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:

(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: $[0, 100]$

(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.

(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 $x = 300$.

(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 $x = 250$

Would you like more details on any part of this solution?

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.