Problem 118 - College Algebra with Modeling and Visualization

Question

Business \(A\) company manufactures compact discs with recorded music. The master disc costs \(\$ 2000\) to produce and copies cost \(\$ 0.45\) each. If a company spent \(\$ 2990\) producing compact discs, how many copies did the company manufacture?

Step-by-Step Solution

Answer

The company manufactured 2200 copies.

Step 1: Identify Fixed and Variable Costs

First, recognize that producing the master disc is a fixed cost of $2000, while the variable cost is $0.45 per copy. The total production cost was $2990.

Step 2: Calculate Remaining Budget After Fixed Costs

Subtract the fixed cost of the master disc from the total cost to determine the budget available for producing copies. \[\text{Remaining Budget} = 2990 - 2000 = 990 \]

Step 3: Determine Number of Copies

Use the remaining budget to calculate the number of copies produced. Divide the remaining budget by the cost of each copy. \[\text{Number of copies} = \frac{990}{0.45}\] Simplify the division:\[ \text{Number of copies} = 2200 \]

Key Concepts

Cost Analysis

Cost analysis is a method used to understand the cost components involved in producing a product or service. In the context of the exercise, we have two main types of costs: fixed and variable costs.

• Fixed Costs: These are costs that do not change regardless of the number of items produced. In this problem, the fixed cost is the cost of producing the master disc, which is $2000.
• Variable Costs: These vary with the level of output, meaning they change based on the number of items produced. Here, each additional compact disc costs $0.45 to make.

By performing a cost analysis, you can determine how these fixed and variable costs add up to form the total cost. Understanding the relationship between these costs helps in deciding the number of items that can be produced within a given budget. This forms the basis for more complex financial planning and decision-making in business.

Budgeting

Budgeting involves planning how to allocate resources, such as money, effectively. In this scenario, the company has a total budget of $2990 for producing compact discs. Budgeting requires careful management of both fixed and variable expenses to ensure that financial goals are met without overspending. When budgeting, it is crucial to remember the following steps:

• Identify and separate your fixed costs. In this example, that's the $2000 for the master disc, which must be budgeted first as it doesn't change.
• Determine the remaining funds available for variable costs (i.e., making additional copies).
• Use these remaining funds strategically to maximize production.

Sound budgeting is about balance and foresight, helping companies understand how much product they can afford to produce while keeping an eye on both fixed and variable costs.

Linear Equations

Linear equations are mathematical expressions that show the relationship between variables by generating a straight line when graphed. In this exercise, determining how many copies can be produced is a problem of balancing a linear equation with one variable: the number of copies, denoted here as \( x \).The equation that models our situation is:\[2000 + 0.45x = 2990\]Here's how you solve the linear equation:

• First, recognize that 2000 represents fixed costs and 0.45x represents the variable costs of producing \( x \) copies.
• Subtract the fixed costs from the total budget to isolate the variable part: \( 0.45x = 990 \).
• Solve for \( x \) by dividing both sides by 0.45: \( x = \frac{990}{0.45} \).

The result, \( x = 2200 \), tells us the company can produce 2200 discs with the remaining budget. Understanding linear equations is fundamental to solving real-world problems involving budgeting and resource allocation.