Problem 14 - 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 Mobile Share Plan.

Step-by-Step Solution

Answer

The total cost model for the Mobile Share Plan is \( C = 120 + 35P \).

Step 1: Identify Fixed Costs

The Mobile Share Plan has a fixed monthly fee of $120. This fee does not change regardless of the number of devices, making it a constant in our equation.

Step 2: Determine Variable Costs

For each device in the Mobile Share Plan, there is a data charge of $35. If we have \( P \) devices needing data, the total data charge will be \( 35P \). This makes it the variable component of our cost.

Step 3: Combine Fixed and Variable Costs

Now, combine the fixed cost (monthly fee) and the variable cost (data charges per device). The total cost \( C \) of the Mobile Share Plan can be modeled by the equation: \[ C = 120 + 35P \]

Key Concepts

Fixed Costs

Fixed costs are those that do not change with the quantity of items or services you are using. In the context of the Mobile Share Plan, the fixed cost is the monthly fee of $120.
This means that whether you have one device or ten, you're going to pay this $120 regardless.
Fixed costs provide stability and predictability in your budget. Understanding these costs helps in planning and comparing different plans.
When comparing fixed costs in mobile plans, consider factors such as how the service meets your needs for a number of lines or devices.
Some key considerations about fixed costs include:

• Fixed costs are constant and do not change with usage.
• Have a base fee that stays the same month to month.
• Helps to determine the minimum expense with no additional usage.

Recognizing fixed costs is crucial when analyzing and deciding between several mobile plans.

Variable Costs

Variable costs in a pricing model are expenses that change depending on the quantity of goods or services.
In the Mobile Share Plan, variable costs come from the data charges for each device.
Specifically, for every device requiring data, there is a charge of $35 per month.
So, if you have \( P \) devices, your variable cost is \( 35P \).
This aspect of costing encourages individuals to think about how the number of data devices impacts their total bill.
It's important to understand how variable costs can fluctuate:

• They change with the number of devices using data.
• They can lead to higher overall costs if not controlled.
• Allow flexibility based on needs, but require monitoring to avoid unexpected high charges.

Grasping the concept of variable costs helps in making informed decisions when adjusting the number of devices in your plan.

Linear Equation

A linear equation is a mathematical way to represent a constant relationship between two variables.
In the scenario of the Mobile Share Plan, the total cost equation is a linear equation.
It combines both fixed and variable costs, providing a clear and straightforward way to calculate costs.
The equation is \[ C = 120 + 35P \], where \( C \) is the total cost and \( P \) is the number of devices.
This formulation helps users quickly determine their monthly bill based on how many devices need data.
Some aspects to understand about linear equations include:

• They show a direct proportionality between variables.
• Have a straight-line graph representation.
• Keep computations simple, aiding in easy expense calculations.

Linear equations are practical for everyday use, like budgeting for mobile plans, since they offer an uncomplicated way to project total costs based on changing needs.

Mobile Share Plan

The Mobile Share Plan is a type of wireless service plan that offers a comprehensive cost structure for managing multiple devices.
This plan is designed to provide flexibility and ease-of-use for those with multiple devices, such as families or small businesses.
Its standout features include a flat $120 monthly fixed cost for up to 10 devices, with unlimited talk and text.
Add to this, a variable cost of $35 per device for data, and you are looking at a plan that can be tailored to fit various user sizes.
Key factors that make a Mobile Share Plan worthwhile are:

• Cost-effectiveness for large groups.
• Simplified billing with one common bill for all devices.
• Flexibility in adjusting the number of data devices without changing the base cost significantly.

Understanding all these elements helps users to make better decisions about how their service needs can best be met while keeping costs in check.