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The given exponential function for the car's value is:

\[
v(t) = 27,500(0.90)^t
\]

Where:
- $ v(t) $ represents the dollar value of the car after $ t $ years.
- $ t $ is the number of years.
- The initial value of the car is when $ t = 0 $.

### Step 1: Finding the Initial Value
The initial value is found by plugging $ t = 0 $ into the function:

\[
v(0) = 27,500(0.90)^0 = 27,500 \times 1 = 27,500
\]

Thus, the initial value is **$27,500**.

### Step 2: Finding the Value After 10 Years
Now, calculate the value after 10 years by substituting $ t = 10 $ into the equation:

\[
v(10) = 27,500(0.90)^{10}
\]

I'll calculate that for you.The value of the car after 10 years is approximately **$9,589** (rounded to the nearest dollar).

### Summary:
- **Initial value**: $27,500
- **Value after 10 years**: $9,589

Would you like any further details or clarifications?

Here are 5 related questions:
1. How does depreciation affect the value of a car over time?
2. What is the significance of the 0.90 factor in the function?
3. How would the car’s value change if the depreciation rate was different?
4. What would the car’s value be after 5 years instead of 10?
5. Can you graph the depreciation of the car's value over time?

**Tip**: Exponential functions are often used to model depreciation because they represent a consistent percentage decrease over time.

The dollar value v(t) of a certain car model that is t years old is given by the following exponential function: v(t) = 27,500(0.90)^t. Find the initial value of the car and the value after 10 years. Round your answers to the nearest dollar as necessary.

Learn how to calculate the initial value of a car and its value after 10 years using an exponential decay model. The problem uses the formula v(t) = 27,500(0.90)^t. Perfect for high school students studying algebra and exponential functions.

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