Let’s work through this step by step.

Key Details:

1. Starting Price of Car: $20,000
2. Constant Rate of Increase: $820 per year
3. Find Price after 3.1 Years: Use the linear equation model.

Linear Function:

The situation is modeled by a linear function where:

• Independent Variable: Time ($t$) in years.
• Dependent Variable: Price ($p$) in dollars.

The equation follows the slope-intercept form:
$p = mt + b$ Where:

• $b = 20,000$ (initial price at $t = 0$)
• $m = 820$ (rate of increase per year)

Thus, the linear equation is: $p = 820t + 20,000$

Part 1: Compute Price After 3.1 Years:

Substitute $t = 3.1$ into the equation: $p = 820(3.1) + 20,000$ $p = 2,542 + 20,000 = 22,542$

The price of the car after 3.1 years is $\mathbf{\$22,542}$.

Part 2: Is a Linear Model Reasonable?

Evaluate whether a linear model is reasonable for this context:

• Linear models assume constant rates of change. While this may be true in the short term, car prices typically do not rise indefinitely at a constant rate. Other factors like market conditions, inflation, or depreciation might affect the price.

Answer: The most reasonable choice is:

• B: The linear model is most likely not reasonable because the price of a new car of the same model never changes regardless of how much time passes.

Final Answers:

1. Correct Option for Variables:
   B: Independent variable is time ($t$), dependent variable is price ($p$). The equation is $p = 820t + 20,000$.

2. Price After 3.1 Years:
   $22,542

3. Reasonableness of Linear Model:
   B: The linear model is most likely not reasonable because the price of a new car typically doesn't increase indefinitely.

Questions for Further Exploration:

1. How would the model change if the price of the car decreased over time instead of increasing?
2. What factors might cause the price of a car to change in a nonlinear manner?
3. Can a quadratic or exponential model better capture real-world pricing trends for cars?
4. How does depreciation differ from the linear increase described here?
5. Could this model apply to other assets that appreciate over time, like real estate?

Tip:

When solving problems involving models, always consider the real-world context to determine if the model assumptions hold true over time.