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Let's go through each question in Task 2:

### Part (a)
1. **Average Rate of Change on [0,4]**:
   - At \( t = 0 \), the water level is \( 12,000 \) gallons.
   - At \( t = 4 \), the water level is \( 4,000 \) gallons.
   - The average rate of change is calculated as:
     \[
     \frac{\text{Change in water level}}{\text{Change in time}} = \frac{4,000 - 12,000}{4 - 0} = \frac{-8,000}{4} = -2,000 \text{ gallons per hour}
     \]
   - **Interpretation**: This negative rate of change indicates that the water level in the tank is decreasing at a rate of \( 2,000 \) gallons per hour from \( t = 0 \) to \( t = 4 \).

2. **Average Rate of Change on [6,10]**:
   - At \( t = 6 \), the water level is \( 4,000 \) gallons.
   - At \( t = 10 \), the water level is \( 8,000 \) gallons.
   - The average rate of change is:
     \[
     \frac{8,000 - 4,000}{10 - 6} = \frac{4,000}{4} = 1,000 \text{ gallons per hour}
     \]
   - **Interpretation**: This positive rate of change means that the water level is increasing at \( 1,000 \) gallons per hour from \( t = 6 \) to \( t = 10 \).

### Part (b)
- **Explanation for What Occurred Between 4 and 6 Hours**:
   - Between \( t = 4 \) and \( t = 6 \), the water level remains constant at \( 4,000 \) gallons. This suggests that there was no water being added or removed from the tank during this time interval, implying a pause in the inflow or outflow.

### Part (c)
1. **Average Rate of Change on [0,10]**:
   - At \( t = 0 \), the water level is \( 12,000 \) gallons.
   - At \( t = 10 \), the water level is \( 8,000 \) gallons.
   - The average rate of change over this entire interval is:
     \[
     \frac{8,000 - 12,000}{10 - 0} = \frac{-4,000}{10} = -400 \text{ gallons per hour}
     \]
   - **Interpretation**: This negative rate of change indicates that, on average, the water level is decreasing by \( 400 \) gallons per hour from \( t = 0 \) to \( t = 10 \).

2. **Does This Rate of Change Provide a Good Indication of What is Happening?**:
   - Not entirely. Although the average rate is \( -400 \) gallons per hour, this rate does not fully capture the changes happening within each segment of the interval. Specifically, there is a sharp decrease from \( t = 0 \) to \( t = 4 \), a period of no change from \( t = 4 \) to \( t = 6 \), and then an increase from \( t = 6 \) to \( t = 10 \). The average rate gives a general trend but misses these detailed variations.

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Would you like further clarification on any part? Here are some related questions to deepen your understanding:

1. How would you calculate the instantaneous rate of change at a specific time, if possible?
2. What does a constant rate of change imply in real-world scenarios, such as water flow in tanks?
3. How can you interpret zero rates of change in real-world contexts?
4. Could the pattern in this graph suggest a malfunction in a system that controls water levels?
5. How does analyzing intervals of change help in predicting future water usage or patterns?

**Tip:** The average rate of change provides an overview, but examining intervals helps to reveal specific patterns in data trends.

The graph shows the amount of water in a water tank over several hours. Answer the following questions:

a. What is the average rate of change on the interval [0,4] and on the interval [6,10]? What is a possible explanation for what each rate of change indicates?

b. What is a possible explanation for what occurred between 4 and 6 h?

c. What is the average rate of change on the interval [0,10]? What does the rate of change mean? Does this rate of change give a good indication as to what is happening with the water in the cistern from 0 h to 10 h? Explain.

Explore how to calculate and interpret the average rate of change in a water tank over time, using intervals [0,4], [6,10], and [0,10]. Understand what each rate of change represents and the implications of constant water levels.

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