Question

A pretzel stand has fixed costs for the facility and cooking supplies of \(\$ 500\). The cost for the labor and supplies to cook one pretzel after the pretzel stand has been set up is \(\$ 2\) per pretzel. What is the graph of the cost function \(c(x)\) given \(x\) pretzels?

Short answer

The graph of the cost function is a line starting at (0, 500) with a slope of 2.

Step-by-step solution

Understand Fixed and Variable Costs

Identify and understand the fixed and variable costs in the problem. The fixed cost (costs that do not change with the number of pretzels) is $500 for the facility and cooking supplies. The variable cost (costs that change with the number of pretzels) is $2 per pretzel.

Create the Cost Function

The total cost is composed of the fixed costs plus the variable costs. The cost function can be expressed as \( c(x) = 500 + 2x \), where \( c(x) \) is the total cost and \( x \) is the number of pretzels.

Determine the Graph's Features

The cost function \( c(x) = 500 + 2x \) is a linear equation. The y-intercept of the graph is 500, meaning if no pretzels are made, the cost is \(500. The slope of the graph is 2, indicating the cost increases by \)2 for every additional pretzel made.

Graph the Cost Function

Plot the graph with the x-axis representing the number of pretzels, \( x \), and the y-axis representing the total cost, \( c(x) \). Start at the y-intercept (0, 500) and use the slope to plot additional points, like (1, 502), (2, 504), continuing in this pattern.

Key concepts

Fixed Costs

Fixed costs are the expenses that remain constant regardless of the number of goods or services produced. In the case of the pretzel stand, fixed costs amount to $500. These include expenses like rent for the facility and initial cooking supplies that do not fluctuate with the number of pretzels produced.

Understanding fixed costs is crucial because even if no pretzels are made, the business still incurs these $500 costs.

• Fixed costs are often time-bound (e.g., monthly rent).
• These costs do not change with production volume.

The fixed cost component helps determine the base level of funding needed to start and maintain operations independent of production output.

Variable Costs

Unlike fixed costs, variable costs change in direct proportion to the level of production or operations. For the pretzel stand, variable costs are $2 per pretzel. Each time a pretzel is made, the cost goes up by $2, covering additional supplies and labor needed for each pretzel.

Variable costs are critical in calculating the total cost of production as they directly affect profitability depending on how many items are produced.

• Variable costs increase with each product unit created.
• They are essential in economic decision-making and cost-benefit analysis.

Knowing variable costs helps in pricing strategies and cost management, influencing how businesses scale up their operations efficiently.

Linear Equations

A linear equation forms a straight line when plotted on a graph. It describes a direct relationship between two variables. In this case, the cost function equation for the pretzel stand is represented as:\[ c(x) = 500 + 2x \] Here, \( c(x) \) is the total cost, and \( x \) represents the number of pretzels. The formula highlights how each pretzel incrementally adds \(2 to the base fixed cost of \)500.

Linear equations are essential in understanding relationships between different factors in a predictable manner. The following elements are notable in a linear equation:

• The y-intercept, here $500, signifies the starting point or fixed cost irrespective of units produced.
• The slope, here 2, indicates how steeply costs rise with each additional unit of production, i.e., each pretzel.

Analyzing linear equations allows businesses to predict future trends, manage resources, and understand the impacts of scaling production.