Let's go through each part of the problem.

Problem Overview:

You are given a graph of a function $W(x)$, where $W(x)$ represents the depth of water in a reservoir $x$ days since the beginning of the year. The graph spans the interval $0 \leq x \leq 365$.

(a) [2 marks] Domain and Range

• Domain: This is the set of possible $x$-values. From the graph, $x$ represents the number of days since the beginning of the year, and it is defined from $0$ to $365$.

  • Domain: $[0, 365]$.

• Range: This is the set of possible $W(x)$-values, which represent the water depth. From the graph, the depth $W(x)$ seems to vary between approximately $20 \, \text{ft}$ and $100 \, \text{ft}$.

  • Range: Approximately $[20, 100] \, \text{ft}$.

(b) [3 marks] Find $W(100)$ and $W(200)$ and Explain

• From the graph:

  • $W(100) \approx 90 \, \text{ft}$.
  • $W(200) \approx 40 \, \text{ft}$.

  These values represent the depth of the water in the reservoir on the 100th day and 200th day of the year.

(c) [1 mark] Find $x$ such that $W(x) = 50$

• From the graph, $W(x) = 50$ appears to happen at two points:
  • $x \approx 60$ days.
  • $x \approx 320$ days.

(d) [2 marks] Intervals of Increase and Decrease

• Increasing: The function $W(x)$ increases from day $x = 0$ to about $x = 110$ days.
• Decreasing: The function $W(x)$ decreases from about $x = 110$ days to $x = 270$ days.
• Increasing again: The function $W(x)$ increases slightly again from about $x = 270$ to $x = 365$ days.

Would you like further details or have any questions about the solution?

5 Additional Related Questions:

1. What is the maximum depth of water in the reservoir according to the graph?
2. Estimate the day when the depth of the water is the highest.
3. How would you interpret the meaning of the decreasing intervals in terms of reservoir activity?
4. Can you determine the average depth of the reservoir over the year based on the graph?
5. How would you find the total amount of water in the reservoir over a period of time using this graph?

Tip:

When interpreting graphs, always pay attention to the units and labels on both axes. This will help provide accurate context for domain, range, and specific points.