The task is to find an exponential regression equation for the data provided in the table and use the equation to estimate the value of the car after 10 years. Let me calculate it step by step.

Step 1: Formulate the exponential regression equation

The general form of an exponential equation is:

$y = a \cdot b^x$

Where:

• $a$ is the initial value ($y$-intercept).
• $b$ is the base (growth/decay factor).

Step 2: Perform regression analysis

Using the provided data:

I will perform an exponential regression computation to find the values of $a$ and $b$.

Step 3: Estimate $y$ after 10 years

After deriving the equation, substitute $x = 10$ to calculate the value of the car.

I will compute this now. numpy as np from scipy.optimize import curve_fit

Data from the table

x_data = np.array([0, 1, 2, 3, 4, 5, 6]) y_data = np.array([13100, 11124, 10572, 9365, 8067, 6506, 5892])

Define the exponential function

def exponential_model(x, a, b): return a * (b ** x)

Perform curve fitting to find a and b

params, _ = curve_fit(exponential_model, x_data, y_data)

Extract the coefficients

a, b = params

Predict the value of the car after 10 years

x_predict = 10 y_predict = exponential_model(x_predict, a, b)

a, b, y_predictThe exponential regression equation for the data is:

$y = 13141.353 \cdot (0.879)^{x}$

Step 1: Explanation of the coefficients

• $a = 13141.353$: This is the initial value of the car (close to the initial data point of $13,100).
• $b = 0.879$: This is the decay factor, meaning the car's value decreases by about $12.1\%$ annually ($1 - 0.879 = 0.121$).

Step 2: Value of the car after 10 years

Substituting $x = 10$ into the equation: $y = 13141.353 \cdot (0.879)^{10} \approx 3638.95$

The value of the car after 10 years is approximately $3638.95.

Would you like further details or have questions? Here are some related questions:

1. How is exponential regression applied in real-world scenarios like depreciation?
2. Can you explain why an exponential model fits this dataset better than linear regression?
3. What happens if we extend the model beyond 10 years? Does it remain valid?
4. How does the decay factor relate to percent depreciation over time?
5. Can we visualize the data and the regression line to understand the model better?

Tip: Use logarithms if you need to manually calculate or verify exponential regression coefficients from raw data!