Problem 53 - Algebra 1: Concepts and Skills

Question

In Exercises \(52-54,\) use the following information. The number of people who worked for the railroads in the United States each year from 1989 to 1995 can be modeled by the equation \(y=-6.6 x+229,\) where \(x\) represents the number of years since 1989 and \(y\) represents the number of railroad employees (in thousands). About how many people worked for the railroads in \(1995 ?\)

Step-by-Step Solution

Answer

In 1995, an estimated 189,400 people worked for the railroads in the U.S.

Step 1: Calculate the Number of Years Since 1989

Firstly, subtract the base year, 1989, from the year we're interested in, 1995. This results in \(1995-1989 = 6\). So, \(x = 6\).

Step 2: Substitute x into the equation

Now, substitute \(x = 6\) in the given linear equation \(y=-6.6 x+229\). This leads to \(y=-6.6 * 6+229\).

Step 3: Solve the equation for y

Calculate the value of y by performing the multiplication and addition. This results in \(y = -39.6 + 229 = 189.4\) thousand.

Key Concepts

Railroad Employees

The concept of railroad employees in this exercise refers to the people employed by railroad companies over a certain period. In the exercise, we are looking at data from 1989 to 1995. During this time, the number of railroad employees is modeled using a linear equation. This approach lets us estimate the count of employees for any given year within that period by substituting the year into the equation as a variable value.

Understanding the number of railroad employees is important for multiple reasons:

• It shows trends in employment, revealing economic shifts or technological advancements.
• It helps in planning for workforce management, including recruitment and layoffs.
• Railroad employment impacts related industries, influencing economic health broadly.

By evaluating employment trends, stakeholders can make informed decisions related to operational, regulatory, and economic strategies.

Modeling Data

Modeling data involves creating a mathematical equation or formula that captures and predicts trends from past to future values. In this exercise, the linear equation \(y = -6.6x + 229\) is used to model the number of railroad employees over time.

A few key elements of data modeling include:

• Trend Analysis: The exercise reveals a decline (as seen by the negative slope \(-6.6\)) in employment.
• Predictive Capability: The equation allows us to predict employee counts for years beyond the given dataset.
• Simplicity: Linear models, while basic, provide clear insights and are easy to interpret.

In education and real-world applications, effective data modeling helps to draw conclusions and plan for future scenarios based on historical trends. It simplifies complex datasets into understandable and quantifiable insights.

Years Since Base Year

The base year in an analysis provides a starting point for measuring change over time. For this exercise, the base year is 1989. All calculations of time-related change are in reference to this year.

The concept of "years since base year" establishes a consistent framework to track developments.

• It allows for simplified calculations as all subsequent years are viewed as intervals from the base year.
• It makes interpreting results easier, as changes in variables are tracked in an orderly, familiar format.
• It standardizes comparisons across datasets, ensuring uniformity in data analysis.

In the context of the given exercise, "1995 - 1989 = 6" means that 1995 is six years from the base year, making it easy to substitute \(x = 6\) into the equation. This method provides clarity and accuracy in the time analysis involved in linear equations.