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Let's break down the problem in the image you uploaded:

### Part a: Is the table linear or nonlinear?

To determine if the table represents a linear or nonlinear function, we need to check if the distance \( d \) changes at a constant rate as time \( t \) increases. In other words, check the rate of change between consecutive points.

- Between \( t = 1 \) and \( t = 3 \), the change in distance is \( 180 - 60 = 120 \) miles, and the change in time is \( 3 - 1 = 2 \) hours. So, the rate of change is \( \frac{120}{2} = 60 \) miles per hour.
  
- Between \( t = 3 \) and \( t = 5 \), the change in distance is \( 310 - 180 = 130 \) miles, and the change in time is \( 5 - 3 = 2 \) hours. The rate of change is \( \frac{130}{2} = 65 \) miles per hour.
  
- Between \( t = 5 \) and \( t = 7 \), the change in distance is \( 450 - 310 = 140 \) miles, and the change in time is \( 7 - 5 = 2 \) hours. The rate of change is \( \frac{140}{2} = 70 \) miles per hour.
  
- Between \( t = 7 \) and \( t = 9 \), the change in distance is \( 540 - 450 = 90 \) miles, and the change in time is \( 9 - 7 = 2 \) hours. The rate of change is \( \frac{90}{2} = 45 \) miles per hour.

Since the rate of change is not constant (it varies from 60 to 65 to 70 to 45 miles per hour), the table represents a **nonlinear function**.

### Part b: Which car is moving faster?

- Car A has a linear function \( d = 50t \), which means its speed is constant at 50 miles per hour.
  
- For Car B, we observe that the speed varies, as seen from the rate of change calculations above. However, to compare the overall speed, let's take an average rate for Car B from \( t = 1 \) to \( t = 9 \):
  
  \[
  \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{540 \text{ miles}}{9 \text{ hours}} = 60 \text{ miles per hour}.
  \]
  
Since the average speed of Car B (60 miles per hour) is greater than the constant speed of Car A (50 miles per hour), **Car B is moving at a faster rate on average**.

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Would you like more details on any specific part of the solution? Here are some further related questions you might consider:

1. How can we find the exact function that represents Car B's movement?
2. What is the definition of a linear function, and how does it differ from nonlinear?
3. Could we compute the instantaneous speed of Car B at specific times?
4. How do different rates of change impact the shape of a graph?
5. How can the average speed over different intervals affect decision-making in real-life scenarios?

**Tip:** To determine if a table represents a linear function, always check if the rate of change (slope) remains constant across the intervals.

The linear function d = 50t represents the distance (in miles) Car A is from a car rental store after t hours. The table shows the distances Car B is from the rental store. Is the table representing a linear or nonlinear function? Which car is moving faster?

Explore the comparison between a linear and nonlinear function through the distance-time relation of two cars. Learn how to calculate the rate of change and determine which car moves faster using average speed. Suitable for Grade 8-10 students.

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