Final answer:
In the theater group's fundraiser, the independent variable is the number of cars washed (c), and the dependent variable is the profit (p). The relationship between c and p is a function, described by the equation p = 5c - 34. The domain and range are non-negative integers and non-negative real numbers, respectively.
Step-by-step explanation:
Understanding Variables, Functions, and Equations
To address the theater group's carwash fundraiser scenario, we need to identify the independent and dependent variables, determine if the relationship between the number of cars washed (c) and the profit (p) is a function, and write the corresponding equation. Additionally, we'll establish the reasonable domain and range for this context.
a. In this situation, the independent variable is the total number of cars washed (c) because it is the variable that is controlled or changed in the experiment. The dependent variable is the profit (p), which depends on the number of cars washed.
b. The relationship between c (the independent variable) and p (the dependent variable) is a function because for each number of cars washed, there is exactly one corresponding profit amount.
c. The equation showing the relationship is p = 5c - 34. This represents the profit (p) made from washing c cars, where 5 dollars is charged per car, minus the initial 34 dollars spent on soap.
d. A reasonable domain for c would be the set of non-negative integers since you cannot wash a negative amount of cars, and it must be a whole number. A suitable range for p would be non-negative real numbers because the group can either break even or make a profit, but not incur extra costs outside the initial 34 dollars spent on soap.