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To solve the problem, we need to fit an **exponential regression equation** to the given data. An exponential regression equation takes the form:

\[
y = ab^x
\]

Where:
- \( y \) is the value of the car.
- \( x \) is the number of years.
- \( a \) is the initial value (when \( x = 0 \)).
- \( b \) is the base that represents the rate of decay or growth.

### Step 1: Identifying known values
From the table, the initial value \( y_0 \) is 16300 when \( x = 0 \). Therefore, \( a = 16300 \).

### Step 2: Using regression to find \( b \)
We will now calculate the value of \( b \) based on the data. To do this properly, I would need to perform a logarithmic transformation or use a statistical software tool to perform exponential regression.

However, based on the data provided:

| Years (x) | Value (y) |
| --------- | --------- |
| 0         | 16300     |
| 1         | 13901     |
| 2         | 12417     |
| 3         | 10865     |
| 4         | 9604      |
| 5         | 8520      |

I will estimate \( b \) and perform the regression step. Let me handle the calculation for you.From the exponential regression calculation, the equation that models the car's value over time is approximately:

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

Using this equation, the predicted value of the car after 15 years is approximately:

\[
y(15) \approx 2295.51 \text{ dollars}
\]

Would you like further details on any part of the process, or is there anything you'd like to clarify?

Here are 5 related questions that expand upon this topic:
1. How is exponential decay different from linear depreciation?
2. What other types of regression can be used for modeling data?
3. How would the model change if the car's value increased instead of decreased?
4. Can exponential regression be used for other types of assets or investments?
5. How do small changes in the base \( b \) affect the long-term predictions?

**Tip**: Always remember to round your coefficients to appropriate significant figures based on the precision of your data.

The accompanying table shows the value of a car over time that was purchased for 16300 dollars, where x is years and y is the value of the car in dollars. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, determine the value of the car, to the nearest cent, after 15 years.

Solve an exponential regression problem where a car's value depreciates over time. The car is purchased for 16300 dollars, and the task is to find the exponential decay equation and predict its value after 15 years. Detailed step-by-step solution provided for students in Grades 10-12.

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