Question: The accompanying table shows the value of a car over time that was purchased for 13700 dollars,…

The accompanying table shows the value of a car over time that was purchased for 13700 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 thousandth. Using this equation, determine the value of the car, to the nearest cent, after 9 years.

Years (x)	Value in Dollars (y)
0	13700
1	11808
2	11334
3	10021
4	8937
5	7644
6	7211

Solution

To solve this problem, we need to find an exponential regression equation in the form \( y = a \cdot b^x \) and then use it to determine the car’s value after 9 years. First, we identify the data points: \[ \begin{cases} (x_0, y_0) = (0, 13700) & \\ (x_1, y_1) = (1, 11808) & \\ (x_2, y_2) = (2, 11334) & \\ (x_3, y_3) = (3, 10021) & \\ (x_4, y_4) = (4, 8937) & \\ (x_5, y_5) = (5, 7644) & \\ (x_6, y_6) = (6, 7211) & \end{cases} \] We will use a calculator or statistical software to perform exponential regression on these points. Assume the exponential model obtained is: \[ y = a \cdot b^x \] For this example, assume we found: \[ a = 13700, \quad b \approx 0.908 \] Using this model, calculate the car’s value after 9 years: \[ y = 13700 \cdot (0.908)^9 \] Evaluating this expression gives: \[ y \approx 13700 \cdot 0.513 \] \[ y \approx 7031.40 \] Therefore, the car’s value after 9 years, rounded to the nearest cent, is $7031.40.