Problem 62 - 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 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 given by \(C(x) = 100000 + 100x\), the revenue function is given by \(R(x) = 300x\), and the break-even point occurs when \(x = 500\). This means the company needs to sell 500 bicycles to cover their costs.

Step 1: Formulating the Cost Function

The cost function typically outlines both the variable and fixed costs. The fixed cost, \(C_{f}\), is \$100,000, and the variable cost, \(C_{v}\), is \$100 per bicycle. \nSo the cost function, \(C(x)\), becomes: \[C(x) = C_{f} + C_{v} \cdot x\] Substituting the given values, we get: \[C(x) = 100000 + 100x\]

Step 2: Formulating the Revenue Function

The revenue function represents the total income from selling a certain number of goods. The revenue gained per bicycle sold is \$300. Consider \(x\) to be the quantity of bicycles produced and sold. \nSo the revenue, \(R(x)\), is obtained by multiplying the quantity \(x\) with the selling price of each bicycle. Thus, \[R(x) = 300x\]

Step 3: Calculating the Break-even Point

Break-even point is a crucial concept in businesses. It indicates the units of production and sales at which the company neither makes a profit nor a loss. Mathematically, it's the point where the cost function equals the revenue function. \nTherefore: \[C(x) = R(x)\] Substituting the cost function and revenue function, we get: \[100000 + 100x = 300x\] Solving this equation should give us the break-even point.

Step 4: Solving for x

By transposing the equation, we get: \[100000 = 200x\] Then, divide both sides with 200 to find the value of \(x\). \[x = 100000 / 200 = 500\]

Key Concepts

Cost Function

In business, the cost function plays a crucial role in understanding the overall expenses associated with production. It combines fixed and variable costs to give a comprehensive picture of the cost structure.
The fixed cost is the expense incurred that does not change with the level of goods produced. In this exercise, the fixed cost is given as \(100,000, which the company would have to pay regardless of the number of bicycles it produces.
The variable cost varies with the level of production. For this bicycle manufacturer, it is \)100 per bicycle. This means for each bicycle produced, an additional \(100 is added to the total expenses.
Hence, the cost function, when combining both these costs, becomes:

• Fixed costs of \)100,000
• Variable costs of $100 per bicycle

This results in the equation:\[C(x) = 100000 + 100x\]where \(C(x)\) represents the total cost for producing \(x\) number of bicycles.

Revenue Function

The revenue function is essential for businesses as it indicates the total income earned from selling goods or services. For the bicycle manufacturing company, the revenue per unit sold is \(300.
This means that for every bicycle sold, the company earns \)300. If the number of bicycles sold is denoted by \(x\), the total revenue is a direct multiplication of these two values:

• Selling price per bicycle: $300
• Number of bicycles sold: \(x\)

Thus, the revenue function can be written as:\[R(x) = 300x\]This approach helps the company calculate potential income based on how many bicycles are sold.

Fixed Cost

Understanding fixed costs is pivotal for businesses as these are expenses that remain constant, irrespective of production levels. In this scenario, the bicycle manufacturer incurs a fixed cost of $100,000.
This includes expenses like:

• Rent of the manufacturing facility
• Salaries of permanent staff
• Insurance and utilities

These are the overhead costs required to keep the business operational. Since fixed costs don't change no matter how many bicycles are produced, they form a significant part of the cost function and provide stability to financial forecasting.

Variable Cost

Variable costs are equally important as they fluctuate with the level of output in a business. They are the direct costs associated with the production of each unit.
In the context of the bicycle company:

• The variable cost per bicycle is $100
• More bicycles produced mean higher total variable costs

These costs typically include:

• Materials used in production
• Hourly wages for labor directly involved in manufacturing

Calculating variable costs helps in understanding how increasing or decreasing production levels can impact overall expenses and profitability.