Problem 57 - Thinking Mathematically

Question

Exercises 57-60 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 is \(C(x) = 18000 + 20x\), the revenue function is \(R(x) = 80x\), and the break-even point is at 300 canoes.

Step 1: Write the cost function, C

The total cost, C, to produce \(x\) canoes is equal to the fixed cost plus the variable cost per canoe times the number of canoes produced. So, the cost function, \(C(x)\), is \(C(x) = 18000 + 20x\).

Step 2: Write the revenue function, R

Total revenue, R, from selling \(x\) canoes is equal to the selling price per canoe times the quantity of canoes sold. As such, the revenue function, \(R(x)\), is \(R(x) = 80x\).

Step 3: Determine the break-even point

The break-even point happens when total revenue \(R(x)\) equals total cost \(C(x)\). To find it, therefore, set \(R(x) = C(x)\) and solve for \(x\). So, the equation becomes \(80x = 18000 + 20x\). Solving this yields \(x = 300\). This means the company needs to sell 300 canoes to cover their costs and start making a profit.

Key Concepts

Cost Function

The cost function represents the total expenses incurred by a company in the production process. For the manufacture of small canoes, the cost is split into fixed costs and variable costs. The fixed cost in this scenario is \(18,000, which is a constant overhead cost regardless of the number of canoes produced. Then there's the variable cost, which is \)20 per canoe.

The cost function, represented as \(C(x)\), combines these costs into an equation:

• Fixed Cost: This is the expense that remains the same no matter the level of production. Here, it's \(18,000.
• Variable Cost: This varies with the production quantity, in this case, \)20 for each canoe.
• Total Cost Formula: \(C(x) = 18000 + 20x\)

Thus, to find the total cost of producing \(x\) canoes, simply plug in the number of canoes for \(x\) in the cost function. This expression effectively showcases how both constant and changing costs are woven into the fabric of production expenses.

Revenue Function

The revenue function is a mathematical expression of the total income generated from selling a product. For the business making canoes, the income stream comes from the selling price per unit multiplied by the number of units sold.

The given selling price is \(80 per canoe, so the revenue function, \(R(x)\), is the product of this price and the quantity sold.

• Selling Price: The amount the company charges for each canoe, here it is \)80.
• Volume Sold: The number of canoes sold, represented by \(x\).
• Revenue Formula: \(R(x) = 80x\)

To calculate the revenue, you insert the number of canoes sold into \(x\) in this function. This equation helps businesses understand how changes in sales volume directly influence their income.

Fixed Cost

Fixed cost in business denotes expenses that do not change with production volume. They are the costs that remain steady irrespective of how many products a company makes, such as rent, salaries, or initial investments in machinery.

In the canoe manufacturing example, the fixed cost amounts to $18,000.

• Unchanging Expenses: These won't fluctuate even if zero canoes are produced.
• Initial Financial Outlay: Must be accounted for before any production begins.
• Economic Stability: Knowing fixed costs helps in tracking really variable expenses.

Understanding fixed costs is crucial for forecasting and planning, as these outlays form the base expenditure of the company and must be covered by any revenue generated.

Variable Cost

Variable costs are those expenses that change with production volume. Essentially, they rise with more production and decrease with less. Each additional canoe produced in our problem incurs a \(20 charge, directly linking variable cost to production quantity.

Given this cost structure:

• Cost Per Unit: Reflecting cost changes in relation to activity levels, here \)20 per canoe.
• Direct Correlation: More units produced results in higher variable costs.

Hence, the total variable costs vary linearly with production, represented by the term \(20x\) in the cost function. Mastery of variable cost behavior is essential for accurately scaling production aims and understanding financial impacts of scaling operations.

Selling Price

The selling price is the amount a company charges customers for a unit of product. It's a significant factor in determining both revenues and profits. For the canoe business, the selling price is set at $80 per canoe, designed to cover various costs and yield a profit.

Here’s why it matters:

• Pricing Strategy: Optimal pricing accommodates costs, desired profit, and market competition.
• Consumer Choice: Influences purchase decisions and brand perception.
• Margin Calculation: Larger profit margins arise from smart pricing, ensuring financial viability.

Setting the right selling price is crucial since it influences the revenue function and overall profitability. It's a balancing act between gaining competitive advantage and maintaining sound financial health.