Problem 71 - College Algebra and Calculus: An Applied Approach

Question

A company produces a product for which the variable cost is $$\$ 11.75$$ per unit and the fixed costs are $$\$ 112,000$$. The product sells for $$\$ 21.95$$ per unit. Let \(x\) be the number of units produced and sold. (a) Add the variable cost and the fixed costs to write the total cost \(C\) as a function of the number of units produced. (b) Write the revenue \(R\) as a function of the number of units sold. (c) Use the formula \(P=R-C\) to write the profit \(P\) as a function of the number of units sold.

Step-by-Step Solution

Answer

The total cost function is given by \(C(x) = 11.75x + 112000\), the revenue function is \(R(x) = 21.95x \), and the profit function is \(P(x) = 10.2x - 112000\).

Step 1: Formulate the Total Cost

The total cost \(C(x)\) of producing \(x\) units is the sum of the variable cost (per unit) and the fixed costs. Therefore, \(C(x) = 11.75x + 112000\)

Step 2: Compute the Revenue

The revenue \(R(x)\) from selling \(x\) units is simply the selling price times the number of units sold. So, \(R(x) = 21.95x \)

Step 3: Determine the Profit

The profit \(P(x)\) is the revenue minus the total cost. Therefore, we have \(P(x) = R(x) - C(x) = 21.95x - (11.75x + 112000) = 10.2x - 112000 \)

Key Concepts

Variable Cost

The variable cost is an essential concept for understanding business expenses. It refers to the costs that fluctuate with the level of production. In this exercise, the variable cost per unit is \( \$ 11.75 \). This means that for every additional unit produced, this cost is added to the total production cost.
To put it simply, the more you produce, the higher the variable costs accumulate, as they are incurred on a per-item basis.

• They are directly proportional to the number of units produced.
• Common examples include raw materials and direct labor costs.

Understanding this concept is crucial because it directly impacts pricing, budgeting, and profit planning strategies.

Fixed Costs

Fixed costs are the costs that do not change with the level of production. Regardless of how many units are produced, these costs remain constant. For the company in our exercise, the fixed costs amount to \( \$ 112,000 \). This stays the same whether the company produces one unit or thousands of units.
Examples of fixed costs include rent, salaries, and insurance.

• They provide stability and predictability to a business's financial planning.
• Important for calculating break-even points and profits.

\(C(x)\) as a total cost function comprises both fixed and variable costs, and it is key to understanding the full financial picture of any production process.

Profit Function

The profit function is a fundamental tool for assessing a business's financial health. It is determined by subtracting the total cost from the total revenue, represented by \( P(x) = R(x) - C(x) \). Here, \( P(x) \) is the profit for \( x \) units produced and sold.
In the given problem:

• Revenue Function \( R(x) = 21.95x \)
• Total Cost Function \( C(x) = 11.75x + 112000 \)

Thus, the profit function becomes \( P(x) = 10.2x - 112000 \).
This equation quickly conveys how the number of units sold affects the profit. If the units sold don't cover both variable and fixed costs, the business will incur a loss. Knowing this helps in predicting future profits or losses depending on changes in either sales volume or costs.

Revenue Function

The revenue function is a representation of a company's income based on sales. It outlines how much money is brought in before any costs are deducted. For this exercise, the revenue \( R(x) \) is given by the equation \( R(x) = 21.95x \), where \( x \) signifies the number of units sold.
This equation indicates that each unit sold contributes \( \$ 21.95 \) to the revenue. The simplicity of this function lies in its straightforward formula, showing a linear relationship between the sales volume and revenue.

• Directly influences both the profit and break-even analysis.
• Requires a deeper understanding when setting pricing strategies to ensure profitability.

With each increase in units sold, the revenue rises proportionally, which is crucial for maintaining a positive cash flow.