Problem 38 - Elementary Algebra Within Reach

Question

A company produces hockey sticks that sell for \(\$ 79\) per unit. The cost of producing each unit is \(\$ 53.25\), and the company has fixed costs of \(\$ 1000\). (a) Use a verbal model to show that the cost \(C\) of producing \(x\) units is \(C=53.25 x+1000\) and the revenue \(R\) from selling \(x\) units is \(R=79 x\). (b) Use a graphing calculator to graph the cost and revenue functions in the same viewing window. Approximate the point of intersection of the graphs and interpret the result.

Step-by-Step Solution

Answer

The break-even point obtained from the point where the Cost and Revenue lines intersect represents the number of units that must be sold in order for the company to cover its production costs. This is a critical value that helps the company understand its minimum sales requirements to avoid any losses.

Step 1: Formulate Cost and Revenue Equations

The cost \(C\) of producing \(x\) units is derived from the fixed cost of production and the variable cost per unit: \(C = 53.25x + 1000\). The revenue \(R\) gained from selling \(x\) units is directly proportional to the number of units sold: \(R = 79x\).

Step 2: Graph the Equations

Using a graphing calculator, graph the cost and revenue equations on the same graph. The y-axis represents the cost and revenue, while the x-axis represents the number of units. The cost function will be a straight line starting from \(y = 1000\) (fixed costs), and the revenue function will be a straight line starting from the origin.

Step 3: Find and Interpret the Point of Intersection

The point where both lines intersect represents the break-even point i.e., the number of units that need to be sold to cover the costs of production. Use the graphing calculator to find this point and interpret it.

Key Concepts

Revenue Function

In a business context, the revenue function represents the total income generated from selling goods or services. For the case of the hockey sticks, this function helps determine income based on the number of units sold. Simply put, the revenue function can be expressed as the product of the price per unit and the number of units sold.

The formula for the revenue function given in this problem is:

• \( R = 79x \)

Where:

• \( R \) is the total revenue.
• \( 79 \) is the selling price per hockey stick.
• \( x \) is the number of hockey sticks sold.

This linear equation shows that revenue increases directly with every additional unit sold. In this case, for each stick sold, the revenue increases by $79.

Cost Function

The cost function is crucial for determining the total expenses associated with producing a certain number of products. It encompasses both variable and fixed costs, thereby giving a complete picture of the financial outlay needed for production. In this example, each hockey stick has a specific production cost or variable cost.The cost function for producing hockey sticks can be defined as:

• \( C = 53.25x + 1000 \)

Where:

• \( C \) is the total cost of production.
• \( 53.25 \) is the cost to produce one unit, or hockey stick.
• \( x \) represents the number of units produced.
• \( 1000 \) represents the fixed costs, which we will discuss more in the next section.

The presence of fixed costs means that even if no products are made, there's still a minimum cost to pay.

Fixed Costs

Fixed costs refer to the expenses that do not change with the level of production or sales activity. These costs remain constant in the short term regardless of how much is produced. In the given problem, the fixed costs are $1000. You incur these costs even if you don't produce or sell any items. Here's how fixed costs affect the cost function:

• The term \( + 1000 \) in the cost function \( C = 53.25x + 1000 \) signifies the fixed costs.

Understanding fixed costs is crucial as they contribute to the break-even analysis, where the goal is to assess how many units are needed to cover both fixed and variable costs effectively. This is covered by the intersection in the graph mentioned briefly in upcoming parts.

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical equations and functions. It allows you to plot and explore complex equations seamlessly. In this exercise, a graphing calculator is used to plot both the revenue and cost functions on the same axis. When you enter the equations:

• Revenue: \( R = 79x \)
• Cost: \( C = 53.25x + 1000 \)

into a graphing calculator, the cost graph will start at $1000 on the y-axis due to the fixed costs, while the revenue graph will start at the origin (0,0). The intersection of these lines indicates the break-even point, which is the number of units that must be sold to exactly cover all costs without making a profit or a loss. This point is crucial for businesses to identify the minimum sales target needed to avoid losses.