Problem 61 - College Algebra

Question

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 he number of canoes produced and sold, )

Step-by-Step Solution

Answer

The cost function is \(C(x) = \$18,000 + \$20x.\) The revenue function is \(R(x) = \$80x.\) The break-even point is when the company produces and sells 300 canoes.

Step 1: Identification of Variables

The given constants are: Fixed cost, \(F = \$18,000\), Variable cost per unit, \(V = \$20\), Selling price per unit, \(P=\$80\), Quantity of units produced and sold, \(x\).

Step 2: Formulation of Cost Function

The cost function, \(C(x)\), represents the total cost for producing and selling \(x\) number of units. It comprises of the fixed cost (\$18000) and the variable cost. Variable cost is the product of the cost for producing one unit and the quantity of units. Hence, the cost function is \(C(x) = F + Vx = \$18,000 + \$20x.\)

Step 3: Formulation of Revenue Function

The revenue function, \(R(x)\), represents the total revenue earned by selling \(x\) quantity of units. It's the product of the quantity of units and the selling price per unit. Hence, the revenue function is \(R(x) = Px = \$80x.\)

Step 4: Determination of Break-even point

The break-even point can be found by equating the cost function to the revenue function. This gives us the quantity at which the revenue generated equals the cost incurred, resulting in neither profit nor loss. So, solve for \(x\) in \(C(x) = R(x)\), which results in \$18,000 + \$20x = \$80x. Solving the equation: \$20x = \$80x - \$18,000, gives us \(x = \$18,000 / \$60 = 300\).

Key Concepts

Cost Function

A cost function is used to calculate the total cost incurred by a company in the production and selling of products. It combines both fixed and variable costs. Fixed costs are the expenses that do not change, such as rent or manager salaries, regardless of how many units are produced. Variable costs, however, fluctuate based on the number of units produced. These include materials and labor costs.

For our canoe manufacturing example, the cost function is denoted by

• Total Fixed Costs = \( \\(18,000 \)
• Variable cost per canoe = \( \\)20 \)
• Thus, the cost function is: \( C(x) = \\(18,000 + \\)20x \)

This function gives you the total cost for producing and selling \( x \) number of canoes. By understanding this function, businesses can predict how costs will rise with increased production.

Revenue Function

Creating a revenue function is crucial for understanding how much money a company can generate from selling its products. This function multiplies the selling price per unit by the number of units sold. It’s important to accurately set the selling price, as it impacts profitability and competitiveness in the market.

For the small canoe business:

• Selling price per canoe = \( \$80 \)
• The revenue function is: \( R(x) = 80x \)

This function helps the company estimate earnings based on different levels of sales. Revenue is fundamental as it indicates the income generated from core business activities.

Break-even Point

The break-even point is vital as it tells a business when it will start making profits. This is the stage where total cost equals total revenue, resulting in no net profit or loss. It’s calculated by setting the cost function equal to the revenue function.

In our example, the break-even occurs when:

• Cost function: \( C(x) = \\(18,000 + \\)20x \)
• Revenue function: \( R(x) = \\(80x \)
• Setting them equal: \( \\)18,000 + \\(20x = \\)80x \)
• Solving gives \( x = 300 \)

This means the company needs to produce and sell 300 canoes to cover all costs and not make a loss. Beyond this, every sale contributes to profit. Understanding this point is essential for business planning and financial forecasts.

Fixed and Variable Costs

Fixed and variable costs are pivotal in understanding a business's overall cost structure. Fixed costs remain constant regardless of the production volume. Examples include rent or salaried employees’ wages. They don't change with the level of production.

Variable costs, however, change with the production volume. For each additional unit made, these costs increase. They assimilate costs such as raw materials and direct labor.

In the pigeon canoe company:

• Fixed costs = \( \\(18,000 \)
• Variable cost per canoe = \( \\)20 \)

By distinguishing between fixed and variable costs, management can make informed decisions about scaling production and improving cost-efficiency. Understanding these concepts ensures a business can appropriately allocate resources to maximize profitability.