A company makes three sizes of cardboard boxes: small,...

A company makes three sizes of cardboard boxes: small, medium, and large. It costs $$\$ 2.50$$ to make a small box, $$\$ 4.00$$ for a medium box, and $$\$ 4.50$$ for a large box. Fixed costs are $$\$ 8000$$. (a) Express the cost of making $x$ small boxes, $y$ medium boxes, and $z$ large boxes as a function of three variables: $C=f(x, y, z)$. (b) Find $f(3000,5000,4000)$ and interpret it. (c) What is the domain of $f$ ?

A company makes three sizes of cardboard boxes: small, medium, and large. It costs $$\$ 2.50$$ to make a small box, $$\$ 4.00$$ for a medium box, and $$\$ 4.50$$ for a large box. Fixed costs are $$\$ 8000$$.
(a) Express the cost of making $x$ small boxes, $y$ medium boxes, and $z$ large boxes as a function of three variables: $C=f(x, y, z)$.
(b) Find $f(3000,5000,4000)$ and interpret it.
(c) What is the domain of $f$ ?

Key Concepts

Fixed Costs

Fixed costs refer to expenses that do not change with the level of production or the number of items produced. They are incurred regardless of how many units are manufactured and include items such as rent, salaries, or initial setup costs.

Variable Costs

Variable costs are expenses that depend directly on the production level. They change in proportion to the number of units produced, such as the cost to make each individual box, where each additional box increases the total cost by a fixed amount.

Total Cost Function

A total cost function represents the sum of the fixed costs and variable costs. In a production setting, it combines a constant term (the fixed cost) with terms involving the number of items produced (representing the variable costs), often resulting in a linear function when per-unit costs are constant.

Domain of a Function

The domain of a function consists of all the possible input values for which the function is defined. In the context of production cost functions, the domain typically includes non-negative integers since it only makes sense to produce a non-negative number of units.

Function Evaluation and Interpretation

Evaluating the cost function at specific production levels involves substituting numerical values for each variable to calculate the total cost. This process provides a concrete measure of expense for producing a determined number of items, aiding in budget planning and financial decision-making.

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