To find the exponential regression equation for the value of the car over time, we’ll start with an exponential model of the form:

$y = a \cdot e^{bx}$

where:

• $y$ is the value of the car,
• $x$ is the time in years,
• $a$ is the initial value of the car,
• $b$ is the decay rate.

Given data points:

• $x = 0, y = 20800$
• $x = 1, y = 17860$
• $x = 2, y = 15800$
• $x = 3, y = 12984$
• $x = 4, y = 11437$
• $x = 5, y = 10566$

Step 1: Determine Exponential Regression Equation

Using statistical methods (or a calculator with regression functions) to find $a$ and $b$:

After performing the calculations, the regression equation for this data set is approximately:

$y = 20800 \cdot e^{-0.134x}$

where $a = 20800$ (the initial value of the car) and $b = -0.134$ (the decay rate).

Step 2: Predict Value After 15 Years

To find the value of the car after 15 years ($x = 15$), we substitute $x = 15$ into the equation:

$y = 20800 \cdot e^{-0.134 \cdot 15}$

Calculating this gives:

$y \approx 20800 \cdot e^{-2.01} \approx 20800 \cdot 0.134 \approx 2787.20$

Final Answers

1. Regression Equation: $y = 20800 \cdot e^{-0.134x}$
2. Value after 15 years: $2787.20

Would you like details on any steps? Here are related questions you might explore:

1. How would the regression equation change if the initial value of the car were different?
2. What factors might affect the rate of depreciation in real-world situations?
3. How would this calculation change if we assumed linear depreciation instead of exponential?
4. How accurate is exponential regression for other assets like property?
5. What are alternative models for asset depreciation?

Tip: Exponential decay models are effective for items losing value at a percentage rate, such as electronics or vehicles.