In Exercises 53 and 54, the average cost per unit at...

In Exercises 53 and $54,$ the average cost per unit at production level $x$ is defined as $C_{\mathrm{avg}}(x)=C(x) / x,$ where $C(x)$ is the cost function Average cost is a measure of the efficiency of the production process. The cost in dollars of producing alarm clocks is $C(x)=50 x^{3}-$ $750 x^{2}+3740 x+3750$ where $x$ is in units of $1000 .$ (a) Calculate the average cost at $x=4,6,8,$ and 10 . (b) Use the graphical interpretation of average cost to find the production level $x_{0}$ at which average cost is lowest. What is the relation between average cost and marginal cost at $x_{0}($ see Figure 16$) ?$

In Exercises 53 and $54,$ the average cost per unit at production level $x$ is defined as $C_{\mathrm{avg}}(x)=C(x) / x,$ where $C(x)$ is the cost function Average cost is a measure of the efficiency of the production process.
The cost in dollars of producing alarm clocks is $C(x)=50 x^{3}-$ $750 x^{2}+3740 x+3750$ where $x$ is in units of $1000 .$
(a) Calculate the average cost at $x=4,6,8,$ and 10 .
(b) Use the graphical interpretation of average cost to find the production level $x_{0}$ at which average cost is lowest. What is the relation between average cost and marginal cost at $x_{0}($ see Figure 16$) ?$

Key Concepts

Optimization in Production

Optimization in production involves determining the production level that minimizes costs or maximizes profit. In the context of average cost, it often means finding the output level where the average cost is at its lowest, which is also the point where the marginal cost equals the average cost.

Graphical Interpretation in Cost Analysis

Graphical interpretation involves plotting cost functions such as the average cost and marginal cost curves on a graph. This helps illustrate relationships between different cost measures, such as the tangency point where marginal cost intersects average cost at its minimum, visually confirming the optimal production level.

Average Cost Function

The average cost function is obtained by dividing the total cost of production by the number of units produced. It measures the cost per unit and provides insight into the efficiency of production, helping to identify the level of output that minimizes cost per unit.

Cost Function

A cost function represents the total expenditure incurred in the production process as a function of the number of units produced. It typically includes both fixed and variable components, illustrating how costs scale with production volume.

Marginal Cost

Marginal cost is the additional cost of producing one extra unit of output. It is calculated as the derivative of the total cost function with respect to production quantity and plays a key role in production decisions by indicating how total cost changes with incremental production.

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