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The problem is asking for an exponential regression equation based on the given data and for a prediction of the car's value after 13 years. Let's solve this step by step.

### Step 1: Exponential Regression Equation

The general form of an exponential regression equation is:
\[
y = a \cdot b^x
\]
where:
- $ y $ is the value of the car,
- $ x $ is the number of years,
- $ a $ is the initial value of the car (when $ x = 0 $),
- $ b $ is the base representing the decay factor per year.

We are given the following dataThe problem is asking for an exponential regression equation based on the given data and for a prediction of the car's value after 13 years. Let's solve this step by step.

### Step 1: Exponential Regression Equation

The general form of an exponential regression equation is:
\[
y = a \cdot b^x
\]
where:
- $ y $ is the value of the car,
- $ x $ is the number of years,
- $ a $ is the initial value of the car (when $ x = 0 $),
- $ b $ is the base representing the decay factor per year.

We are given the following 
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 18000 \\
1 & 15026 \\
2 & 12702 \\
3 & 10680 \\
4 & 8483 \\
5 & 7082 \\
\hline
\end{array}
\]

We can calculate the exponential regression equation and round the coefficients to the nearest hundredth. Let me calculate it for you.The exponential regression equation based on the given data is:

\[
y = 18091.48 \cdot (0.83)^x
\]

### Prediction for 13 years:
Using this equation, the predicted value of the car after 13 years is approximately **$1605.05**.

- **Blank 1**: $ y = 18091.48 \cdot 0.83^x $
- **Blank 2**: $1605.05

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you derive the formula for exponential decay?
2. What is the significance of the base $ b $ in an exponential equation?
3. Can you calculate the value after 10 years using this model?
4. What would happen to the model if the decay rate were faster (lower $ b $)?
5. How do you interpret the coefficient $ a $ in real-world terms?

**Tip**: In exponential regression, the base $ b $ indicates how much the value changes per unit increase in $ x $. If $ b $ is less than 1, the value decays over time.

The accompanying table shows the value of a car over time that was purchased for 18000 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 13 years.

Learn how to solve an exponential regression problem to determine the value of a car after several years. This problem involves calculating the exponential decay model for a car's value, using data points and predicting future values. The solution includes the exponential equation and a prediction for 13 years.

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