Problem 16 - Applied Calculus

Question

A cell phone company charges a monthly fee of \(\$ 25\) plus \(\$ 0.05\) per minute. Find a formula for the monthly charge, \(C,\) in dollars, as a function of the number of minutes, \(\bar{m}\) the phone is used during the month.

Step-by-Step Solution

Answer

The formula is \\(C(\bar{m}) = 25 + 0.05\bar{m}\\).

Step 1: Understanding the Problem

The problem requires us to create a formula for the monthly charge for a cell phone plan based on the number of minutes used. We know that there is a fixed cost of $25 and a variable cost depending on the number of minutes, \(\bar{m}\).

Step 2: Identifying the Fixed Cost

The fixed cost is the part of the charge that does not change with the number of minutes used. According to the problem, the fixed cost is \(\$25\).

Step 3: Determining the Variable Cost

The variable cost depends on the number of minutes used. The cost per minute is given as \(\$0.05\). Therefore, the total variable cost for \(\bar{m}\) minutes is \(0.05 \times \bar{m}\).

Step 4: Formulating the Total Cost

To find the total monthly charge \(C\), we add the fixed cost and the variable cost. So, the formula becomes \(C = 25 + 0.05 \times \bar{m}\).

Step 5: Simplifying the Formula

The formula for the monthly charge is already simplified. Thus, \(C(\bar{m}) = 25 + 0.05\bar{m}\). This formula calculates the total cost when using \(\bar{m}\) minutes per month.

Key Concepts

Fixed Cost

When dealing with costs in businesses like a cell phone company, the concept of a *fixed cost* is important. Fixed costs are expenses that remain constant regardless of the amount of product or service being used.

In our example of a cell phone company's monthly plan:

• The fixed cost is a flat fee of \(\$25\) charged every month.
• This cost does not change no matter how many minutes are used.

The significance of fixed costs lies in their predictability. Companies can rely on them as regular, unchanging expenses each month. For consumers, understanding what part of their bill comes from a fixed cost can help them manage their budget because they know this cost will stay the same month after month.

Fixed costs, like rent or salaries in a business, provide a stable base in financial planning.

Variable Cost

Variable costs differ from fixed costs as they fluctuate based on usage. In our cell phone plan example, the *variable cost* is defined by the number of minutes the customer uses.

Here’s how it works:

• The company charges \(\$0.05\) for each minute used.
• The total variable cost for the month is \(0.05 \times \bar{m}\), where \(\bar{m}\) represents the number of minutes used.

These costs move up or down with the level of activity and are variable because they depend directly on usage.

For both companies and consumers, variable costs can be more challenging to predict since they depend on behavior, such as how much someone calls in this case. Managing these costs requires tracking and adjusting behavior to meet financial goals.

Cost Function

Combining fixed and variable costs, we can construct a *cost function* to represent the total expense. A cost function provides a formulaic way to calculate the total cost as a function of usage.

In the case of our cell phone plan:

• The cost function is represented by \(C(\bar{m}) = 25 + 0.05\bar{m}\).
• This formula accounts for the \(\\(25\) fixed cost and the \(\\)0.05\) per minute variable cost.

Your total monthly charge (\(C\)) is determined by adding the fixed cost and the product of the variable cost and usage (\(\bar{m}\)).

This cost function is a practical tool because it allows both companies to estimate expected revenues and individuals to forecast their expenses based on minute usage. Understanding how to utilize this formula can aid in better financial planning, ensuring costs are aligned with usage and budget capabilities.