Problem 4 - Intermediate Algebra

Question

A company charges a \(\$ 75\) setup fee plus \(\$ 5.25\) per shirt to silkscreen a design on specialty t-shirts. Write an equation that gives the cost of purchasing \(x\) shirts.

Step-by-Step Solution

Answer

The equation is \(C = 75 + 5.25x\).

Step 1: Identify the Fixed Cost

The company charges a fixed setup fee of $75 for the service, regardless of the number of shirts ordered. This is a constant value that will be included in the total cost equation.

Step 2: Recognize the Variable Cost

The cost per shirt is $5.25. Therefore, the total variable cost will depend on the number of shirts, represented by \(x\). The variable cost can be expressed as \(5.25x\).

Step 3: Combine Fixed and Variable Costs into an Equation

To find the total cost \(C\), you combine the fixed cost and the variable cost: \[C = 75 + 5.25x\]. This equation models the total cost for silkscreening \(x\) shirts.

Key Concepts

Fixed Cost

In business, a fixed cost is an expense that does not change with the level of goods or services produced by the business. These costs are typically associated with time and exist whether or not any products are being produced. Examples include rent, salaries, or setup fees like the one in our exercise.

In the scenario provided, the company imposes a fixed setup fee of \(\\(75\) for their silkscreen services. This fee remains constant regardless of the number of shirts a customer orders. Thus, even if they decide to purchase just one t-shirt or hundreds, the setup cost does not fluctuate, it always starts with \(\\)75\).

Understanding fixed costs is crucial for businesses because it helps in budgeting and financial planning. It represents the baseline cost that must be covered before any profit is made from the variable part of production.

Variable Cost

Variable costs are expenses that change directly and proportionally with the level of production. They increase as more goods are produced and decrease when fewer goods are produced. In the context of the exercise, the cost of silkscreening each shirt is \(\\(5.25\).

Therefore, for each additional shirt ordered, an extra \(\\)5.25\) is added to the total cost. This makes the total variable cost dependent on the number of shirts, represented by \(x\). As a formula, it can be expressed as \(5.25x\).

Understanding variable costs is essential, particularly in making decisions about pricing, production, and evaluating cost behavior over different levels of production. By managing these costs effectively, a company can optimize its profitability.

Cost Modeling

Cost modeling is a powerful financial tool that combines fixed and variable costs into a comprehensive framework to predict how total costs will change with varying levels of production or sales. It's essential for setting prices, managing costs, and forecasting profits.

In our exercise, the cost model is represented by the equation \(C = 75 + 5.25x\), which is a linear equation. Here, \(C\) denotes the total cost, \(75\) is the fixed cost, and \(5.25x\) represents the variable costs, where \(x\) is the number of shirts. This equation is linear because it graphs as a straight line, showing a direct relationship between the number of shirts and the total cost.

Linear cost models like this one are straightforward yet effective. They enable businesses to predict expenditures and plan for different scenarios, whether scaling up production or adjusting pricing strategies. Moreover, they offer valuable insights into how different cost elements interact as operational scales change.