Problem 12 - 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, \(m .\) the phone is used during the month.

Step-by-Step Solution

Answer

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

Step 1: Identify the Fixed Cost

The company has a fixed monthly fee of $25. This is the base cost of using the service, regardless of the number of minutes used.

Step 2: Determine the Variable Cost

The variable cost is calculated based on the number of minutes the phone is used. The company charges $0.05 per minute. Therefore, if the phone is used for \( m \) minutes, the variable cost is \( 0.05m \) dollars.

Step 3: Develop the Total Cost Formula

The total monthly charge, \( C \), is the sum of the fixed and variable costs. The fixed cost is $25, and the variable cost is \( 0.05m \). Thus, the formula for the monthly charge is \( C = 25 + 0.05m \).

Key Concepts

Fixed Costs

Fixed costs are the consistent and unchanging costs that a business incurs over a given period, regardless of the level of goods or services produced. For a cell phone company, this might be a basic monthly fee that customers have to pay whether they use their phones or not. In this case, there is a $25 fixed monthly charge. This stays the same no matter how many minutes are used.

It is important to understand that fixed costs help cover ongoing expenses, such as maintaining the network infrastructure and customer support. They do not fluctuate with the amount of usage or production.

In economic terms, fixed costs are crucial because they provide a predictable base that helps the company maintain operations and plan for revenue generation. For example, this cell phone company's fixed cost structure allows them to ensure their baseline costs are met, simplifying their financial planning.

Variable Costs

Variable costs are expenses that change in direct proportion to the activity level of a business. These costs are variable because they depend on the number of goods or services produced or consumed. For the cell phone company, the variable cost is the charge incurred based on the minutes used, which is $0.05 per minute.

This means that unlike fixed costs, the more you use the service, the higher the variable cost becomes. If you talk for 100 minutes, the variable cost is 100 minutes times $0.05 per minute, totaling $5.

• If you talk for 200 minutes, the cost doubles to 200 minutes times $0.05 per minute equals $10.
• The key point is that variable costs rise and fall directly with the level of usage.

This makes variable costs flexible, but potentially unpredictable as they fluctuate with customer usage.

Cost Function

The cost function is a mathematical formula that helps to calculate the total cost incurred by a business regarding fixed and variable costs. It offers a straightforward way to understand how costs change with varying levels of output or usage.

For the cell phone company, the cost function combines both fixed and variable costs. The formula given in the original exercise is: \[ C = 25 + 0.05m \] where:

• \( C \) is the total monthly charge in dollars.
• \( 25 \) is the fixed cost in dollars.
• \( 0.05m \) is the variable cost depending on the number of minutes \( m \) used.

To use this function, simply plug in the number of minutes the phone was used (\( m \)), and the formula will calculate the total cost \( C \).

This cost function allows customers to anticipate their monthly phone bill and helps the phone company provide clear pricing to its users. It is a crucial tool for understanding and planning for financial expenses both for consumers and businesses.