Problem 16 - Applied Calculus

Question

The cost \(C\), in millions of dollars, of producing \(q\) items is given by \(C=5.7+0.002 q\). Interpret the \(5.7\) and the \(0.002\) in terms of production. Give units.

Step-by-Step Solution

Answer

5.7 million is the fixed cost; 0.002 million is the cost per additional item.

Step 1: Identify the Components of the Expression

The cost function given is \(C = 5.7 + 0.002q\). This is a linear equation where \(5.7\) is the y-intercept and \(0.002\) is the slope. These numerical values have specific interpretations in the context of cost and production.

Step 2: Interpret the Y-Intercept

The \(5.7\) in the equation represents the fixed costs in millions of dollars. This is the cost incurred when no items are produced, meaning it's a constant production fee or initial setup cost that does not vary with the number of items produced. Therefore, \(5.7\) million dollars is the fixed or initial cost of production.

Step 3: Interpret the Slope

The \(0.002\) in the equation represents the variable cost per item produced in terms of millions of dollars. Specifically, for each additional item produced, the total cost increases by \(0.002\) million dollars, or \(2000\) dollars per item since \(0.002\) million dollars equals \(2000\) dollars. Thus, \(0.002\) is the marginal cost per item.

Key Concepts

Fixed Costs

In any business, understanding fixed costs is crucial. Fixed costs are expenses that do not change with the level of goods or services produced. In other words, they remain constant regardless of production output. For a factory, these costs could include:

• Rent for the building
• Insurance premiums
• Salary of employees
• Machinery and equipment maintenance

In the context of the given cost function, the fixed cost is represented by the number 5.7. This means, even if no additional products are produced, a constant cost of 5.7 million dollars will always be present. This amount is the base cost required to keep the operation running.
Recognizing fixed costs helps businesses understand the baseline expenses they need to cover before making a profit, highlighting the importance of producing enough items to surpass these constant expenditures.

Variable Costs

Unlike fixed costs, variable costs fluctuate with the level of production output. These are the costs incurred for producing each additional unit of a product. As the number of items produced grows, so do the total variable costs. These costs typically include:

• Raw materials and supplies
• Hourly wages for production staff
• Utility costs that increase with production usage

In the provided equation, the variable cost is indicated by the numeric coefficient 0.002, which is the amount in millions of dollars. This tells us that for each additional item produced, the cost goes up by \(0.002\) million dollars, equivalent to $2,000 per item. By closely monitoring variable costs, businesses can make informed decisions about pricing and production levels to maximize profitability.

Linear Equations

Linear equations like the one provided in the exercise are used widely in representing cost functions. A linear equation is any equation that, when graphed, produces a straight line. The general form is:

• \(y = mx + b\)

where:

• \(y\) is the dependent variable
• \(m\) is the slope of the line
• \(x\) is the independent variable
• \(b\) is the y-intercept

In our cost equation, \(C = 5.7 + 0.002q\), \(C\) represents the total cost in millions of dollars, \(q\) represents the quantity of items produced, \(5.7\) is the y-intercept indicating fixed costs, and \(0.002\) is the slope or variable cost per item.
Understanding linear equations in cost functions helps in anticipating how costs change with production levels, thus enabling better budgeting and financial forecasting.