https://file.aiquickdraw.com/mathbot/file/1726806417013f9xze6or.jpg

Let's analyze the graph and solve the problems based on the provided image.

### General Graph Information:
- The x-axis represents time in hours (0 to 10 hours).
- The y-axis represents the amount of water in the cistern in thousands of gallons (0 to 12 thousand gallons).
- The graph shows three main intervals:
  - From 0 to 4 hours, there is a decline in the water level.
  - From 4 to 6 hours, the water level remains constant.
  - From 6 to 10 hours, the water level increases.

### Let's answer the questions:

#### a. **Average rate of change on the interval [0,4] and [6,10]?**
- **[0, 4]:** The water level changes from 12 thousand gallons to 2 thousand gallons.
  \[
  \text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{2 - 12}{4 - 0} = \frac{-10}{4} = -2.5 \text{ thousand gallons per hour}.
  \]
  This means the water level decreases by 2.5 thousand gallons per hour between 0 and 4 hours.

- **[6, 10]:** The water level changes from 2 thousand gallons to 8 thousand gallons.
  \[
  \text{Average Rate of Change} = \frac{8 - 2}{10 - 6} = \frac{6}{4} = 1.5 \text{ thousand gallons per hour}.
  \]
  This means the water level increases by 1.5 thousand gallons per hour between 6 and 10 hours.

#### Explanation:
- The negative rate on [0, 4] indicates that water is being removed from the tank.
- The positive rate on [6, 10] indicates that water is being added to the tank.

#### b. **Possible explanation for what occurred between 4 and 6 hours?**
Between 4 and 6 hours, the graph shows a flat line (no change in the water level), indicating that no water is being added or removed. This could suggest that:
- The tank remained idle during this period, meaning no activity (filling or draining) took place.

#### c. **Average rate of change on the interval [0,10]?**
From 0 to 10 hours, the water level changes from 12 thousand gallons to 8 thousand gallons.
\[
\text{Average Rate of Change} = \frac{8 - 12}{10 - 0} = \frac{-4}{10} = -0.4 \text{ thousand gallons per hour}.
\]
This means, on average, the water level decreases by 0.4 thousand gallons per hour over the 10-hour period.

#### Does this rate give a good indication of what happens in the cistern?
No, this average rate does not fully describe the behavior of the water level over time. It indicates a slow overall decrease, but it overlooks significant variations in the rate of change:
- The sharp decrease between 0 and 4 hours.
- The constant level between 4 and 6 hours.
- The increase between 6 and 10 hours.

A better analysis would involve looking at the individual intervals rather than just the overall average rate.

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### Do you want more details or explanations on any part?

Here are five follow-up questions to consider:
1. What would the total change in water level be if the pattern repeated after 10 hours?
2. How would the average rate of change differ if the graph continued for another 10 hours with the same behavior?
3. Could the tank be emptied entirely if the draining rate in the first interval [0, 4] continued indefinitely?
4. If the constant period between 4 and 6 hours had been longer, how would it affect the average rate of change from [0, 10

The graph shows the amount of water in a water tank over several hours. 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 to 10 h? Explain.

Analyze the rate of change of water in a cistern over 10 hours. This graph covers multiple intervals with differing slopes, offering insights into positive, negative, and zero rates of change. Key topics include calculating average rates of change using slope formulas. Suitable for Grades 9-11.

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