Problem 99 - Algebra and Trigonometry

Question

A company is planning to manufacture mountain bikes. The fixed monthly cost will be \(\$ 100,000\) and it will cost \(\$ 100\) to produce each bicycle. A. Write the cost function, \(C,\) of producing \(x\) mountain bikes B. Write the average cost function, \(\bar{C},\) of producing x mountain bikes. C. Find and interpret \(\bar{C}(500), \bar{C}(1000), \bar{C}(2000),\) and \(\bar{C}(4000)\) D. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C}\) ? Describe what this means in practical terms.

Step-by-Step Solution

Answer

A. The cost function is \(C(x) = 100000 + 100x\), B. The average cost function is \(\bar{C}(x) = 100000/x + 100\). C. Average costs for 500, 1000, 2000, 4000 bikes are $300, $200, $150, and $125 respectively. D. The horizontal asymptote is \(y = 100\) which means that as the production volume increases indefinitely, the average cost per bike approaches $100.

Step 1: Write the cost function

We have a fixed cost of $100,000 and a variable per unit cost of $100. This forms a linear equation \(C(x) = 100000 + 100x\). So the cost of producing \(x\) bikes is represented by \(C(x) = 100000 + 100x\) where \(C\) stands for cost and \(x\) stands for the quantity of bikes.

Step 2: Write the average cost function

The average cost function is derived by dividing total cost \(C(x)\) by the number of bikes \(x\). Therefore we get, \(\bar{C}(x) = C(x) / x = {100000\/x + 100}\), which represents the average cost of producing \(x\) number of bikes.

Step 3: Find and interpret the average cost

We use the average cost function to find \(\bar{C}(500), \(\bar{C}(1000), \(\bar{C}(2000), and \(\bar{C}(4000). The results will be \(\bar{C}(500) = 100000/500 + 100 = 300\), \(\bar{C}(1000) = 100000/1000 + 100 = 200\), \(\bar{C}(2000) = 100000/2000 + 100 = 150\), \(\bar{C}(4000) = 100000/4000 + 100 = 125\). This means that the average cost decreases as we produce more bikes.

Step 4: Determine the horizontal asymptote and provide an interpretation

The horizontal asymptote of \(\bar{C}(x)\) is calculated by considering the limit of the function as \(x\) approaches \(\infty\). In this case, as \(x\) approaches infinity, the \(100000 / x\) term approaches 0, leaving only the \(100\) term. So the horizontal asymptote is \(y = 100\). In practical terms, this means that as production volume increases, the average cost per bike approaches $100, regardless of the initial fixed cost.

Key Concepts

Fixed Costs

Fixed costs are the consistent expenses that a business incurs regardless of its production level or sales volume. For this mountain bike manufacturing scenario, the fixed costs amount to $100,000 each month. This expense does not fluctuate with the number of bikes produced; it remains constant whether the company produces one bike or a million.

The primary components of fixed costs often include:

• Rent or lease payments for facilities
• Salaries of permanent staff
• Insurance premiums
• Equipment depreciation

Understanding fixed costs is crucial as they are not impacted by production levels, thus affecting the overall cost structure and pricing strategy.

Variable Costs

Variable costs change depending on the amount of production output. In our example, each mountain bike costs $100 to produce, which represents the variable cost per unit. This type of cost fluctuates with the level of production, meaning that more bikes mean higher total variable costs.

Variable costs are typically associated with:

• Raw materials
• Direct labor costs, depending on hours worked or units produced
• Utility costs tied to production processes

Variable costs are essential for businesses to monitor as they have a direct impact on profit margins and operational flexibility.

Average Cost Function

The average cost function is an important concept in economics. It helps businesses determine the cost per unit of production by dividing the total costs by the number of units produced. In this case, the average cost function is given by \( \bar{C}(x) = \frac{C(x)}{x} = \frac{100000}{x} + 100 \), which shows how costs per bike decrease as the production increases.

The average cost function provides insights into:

• Efficiency of production processes
• Optimal pricing strategies for cost coverage
• Initial cost burden per unit at lower production levels

By analyzing this function, companies can find the most economical production levels, ensuring better resource allocation and financial planning.

Horizontal Asymptote

The horizontal asymptote of the average cost function \( \bar{C}(x) \) is a line that the graph approaches as the number of produced units becomes very large. In our scenario, the asymptote is \( y = 100 \), which can be understood by analyzing the function's behavior as \( x \) (the number of bikes) tends towards infinity.

This horizontal asymptote indicates:

• Long-term cost efficiency, showing the minimum achievable cost per unit
• Impact of fixed costs diminishing over high production volumes
• The role of scaling in reducing the average cost dramatically

Understanding the horizontal asymptote helps businesses envision their cost structure in the long run, guiding important decisions about scaling and expansion.

Economies of Scale

Economies of scale refer to the cost advantages that enterprises obtain due to their scale of operation, with cost per unit of output generally decreasing with increasing scale as fixed costs are spread out over more units of output. This is clearly visible in the mountain bike example, where increasing production reduces the average cost from $300 at 500 units to $125 at 4000 units.

Through economies of scale, businesses can achieve:

• Lower average costs and increased competitive advantage
• Enhanced efficiency through optimized resource utilization
• Better pricing power and market influence

These advantages make economies of scale an essential consideration for businesses aiming for growth and sustainability.