Problem 79 - College Algebra and Calculus: An Applied Approach

Question

The inventor of a new game determines that the variable cost of producing the game is $$\$ 2.95$$ per unit and the fixed costs are $$\$ 8000$$. The inventor sells each game for $$\$ 8.79 .$$ Let \(x\) be the number of games sold. (a) Write the total cost \(C\) as a function of the number of games sold. (b) Write the average cost per unit \(\bar{C}=C / x\) as a function of \(x\) (c) Complete the table. $$\begin{array}{|l|l|l|l|l|}\hline x & 100 & 1000 & 10,000 & 100,000 \\\\\hline \bar{C} & & & & \\ \hline\end{array}$$ (d) Write a paragraph analyzing the data in the table. What do you observe about the average cost per unit as \(x\) gets larger?

Step-by-Step Solution

Answer

The total cost function is \(C(x) = \$8000 + \$2.95x\). The average cost function is \(\overline{C}(x) = (\$8000 + \$2.95x) / x\). Calculating \(\overline{C}(x)\) for given values of \(x\) will populate the table. Analysis will generally show that as \(x\) gets larger, \(\overline{C}(x)\) tends to get lower, which is due to the fact that variable costs become a smaller percentage of total costs.

Step 1: Calculate Total Cost Function

Given that the variable cost is $2.95 per unit and the fixed costs are $8000, the total cost function \(C\) can be written as: \(C(x) = \$8000 + \$2.95x\), where \(x\) is the number of games sold.

Step 2: Calculate Average Cost per Unit Function

The average cost per unit function \(\overline{C}\) can be written as a ratio of the total cost \(C\) to the number of units sold \(x\). Thus, \(\overline{C}(x) = C(x) / x = (\$8000 + \$2.95x) / x\).

Step 3: Fill in the Table

We now substitute \(x\) with the given table values (100, 1000, 10000, 100000) into the average cost function \(\overline{C}(x)\). This will give us average cost per unit associated with each quantity.

Step 4: Analyze the Data

At this step, the data in the table is observed and the trend is analyzed. Specifically, the change in average cost per unit as \(x\) gets larger is determined.

Key Concepts

Cost Function

Understanding the cost function is essential for any business to effectively manage its expenses and determine pricing strategies. The cost function represents the total cost that a company incurs in the production of goods or services. This function typically includes both fixed and variable costs.

In the given exercise, the total cost function, denoted as \(C(x)\), is calculated by adding the fixed costs to the product of the variable cost per unit and the number of units produced, symbolically expressed as \(C(x) = fixed costs + (variable cost \times x)\). For the new game, fixed costs are \(\$8000\) and the variable cost per unit is \(\$2.95\), leading to the cost function \(C(x) = \$8000 + \$2.95x\), where \(x\) is the number of games produced and sold.

Variable Cost

Variable costs are expenses that change in direct proportion to the amount of goods or services that a business produces. Unlike fixed costs, variable costs fluctuate with production volume. Common examples of variable costs include raw materials, direct labor, and energy usage. The more you produce, the higher the variable costs will be.

In our textbook example, the variable cost per unit is constant at \(\$2.95\) for each game sold. Therefore, for every additional game produced, the total cost increases by this amount. Variable costs are vital in the calculation of total costs and in determining the break-even point where total costs and total revenue are equal.

Fixed Costs

Fixed costs, in contrast to variable costs, do not change with the level of production or sales. These are expenses that remain constant regardless of business activity. Examples include rent, insurance, salaries for permanent staff, and depreciation of assets. Fixed costs are necessary to consider when setting prices and planning for profitability.

The exercise presents a fixed cost of \(\$8000\), which is a one-time expense that will not fluctuate with the number of games sold. It's an essential component of the total cost function, as it must be covered by sales revenue before any profit can be realized.

Function Analysis

Function analysis involves examining the properties and behavior of functions. In a business context, this often means analyzing cost, revenue, and profit functions to make informed financial decisions and predict future performance. By conducting a thorough function analysis, businesses can understand how changes in production or sales levels affect their overall costs and profitability.

For the given exercise, analyzing the average cost per unit function \(\overline{C}(x) = (\$8000 + \$2.95x) / x\) reveals how average costs change with different production levels. As the production quantity increases, our analysis would typically show that the average cost per unit decreases due to the fixed costs being spread out over more units, demonstrating the concept of economies of scale. This insight helps in strategizing pricing, budgeting, and scaling operations.