Problem 67 - Brief Applied Calculus

Question

SOCIAL SCIENCE: Age at First Marriage Americans are marrying later and later. Based on data for the years 2000 to 2007 , the median age at first marriage for men is \(y_{1}=0.12 x+26.8\), and for women it is \(y_{2}=0.12 x+25\), where \(x\) is the number of years since 2000 . a. Graph these lines on the window \([0,30]\) by \([0,35] .\) b. Use these lines to predict the median marriage ages for men and women in the year 2020 . [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the median marriage ages for men and women in the year 2030 .

Step-by-Step Solution

Answer

In 2020, men: 29.2 years, women: 27.4 years. In 2030, men: 30.4 years, women: 28.6 years.

Step 1: Understand the Linear Equations

The equations given for the median age at first marriage are linear functions where \(y_1 = 0.12x + 26.8\) for men and \(y_2 = 0.12x + 25\) for women. Here, \(x\) represents the number of years since 2000.

Step 2: Identify the Year Parameter

To solve the problem for specific years, identify \(x\) for each. For 2020, \(x = 2020 - 2000 = 20\). For 2030, \(x = 2030 - 2000 = 30\).

Step 3: Calculate Median Ages for 2020

Substitute \(x = 20\) into the equations to find the median marriage ages in 2020. For men:\[y_1 = 0.12(20) + 26.8 = 2.4 + 26.8 = 29.2\]For women:\[y_2 = 0.12(20) + 25 = 2.4 + 25 = 27.4\]

Step 4: Calculate Median Ages for 2030

Substitute \(x = 30\) into the equations to find the median marriage ages in 2030. For men:\[y_1 = 0.12(30) + 26.8 = 3.6 + 26.8 = 30.4\]For women:\[y_2 = 0.12(30) + 25 = 3.6 + 25 = 28.6\]

Step 5: Draw the Graph (Optional)

To graph these equations over the window \([0,30]\) by \([0,35]\), plot the lines based on the equations \(y_1 = 0.12x + 26.8\) and \(y_2 = 0.12x + 25\) over the range of \(x\) from 0 to 30 to show the trend in increasing age.

Key Concepts

Linear Equations

In linear algebra, a linear equation is a powerful tool to represent relationships between variables. In the context of social sciences, especially relating to statistical trends, linear equations can be applied to forecast data patterns. A linear equation like \( y = mx + b \) consists of:

• \( y \): the output or dependent variable. In our case, the median age at first marriage.
• \( m \): the slope, indicating the rate of change. Here, a slope of 0.12 suggests a gradual increase in the median marriage age over years.
• \( x \): the input or independent variable. Here, it represents the number of years since 2000.
• \( b \): the y-intercept, or the starting value. For example, 26.8 and 25 for men and women respectively, denote the median age at first marriage in the year 2000.

Linear equations in social sciences serve to predict future events, helping policymakers make informed decisions. Understanding the components and their roles allows for effective application in real scenarios.

Data Interpretation

Interpreting data through linear equations necessitates converting real-world questions into mathematical expressions. This framework simplifies complex data patterns into understandable insights.
In the provided exercise, identifying the year parameter \(x\) and substituting it into the linear equations \(y_1 = 0.12x + 26.8\) and \(y_2 = 0.12x + 25\) is key. For instance, to find the median ages in 2020:

• Calculate \(x\) as the difference between the target year and 2000. For 2020, \(x = 20\).
• Plug \(x = 20\) into both equations.
• For men: calculate \(y_1 = 0.12(20) + 26.8 = 29.2\).
• For women: compute \(y_2 = 0.12(20) + 25 = 27.4\).

Data interpretation assists in understanding trends and drawing conclusions. It allows scientists, economists, and social planners to anticipate changes over time.

Graphing

Graphing is an essential tool in visualizing linear equations and understanding trends over a specified range. By plotting these linear equations, we can see the projected increase in the median age at first marriage over time.
To effectively graph, follow these steps:

• Determine the graph's axis limits. In this case, the x-axis ranges from 0 to 30, representing 2000 to 2030, while the y-axis ranges from 0 to 35, allowing sufficient space for the age values.
• Plot the linear equations \(y_1 = 0.12x + 26.8\) and \(y_2 = 0.12x + 25\) for each x value in the range.
• Observe the lines' slope, reflecting the consistent increase in median age each year.

Graphs allow us to visually interpret data, making it easier to predict future values at a glance. They clarify complex data? trends into a form that can be quickly understood.