A company manufacturing snowboards has fixed costs of 200...

A company manufacturing snowboards has fixed costs of $\$ 200$ per day and total costs of $\$ 3,800$ per day for a daily output of 20 boards. (A) Assuming that the total cost per day $C(x)$ is linearly related to the total output per day $x,$ write an equation for the cost function. (B) The average cost per board for an output of $x$ boards is given by $\bar{C}(x)=C(x) / x .$ Find the average cost function. (C) Sketch a graph of the average cost function, including any asymptotes, for $1 \leq x \leq 30$. (D) What does the average cost per board tend to as production increases?

A company manufacturing snowboards has fixed costs of $\$ 200$ per day and total costs of $\$ 3,800$ per day for a daily output of 20 boards.
(A) Assuming that the total cost per day $C(x)$ is linearly related to the total output per day $x,$ write an equation for the cost function.
(B) The average cost per board for an output of $x$ boards is given by $\bar{C}(x)=C(x) / x .$ Find the average cost function.
(C) Sketch a graph of the average cost function, including any asymptotes, for $1 \leq x \leq 30$.
(D) What does the average cost per board tend to as production increases?

Key Concepts

Fixed Costs

Fixed costs refer to the expenses that remain constant regardless of the level of production. In cost analysis, these are costs that must be paid even if no goods are produced, and they appear as the constant term in a cost function. Understanding fixed costs is crucial in determining the base expense from which additional variable costs will be added.

Variable Costs

Variable costs are the expenses that vary directly with the level of production. In the context of a linear cost function, these costs are represented as the slope, indicating the additional cost incurred for each extra unit produced. Distinguishing variable costs from fixed costs helps in understanding how overall costs adjust as production levels change.

Linear Cost Function

A linear cost function models the total cost as a straight line, expressed as C(x) = fixed cost + (m × x), where 'm' is the marginal (or variable) cost per unit. This type of function simplifies analysis by assuming a constant variable cost per additional unit, allowing for straightforward calculations and predictions about total production costs.

Average Cost Function

The average cost function calculates the cost per unit produced by dividing the total cost by the number of units. It is typically expressed as ?(x) = C(x)/x. This function is critical for understanding the efficiency and scalability of production, as well as for making pricing decisions in relation to production volume.

Asymptotic Behavior

Asymptotic behavior in cost functions refers to the trend observed when the production volume becomes very large. For an average cost function derived from a linear total cost function, the average cost per unit tends to approach the variable cost per unit, since the fixed cost becomes increasingly insignificant as output grows. This concept is important for long-term cost analysis and strategic business planning.

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