For a certain model of fax machine, the manufacturing cost C...

For a certain model of fax machine, the manufacturing cost $C$ per machine is given by the equation $$ C=\frac{250 x+10,000}{x} $$ where $x$ is the number of fax machines manufactured and $\operatorname{cost} C$ is in dollars per machine. a. Find the cost per fax machine when manufacturing 100 fax machines. b. Find the cost per fax machine when manufacturing 1000 fax machines. c. Does the cost per machine decrease or increase when more machines are manufactured? Explain why this is so.

For a certain model of fax machine, the manufacturing cost $C$ per machine is given by the equation
$$
C=\frac{250 x+10,000}{x}
$$
where $x$ is the number of fax machines manufactured and $\operatorname{cost} C$ is in dollars per machine.
a. Find the cost per fax machine when manufacturing 100 fax machines.
b. Find the cost per fax machine when manufacturing 1000 fax machines.
c. Does the cost per machine decrease or increase when more machines are manufactured? Explain why this is so.

Key Concepts

Cost Function

In mathematics, a cost function expresses the total cost of production as a function of the quantity produced. It typically includes both fixed costs, which do not change with the level of production, and variable costs, which do. Understanding how each component contributes to the total cost helps in analyzing production efficiency.

Fixed and Variable Costs

Fixed costs are expenses that remain constant regardless of production volume, while variable costs change proportionally with the quantity produced. In cost functions, the fixed cost is often divided by the number of units produced, leading to a decrease in per-unit fixed cost as production increases, and this influences decision making regarding economies of scale.

Substitution in Cost Functions

Evaluating a cost function by substituting specific production values allows for the determination of per-unit cost at different production levels. This method is a fundamental technique in algebra, enabling one to analyze and compare cost implications as operational scales change.

Economies of Scale

Economies of scale refer to the principle that as the scale of production increases, the per-unit cost typically decreases due to the spreading of fixed costs over a larger number of units. This concept is crucial in production planning and business strategy, as it highlights the cost advantages of increased production volumes.

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