Problem 11 - College Algebra

Question

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of \(\$ 20\) and charges of \(\$ .05 / \mathrm{min}\) for calls. Company \(\mathrm{B}\) has a monthly fee of \(\$ 5\) and charges \(\$ .10 / \mathrm{min}\) for calls. Find out how many minutes of calling would make the two plans equal.

Step-by-Step Solution

Answer

The two plans are equal at 300 minutes.

Step 1: Define the Variables

Let's define \( x \) as the number of calling minutes in a month. This will help us express the costs from each company in terms of \( x \).

Step 2: Write the Cost Equation for Company A

Company A's monthly cost is computed by its fixed fee and per-minute charge. The total cost equation is: \( c_A = 20 + 0.05x \), where \( c_A \) is the total monthly cost for \( x \) minutes.

Step 3: Write the Cost Equation for Company B

Company B's monthly cost is also computed by its fixed fee and per-minute charge. The total cost equation is: \( c_B = 5 + 0.10x \), where \( c_B \) is the total monthly cost for \( x \) minutes.

Step 4: Set the Equations Equal to Each Other

To find the number of minutes where the costs are equal, set the two equations equal to each other: \( 20 + 0.05x = 5 + 0.10x \).

Step 5: Solve the Equation for \( x \)

Isolate \( x \) by first subtracting \( 0.05x \) from both sides, yielding: \( 20 = 5 + 0.05x \). Then, subtract 5 from both sides: \( 15 = 0.05x \). Finally, divide both sides by 0.05: \( x = 300 \).

Step 6: Conclusion

The number of minutes where the two plans cost the same is \( x = 300 \). Therefore, if you make 300 minutes worth of calls, both plans will cost the same amount per month.

Key Concepts

Cost Analysis

Cost analysis is a fundamental process in determining the economic impact of different decisions. Here, we used cost analysis to compare two phone plans based on monthly usage.
In this context, each company has different fee structures, which include:

• A fixed monthly fee,
• And a variable charge per minute of usage.

The purpose of cost analysis is to evaluate which plan offers better value at various levels of usage. By analyzing the cost structure of each company, we can determine the point at which both plans offer the same financial benefit (or cost). In our exercise, this was achieved by setting the costs equal to each other to establish the breakeven point. Understanding these concepts can aid in making more informed financial decisions.

Linear Equations

Linear equations are mathematical expressions that describe a constant rate of change. In our phone plan problem, we utilize linear equations to represent the cost as a function of minutes used.
Linear equations typically have the form:

• \( y = mx + b \)

In this form, \( m \) represents the per-minute charge, or slope, while \( b \) is the fixed monthly fee, or y-intercept.

For Company A, the equation \( c_A = 20 + 0.05x \) reflects a base cost of \(20 and a charge of \)0.05 per minute. Similarly, Company B's equation, \( c_B = 5 + 0.10x \), reflects a \(5 base fee with a \)0.10 charge per minute. By using these linear equations, we can calculate and compare the costs effectively at any number of minutes. Solving such equations helps uncover relationships and intersections between different cost structures.

Problem Solving

Problem solving is an essential skill, especially when evaluating various options and making decisions. In the phone plan scenario, problem solving involves:

• Identifying the variables and constraints,
• Formulating equations that represent these constraints,
• And solving the equations to find practical solutions.

The process begins with defining the problem clearly. In our case, this meant setting up equations to understand at what minute usage both phone plans cost the same.
By equating the respective cost equations, we can isolate the variable, \( x \). Solving the equation \( 20 + 0.05x = 5 + 0.10x \) allowed us to pinpoint the exact usage level where the costs equal. Hence, problem solving is about using mathematical tools and logical reasoning to tackle real-life questions efficiently. Developing these skills can significantly enhance critical thinking and decision-making abilities.