Average cost. A company manufacturing surfboards has fixed...

Average cost. A company manufacturing surfboards has fixed costs of 300 per day and total costs of 5,100 per day at 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?

Average cost. A company manufacturing surfboards has fixed costs of 300 per day and total costs of 5,100 per day at 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?

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Key Concepts

Asymptotic Behavior

In the context of average cost functions, asymptotic behavior refers to the tendency of the average cost per unit to approach a particular value as production levels increase indefinitely. Typically, as output increases, the impact of the fixed cost diminishes, and the average cost tends to stabilize or approach the variable cost per unit.

Fixed and Variable Costs

Fixed costs are expenses that do not change with the level of production, while variable costs change in proportion to the level of output. Understanding how these two components contribute to the total cost is essential in analyzing cost behaviors in production and setting up the cost functions.

Linear Cost Functions

A linear cost function represents the total cost as a straight-line relationship between the number of units produced and the cost incurred. It is typically expressed in the form C(x) = fixed cost + (variable cost per unit)x, where the fixed cost remains constant and the variable cost per unit is the slope of the line.

Average Cost per Unit

The average cost function is obtained by dividing the total cost function by the number of units produced. It represents the cost incurred on average for each unit and is useful for assessing efficiency, pricing decisions, and economies of scale as production levels change.

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