Question

MULTI-STEP PROBLEM 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 \(y=-6.61 x+229,\) where \(x\) represents the number of years since 1989 and \(y\) represents the number of railroad employees (in thousands). a. Find the \(y\) -intercept of the line. What does it represent? b. Find the \(x\) -intercept of the line. What does it represent? c. About how many people worked for the railroads in \(1995 ?\) d. Writing Do you think the line in the graph will continue to be a good model for the next 50 years? Explain.

Short answer

a) The y-intercept is 229, representing the number of railroad employees (in thousands) in 1989. b) The x-intercept is approximately 34.65, indicating that the number of railroad employees is projected to reach zero around the year 2024. c) The model estimates that about 189,340 people worked for the railroads in 1995. d) The model may not accurately predict employee count for the next 50 years due to potential significant impacts by factors such as economic changes, technological advancements, policy changes, etc.

Step-by-step solution

Find the Y-Intercept

The y-intercept is the constant term in the given equation, which is 229. It represents the number of railroad employees (in thousands) in the year 1989.

Find the X-Intercept

The x-intercept can be found by setting \( y = 0 \) in the equation and solving for 'x'. This gives \( x = -229 / -6.61 \) which approximates to 34.65. This represents the year (after 1989) when the number of railroad employees is projected to reach zero, or roughly in 2024.

Find the Number of Employees in 1995

Substitute \( x = 1995 - 1989 = 6 \) into the equation to find 'y', which is roughly 189.34. So, about 189,340 people worked for the railroads in 1995.

Evaluating the Model's Validity Over 50 Years

The linear model is an approximation and may not accurately predict employee count for the next 50 years. Factors like economic changes, technological advancements, policy changes, etc., can significantly impact the number of employees, which this model does not account for.