Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $$\$ 90$$ monthly fee, unlimited talk and text on up to 8 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 cost model is \( C = 120 + 35P \).

Step 1: Identify the Fixed Costs

The fixed cost for the Mobile Share Plan is the monthly fee. For this plan, the fixed monthly fee is \( \$120 \).

Step 2: Determine the Variable Costs

The variable cost is the cost associated with data for each device. The Mobile Share Plan charges \( \$35 \) for each device. If \( P \) represents the number of devices, then the data cost for these devices is given by \( 35P \).

Step 3: Construct the Overall Cost Model

The total cost, \( C \), of the Mobile Share Plan is the sum of the fixed and variable costs. Thus, the expression for the total cost is \( C = 120 + 35P \), where \( P \) is the number of devices.

Key Concepts

Cost Model

A cost model is a mathematical equation or expression that helps in determining the total cost associated with a particular product or service. It takes into consideration both fixed and variable costs to arrive at an overall cost determination.

In the context of the Mobile Share Plan, the cost model can be used to calculate the total cost of having multiple devices on a mobile plan. It's a tool to help in understanding how costs change based on the number of devices.

With the given problem, the cost model is formed by taking the fixed monthly fee and adding any additional variable costs based on device utilization.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a value or a set of values. They are fundamental in constructing cost models, like the one used in the mobile plan example. Understanding these expressions is crucial for analyzing and interpreting how different factors affect total costs.

In the solution, the expression for the Mobile Share Plan's total cost is given by:

• The fixed fee: \( 120 \)
• The variable cost for \( P \) devices: \( 35P \)

Together, they form the algebraic expression that represents the total cost:\[ C = 120 + 35P \]Here, \( C \) is the total cost, and \( P \) is the number of devices. Algebraic expressions like this help in easily understanding and calculating costs.

Variable Costs

Variable costs are expenses that change in proportion to the volume of goods or services produced. In our wireless plan example, these costs vary with the number of devices using data services.

For the Mobile Share Plan, each device incurs an additional cost of \( 35 \) if it uses the data service. Therefore, if you have \( P \) devices, the variable cost can be expressed as \( 35P \).

Variable costs are important because they allow the flexibility to predict changes in total expenses as usage fluctuates. Understanding them helps in making informed decisions based on potential growth or reduction in usage.