Problem 13 - College Algebra

Question

For the following exercises, use this scenario: A wiress carrier offers the following plans that a person is considering. The Family Plan: s90 monthly fee, unlimited talk and text on up to 5 lines, and data charges of \(\$ 40\) for each device for up to 2 \(\mathrm{GB}\) of data per device. The Mobile Share Plan: \(\$ 120\) monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of \(\$ 35\) for each device up to a shared total of 10 \(\mathrm{GB}\) of data. Use \(P\) for the number of devices that need data plans as part of their cost. Find the model of the total cost of the Family Plan.

Step-by-Step Solution

Answer

The total cost model for the Family Plan is \( C_F = 90 + 40P \).

Step 1: Understanding the Elements of the Family Plan

The Family Plan consists of a fixed monthly fee of $90 and data charges that apply per device. For each device that requires data, there's an additional charge of $40, specifically for up to 2 GB of data per device.

Step 2: Introducing the Variable for Devices

Let the variable \( P \) represent the number of devices that require a data plan. Each device incurs a charge of $40 according to the Family Plan structure.

Step 3: Formulating the Cost Equation

Combine the fixed monthly fee with the variable data charges. The formula for total monthly cost, \( C_F \), of the Family Plan is given by:\[ C_F = 90 + 40P \]Where 90 is the monthly fee and \( 40P \) represents the total data charges for \( P \) devices.

Key Concepts

Family Plan Equation

When dealing with mobile or wireless plans, it's important to have an equation or formula that helps us understand and calculate the total costs involved. The Family Plan is a great example to explore this concept. In this specific plan, you need to account for both a fixed monthly fee and additional charges per device for data usage.

Here's a breakdown of how to create the equation. Suppose you're considering the Family Plan with a fixed monthly fee of \(90. To this, you add the data charges per device, which is \)40 for each device needing up to 2GB of data. In mathematical terms, if the number of devices needing a data plan is represented by the variable \( P \), then the total cost of the Family Plan, \( C_F \), is given by the equation:

• The fixed monthly cost of \(90 is constant.
• The additional cost per device is \)40, multiplied by the number of devices \( P \).

Now, putting this together:\[ C_F = 90 + 40P \]
This equation allows you to easily determine the Family Plan's total monthly cost based on how many devices need data coverage.

Data Plan Charges

Data charges are an important consideration when choosing a mobile plan. These charges can significantly affect the overall cost, depending on the number of devices and the amount of data required. In the Family Plan, we're focused on a specific pricing structure: \(40 for each device that uses up to 2GB of data.

Let’s illustrate how these charges apply. Consider that you have several devices, each needing data. If each device costs \)40 a month for its data, then the total data charges can be calculated based on the number of devices. Using the variable \( P \) to represent the number of devices:

• The cost for one device is $40.
• The total cost for \( P \) devices is \( 40P \).

This structure clear simplifies the budgeting process, as it allows you to create a direct proportional relationship between the number of devices and the total data charges.

Algebraic Expressions

Algebraic expressions are key in setting up equations like the Family Plan's cost model. By using algebra, we can conveniently summarize and solve for costs involving variables such as the number of devices. An algebraic expression combines numbers, variables, and arithmetic operations to represent a quantity mathematically.

In the context of the Family Plan, we use an algebraic expression to model costs. We introduce a variable \( P \) to represent the number of devices needing data plans. Combine fixed costs and variable costs, and here’s what the expression looks like:

• The fixed part, \(90, is an unchanging number in the expression.
• The variable part, \( 40P \), involves multiplying the per-device cost (\)40) by the number of devices \( P \).

The expression used is:\[ 90 + 40P \]
This equation effectively captures the complete cost structure of the Family Plan. It shows how algebra can take complex pricing information and turn it into a simple method for calculation and analysis.