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Let's analyze the problem you uploaded. It consists of several questions about a function \( W(x) \), where \( W(x) \) represents the depth of water in a reservoir measured over time \( x \), with the graph of \( W(x) \) provided.

### Breakdown of the Questions:

1. **(a)** Find the domain and range of the function \( W(x) \).
2. **(b)** Find \( W(100) \) and \( W(200) \) and explain what these values represent.
3. **(c)** Find the values of \( x \) such that \( W(x) = 50 \).
4. **(d)** Determine the intervals on which \( W(x) \) is increasing and decreasing.

Now, let's go through each question:

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### (a) **Domain and Range**:
- **Domain**: The domain refers to the possible values of \( x \), which represents the number of days since the beginning of the year. Based on the graph, the \( x \)-axis runs from \( 0 \leq x \leq 365 \), meaning the function is defined for all values of \( x \) from 0 to 365 days.
  - **Domain**: \( [0, 365] \)

- **Range**: The range represents the possible values of \( W(x) \), which is the depth of water in feet. Based on the graph, the \( y \)-values (depth) range approximately from 0 feet to about 100 feet.
  - **Range**: \( [0, 100] \)

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### (b) **Find \( W(100) \) and \( W(200) \)**:
Looking at the graph:

- **\( W(100) \)**: At \( x = 100 \) days, it appears that the value of \( W(x) \) is around 75 feet.
- **\( W(200) \)**: At \( x = 200 \) days, it seems \( W(x) \) is near 25 feet.

**What do these values represent?**  
These values represent the depth of water in the reservoir on the 100th and 200th days of the year. Specifically:
- \( W(100) \approx 75 \) feet indicates the water level was around 75 feet deep on day 100.
- \( W(200) \approx 25 \) feet means the water level was around 25 feet deep on day 200.

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### (c) **Find the value(s) of \( x \) such that \( W(x) = 50 \)**:
From the graph, we observe the points where \( W(x) = 50 \). These occur approximately at:
- \( x \approx 50 \) days (the water is rising),
- \( x \approx 300 \) days (the water is rising again).

Thus, the values of \( x \) for which \( W(x) = 50 \) are approximately \( x = 50 \) and \( x = 300 \).

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### (d) **Intervals of Increase and Decrease**:
To determine the intervals where \( W(x) \) is increasing or decreasing, we look at the slope of the graph:

- **Increasing**: The graph is increasing on the intervals:
  - From \( x = 0 \) to about \( x = 150 \)
  - From \( x = 250 \) to \( x = 365 \)

- **Decreasing**: The graph is decreasing on the intervals:
  - From \( x = 150 \) to \( x = 250 \)

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### Related Questions:
1. What is the significance of the maximum point on the graph of \( W(x) \)?
2. How would the function \( W(x) \) change if the water reservoir experienced an unusually dry year?
3. Can \( W(x) \) be modeled by a trigonometric function based on its periodic nature?
4. How would you find the derivative of \( W(x) \) to confirm the intervals of increase and decrease?
5. What practical factors could affect the depth of water as described by \( W(x) \)?

**Tip**: When analyzing a graph for increasing and decreasing intervals, always look for where the slope changes sign.

Consider the graph of y = W(x), 0 ≤ x ≤ 365, where W(x) represents the depth of water in a reservoir x days since the beginning of the year. Answer the following questions about W(x): (a) What is the domain and range? (b) Find W(100) and W(200). What do these values represent? (c) Find the value(s) of x such that W(x) = 50. (d) Determine the intervals on which W(x) is increasing and decreasing.

Learn how to analyze a function representing the depth of water in a reservoir. We find the domain, range, specific values like W(100) and W(200), and determine where the function is increasing or decreasing. Suitable for high school math students.

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