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Chapter 0: Problem 42

A woman decides to start a small business making monogrammed cocktail napkins. She can set aside \(\$ 1,870\) for monthly costs. Fixed costs are \(\$ 1,329.50\) per month and variable costs are \(\$ 3.70\) per set of napkins. How many sets of napkins can she afford to make per month?

Short Answer

Expert verified
She can produce approximately 146 sets of napkins per month.

Step by step solution

01

Understand the Problem

The woman has a total budget of \(\\( 1,870\) for monthly costs, which include both fixed and variable costs. Fixed costs are \(\\) 1,329.50\) per month, and variable costs are \(\$ 3.70\) per set of napkins. We need to find out how many sets of napkins she can produce within her budget each month.
02

Set Up the Equation for Total Costs

First, calculate the remaining budget after covering fixed costs. Then, use this remaining budget to determine how many sets of napkins she can afford to make. The total costs equation can be represented as:\[\text{Remaining Budget} = \text{Total Budget} - \text{Fixed Costs}\]The equation for total costs based on the number of napkins (\(n\)) is:\[\text{Remaining Budget} = 3.70n\]
03

Calculate the Remaining Budget

Subtract the fixed costs from the total monthly budget to find the remaining budget available for variable costs:\[\text{Remaining Budget} = 1,870 - 1,329.50 = 540.50\]This is the amount she can spend on variable costs (napkins).
04

Solve for the Number of Napkins

Use the remaining budget to solve for the number of napkins (\(n\)) that can be produced. Substitute the remaining budget into the equation for variable costs:\[540.50 = 3.70n\]Solve for \(n\) by dividing both sides of the equation by \(3.70\):\[n = \frac{540.50}{3.70} \]Calculate the value to find:\[n \approx 146\]Thus, she can produce approximately 146 sets of napkins.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Costs
Fixed costs are expenses that do not change in response to the production level or business output. In the context of the monogrammed napkin business, the woman has fixed costs of \(1,329.50\) each month. These costs might include rent for workspace, utility bills, or perhaps a monthly salary for employees.

Fixed costs are crucial because they must be paid regardless of how many napkins are produced, and they are consistent from month to month. Understanding these costs helps in assessing the minimum budget needed to keep the business operational before even considering variable costs.
Variable Costs
Variable costs fluctuate with the level of production or business activity. In this case, the variable costs are related to the number of napkin sets produced, costing \(3.70\) per set.

This means that as the production of napkin sets increases, the variable costs will also increase in direct proportion. For instance, if 10 napkin sets are produced, the variable cost would be \(3.70 \times 10 = 37.00\). On the other hand, if no napkins are produced, the variable costs would be zero. These costs are essential for understanding how production levels affect overall spending.
Budget Constraint
A budget constraint represents the limitations on the spending capacity due to budgetary restrictions. In our scenario, the woman has a total monthly budget of \(1,870\).

The fixed costs of \(1,329.50\) must be covered first, leaving a total of \(540.50\) for variable costs. This constraint dictates the maximum number of napkin sets that can be produced within this budget. By analyzing these constraints, individuals can make informed decisions to optimize production without overspending.
Linear Equations
Linear equations are mathematical expressions that represent a straight line when plotted on a graph. In business, they are useful for modeling relationships between costs and production.

In the given scenario, the relation between the number of napkin sets (\(n\)) and their total variable costs is linear: \( 3.70n = 540.50\). This equation allows us to determine \(n\), the number of napkin sets that can be produced given the remaining budget.

Solving this equation by dividing both sides by \(3.70\), we find that \(n \approx 146\). Linear equations provide a straightforward method to calculate production levels, ensuring businesses remain within their budget.

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