A car loses it value at a rate of 4.5% annually. If a car is purchased for $24,500, which equation can be used to determine the value of the car, V, after 5 years?

Options

A. `V=24,500(0.045)^(5)`
B. `V=24,500(0.55)^(5)`
C. `V=24,500(1.045)^(5)`
D. `V=24,500(0.955)^(5)`

Answer

To determine the value of the car, V, after 5 years, we can use the concept of exponential decay due to depreciation. The car loses value at a rate of 4.5% annually.

### Step-by-Step Solution:

1. **Identify the Initial Value**: The initial purchase price of the car is $24,500.

2. **Determine the Depreciation Rate**: The car loses value at a rate of 4.5% annually. This means that each year, the car retains 100% - 4.5% = 95.5% of its value. In decimal form, this is 0.955.

3. **Set Up the Value After One Year**: After one year, the value of the car, V1, can be calculated as:
\[ V_1 = 24500 \times (1 - 0.045) = 24500 \times 0.955 \]

4. **Calculate Value After Two Years**: For the second year, the value of the car, V2, will be:
\[ V_2 = V_1 \times (1 - 0.045) = 24500 \times 0.955 \times 0.955 = 24500 \times (0.955)^2 \]

5. **Continue This Pattern**: Continuing this pattern, we can express the value of the car after n years. After 3 years:
\[ V_3 = 24500 \times (0.955)^3 \]
After 4 years:
\[ V_4 = 24500 \times (0.955)^4 \]
After 5 years:
\[ V_5 = 24500 \times (0.955)^5 \]

6. **Final Equation**: Thus, the equation to determine the value of the car after 5 years is:
\[ V = 24500 \times (0.955)^5 \]

### Conclusion:
The equation that can be used to determine the value of the car, V, after 5 years is:
\[ V = 24500 \times (0.955)^5 \]