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Chapter 1: Problem 4

Suppose that the price of gasoline is \(\$ 3.09\) per gallon. a. Generate a formula that describes the cost, \(C\), of buying gas as a function of the number of gallons of gasoline, \(G,\) purchased. b. What is the independent variable? The dependent variable? c. Does your formula represent a function? Explain. d. If it is a function, what is the domain? The range? e. Generate a small table of values and a graph.

Short Answer

Expert verified
The formula is \(C = 3.09G\). Independent variable: \(G\). Dependent variable: \(C\). It is a function with domain \([0, \infty)\) and range \([0, \infty)\).

Step by step solution

01

Generate the formula

The price per gallon of gasoline is given as \$3.09\. To find the cost \(C\) as a function of the number of gallons \(G\), multiply the price per gallon by the number of gallons: \[C = 3.09G.\]
02

Identify the variables

The independent variable is \(G\) (the number of gallons of gasoline), while the dependent variable is \(C\) (the cost of gasoline).
03

Check if it's a function

Yes, the formula \(C = 3.09G\) represents a function because for each input value of \(G\), there is exactly one output value of \(C\).
04

Determine the domain

The domain of the function is all non-negative real numbers \(G \geq 0\) because you cannot purchase a negative amount of gasoline.
05

Determine the range

The range of the function is all non-negative real numbers \(C \geq 0\) since the cost cannot be negative.
06

Generate a table of values

Let's generate a small table of values for the function \(C = 3.09G\):table\begin{array}{c|c}G \ & C \ \hline0 \ & 0 \ \hline1 \ & 3.09 \ \hline2 \ & 6.18 \ \hline3 \ & 9.27 \ \hline4 \ & 12.36 \ \hline5 \ & 15.45 \ \end{array}
07

Plot the graph

The graph of \(C = 3.09G\) is a straight line passing through the origin (0, 0). Each point \( (G, C) \) on the graph represents the cost for a given number of gallons purchased.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Function
A cost function describes the total cost incurred as a function of some variable.
In our exercise, it helps us understand how much it costs to buy a certain amount of gasoline, based on the price per gallon.
Since the price of gasoline is constant at $3.09 per gallon, the total cost can be calculated by multiplying the price per gallon by the number of gallons purchased.
This relationship is expressed as: \(C = 3.09G\), where: \(C\) is the cost, and \(G\) is the number of gallons.
Knowing this formula can help you easily budget for gas expenses and make better financial plans.
Independent Variable
In any function, the independent variable is the variable that you can control or choose its value.
In the context of our gasoline cost function, the independent variable is \(G\), which represents the number of gallons of gasoline you decide to purchase.
This means that the cost (dependent variable) directly depends on how many gallons (\

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Most popular questions from this chapter

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Suppose that a church congregation has 100 members, each of whom donates \(10 \%\) of his or her income to the church. The church collected \(\$ 250,000\) last year from its members. a. What was the mean contribution of its members? b. What was the mean income of its members? c. Can you predict the median income of its members? Explain your answer.

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