A new car is purchased for 22400 dollars. The value of the car depreciates at 6.75% per year. What will the value of the car be, to the nearest cent, after 9 years?


To determine the value of the car after 9 years, we can use the formula for exponential depreciation:

\[ V = P \times (1 - r)^t \]

where:
- $ V $ is the value of the car after $ t $ years,
- $ P $ is the initial purchase price of the car,
- $ r $ is the annual depreciation rate (expressed as a decimal),
- $ t $ is the number of years.

Given:
- $ P = 22400 $ dollars,
- $ r = 6.75\% = 0.0675 $,
- $ t = 9 $ years.

Let's plug these values into the formula:

\[ V = 22400 \times (1 - 0.0675)^9 \]

First, calculate $ 1 - 0.0675 $:

\[ 1 - 0.0675 = 0.9325 \]

Next, raise 0.9325 to the 9th power:

\[ 0.9325^9 \approx 0.5337 \]

Finally, multiply this result by the initial price:

\[ V \approx 22400 \times 0.5337 \approx 11955.68 \]

Therefore, the value of the car after 9 years, to the nearest cent, is approximately **$11,955.68**.

Would you like any further details or have any questions?

Here are five related questions to expand on this topic:
1. How would the car's value change if the depreciation rate was 5% instead of 6.75%?
2. What would the car's value be after 12 years at the same depreciation rate?
3. How does the depreciation rate affect the long-term value of assets?
4. Can you derive the formula for exponential depreciation from the compound interest formula?
5. What are the practical implications of vehicle depreciation for car owners and buyers?

**Tip:** Always keep track of how depreciation affects the value of your assets, as this can significantly impact long-term financial planning.

A new car is purchased for 22400 dollars. The value of the car depreciates at 6.75% per year. What will the value of the car be, to the nearest cent, after 9 years?

Discover how to calculate the depreciation of a car valued at 22400 dollars with a 6.75% annual rate over 9 years. This problem explores exponential decay concepts, using the depreciation formula to find the car's value to the nearest cent. Ideal for high school students in Grades 9-12.

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