Problem 5 - 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 is the fixed production cost in millions of dollars; 0.002 is the cost per item in millions of dollars.

Step 1: Identify the Formula Components

The formula given for cost is \(C = 5.7 + 0.002q\). Here, \(C\) represents the cost in millions of dollars, and \(q\) represents the number of items produced.

Step 2: Interpret the Constant Term

The constant term 5.7 in the equation represents the fixed cost of production when no items are made. Thus, it can be interpreted as the initial cost of production in millions of dollars, regardless of the number of items produced.

Step 3: Interpret the Coefficient of q

The coefficient of \(q\), which is 0.002, indicates the variable cost per item. It represents the additional cost, in millions of dollars, required to produce each additional item.

Key Concepts

Linear Equation

A linear equation is a mathematical expression where each term is either a constant or the product of a constant and a single variable. In the context of cost functions, linear equations are used to model and predict the cost associated with producing a certain number of items.
Here's how it works in the given problem:

• The equation is expressed as: \( C = 5.7 + 0.002q \), where \( C \) is the dependent variable representing cost, and \( q \) is the independent variable representing the quantity of items produced.
• The term "linear" comes from the fact that the graph of a linear equation is a straight line. When you plot \( C \) against \( q \), the points will form a straight line.
• Linear equations have the general form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

Understanding the components of a linear equation helps in identifying how changes in the variable \( q \) affect the overall cost \( C \). A key skill is interpreting the fixed and variable parts of the equation, which are crucial in understanding cost behavior.

Fixed Cost

Fixed costs are expenses that do not change with the level of production or sales activity. In other words, whether you produce 0 items or hundreds, your fixed costs remain constant.
In the equation \( C = 5.7 + 0.002q \), the fixed cost is represented by the constant term 5.7. This means:

• The fixed costs are 5.7 million dollars, which are incurred even if no items are produced.
• These costs could include expenses such as factory rent, salaries of permanent staff, or machinery depreciation.

Having a good grasp of fixed costs helps businesses budget effectively and make strategic decisions. Recognizing these costs means understanding the constant financial obligations that exist, regardless of production output.

Variable Cost

Variable costs are costs that vary directly with the level of production. Unlike fixed costs, they change as the production level changes. This flexibility allows businesses to manage these costs based on the production demands.
In our cost function, \( C = 5.7 + 0.002q \), the variable cost is represented by the coefficient 0.002. This means:

• Each additional item produced adds 0.002 million dollars (or 2,000 dollars) to the total cost.
• Variable costs could include raw materials, direct labor, or utility expenses directly associated with production.

The ability to calculate and adjust variable costs gives businesses insights into how production changes impact overall costs. This understanding is essential for setting prices, managing budgets, and optimizing production levels to maximize profit.