Question

Use the following information. You are moving to Houston, Texas, and are switching your cellular phone company. Your new peak air time rate in Houston is \(\$ .23\) per minute. Your bill also includes a monthly access charge. For 110 minutes of peak air time your bill is \(\$ 51.30\). Write an equation that models the cost \(C\) of your monthly bill in terms of the number of minutes \(m\) used. (All of your minutes are during peak air time.)

Short answer

The cost \(C\) of the monthly bill in terms of the number of minutes \(m\) used is given by the equation \(C = 0.23m + 25.7\) dollars.

Step-by-step solution

Understand the problem

You are given the cost for 110 minutes of peak air time along with the rate per minute. The goal of the exercise is to find an equation that models the cost of the phone bill per month with respect to the number of minutes used.

Write down the known values

The known values in this case are the rate per minute \(r = 0.23\) and the cost for 110 minutes \(C_{110} = 51.30\) dollars.

Formulate the equation

We can express the cost \(C\) in terms of minutes \(m\) as \(C = rm + a\), where \(r\) is the rate per minute, \(m\) is the number of minutes and \(a\) is the monthly access charge which we aim to find.

Solve the equation for access charge

Substituting \(r = 0.23\) and \(m = 110\) in the above equation, we get \(51.30 = 0.23 × 110 + a\). Simplifying this gives \(a = 51.30 - (0.23 × 110)\), thus \(a = 25.7\) dollars.

Write the final model equation

Substituting the value of \(a\) back in the original equation we get the final model as \(C = 0.23m + 25.7\)

Key concepts

Linear Equations

Linear equations are fundamental in algebra and represent relationships between two quantities that have a constant rate of change. This constant relationship is graphically represented by a straight line on a coordinate plane and algebraically by an equation of the form \(y = mx + b\) where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope of the line, and \(b\) is the y-intercept.

In the case of the phone bill problem, the linear equation is used to model the relationship between the cost of the bill (\(C\)) and the number of peak air time minutes used (\(m\)). The slope (\(r\)) represents the rate per minute, and the y-intercept (\(a\)) represents the monthly access charge. For every additional minute of air time, the cost increases by a fixed amount, showing a direct variation and thus creating a linear model.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario. The goal is to use mathematics to analyze and make predictions about real-life situations. In mathematical modeling, you commonly identify variables, constants, and relationships that can be written in the form of equations or inequalities.

In our cellular phone bill example, the real-world scenario is the monthly expense based on usage. Mathematical modeling is used to write an equation that helps to predict your monthly phone bill based on the minutes of peak air time. This model is a tool for understanding how different factors, such as the number of minutes used, affect the total cost.

Solving for a Variable

When solving for a variable in a mathematical equation, the aim is to isolate that variable on one side of the equation. This process often involves a series of algebraic manipulations like addition, subtraction, multiplication, division, and factoring.

In the textbook example, solving for the variable involved finding the value of the monthly access charge (\(a\)). Once the rate (\(r\)) and the cost for a known number of minutes (\(C_{110}\)) were substituted into the model equation, algebra was used to isolate \(a\) and solve for its value. Knowing \(a\) enables you to understand the fixed cost element of the billing, separate from the variable cost per minute of usage.