Problem 58 - Thinking Mathematically

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 \(C = 100x + 100,000\), the revenue function is \(R = 300x\), and the break-even point occurs when 500 bicycles are produced and sold.

Step 1: Write Cost Function

By using the cost function formula \(C = m*x + b\), we can insert the given values for producing bicycles which are \(\$ 100\) per bike (\(m\)) and fixed costs of \(\$ 100,000\) (\(b\)). So the cost function becomes: \(C = 100x + 100,000\).

Step 2: Write Revenue Function

Now, we use the revenue function formula \(R = p*x\). The given selling price per bike is \(\$ 300\) (\(p\)), so the function becomes: \(R = 300x\).

Step 3: Determine Break-even Point

The break-even point is where revenue equals cost (\(R = C\)), so we set the equations equal to each other and solve for \(x\): \(300x = 100x + 100,000\). Solving this gives: \(200x = 100,000\, \Rightarrow\, x = 500\). This means that the company breaks even when it produces and sells 500 bicycles.

Key Concepts

Cost Function

The cost function is a vital component in understanding business expenses. It provides a mathematical model of how total production costs accumulate. In our bicycle manufacturing example, the cost function is given by:

• Fixed costs: These are expenses that do not change regardless of the number of items produced. Our fixed cost here is \( \\(100,000 \), which covers overheads like rent, salaries, and utilities.
• Variable costs: These costs vary directly with the quantity of production. For our company, it's \( \\)100 \) per bicycle.

The cost function equation is represented as \(C = 100x + 100,000\), where \(x\) is the number of bicycles produced and sold. This clearly accounts for both fixed and variable costs, allowing any manufacturer to predict future expenses with production changes.

Revenue Function

The revenue function represents the total income generated from selling goods or services. It's crucial for understanding potential earnings. For a business, maximizing revenue often translates to increasing sales volume at an optimal price.

In our scenario, the revenue function is defined as \(R = 300x\). Here, \(x\) is the number of bicycles sold, and \(\$300\) is the price at which each bicycle is sold.

This straightforward linear function highlights how revenue increases proportionally with the number of units sold. Therefore, the more bicycles the company sells, the greater the revenue, guiding strategic pricing and sales tactics.

Fixed Cost

Fixed costs are an inevitable part of any business structure, representing expenses that remain constant regardless of production output. These costs are essential to consider when planning finances, as they are incurred even when no products are being made.

In the bicycle company example, fixed costs total \( \$100,000 \). This covers consistent expenses such as:

• Rent for the production facility,
• Salaries for permanent staff,
• Insurance and utility bills.

Since fixed costs stay the same, they must be managed efficiently to ensure they are well covered by sales.

Variable Cost

Variable costs fluctuate with the production volume. They increase as more goods are produced and decrease when less is output. Managing these costs is important for maintaining profitability as production scales.

In our case, each bicycle incurs a variable cost of \( \$100 \). This includes materials, labor specific to bicycle assembly, and any other expenses that vary with production volume.

• Material costs like metal and rubber,
• Hourly wages for assembly line workers,
• Utility costs associated with running machinery.

Businesses aim to reduce variable costs per unit to boost margins, making the pricing of products and supplier negotiations vital.

Profit Analysis

Profit analysis is integral in assessing a company's financial health by understanding how costs and revenues impact the bottom line. The break-even analysis is a significant part of this, identifying the point where a business neither loses nor gains money.

In this bicycle business, the break-even point is reached when the company sells 500 bicycles, where the revenue matches the total costs. This is calculated by setting the cost function equal to the revenue function \(300x = 100x + 100,000\), solving to find \(x = 500\).

• At this point, revenue covers all fixed and variable costs.
• Producing and selling more than 500 bicycles results in profit.
• Understanding this helps businesses make informed decisions about scaling production.

Profit analysis, therefore, helps forecast financial outcomes and guide strategic planning to enhance profitability.