(\(V(t)\)\) is the value of the car after \(\(t\)\) years,
- \(\(P\)\) is the initial principal value (purchase price),
- \(\(r\)\) is the annual depreciation rate,
- \(\(t\)\) is the number of years.
- For this problem:
\(\[
V(t) = 15000 \times (1 - 0.05)^t
\]\)
Which simplifies to:
\(\[
V(t) = 15000 \times (0.95)^t
\]\)
3. Calculate the Car's Value After 3 Years:
- Plug in \(\(t = 3\)\) into the function:
\(\[
V(3) = 15000 \times (0.95)^3
\]\)
4. Perform the Calculation:
- Compute \(\((0.95)^3\)\):
\(\[
(0.95)^3 = 0.857375
\]\)
- Multiply by the principal value:
\(\[
V(3) = 15000 \times 0.857375 = 12860.625
\]\)
So, the value of the car 3 years later is $12,860.63 (rounded to two decimal places).
