Problem 62 - 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 bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

Step-by-Step Solution

Answer

The cost function is \(C(x) = \$100,000 + \$100x\), the revenue function is \(R(x) = \$300x\), and the break-even point is 500 bicycles.

Step 1: Write the cost function, C

The cost function, \(C(x)\), for producing x bicycles is made up of the fixed costs plus the variable costs per bicycle multiplied by the number of bicycles produced. Therefore, \(C(x) = \$100,000 + \$100x\).

Step 2: Write the revenue function, R

The revenue function, \(R(x)\), is the selling price per bike multiplied by the number of bicycles sold. Therefore, \(R(x) = \$300x\).

Step 3: Determine the break-even point

The break-even point occurs when cost equals revenue, i.e., when \(C(x) = R(x)\). This gives the equation \$100,000 + \$100x = \$300x. Solving this equation for x gives \(x = \$100,000 / (\$300 - \$100) = 500\).

Step 4: Interpret the break-even point

The break-even point, x = 500, means that the company needs to produce and sell 500 bicycles in order to cover their costs. Any bicycles sold above this number will result in a profit, while selling fewer than this number will result in a loss.

Key Concepts

Cost Function

The cost function is a mathematical representation of the total cost incurred by a business in the production of goods. In our bicycle manufacturing example, the cost function, denoted by \( C(x) \), is composed of two parts:

• Fixed costs: These are the costs that remain constant regardless of the number of bicycles produced. In this case, the fixed costs are \( \\(100,000 \).
• Variable costs: These costs vary with the level of production. Here, the variable cost is \( \\)100 \) per bicycle.

Together, the cost function is expressed as: \( C(x) = \\(100,000 + \\)100x \), where \( x \) is the number of bicycles produced. This formula helps determine the total cost for producing any given number of bicycles.

Revenue Function

The revenue function represents the total income a company earns from selling its products. For our bicycle company, the revenue function is represented as \( R(x) \).

• Selling price per bicycle: Each bicycle sells for \( \\(300 \).
• Number of bicycles sold: This is represented by \( x \), the same variable used in the cost function.

Hence, the revenue function is \( R(x) = \\)300x \). This formula indicates the total revenue based on the number of bicycles sold. It helps to compare income against costs to determine profit or loss.

Fixed Costs

Fixed costs are a critical component of cost analysis in any business. These are expenses that do not change with the volume of products manufactured. In the bicycle manufacturing scenario, fixed costs total \( \$100,000 \).

• Fixed costs include expenses like rent for factory space, salaries of permanent staff, and machinery, assuming they do not fluctuate with output.
• Since fixed costs are constant, they impact the initial threshold of units that need to be sold to cover these costs, influencing the break-even point.

Understanding fixed costs is essential for planning and forecasting, as they can significantly affect the financial health of a business.

Variable Costs

Variable costs change in direct proportion to the quantity of goods produced. For our bicycle manufacturer, each bicycle incurs a variable cost of \( \$100 \).

• Variable costs typically include materials, direct labor, and utilities proportional to production activity.
• Unlike fixed costs, variable costs adjust with production levels, so producing more bicycles increases these costs correspondingly.

Acknowledging the nature of variable costs helps businesses in setting the right pricing strategies and ensuring efficient production processes to maintain profitability. Together with fixed costs, they form the broader cost structure that influences overall financial management.