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Chapter 3: Problem 4

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of women enrolled in the fall in degreegranting institutions of higher education increased from \(10,184,000\) women in 2006 to \(11,658,000\) women in 2009. Round to the nearest thousand. (Source: nces .ed.gov, 2011)

Short Answer

Expert verified
The ordered pairs are \( (2006, 10,184,000) \) and \( (2009, 11,658,000) \). The average rate of change is 491,333.33 women per year.

Step by step solution

01

- Identify the Ordered Pairs

Represent the years and the number of women enrolled as ordered pairs. For the year 2006 with 10,184,000 women, the ordered pair is \( (2006, 10,184,000) \) and for the year 2009 with 11,658,000 women, the ordered pair is \( (2009, 11,658,000) \).
02

- Write the Ordered Pairs

Express the ordered pairs identified: First pair is \( (2006, 10,184,000) \) and the second pair is \( (2009, 11,658,000) \).
03

- Calculate the Average Rate of Change

Use the formula for the average rate of change \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \).\For our points, \( x_1 = 2006 \), \( y_1 = 10,184,000 \), \( x_2 = 2009 \), and \( y_2 = 11,658,000. \). Substituting the values, we get: \ m = \frac{{11,658,000 - 10,184,000}}{{2009 - 2006}} = \frac{{1,474,000}}{{3}} = 491,333.33 \ women per year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ordered pairs
In this exercise, we identify the number of women enrolled in degree-granting institutions over two different years: 2006 and 2009. We can express the information as ordered pairs, which are a fundamental concept in algebra.

An ordered pair consists of two elements within parentheses, separated by a comma鈥攖he first element representing the 'x' value, and the second element representing the 'y' value. In our exercise:

  • The year 2006 is paired with the corresponding number of women enrolled, 10,184,000, forming the ordered pair (2006, 10,184,000).
  • Similarly, for the year 2009, the pair is (2009, 11,658,000).
Understanding how to create and interpret ordered pairs helps us in visualizing and solving many algebraic problems. These pairs help map real-world problems into a mathematical format for easier analysis.
rate of change
The average rate of change gives us an understanding of how a quantity increases or decreases over time. In algebra, it is used extensively to analyze trends and make predictions.

From our ordered pairs (2006, 10,184,000) and (2009, 11,658,000), we want to find how rapidly the enrollment of women in degree-granting institutions has changed over these years. The formula to calculate the average rate of change, often denoted as 'm', is:

\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]

Applying our values, we have:

\[ m = \frac{{11,658,000 - 10,184,000}}{{2009 - 2006}} = \frac{{1,474,000}}{{3}} = 491,333.33 \text{ women per year} \]

This tells us that, on average, the number of women enrolled increased by 491,333.33 women per year from 2006 to 2009.
algebra equations
Algebra equations help solve many real-world problems by transforming them into mathematical statements. By solving these equations, we find unknown values and answers.

In this exercise, we used the average rate of change formula, which is an algebraic equation, to understand how the number of women enrolled changed over time. The equation we used was:

\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]

Each part of this equation has its significance:

  • \( y_2 \) and \( y_1 \) are the values of women enrolled in different years.
  • \( x_2 \) and \( x_1 \) are the years.
Substituting these values into the equation allows us to calculate the rate of change, giving us a clear perspective of the trend.

Understanding how to manipulate and solve algebraic equations is crucial for analyzing data, making predictions, and solving various mathematical problems.

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Most popular questions from this chapter

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=\frac{1}{2} x-8\)

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((6,2)\) and \((6,7)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{6-6}{7-2} \\ m &=\frac{0}{5} \\ m &=0 \end{aligned} $$

For exercises 97-98, some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any of the strategies listed above that you currently use to study math.

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((5,-9)\) and \((-2,-11)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{-11-9}{-2-5} \\ m &=\frac{-20}{-7} \\ m &=\frac{20}{7} \end{aligned} $$

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-3,-4) ;(1,3)\)
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