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To solve this problem, we need to determine an exponential regression equation for the given data and then use that equation to estimate the car's value after 15 years.

### Step 1: Understanding the exponential regression equation
The general form of an exponential decay model is:

\[
y = a \cdot b^x
\]

Where:
- $ y $ is the value of the car after $ x $ years,
- $ a $ is the initial value (when $ x = 0 $, $ y = a $),
- $ b $ is the decay factor, representing how the value decreases each year,
- $ x $ is the number of years.

From the table, the initial value when $ x = 0 $ is given as $ y = 16300 $, so we know that $ a = 16300 $.

### Step 2: Setting up the regression model
We can use the data points to calculate the decay factor $ b $ by applying exponential regression. I'll compute the regression coefficients (rounded to two decimal places) for the equation $ y = a \cdot b^x $.

### Step 3: Solving for the car's value after 15 years
Once we have the equation, we'll substitute $ x = 15 $ to find the value of the car after 15 years.

Let me compute this regression for you and the final result.The exponential regression equation that models the car's value over time is approximately:

\[
y = 16111.81 \cdot (0.88)^x
\]

Where $ y $ is the value of the car after $ x $ years.

### Step 4: Predicting the car's value after 15 years
Using this equation, the predicted value of the car after 15 years is:

\[
y(15) \approx 16111.81 \cdot (0.88)^{15} \approx 2295.51
\]

Thus, the estimated value of the car after 15 years is **$2295.51**.

Would you like more details or have any questions about the process?

Here are some related questions you can explore:
1. How is the decay factor $ b $ related to the depreciation rate?
2. What would the car's value be after 10 years?
3. Can this exponential model be applied to other assets that depreciate over time?
4. How accurate is this model if depreciation patterns change in reality?
5. How would the equation change if the car's value increased over time?

**Tip**: Exponential decay models are widely used to describe depreciation, population decline, and radioactive decay.

The table shows the value of a car over time that was initially purchased for $16,300. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth. Then, determine the value of the car to the nearest cent after 15 years.

Learn how to find the exponential regression equation for a car's value depreciating over time, starting from $16,300. This guide includes step-by-step calculations for predicting the car's value after 15 years. Suitable for high school students.

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