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Let's go through the problem based on the table provided and the three sub-questions.

### Part a) 
**Determine the average rate of change in the size of the crowd for each half-hour of the rally.**

The average rate of change is calculated as the change in the number of people divided by the change in time, which is \( \Delta y / \Delta x \).

From the table:
- For the interval from 0.0 to 0.5 hours:
  \[
  \frac{176 - 0}{0.5 - 0} = \frac{176}{0.5} = 352 \text{ people/hour}
  \]
- For the interval from 0.5 to 1.0 hours:
  \[
  \frac{245 - 176}{1.0 - 0.5} = \frac{69}{0.5} = 138 \text{ people/hour}
  \]
- For the interval from 1.0 to 1.5 hours:
  \[
  \frac{388 - 245}{1.5 - 1.0} = \frac{143}{0.5} = 286 \text{ people/hour}
  \]
- For the interval from 1.5 to 2.0 hours:
  \[
  \frac{402 - 388}{2.0 - 1.5} = \frac{14}{0.5} = 28 \text{ people/hour}
  \]
- For the interval from 2.0 to 2.5 hours:
  \[
  \frac{432 - 402}{2.5 - 2.0} = \frac{30}{0.5} = 60 \text{ people/hour}
  \]
- For the interval from 2.5 to 3.0 hours:
  \[
  \frac{415 - 432}{3.0 - 2.5} = \frac{-17}{0.5} = -34 \text{ people/hour}
  \]

### Part b) 
**What do these numbers represent?**

The calculated values in part (a) represent the average rate at which the number of people in the crowd changes over each half-hour interval. Positive values indicate that the crowd is growing, while negative values indicate that the crowd is shrinking during that time period.

### Part c) 
**What do positive and negative rates of change mean in this situation?**

- A **positive rate of change** means that the crowd is increasing during that particular half-hour period. For instance, from 0.0 to 0.5 hours, the crowd is growing at 352 people per hour.
- A **negative rate of change** (such as in the last interval, from 2.5 to 3.0 hours) indicates that the crowd is decreasing, meaning people are leaving the rally at that time.

Would you like further clarification or additional details on any of the calculations?

Here are some related questions to further explore the topic:
1. What is the total increase in the number of people over the 3-hour period?
2. How does the crowd's growth rate change over time? Is it steady or fluctuating?
3. Could we use this data to predict the number of people after 3.5 hours?
4. Why do you think the crowd started to shrink after 2.5 hours?
5. How would you interpret the average rate of change over the entire 3-hour period?

**Tip:** The average rate of change formula is useful in many contexts, especially when analyzing trends or motion over time.

Determine the average rate of change in the size of the crowd for each half-hour of the rally. What do these numbers represent? What do positive and negative rates of change mean in this situation?

Learn how to calculate the average rate of change in crowd size over a 3-hour rally with data provided at each half-hour interval. Explore the meaning of positive and negative rates of change and how they reflect changes in the crowd's size.

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