Problem 100 - College Algebra

Question

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000 .\) It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C\), of producing \(x\) pairs of shoes. b. Write the average cost function, \(C\), of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), C(10,000),\) and \(C(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(C ?\) Describe what this represents for the company.

Step-by-Step Solution

Answer

The total cost function is \(C(x) = 300,000 + 30x\) and the average cost function is \(\bar{C}(x) = \frac{300000 + 30x}{x}\). Costs for specific values are as follows: \(\bar{C}(1000) = 330\), \(C(10,000) = 600,000\), and \(C(100,000) = 3300000\). The horizontal asymptote is at $30, which represents the cost per shoe when the company produces a very large number of shoes, effectively spreading out the fixed cost over more units and minimising the average cost per unit.

Step 1: Write the Total Cost Function

The total cost function, \(C(x)\), consists of a fixed element, which is $300,000 monthly, and a variable element that changes with the number of produced shoes which costs $30 per pair. Therefore, the total cost function becomes \(C(x) = 300000 + 30x\).

Step 2: Write the Average Cost Function

The average cost function, \(\bar{C}(x)\), is the total cost function divided by the number of units produced. Thus, the average cost function becomes \(\bar{C}(x) = \frac{C(x)}{x} = \frac{300000 + 30x}{x}\).

Step 3: Calculate the specified cost values

For \(\bar{C}(1000)\), simply substitute \(x = 1000\) into the average cost function to get \(\bar{C}(1000) = \frac{300000 + 30*1000}{1000} = 330\). This means that it costs the company $330 per shoe when it produces 1000 shoes. For \(C(10,000)\) and \(C(100,000)\), substitute \(x = 10,000\) and \(x = 100,000\) respectively into the total cost function to find the total cost when producing these numbers of shoes: \(C(10,000) = 300000 + 30*10000 = 600000\) and \(C(100,000) = 300000 + 30*100000 = 3300000\). These results tell us that the total cost for the company to produce 10,000 shoes is $600,000, and $3,300,000 for 100,000 shoes.

Step 4: Find the Horizontal Asymptote and Interpret

The horizontal asymptote of the average cost function, \(\bar{C}(x)\), is the value that the function approaches as \(x\) approaches infinity. Looking at the average cost function, it can be seen that as \(x\) approaches infinity, the 300000/x term vanishes, so the function approaches $30. This asymptote represents the cost per shoe when the company produces a very large number of shoes, in this case the variable cost of $30. This means as they increase production, the fixed costs get spread out over more shoes, reducing the average cost per shoe towards the variable cost per shoe.

Key Concepts

Fixed and Variable Costs

Understanding cost structures is crucial for any business. Fixed costs are expenses that do not change regardless of the production output. For example, rent for factory space, salaries for management, or other overhead costs fall into this category. In the running shoes company case, the fixed cost is given as \(300,000 per month. This cost will incur even if no shoes are produced.

On the other hand, variable costs are expenses that vary directly with the level of production. The more units produced, the higher these costs will be. In the exercise, producing each pair of shoes incurs a variable cost of \)30. It's essential to recognize that the total cost function of a business incorporates both fixed and variable costs and that the balancing act between them can significantly influence profitability.

Average Cost Function

The average cost function is a vital tool for a company to assess its cost efficiency. It is obtained by dividing the total cost, which includes both fixed and variable costs, by the number of units produced. In algebraic terms, for the shoe company, the average cost function, denoted as \bar{C}(x)\, is calculated as \frac{300,000 + 30x}{x}\, where \(x\) is the number of shoes produced.

This function provides insights into how costs per unit change with production levels. When a company produces a small number of goods, the fixed costs significantly impact the average cost. However, as production increases, the influence of fixed costs decreases on a per-unit basis, potentially leading to a more favorable economy of scale.

Horizontal Asymptote Interpretation

In the context of the average cost function, a horizontal asymptote provides valuable business insight. A horizontal asymptote occurs at the value the function approaches as the number of units produced goes infinitely large. For the company in the exercise, as the quantity of shoes produced increases without bound, the average cost per shoe approaches the variable cost per unit, which is $30.

This mathematical limit symbolizes the point where fixed costs have been distributed across enough units to no longer significantly impact the average cost. Reaching this asymptote means the company has achieved the lowest possible average cost per shoe, provided that the variable costs remain unchanged. Understanding this asymptotic value can be instrumental for businesses in planning and pricing strategies to optimize profit margins.