Problem 72 - College Algebra and Calculus: An Applied Approach

Question

A company produces a product for which the variable cost is $$\$ 9.85$$ per unit and the fixed costs are $$\$ 85,000$$. The product sells for $$\$ 19.95$$ 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 \(C(x) = 85000 + 9.85x\), the revenue function is \(R(x) = 19.95x\), and the profit function is \(P(x) = 10.1x - 85000\).

Step 1: Formulate the Total Cost Function

The total cost \(C\) that the company incurs is a combination of fixed costs and variable costs. The variable costs are dependent on the quantity of units produced and sold, represented by \(x\). Here, the fixed cost is highlighted as $85,000$ and the variable cost per unit is $9.85$. Therefore, the total cost function can be formulated as: \(C(x) = 85000 + 9.85x\).

Step 2: Formulate the Revenue Function

The revenue \(R\) generated is a product of the selling price per unit and the number of units sold, \(x\). Given that the selling price per unit is $19.95$, the revenue function is then: \(R(x) = 19.95x\).

Step 3: Formulate the Profit Function

Profit \(P\) is determined by subtracting the total cost from the total revenue. From step 1, we found that \(C(x) = 85000 + 9.85x\) and from step 2, \(R(x) = 19.95x\). Hence, using the formula \(P = R - C\) gives: \(P(x) = R(x) - C(x) = 19.95x - (85000 + 9.85x) = 10.1x - 85000\).

Key Concepts

Variable Cost

Variable costs are expenses that change in proportion to the activity or production output of a business. In the context of the given exercise, the variable cost is the amount it costs to produce one additional unit of a product. Here, it is stated to be \(\$9.85\) per unit. As a company produces and sells more units, the total variable cost increases linearly based on the number of units, represented mathematically as \(9.85x\), where \(x\) is the number of units produced and sold.

Understanding variable costs is crucial for businesses as it impacts pricing and profit margins, especially when considering how to scale production and manage expenses to maintain profitability.

Fixed Costs

Fixed costs are business expenses that remain constant regardless of the number of units produced or sold. In this exercise, the company's fixed costs are \(\$85,000\). These costs are incurred even when production levels are zero and remain unchanged as production increases.

Common examples of fixed costs include rent, salaries, insurance, and utilities. For a business, knowing their fixed costs is essential for budgeting and setting financial targets. A high fixed cost structure might require higher volumes of production or sales to break even and achieve profitability.

Revenue Function

The revenue function, symbolized as \(R(x)\), expresses the total amount of money a company brings in from selling its products. It's direct, depending on the number of units sold, \(x\), and the selling price per unit.

In our exercise, the product is sold for \(\$19.95\) per unit, making the revenue function \(R(x) = 19.95x\). This function is linear, indicating a direct relationship where revenue increases with every additional unit sold. For businesses, analyzing this function helps in understanding sales performance and forecasting future revenue based on selling prices and anticipated sales volumes.

Profit Function

Profit, the cornerstone of every business, is what remains after accounting for all costs from total revenue. The profit function is denoted as \(P(x)\) and it's defined by subtracting total cost from total revenue. From our problem, the profit function is \(P(x) = 19.95x - (85000 + 9.85x) = 10.1x - 85000\).

This simplified formula tells us that for each unit sold, \(\$10.1\) contributes to profit after covering variable costs. However, for the company to start making a profit, they need to first cover the \(\$85,000\) in fixed costs. This function helps businesses determine the 'break-even point', which is the minimum number of units that need to be sold to start earning a profit.

Functions of the number of units

The concept of functions of the number of units is all about relating how different economic measures such as cost, revenue, and profit change with respect to the number of units produced and sold. These functions help in planning and operational decisions.

For instance, with the total cost function \(C(x) = 85000 + 9.85x\), leaders can estimate the cost at different production levels. Similarly, the revenue and profit functions, \(R(x) = 19.95x\) and \(P(x) = 10.1x - 85000\), provide insights into potential earnings and financial health at varying sales volumes. These functions underpin key business activities like budgeting, pricing strategies, and financial forecasting.