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Chapter 1: Problem 5

The cost of driving a car to work is estimated to be \(\$ 2.00\) in tolls plus 32 cents per mile. Write an equation for computing the total cost \(C\) of driving \(M\) miles to work. Does your equation represent a function? What is the independent variable? What is the dependent variable? Generate a table of values and then graph the equation.

Short Answer

Expert verified
The equation is \(C = 2.00 + 0.32M\). It represents a function. The independent variable is miles \(M\), and the dependent variable is cost \(C\).

Step by step solution

01

Form the cost equation

Given the cost includes a fixed toll of \(2.00\) and a variable cost of \(0.32\) per mile. The total cost equation is \(C = 2.00 + 0.32M\).
02

Determine if the equation represents a function

In the equation \(C = 2.00 + 0.32M\), for each value of \(M\), there is exactly one value of \(C\). Hence, it represents a function.
03

Identify the independent and dependent variables

- The independent variable is \(M\), the number of miles.- The dependent variable is \(C\), the cost which depends on the miles driven.
04

Generate a table of values

Calculate the cost for select values of miles (\(M\)). Using the equation \(C = 2.00 + 0.32M\), fill in the table below:\[\begin{array}{|c|c|}\hlineM (miles) & C (cost) \hline0 & 2.00 \5 & 3.60 \10 & 5.20 \20 & 8.40 \30 & 11.60 \hline\end{array}\]
05

Graph the equation

Plot the points from the table on a coordinate plane, where \(M\) is on the x-axis (independent variable) and \(C\) is on the y-axis (dependent variable). Draw a line through the points extending in both directions.

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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 is a mathematical formula that allows you to calculate the total cost based on certain variables. In this exercise, the cost of driving a car to work includes a fixed toll and a variable cost per mile driven. The equation provided is:

\(C = 2.00 + 0.32M\).

Here:
  • \(C\) represents the total cost of driving.
  • \(2.00\) is the fixed toll charge, which does not change regardless of miles driven.
  • \(0.32M\) represents the variable cost, where \(0.32\) is the cost per mile and \(M\) is the number of miles driven.

This equation is a linear equation because it is in the form \(y = mx + b\), where \(m\) is the slope (0.32, the cost per mile) and \(b\) is the y-intercept (2.00, the fixed toll charge).

Since each value of \(M\) results in a unique value of \(C\), this equation also represents a function.
Dependent and Independent Variables
In any equation, it's essential to understand the roles of dependent and independent variables. In our cost function \(C = 2.00 + 0.32M\):
  • The independent variable is \(M\), the number of miles driven. This is the variable you control or choose.
  • The dependent variable is \(C\), the total cost. This value depends on the number of miles you drive.

To identify these variables, ask yourself: 'What am I changing?' (independent variable) and 'What is changing as a result?' (dependent variable). Knowing which variables are independent and dependent helps you better predict and understand their relationship.
Graphing Linear Equations
Graphing a linear equation like \(C = 2.00 + 0.32M\) helps you visualize the relationship between the variables.
  • Create a table of values: Calculate \(C\) for different values of \(M\). For example, when \(M = 0\), \(C = 2.00\). When \(M = 10\), \(C = 5.20\).
  • Plot the points: On a coordinate plane, plot the calculated points from the table with \(M\) on the x-axis and \(C\) on the y-axis. For instance, (0, 2.00) and (10, 5.20).
  • Draw the line: Connect the points with a straight line. This line represents the cost function, showing how the cost increases with each additional mile driven.

By graphing, you can easily see patterns and predict values. For instance, a steeper line would indicate a higher per-mile cost. Graphing also helps in checking the accuracy of your calculations and understanding the overall behavior of the function.

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