Question

$$ C=0.0045 P+5.22 $$ 28\. A production line manager uses the equation above to predict the production cost per item produced in dollars, \(C\), based on the number of products made, \(P\). In the context of the model, what is the meaning of \(5.22 ?\) A) The initial production cost, in dollars, of each product made B) The approximate production cost increase, in dollars per item, for each item made C) The approximate production cost, in dollars per item, for every \(0.0045\) products made D) The approximate production cost decrease, in dollars per item, for every \(0.0045\) products made

Short answer

A) The initial production cost, in dollars, of each product made

Step-by-step solution

Analyze the given equation

We have the equation relating the production cost per item produced, C, and the number of products made, P: \[C = 0.0045P + 5.22\] Here, 0.0045 represents the slope of the linear equation, which is the rate of change of the production cost per item produced as the number of products made increases. The constant term, 5.22, is the value of C when P=0, i.e., the production cost per item produced when no items have been made.

Understand the meaning of 5.22

Since 5.22 represents the value of C when P=0, this indicates the initial baseline production cost per item made. Therefore, in the context of the model, 5.22 is the initial production cost, in dollars, of each product. Hence, the correct answer is option A: A) The initial production cost, in dollars, of each product made

Key concepts

Linear Equations

Understanding linear equations is crucial for analyzing relationships where one variable changes at a steady rate with respect to another. In our production cost equation, the relationship between the cost per item (\( C \)) and the number of items produced (\( P \)) can be described by such an equation. A linear equation can always be written in the form \[ y = mx + b \] where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope, indicating the rate at which \( y \) changes with \( x \) and \( b \) is the y-intercept, representing the initial value of \( y \) when \( x \) is zero. In our case, \[ C = 0.0045P + 5.22 \] \( C \) is the dependent variable representing the cost, while \( P \) is the independent variable representing the number of products.

In simpler terms, if you know the number of items produced, you can use this linear equation to calculate the production cost for that number of items. This predictability is invaluable for business planning and cost analysis.

Rate of Change

The rate of change in the context of a linear equation refers to how much one variable changes in response to a change in another variable. It's a constant value when we are dealing with linear relationships. In the production cost formula, the rate of change is given by the coefficient of the independent variable (\( 0.0045 \) in our equation).

This coefficient, often represented as the slope in a linear equation, tells us that for each additional product made, the production cost increases by approximately \$0.0045\. To visualize this, imagine a graph where every step to the right represents one more product made, and simultaneously, a slight step up represents the increase in cost. The steady incline of this graph is the visual manifestation of the equation's rate of change, a concept that is foundational in understanding how variables interact within a system.

Initial Cost

The term 'initial cost' in a production cost equation refers to the baseline cost that exists even when the production quantity is zero. It's represented by the constant term in the linear equation, which in this instance, is \( 5.22 \).

The initial cost often includes fixed expenses such as machinery, rent, or labor — costs that are incurred regardless of the quantity produced. It is an essential factor in financial planning as it constitutes the base from which all additional production costs are added. When no items are produced (\( P = 0 \)), the production cost (\( C \) would be \$5.22 per item, suggesting that even without any production, certain costs are inherently present. Recognizing and accounting for initial costs helps businesses in setting pricing strategies and understanding the financial thresholds needed for profitability.