Problem 9 - College Algebra with Corequisite Support

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 $$\$ .05 /$$ min for calls. Company \(B\) has a monthly fee of $$\$ 5$$ and charges $$\$ .10 / \mathrm{min}$$ for calls. Find the model of the total cost of Company A's plan, using \(m\) for the minutes.

Step-by-Step Solution

Answer

The model for Company A's plan is \( C_A = 20 + 0.05m \).

Step 1: Define the Cost Structure

Company A's cost consists of a fixed monthly fee and a variable cost based on calling minutes. Let's outline the components: - Fixed monthly fee: \( \\(20 \)- Cost per minute: \( \\)0.05 \) per minute.

Step 2: Formulate the Cost Equation

The total cost for Company A can be expressed as a function of the number of minutes, \( m \). The formula combines the fixed fee with the variable cost:\[C_A = 20 + 0.05m\]

Step 3: Verify the Model with an Example

To ensure the model works, calculate the total cost for a specific number of minutes, say 100 minutes:\[C_A(100) = 20 + 0.05 \times 100 = 20 + 5 = 25\]This means that for 100 minutes of calls, Company A would charge \$25, which checks out with the model.

Key Concepts

Company A Cost Plan

When considering different telephone plans, it's important to understand what makes up the total cost in each option. In our scenario, **Company A's Cost Plan** gives consumers a mix of fixed and variable pricing. This means you have to pay a certain amount every month no matter how many minutes you use, plus an additional charge for each minute you talk.

To break this down:

• **Fixed Monthly Fee**: Every month, the client pays $20. This fee does not change, regardless of how many minutes they talk.
• **Cost Per Minute**: In addition to the fixed fee, Company A charges $0.05 for each minute of calls.

Think of it as a basic structure for assessing how expenses add up with Company A. Whether you are a heavy or light user, these two factors will determine what you pay monthly.

Mathematical Modeling

**Mathematical Modeling** plays a vital role in understanding and managing costs. By creating a model, we can simulate real-world scenarios with mathematical equations.

In this exercise, the mathematical model helps us calculate how much a customer will have to pay for using Company A's plan. We represent the cost with a formula using variables that reflect real components of the cost structure.

The model uses:

• The fixed monthly fee: $20
• The variable cost depending on the usage: $0.05 per minute
• A variable 'm' representing the number of minutes used

By putting these components together in a mathematical formula, we create a simple model, which helps in predicting costs under different usage scenarios. This approach is practical for decision-making, allowing customers to anticipate their expenses before choosing a plan.

Function Formula

The **Function Formula** plays a crucial role in projecting the cost for Company A's plan. It helps users estimate their total monthly bill based on how many minutes they expect to use. The function is straightforward to understand due to its clear components.

The formula for Company A's plan is:\[C_A = 20 + 0.05m\]
Here:

• \(C_A\) stands for the total cost for Company A.
• The term '20' represents the fixed monthly fee.
• '0.05m' is the variable component, where 'm' denotes the number of minutes used, multiplied by the cost per minute ($0.05).

This function conveniently shows the relationship between the cost and usage, where consumers can easily plug in different values for 'm' (minutes) to see how changes in usage affect their total cost. Using this function, predicting expenses becomes much simpler, aiding better financial planning and avoiding surprise bills.