Question

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Explanation:

Identify variables and relationship

Using the Independent and Dependent Variables knowledge point

• Independent variable: price of gasoline per gallon \(p\) (in dollars)
• Dependent variable: number of tourists in a national park \(t\) (in thousands or absolute count)
• Relationship: As the price of gasoline increases, travel costs rise, leading to a decrease in the number of tourists. Thus, \(t\) is a decreasing function of \(p\).

Analyze the axes of the graphs

• The prompt defines: Let \(p\) represent price of gasoline per gallon and \(t\) represent number of tourists.
• Therefore, the horizontal axis (independent variable) must be \(p\) (price of gasoline, in dollars), and the vertical axis (dependent variable) must be \(t\) (number of tourists).
• Looking at the options:
• Graph A: Vertical axis is labeled "t (people in millions)" and horizontal axis is \(p\) (\$).
• Graph B: Vertical axis is labeled \(p\) (\$) and horizontal axis is "t (people in millions)".
• Graph C: Vertical axis is labeled "t (tourists)" (with values up to 15000) and horizontal axis is \(p\) (\$).
• Graph D: Vertical axis is labeled \(p\) (\$) and horizontal axis is "t (people)".
• Since \(p\) is the independent variable, it must be on the horizontal axis, and \(t\) (tourists) must be on the vertical axis. This leaves Graph A and Graph C.
• A national park's tourist count is more realistically measured in thousands (e.g., up to 15,000 as in Graph C) rather than millions (as in Graph A, where 1 million tourists would be extremely high for a daily/weekly scale shown). Thus, Graph C is the correct representation.

Evaluate the validity of the model

• The graph shows a decreasing curve, meaning that as the price of gasoline increases, the number of tourists decreases.
• This is a valid economic model because higher fuel prices increase the cost of travel, which discourages people from visiting national parks.
• Therefore, the graph is a valid model because the number of tourists decreases as the price of gasoline per gallon increases.

Answer:

Question 1

• (A) Graph A
• (B) Graph B
• (C) Graph C (Correct answer)
• (D) Graph D

Question 2

• (A) The graph is not a valid model of the true function.
• (B) The graph is a valid model of the function because number of tourists decreases as price of gasoline per gallon increases. (Correct answer)
• (C) The graph is a valid model of the function because price of gasoline per gallon decreases as number of tourists increases.
• (D) The validity of the graph as a model of the true function can never be known.