Math Problem Statement

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. Questions include finding domain, range, specific values of W(x), and intervals of increase and decrease.

Solution

Let's go through each part of the problem.

Problem Overview:

You are given a graph of a function W(x)W(x), where W(x)W(x) represents the depth of water in a reservoir xx days since the beginning of the year. The graph spans the interval 0x3650 \leq x \leq 365.

(a) [2 marks] Domain and Range

  • Domain: This is the set of possible xx-values. From the graph, xx represents the number of days since the beginning of the year, and it is defined from 00 to 365365.

    • Domain: [0,365][0, 365].
  • Range: This is the set of possible W(x)W(x)-values, which represent the water depth. From the graph, the depth W(x)W(x) seems to vary between approximately 20ft20 \, \text{ft} and 100ft100 \, \text{ft}.

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

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

  • From the graph:

    • W(100)90ftW(100) \approx 90 \, \text{ft}.
    • W(200)40ftW(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 xx such that W(x)=50W(x) = 50

  • From the graph, W(x)=50W(x) = 50 appears to happen at two points:
    • x60x \approx 60 days.
    • x320x \approx 320 days.

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

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

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Functions
Domain and Range
Increasing and Decreasing Intervals

Formulas

None required for this problem; focuses on graph interpretation.

Theorems

None specific; applies graphical and interval analysis.

Suitable Grade Level

Grades 10-12