Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

The fixed monthly cost of operating a plant that makes Zbars is \(\$ 7000\), while the cost of manufacturing each unit is \(\$ 100\). Write an expression for \(C(x)\), the total cost of making \(x\) Zbars in a month.

Step-by-Step Solution

Answer

\(C(x) = 7000 + 100x\)

Step 1: Understand Fixed Costs

The fixed monthly cost is the amount you have to pay regardless of how many Zbars you make. In this problem, the fixed cost is given as $7000.

Step 2: Understand Variable Costs

The variable cost is the cost associated with making each additional Zbar. Here, the cost per Zbar is $100. Thus, if you make x Zbars, the variable cost will be $100 times x.

Step 3: Formulate the Total Cost Expression

To find the total cost, we add the fixed cost and the variable cost. Hence, the total cost expression \(C(x)\) is the sum of the fixed cost and the variable cost: \(C(x) = 7000 + 100x\).

Key Concepts

Fixed Costs

When running a manufacturing operation like making Zbars, there are always costs you face from the get-go, no matter how much you produce. These are called fixed costs. They do not change with the level of production or sales activities. In the case of our Zbar plant, the fixed cost is $7000 every month. This is the amount that the company has to pay just to keep the doors open and the lights on, regardless of whether they produce any Zbars at all. Fixed costs may include items such as rent, salaries of permanent staff, and insurance. They are crucial for planning, as they help provide a baseline of what you need to cover every month.

Variable Costs

Variable costs, unlike fixed costs, fluctuate with production levels. They rise and fall with how much you decide to produce, which makes them more flexible. In our example, the variable cost per Zbar is set at $100. That means if you produce one Zbar, it costs you an additional $100. If you produce a hundred, this cost scales up to $10,000.
Here's the formula to compute total variable costs:

• Total Variable Costs = Cost per Unit * Number of Units Produced

Variable costs are essential for determining how scalable and flexible the production process is. Understanding these costs helps producers decide on feasible production levels.

Linear Functions

A linear function is a mathematical expression used to show a relationship of direct proportionality between two quantities. For costs, this means as one quantity (like the number of Zbars produced) changes, the total cost changes at a consistent rate. In simpler terms, with linear functions, costs grow in a straight line.
In this example, the total cost function is represented as:

• \( C(x) = 7000 + 100x \)

This equation starts with the fixed cost of \(7000 and adds the variable cost, which is \)100 per Zbar multiplied by the number of Zbars.
Linear functions provide a clear and simplified view of how costs accumulate in a business, making it easier to predict, plan, and make informed financial decisions.