Question

The U.S. Bureau of Labor Statistics projects job growth by using three models to make low, moderate, and high estimates. The equations below model the projected number of auto mechanics \(m\) from 1994 to \(2005.\) In all three models, \(t\) is the number of years since 1994 Model 1: m=13,272 t+736,000\( Model 2: m=9455 t+736,000\) Model 3: m=11,455 t+736,000\( a. For each model, write an equation that enables you to predict the year in which the number of auto mechanics will reach \)800,000\(. b. In the same coordinate plane, graph the related function for each equation that you found in part (a). According to each model, in what year will the number of auto mechanics reach \)800,000 ?$ c. Visual THINKING Which model gives a high estimate of the number of mechanics? a low estimate? How can you tell this from the graphs of the models?

Short answer

In Model 1, the number of auto mechanics will reach 800,000 in approximately the year 1999. In Model 2, it will be around the year 2001. And In Model 3, it will be around the year 2000 (by rounding off to the nearest year). Model 1 gives the high estimate of the number of mechanics and Model 2 gives a low estimate.

Step-by-step solution

Rewrite the Equations

In order to find the year for when the number of auto mechanics will be 800,000, it is necessary to find \(t\) from the given equations. The models can be rewritten to solve for \(t\) by subtracting 736,000 from both sides and dividing by the coefficient of \(t\). \nModel 1: \(t= (m - 736,000) /13,272 \) \nModel 2: \(t= (m - 736,000) /9,455 \) \nModel 3: \(t= (m - 736,000) /11,455 \)

Finding the Year

By substituting \(m = 800,000\) in each model, we calculate the value of \(t\). \nModel 1: \(t= (800,000 - 736,000) /13,272 = 4.83 \) \nSo, the year will be 1994 + 4.83 = 1998.83 \nModel 2: \(t= (800,000 - 736,000) /9,455 = 6.77 \) \nSo, the year will be 1994 + 6.77 = 2000.77 \nModel 3: \(t= (800,000 - 736,000) /11,455 = 5.59 \) \nSo, the year will be 1994 + 5.59 = 1999.59

Graphing the Equations

In part b, a graph is drawn for the equations derived in step 1. The x-axis will represent the number of years after 1994 (i.e., the years from 1994 to 2005) and the y-axis will represent the number of auto mechanics. The line representing each model is plotted.

Interpret the Graph

The intersection of each line with the y-value of 800,000 will show the year when this number of mechanics is predicted according to each model. The graph shows the estimates provided by the models visually and can also be used to identify which model provides the highest and lowest estimates. The model that predicts the maximum number of mechanics fastest would have the steepest slope (highest value). This will be Model 1. The model that predicts the smallest growth (smallest slope or smallest value, and hence reaching 800,000 last) is Model 2.

Key concepts

Algebraic Modeling

Algebraic modeling is a powerful tool used to represent real-world situations mathematically, allowing us to make predictions and informed decisions. In our case, the U.S. Bureau of Labor Statistics uses algebraic models to estimate the future number of auto mechanics.

The models provided are linear equations, where each represents a possible growth scenario based on different assumptions. In Model 1, Model 2, and Model 3, the variable 't' represents the time since 1994, and 'm' stands for the projected number of mechanics. By constructing these models, the bureau can forecast job growth across different years.

To find the year in which the number of auto mechanics will reach 800,000, we set 'm' equal to this number and solve for 't' in each model equation. This technique exemplifies how algebraic equations can be manipulated to extract specific data points that help in planning and resource allocation.

Graphing Linear Functions

Graphing linear functions is a visual way to understand the behavior of algebraic models. For the job projection models, a graph translates the algebraic equations into lines on a coordinate plane, with 't' on the x-axis and 'm' on the y-axis.

To graph the equations derived for the year when the number of auto mechanics will reach 800,000, each line is plotted so that the x-intercept corresponds to the year 't'. This graphical representation allows for a quick comparison between the three growth scenarios. It is immediately apparent which model predicts faster growth in the number of auto mechanics, as the slope of the line indicates the rate of change over time.

Graphical analysis is invaluable for identifying trends and making comparisons swiftly. It highlights the efficiency and utility of graphing linear functions in understanding and interpreting algebraic data.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula \( y = mx + b \) where \( m \) is the slope of the line and \( b \) is the y-intercept. This form is particularly useful for quickly graphing linear equations and understanding their properties.

In the context of the Bureau of Labor Statistics models, the coefficient of \( t \) is the slope, and this represents the rate at which the number of auto mechanics is projected to change per year. The constant term, 736,000 in this case, is where the line crosses the y-axis, reflecting the starting point of the data in 1994.

By using the slope-intercept form, we can see how the different slopes of the models represent different rates of job growth. The steeper the slope, the higher the rate. Therefore, interpreting the slope helps us understand how quickly the number of auto mechanics is expected to reach the targeted figure of 800,000 according to each model.