Problem 61 - College Algebra Essentials

Question

Exercises \(61-64\) describe a number of business ventures. For each exercise a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

Step-by-Step Solution

Answer

The cost function for the company is \(C(x) = 18000 + 20x\), the revenue function is \(R(x) = 80x\), and the break-even point is \(x = 300\). This means the company needs to sell at least 300 canoes to cover all the costs.

Step 1: Compute the Cost Function

The cost function, \(C(x)\), is made up of two components: fixed costs and variable costs. The fixed cost for this company is \( \$ 18000 \), and the cost to produce each canoe (variable cost) is \( \$ 20 \). So, in total, the cost function will be: \(C(x) = 18000 + 20x\) where \(x\) is the number of canoes produced.

Step 2: Compute the Revenue Function

The revenue function, \(R(x)\), is simply the selling price of the product multiplied by the quantity sold. Since the selling price per canoe is \( \$ 80 \), the revenue function will be: \(R(x) = 80x\) where \(x\) is the number of canoes sold.

Step 3: Find the Break-Even Point

The break-even point occurs when \(C(x) = R(x)\). That is when the costs and revenues are equal which implies no profit or loss is being made. This happens when: \n\(18000 + 20x = 80x\). Solving this equation for \(x\) will yield the break-even point.

Step 4: Solve the Equation

To solve the equation \(18000 + 20x = 80x\), first rearrange the equation by subtracting \(20x\) from both sides which results in \(60x = 18000\). Then, divide both sides by 60 to get the value of \(x\), which represents the number of canoes needs to be sold to break-even.

Step 5: Interpret the Result

The break-even point meaning, in this context, is the minimum number of canoes the company has to sell so that the total costs (fixed and variable costs) are covered by the total income gained from selling.

Key Concepts

Cost Function

The cost function is a mathematical expression that calculates the total expenses needed to produce a certain number of goods or services. In this example, we have the production of small canoes. The cost function, denoted as \(C(x)\), includes two key components:

• **Fixed Costs:** These are costs that remain constant regardless of the number of canoes produced. For our canoe company, the fixed cost is \( \\(18,000 \). This covers things like rent and salaries that don't change with production levels.
• **Variable Costs:** These costs depend on the number of canoes produced. Here, the variable cost is \( \\)20 \) per canoe, which covers materials and labor associated with each unit.

Therefore, the cost function combines both fixed and variable costs and is expressed as:\[C(x) = 18000 + 20x\]where \(x\) represents the number of canoes produced. Understanding this function is crucial as it helps businesses anticipate expenses and manage budgets accordingly.

Revenue Function

The revenue function calculates the total income generated from selling goods or services. In our canoe-selling context, the revenue function is represented by \(R(x)\). It simplifies the process of estimating potential earnings from sales.

• The **Selling Price** for each canoe is \( \$80 \), which is the amount charged to customers per unit.

The revenue function is therefore calculated based on this fixed price and the number of canoes sold, which is described by the formula:\[R(x) = 80x\]where \(x\) is the number of canoes sold. This function helps companies project income, setting benchmarks for financial success. By predicting revenue, businesses can strategize to ensure profitability.

Fixed Costs

Fixed costs are essential expenses that do not fluctuate with the level of production or sales. These costs occur whether your business produces one canoe or thousands. In the canoe example:

• The **Fixed Cost** is \( \$18,000 \). This includes overheads such as rent, salaries, and insurance.

Such costs are critical as they remain constant, providing stability but also representing a hurdle that needs to be covered by sales revenue. Calculating fixed costs accurately is crucial for determining the break-even point and overall profitability. A clear understanding of fixed costs allows businesses to make informed pricing and production decisions.

Variable Costs

Variable costs differ depending on the number of goods or services produced. They are directly linked to production process inputs, influencing the cost function significantly.

• For our canoe manufacturing company, the **Variable Cost** is \( \$20 \) per canoe. This includes expenses such as materials and direct labor involved in creating each unit.

In the cost function \(C(x) = 18000 + 20x\), the term \(20x\) reflects these variable costs. If production increases, variable costs rise in a linear manner, affecting total expenses and hence influencing profitability. Efficiently managing variable costs is key to maintaining a healthy profit margin and achieving business growth.