Problem 70 - Applied Calculus

Question

Women's pay has often lagged behind men's, although Title VII of the Civil Rights Act requires equal pay for equal work. Based on data from \(2000-2011\), women's annual earnings as a percent of men's can be approximated by the formula \(y=0.36 x+77,\) where \(x\) is the number of years since 2000 . (For example, \(x=10\) gives \(y=80.6,\) so in 2010 women's wages were about \(80.6 \%\) of men's wages.) a. Graph this line on the window [0,30] by [0,100] . b. Use this line to predict the percentage in the year 2020\. [Hint: Which \(x\) -value corresponds to \(2020 ?\) ] c. Predict the percentage in the year \(2025 .\)

Step-by-Step Solution

Answer

In 2020, women's earnings are 84.2% of men's; in 2025, they are 86%.

Step 1: Identify x for the Years 2020 and 2025

To find the percentages for 2020 and 2025, identify the values of \( x \) corresponding to these years based on the reference year 2000. For 2020, \( x = 20 \) and for 2025, \( x = 25 \).

Step 2: Calculate y for 2020

Use the formula \( y = 0.36x + 77 \) with \( x = 20 \) to find the predicted percentage in 2020. Substitute \( x = 20 \):\[ y = 0.36(20) + 77 = 7.2 + 77 = 84.2 \]Thus, in 2020, women's earnings are predicted to be 84.2% of men's earnings.

Step 3: Calculate y for 2025

Use the same formula \( y = 0.36x + 77 \) with \( x = 25 \) to predict the percentage in 2025. Substitute \( x = 25 \):\[ y = 0.36(25) + 77 = 9 + 77 = 86 \]So, in 2025, women's earnings are predicted to be 86% of men's earnings.

Key Concepts

Civil Rights and Wage Equality

The Civil Rights Act of 1964 was a landmark legislation in the United States. One of its primary objectives was to combat racial discrimination across various spheres of public life, including the workplace. Title VII of the Act specifically addresses employment discrimination and mandates equal pay for equal work regardless of gender, among other factors. Despite this legal framework, historical data reveals that women have consistently earned less than men for similar work.
This wage disparity, often referred to as the 'gender wage gap,' has persisted over decades.
The formula presented in the exercise highlights this gap, showing how women's earnings as a percentage of men's have changed over time.
For many years, women had to fight for their civil rights, including the right to equal pay. Legal mandates are crucial but cultural changes are equally important to bridge this gap entirely.
Efforts to achieve true gender equality in wages continue through awareness campaigns, policy reforms, and grassroots activism.

Understanding Linear Approximation

Linear approximation is a mathematical method used to estimate the value of a function near a certain point using a linear function. This technique is particularly valuable in situations where a simple linear function can effectively mimic the behavior of a more complex function over a small range.
In our context, the formula provided approximates women’s earnings over time as a linear function:

• The equation is: \( y = 0.36x + 77 \) where \( y \) is women's earnings as a percentage of men's.
• \( x \) represents the number of years since 2000.

The slope of the line, 0.36, indicates the increase in women's earnings percentage for each additional year.
The y-intercept, 77, suggests the estimated earnings percentage in the baseline year of 2000.
This simple model helps visualize and predict changes without necessarily capturing all underlying complexities.

Graphing Functions to Visualize Data

Graphing functions is a powerful tool to visualize and interpret data. It helps in understanding relationships between variables and trends over time.
Graphing the linear function \( y = 0.36x + 77 \) on a coordinate plane with the specified window [0,30] by [0,100] allows us to see:

• How women's earnings as a percentage of men's increase each year.
• The pace and direction of wage equality improvements.

To graph the function:

• Plot points for selected \( x \)-values, like 0, 10, 20, and 30, and calculate corresponding \( y \)-values.
• Join these points with a straight line, representing the continuous nature of time and earnings data.

The graph makes it easy to predict future percentages, assess past trends, and understand ongoing progress towards wage equality.

Predictive Modeling with Linear Functions

Predictive modeling is a statistical technique used to forecast future outcomes based on existing data. In this exercise, a linear function is utilized to predict women’s earning percentages for future years (2020 and 2025). This process involves:

• Identifying the correct \( x \)-values for the desired prediction years. For 2020, \( x = 20 \) and for 2025, \( x = 25 \).
• Substituting these \( x \)-values into the linear equation \( y = 0.36x + 77 \).
• Calculating the predicted \( y \)-values, which are the earning percentages.

For instance, in 2020, the model forecasts that women's earnings will be approximately 84.2% of men's, and by 2025, this is expected to be around 86%. These predictions offer valuable insights, highlighting gradual progress toward achieving wage parity. However, relying solely on linear models may not capture potential fluctuations or new changes, so predictions should be used with caution and awareness of their limitations.