Question

Career Choice \(\quad\) An employee has two options for positions in a large corporation. One position pays \(\$ 12.50\) per hour plus an additional unit rate of \(\$ 0.75\) per unit produced. The other pays \(\$ 9.20\) per hour plus a unit rate of \(\$ 1.30 .\) (a) Find linear equations for the hourly wages \(W\) in terms of \(x\), the number of units produced per hour, for each option. (b) Use a graphing utility to graph the linear equations and find the point of intersection. (c) Interpret the meaning of the point of intersection of the graphs in part (b). How would you use this information to select the correct option if the goal were to obtain the highest hourly wage?

Short answer

Option 1: \(W_1 = 12.50 + 0.75x\). Option 2: \(W_2 = 9.20 + 1.30x\). The better option depends on the number of units produced in comparison to the point of intersection. If less than this point, Option 1 is better. If more, Option 2 is the better choice.

Step-by-step solution

Formulating the Equations

The two equations can be formulated as: For option 1, the equation is \(W_1 = 12.50 + 0.75x\). For option 2, the equation is \(W_2 = 9.20 + 1.30x\). Where \(W_1\) and \(W_2\) are the wages for option 1 and option 2 respectively, and \(x\) is the number of units produced per hour.

Graphing the Equations

Plot the two equations on a graphing utility. The \(x\) axis represents the number of units produced per hour and the \(y\) axis represents the wages per hour. Each plotted line will represent one of the payment options.

Find the Point of Intersection

From the graph, observe where the two lines intersect. This intersection will provide us the point of interest.

Interpret the Point of Intersection

The coordinates of the point of intersection give the number of units per hour at which both jobs pay the same. If the number of units produced per hour is less than this, then the first payment option is better. If it's more, then the second option is better.

Key concepts

Graphing Linear Equations

Understanding how to represent and analyze income scenarios through graphing linear equations is vital in comparing different job payment structures. In our exercise, we translated hourly wage scenarios into mathematical expressions. The graph of these equations is a straight line, with the wage per hour on the vertical (y-axis) and the number of units produced per hour on the horizontal (x-axis).

Graphing serves a practical purpose; it visualizes the relationship between hours worked and units produced, showing how an employee's output affects their earnings. By plotting the equations on a graph, students can easily see how their pay changes as their productivity levels vary. This skill is not only practical in a classroom setting but is also essential in real-world decision-making, such as when comparing job offers with different compensation models.

Interpreting Intersection Point

The interpreting intersection point concept is crucial when evaluating options that depend on varying conditions. In the case of the given exercise, the intersection point stands out as the moment of equilibrium -- where two different wage options yield the same income for the employee.

To analyze this, one should find where the lines representing each pay structure meet. At this juncture, the number of produced units does not favor one job over the other. Understanding this point can guide individuals in making economic decisions that best suit their work habits and financial needs. For example, a higher base pay might be more attractive for someone expecting to produce fewer units, while a higher unit rate could benefit a more productive worker.

Formulating Wage Equations

The ability to formulate wage equations is an essential skill for evaluating job offers. It involves translating the terms of an employment contract into mathematical equations that express wages as a function of the number of units produced.

As we see in the exercise, the two job options have different structures -- a fixed hourly wage plus a variable amount per unit produced. By plotting these as linear equations, we have a transparent way to compare the total earnings based on productivity. The equation represents the total earnings potential and becomes a tool for predicting income under different workloads. Formulating these equations can help individuals understand the value of their labor and negotiate better employment terms.

Calculus in Economics

While not directly applied in graphing or interpreting linear wage equations, calculus in economics plays a significant role in more complex situations. Calculus helps in understanding and analyzing changes -- for instance, how a small increase in the units produced could affect the overall wage.

In our context, although calculus is not necessary to analyze the given linear equations, it's the next step for students aiming to deal with non-linear economic models or optimize functions regarding costs, revenues, profits, and other economic variables. The principles of derivatives and integrals enable a deeper investigation into marginal costs and revenues, as well as maximizing profits by finding optimal production levels.