Problem 48 - Intermediate Algebra

Question

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Copiers. \(\quad\) A business is going to rent a copy machine. Under the rental agreement, the company is charged \(\$ 105\) per month and 3 \(\boldsymbol{\alpha}\) for every copy that is made. Describe the relationship between the monthly copier expense and the number of copies made.

Step-by-Step Solution

Answer

The monthly copier expense is modeled by \( C = 105 + 3\alpha x \).

Step 1: Identify Variables

Let's identify the variables involved in the problem. Let \( C \) be the total copier expense in dollars and \( x \) be the number of copies made in a month.

Step 2: Establish Relationship for Fixed Costs

The fixed cost is the rental fee charged every month, which is \( \$105 \). This cost does not depend on the number of copies made.

Step 3: Establish Relationship for Variable Costs

For each copy made, an additional charge of \( 3\alpha \) is incurred. Hence, the variable cost for making \( x \) copies would be \( 3\alpha \times x \).

Step 4: Formulate the Mathematical Model

The total cost \( C \) is the sum of the fixed costs and the variable costs. This relationship can be expressed as the equation: \[ C = 105 + 3\alpha x \]

Step 5: Interpretation

In the equation \( C = 105 + 3\alpha x \), \( 105 \) represents the monthly fixed rental cost, and \( 3\alpha x \) represents the cost associated with the number of copies made. This equation models the total monthly expense for the copier service as a function of the number of copies.

Key Concepts

Fixed Costs

In the context of renting a copier, fixed costs refer to constant expenses that do not change with the number of copies made. They are predictable and occur regularly, providing a stable element in the budgeting process. Here, the fixed cost is the rental fee of \( \$105 \) per month, meaning each month, the business needs to account for this consistent charge, regardless of whether they make one copy or a million.
Fixed costs are essential for businesses to understand and consider, as they impact cash flow and budgeting. By recognizing these regular expenses, businesses can better plan their finances and ensure they are prepared to cover these costs consistently.

Some key features of fixed costs include:

• Unchanging with production levels.
• Predictable and regular.
• Important for financial planning and budgeting.

Variable Costs

Variable costs, unlike fixed costs, change depending on the level of activity. In this copier example, the variable cost is determined by how many copies the business makes. For each copy, there's an extra charge of \( 3\alpha \) dollars. This means that if no copies are made in a month, the business doesn't incur any variable costs.
The concept of variable costs is crucial because it helps businesses anticipate how costs will scale with production levels. Understanding these costs, businesses can make informed decisions about production quantities, pricing strategies, and potential profits.

Characteristics of variable costs include:

• Fluctuating with production levels — higher when more copies are made.
• Impacting the total cost and, consequently, pricing and profit calculations.
• Crucial for cost-volume-profit analysis.

The ability to adjust pricing or production levels according to variable costs is a strategic advantage that can help optimize costs and increase profitability.

Linear Equations

Linear equations are mathematical expressions used to model relationships where one variable changes at a constant rate concerning another. In this case, the linear equation \( C = 105 + 3\alpha x \) represents the total monthly cost \( C \) as a function of the number of copies \( x \).
This equation is linear because the relationship between the total cost and the number of copies is direct and proportional, with no exponents or curves involved. The equation includes two main components: a fixed part (\( 105 \)) and a variable part (\( 3\alpha x \)).

The features and importance of linear equations include:

• Simulating real-world problems where change occurs at a constant rate.
• Providing simplicity in calculations and forecasting.
• Being applicable in finance, economics, and numerous fields for modeling growth, costs, and trends.

Understanding how to create and manipulate linear equations is a valuable skill that enables students and professionals alike to solve practical problems efficiently.