4.14: Common Frameworks for Situational Analysis
- Differentiate among five components of industry environment: customers, competitors, suppliers, regulations, and advocacy groups.
- Explain SWOT.
- Explain the resource-based view of strategy.
After the macro, or PESTEL, level analysis has been completed, the next step in goal setting is to do a situational analysis that looks more closely at external and internal conditions that affect your particular organization. Five key components of the organization’s specific business environment are examined. These are customers, competitors, suppliers, and government and legal issues—including regulations and advocacy or support groups. The analysis looks at what impact these factors may have on a specific organization or business.
Customers
A customer analysis is a critical component of any organizational strategy. You probably would not want to open an extreme sports adventure business in a city where the age of the typical resident is older than fifty. Or, you might be wasting your marketing dollars advertising trendy, designer jeans on the Hallmark TV channel. You might do very well, however, if you opened a combination coffee/book store in a busy college town. If an organization doesn’t know who its customers are or what its customers want, it can’t meet customer needs.
Some of the things a customer analysis should do is to:
- Identify the target customer. Is this customer base growing or is it decreasing? What are your customer demographics (age, income, location, gender, politics, etc.)? What is the revenue of these customers? How much discretionary income do they have?
- Understand the specific customer needs. Why do they buy certain brands? How do make their purchasing decisions? Do they purchase in person or online?
- Show if and how your product or service meets those needs.
Besides determining if a customer base exists in the region, this data can be used in the future to plan effective promotional campaigns, forecast inventory needs, and determine the optimal combination of distribution channels.
Competitors
You read about competitive analysis in the previous section at the macro level. At the situational level, a business needs to identify its specific competitors and assess their potential for taking market share. An organization needs to be aware of future initiatives of the competition (as much as is possible) and examine the competitors’ financial and marketing performances.
Suppliers
Another industry component that will greatly impact an organization is its suppliers. Your business may start by buying raw materials and producing finished goods purchased directly by consumers. Or it may process raw materials into products that are part of another company’s final consumer goods. Some organizations create services rather than goods but still need materials, such as computer software and hardware or office supplies, to provide those services. Whatever the situation, without raw materials or support products, the organization cannot operate.
In the past, it was common for an organization to choose suppliers that were in the same region or at least the same country. In today’s competitive global economy, however, a supplier is likely on the other side of the world. The supply chain is a system comprised of organizations, information, resources, people, technology, and activities that bring products or services from a supplier to a consumer. In larger organizations, entire departments may be dedicated to supply chain logistics. Implementing cooperative alliances with key suppliers is also a popular tactic employed by strategic organizations. Although multiple sources of supply helps to guarantee the availability of supplies, creating a cooperative agreement with one supplier can significantly reduce costs. How to handle suppliers is an extremely important factor in setting goals and generating strategies.
Regulations
Governmental and legal environment are part of the PESTEL analysis discussed in the previous section. At the situational level, however, state and local regulations also need to be part of any analysis. The regulatory burden depends largely upon the type of industry and the specific nature of the business. In some industries, regulation is the single biggest uncertainty affecting investment and spending, corporate image, and risk management. These organizations include airlines, utilities, railways, telecommunications, banking, and pharmaceuticals. Often, the regulations have positive impacts on both consumers and businesses. They provide the public with a high level of confidence in the safety and efficacy of the products. They can also prevent competition from businesses with substandard and low-quality goods from trying to enter an industry. Despite the benefits that regulations can provide, any changes in how the product is manufactured, shipped, tested, or provided will greatly affect unit costs and profit margins.
Regulations are lighter for less risky products and businesses, but every company must comply with federal and state payroll, benefits, tax requirements, and following municipal commercial and building codes can present challenges. For example, a decision to expand the size of a fitness center to attract more customers may be thwarted by a local regulation that fixes a ratio of parking spaces to square footage. Or extra time could be added to delivery schedules if truckers must avoid more direct routes to comply with weight regulations on local highways. Some localities will not permit the shipping of dangerous chemicals through their towns. These factors all affect an organization’s ability to be competitive.
Regulations are a greater burden on small companies unless they are exempted because of size (some companies with fewer than twenty-five employees are exempt from overtime pay provisions, for example.) However, dealing with regulations can be challenging regardless of the size of the organization. Complying with regulations often involves a trade-off between short-term profits and long-term public relations and social responsibility.
Advocacy Groups
Advocacy groups are also known as special interest groups, public interest groups, environmental groups, or political support groups. Whatever their label, their aim is to influence public opinion, public policy, and company behavior. Advocacy groups use a variety of strategies and tactics to draw attention to their causes, including lobbying, promotions using celebrities, and public information campaigns. They provide a type of check on the business community by exposing unethical or unpopular practices. The Internet—especially social media—has greatly strengthened the ability of these groups to impact an organization.
Advocacy groups represent political, economic, and social interests—all of which affect the business environment. (The PESTEL analysis in a previous section looked at the macro effect of these groups.) Today environmental interest groups are extremely varied, and many hold conflicting views about appropriate strategies for pursuing their interests.
People for the Ethical Treatment of Animals (PETA) is one of the more outspoken activist groups and has targeted many organizations it believes abuse animals. In 2016, after years of PETA campaigns aimed at publicizing its marine operations, SeaWorld announced that it would end its orca whale breeding program. In 2017, a combination of low ticket sales and adverse publicity on the general condition of circus animals led Ringling Bros. and Barnum and Bailey Circus to close after almost a century and a half. PETA had filed more than 130 formal complaints against the circus with the U.S. Department of Agriculture. Regardless of criticisms that some advocacy group charges are not scientifically sound, negative publicity alone may be enough to disrupt business.
SWOT: A Situational Analysis Summary
Once external factors have been thoroughly considered, an organization can look at its own internal resources. A SWOT analysis is a method that examines the internal strengths and weaknesses of an organization as well as external opportunities and threats (social, political, economic, legal environmental) that would affect that organization. In fact, a SWOT analysis is really more of a summary of data from various other analyses formulated in a way that allows for comparisons. SWOT stands for s trengths, w eaknesses, o pportunities, and t hreats.
- Strengths are the areas where the organization has particular skills and resources that would allow it to pursue goals effectively. For instance, innovative and collaborative organizational cultures are strengths of Apple and Google.
- Weaknesses are areas where the organization is lacking resources and would be prevented from pursuing some goals. For example, when consumers demanded smaller, fuel-efficient cars, many American manufacturers lacked the designs and equipment to make these cars.
- Opportunities are conditions that are favorable to the organization and would facilitate its efforts to achieve its goals. Building on the earlier example, Japanese automakers saw an opportunity in the American demand for smaller cars to increase their market share.
- Threats are conditions that would prevent the organization from achieving its goals. Many businesses, for instance, are concerned about the threat posed by China’s competitive growing manufacturing capability.
Resource-based View (RBV) Strategic Approach
The resource-based view (RBV) argues that focusing on an organization’s strengths is essential to achieve a sustained competitive advantage. (Remember, the purpose of the SWOT analysis is to help companies identify their strengths.) RBV supporters look for ways to use internal resources (assets) to take advantage of external opportunities. To understand RBV, resources are classified into two groups: tangible and intangible.
- Tangible assets are physical things such as land, equipment and machines, and real estate. Although they are necessary, they aren’t unique and competitors can fairly easily acquire these kinds of assets.
- Intangible assets are anything an organization can own that is not physical. Examples include brand names, intellectual property, and the organization’s reputation and goodwill. These kinds of assets are not easily acquired and usually contribute heavily toward a sustained competitive advantage. For example, Mercedes Benz has a reputation for quality engineering and luxury in its vehicles. People take it as a given that they produce precision, high-quality products
The VRIO Framework
In 1991, J.B. Barney developed the VRIN framework to analyze a firm’s internal resources and capabilities to see if they can be used to sustain a competitive advantage. A few years later, he later revised VRIN to VRIO, the current model. VRIO stands for the four key characteristics that a resource must have if it is to produce sustained competitive advantage.
- Valuable. A resource is valuable if it enables the company to take advantage of opportunities or defend against threats. A valuable resource allows the company to provide value for its customers by making its product better or cheaper. It means the company can continue to add features or lower the price to maintain the perceived value.
- Rare. If only one or two companies can acquire a resource it is considered rare. A resource may be rare because little of it exists, such as workers with particular skills, or because there are few sources for it, such as lithium for batteries. Rare and valuable resources can create a temporary competitive advantage because even if other firms duplicate performance, they don’t have access to the unique resource (tangible or intangible, such as a patent or unique skill). If the resource is not rare, the best a company can achieve is competitive parity (equality).
- Inimitable. If another organization can’t copy, buy, or find a replacement for the resource, it is inimitable . According to Barney, resources can be inimitable if they (1) developed historically over a long period of time, (2) competitors cannot identify the particular resources that are the cause of competitive advantage, and (3) the resource is a result of the corporate culture and personal dynamics of the organization.
- Organized to capture value. The three characteristics listed earlier are “necessary but not sufficient conditions” to achieve a sustained competitive advantage. The missing ingredient is management’s ability to develop the strategy to put it all together. The firm must be organized in a way that can capture value by employing the right strategies.
The VRIO and the SWOT analyses are tools that help companies organize to successfully achieve sustained competitive advantages.
Contributors and Attributions
- Common Frameworks for Situational Analysis. Authored by : John/Lynn Bruton and Lumen Learning. License : CC BY: Attribution
- Young Girl. Authored by : ellemclin. Located at : https://pixabay.com/en/child-with-braids-braids-937658/ . License : CC0: No Rights Reserved
- SWOT analysis diagram in English language. Authored by : Xhienne. Provided by : Wikimedia; Modification by Lumen Learning. Located at : https://commons.wikimedia.org/w/index.php?title=File:SWOT_en.svg . License : CC BY-SA: Attribution-ShareAlike


26: Fluid, Electrolyte, and Acid-Base Balance
Homeostasis, or the maintenance of constant conditions in the body, is a fundamental property of all living things. In the human body, the substances that participate in chemical reactions must remain within narrows ranges of concentration. Too much or too little of a single substance can disrupt your bodily functions. Because metabolism relies on reactions that are all interconnected, any disruption might affect multiple organs or even organ systems.
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- 26.0: Prelude to Fluid, Electrolyte, and Acid-Base Balance
- Water is the most ubiquitous substance in the chemical reactions of life. The interactions of various aqueous solutions—solutions in which water is the solvent—are continuously monitored and adjusted by a large suite of interconnected feedback systems in your body. Understanding the ways in which the body maintains these critical balances is key to understanding good health.
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- 26.1: Body Fluids and Fluid Compartment
- The chemical reactions of life take place in aqueous solutions. The dissolved substances in a solution are called solutes. In the human body, solutes vary in different parts of the body, but may include proteins—including those that transport lipids, carbohydrates, and, very importantly, electrolytes. Often in medicine, a mineral dissociated from a salt that carries an electrical charge (an ion) is called and electrolyte. For instance, sodium ions (Na+) and chloride ions (Cl-) are often referred
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- 26.2: Water Balance
- On a typical day, the average adult will take in about 2500 mL (almost 3 quarts) of aqueous fluids. Although most of the intake comes through the digestive tract, about 230 mL (8 ounces) per day is generated metabolically, in the last steps of aerobic respiration. Additionally, each day about the same volume (2500 mL) of water leaves the body by different routes; most of this lost water is removed as urine.
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- 26.3: Electrolyte Balance
- The body contains a large variety of ions, or electrolytes, which perform a variety of functions. Some ions assist in the transmission of electrical impulses along cell membranes in neurons and muscles. Other ions help to stabilize protein structures in enzymes. Still others aid in releasing hormones from endocrine glands. All of the ions in plasma contribute to the osmotic balance that controls the movement of water between cells and their environment.
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- 26.4: Acid-Base Balance
- Proper physiological functioning depends on a very tight balance between the concentrations of acids and bases in the blood. Acid-balance balance is measured using the pH scale. A variety of buffering systems permits blood and other bodily fluids to maintain a narrow pH range, even in the face of perturbations. A buffer is a chemical system that prevents a radical change in fluid pH by dampening the change in hydrogen ion concentrations in the case of excess acid or base.
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- 26.5: Disorders of Acid-Base Balance
- Normal arterial blood pH is restricted to a very narrow range of 7.35 to 7.45. A person who has a blood pH below 7.35 is considered to be in physiological acidosis, and a continuous blood pH below 7.0 can be fatal. Acidosis has several symptoms, including headache and confusion, and the individual can become lethargic and easily fatigued. A person who has a blood pH above 7.45 is considered to be in alkalosis, and a pH above 7.8 is fatal.


16.6: Learning Activities
The course is divided into Learning Modules. In each module there are several graded learning activities, including a written assignment, a written assignment discussion, 2 or more textbook chapter discussions, and a 2-part learning audit. In addition there is a term paper module to be completed throughout the course. These various learning activities are detailed below.
Written Assignments and Related Discussions:
Each module includes a written assignment and a written assignment discussion activity. In each module, there is a written assignment folder – the dropbox is in that folder, as is the written assignment discussion forum. In addition to leading the discussion on your own assignment, you must also participate in the discussions of other students.
Student-led Discussions:
In every module you will find two or more Student-led Discussions. This is the way we cover the content of the textbook. For each chapter you must ask an original “critical thinking” question about some topic in the chapter. Other students will respond to your question, and you will then reply back to those students. In addition you leading your own discussion thread, you are required to answer some of the questions posed by other students, and they are required to reply to your answers. You are encouraged to keep up these “virtual discussions” as long as they are productive. The idea here is for each student to lead one discussion with the other students about some important and/or controversial issue introduced in each chapter, and participate in a few others. A large percentage of your final grade is determined by your participation in these discussions. I will grade these discussions, but I will not be a participant. If the discussion you are leading gets off track, it is your responsibility to refocus it. You are responsible for maintaining the quality of the discussion threads you lead. Every posting to a discussion should add something substantive to that discussion.
Knowledge Audits:
In each module your appreciation for and understanding of the important content issues and concepts will be assessed two ways:
Module Exams:
There is an objective exam on the textbook chapters for each module. Although you may use your textbook, the exams have a time limit.
Module Recaps:
You are asked to write a brief overview of the most important things that you have learned in each module.
 Part one of this project requires you to interview a person over the age of 75 on their experiences with the agin process. Part two of this project requires you to write a research paper. Your interview will be one of the sources for this paper, but you will be required to do additional research on any conditions your interviewee may have experienced in their lifetime. Part three of the term paper will require you to lead a discussion on your term paper. You are encouraged to keep up these “virtual discussions” as long as they are productive.
Extra Credit / Make-up Work / Incomplete Grades
- The major requirement in this course is to discuss, with other students, the contents of each chapter in the text. There is no substitute for this requirement, and I do not permit “extra credit” or “alternative credit” assignments.
- Also, there is no way to “go back” after a module has ended and “make-up” missed discussion activity, because there are no other students left to learn from your posts and discuss the content with you.
- Finally, an incomplete in the course is not appropriate, as there is no way to complete the course once it has ended and all of the other students are gone.
If you have a question about the learning activities for this course, please post a question in the ” Ask a Question” area located at the bottom of the “Course Information” section of this course. Please be sure to give your post a meaningful subject and to check that area frequently for replies and new posts.
Contributors and Attributions
- Learning Activities. Authored by : Kelly, Jessica. Provided by : Herkimer College. Project : ATD Course. License : Public Domain: No Known Copyright


1.15: Assignment- Lifespan Development in the News
Lifespan Development in the News
| Criteria | Proficient | Developing | Not Evident | Points |
|---|---|---|---|---|
| Finds an appropriate news article and writes a summary | Finds a detailed and relevant news article on a topic related to lifespan development, and provides a coherent summary of the article | Does not provide enough detail in the summary or does not find an appropriate news article | Poor choice of news article, incomplete summary, or difficult to understand | __/4 |
| Finds appropriate research article and writes a summary | Finds a detailed and relevant peer-reviewed article on a psychological topic, and provides a coherent summary of the article. | Does not provide enough detail in the summary or does not find an appropriate journal article | Poor choice of research article, incomplete summary, or difficult to understand. | __/4 |
| Describes the research | Describes the research method, research design, hypothesis, independent and dependent variables (if applicable), and any reliability or validity concerns | Does not fully describe the research method, design, hypothesis, variables, and reliability and validity concerns | Does not describe the research method, design, hypothesis, variables, and reliability and validity concerns | __/4 |
| Compares and contrasts the news and research article and examines the accuracy of the news article | Compares and contrasts the key points, style, and purpose of the news article with that of the research article | Does not fully compare and contrast the key points, style, and purpose of the news article with that of the research article | Does not compare and contrast the key points, style, and purpose of the news article with that of the research article | __/4 |
| Includes correct APA citations | Provides accurate in-text citations AND a reference page for the articles used | Does not fully provide accurate in-text citations AND a reference page for the articles used | Does not provide accurate in-text citations AND a reference page for the articles used | __/4 |
| Total: | __/20 |
Contributors and Attributions
CC licensed content, Original
CC licensed content, Shared previously


14.3: California’s Ethnic and Cultural Identiy
According to the 2020 U.S. census, California's population was 34.7% Non-Hispanic White, 5.7% African American, 1.5% Native American, 16.1% Asian, 0.4% Pacific Islander, 13% two or more races, and 39.4% Hispanic or Latino of any race. Hispanics are the largest racial/ethnic group in California. Non-Hispanic whites have decreased from about 76.3 - 78% of the state's population in 1970 to 36.6% in 2018.While the population of minorities accounts for 100.7 million of 300 million U.S. residents, 20% of the national total live in California (2008).
California has the highest number, and second highest percentage, of Asian Americans by state. Only Hawaii has a higher Asian American percentage than California. While New Mexico and Texas have higher percentages of Hispanics, California has the highest total number of Hispanics of any U.S. state. Hispanics are the largest single ethnic group in the state.
The largest named ancestries in California are Mexican (25%), German (9%), Irish (7.7%), English (7.4%), and Italian (5.8%); there are 65 other ethnicities with sizable populations in California including Arabs, Albanians, Australians, Brazilians, Canadians, Haitians, Iranians/Persians, and Somalis as examples. Both Los Angeles and San Francisco have large numbers of residents with English, French, Italian, Iranian/Persian, German, Russian, and Scandinavian ancestry.
California has the largest population of white Americans in the U.S., totaling 21,453,934 residents as of the 2010 census. The state has the fifth largest population of African Americans in the U.S., an estimated 2,299,072 residents. California's Asian population is estimated at 4.9 million, approximately one-third of the nation's estimated 15 million Asian Americans. California's Native American population of 285,512 is the third largest of any state, behind Arizona and Oklahoma.
Since the 2000 U.S. census, California has been known as the second state in U.S. history (after Hawaii since its statehood in 1959) to have a non-white majority when excluding Hispanic whites, and since 2014, the first state to have a Latino plurality surpass other racial/ethnic groups. The media discussed the possibility of Latinos becoming a majority in the 21st century, for the first time since statehood (1850) when massive immigration of Anglo-Americans during the California Gold Rush of 1848-1849 reduced Spanish speakers to 20% of the population.
Native Americans - Indigenous Peoples of California
As of 2010, California's Native American population of 362,801 was the most of any state. It also has the most Native American tribes, indigenous to the state or not, but the majority of known Californian Indian tribes became extinct in the late 19th century. The U.S. Census includes Latin American Indian, especially immigrants who belonged to indigenous peoples or who have Amerindian heritage from North and South America.
The Cherokee Nation is the largest tribe in the state with a population of 22,000 in 2020, although the number of Cherokee descendants are at 110,000 in the earlier census taken in 2010 and any persons residing in California of Cherokee descent may surpass 600,000 according to demographers. Their tribal nation is in the northeast part of Oklahoma, but they live across the state and neighboring states. They are often descendants of Dust Bowl refugees in the 1930s and 1940s who migrated to the state's farming counties and urban areas for jobs. The largest urban American Indian communities are Los Angeles/Long Beach, San Francisco/Oakland, Sacramento, and San Diego areas.
California also has significant populations of the Apache, Chickasaw, Choctaw, Muscogee, Seminole, Hopi, Zuni, Navajo, Blackfeet, Shoshone, Paiute, Pueblo, Tohono Oʼodham, Cahuilla, Chumash, Karuk, Yurok, and Yokuts tribes. The Cahuilla in the Coachella Valley have profited from real estate land leases, and much of Indio and Palm Springs are tribal-owned lands under legal tribal jurisdiction.
European Americans
California has the largest population of European Americans of any state in the United States. This is due to several factors, including the state's history of immigration, its diverse geography, and its climate.
The state has been a major destination for immigrants from Europe since the early 19 th century. In the 1840s, for example, California was home to many German, Irish, Italian, and English immigrants. These immigrants settled in cities like San Francisco, Los Angeles, and Sacramento, and they helped to shape the state's culture and identity.
Another factor contributing to California's large European American population is its diverse geography. The state is home to a variety of climates, from the Mediterranean climate of Southern California to the Mediterranean climate of Northern California. This diversity of climate has attracted immigrants from all over Europe, who have found places to settle in all parts of the state.
California also has the third largest Greek American population in the United States, behind New York and Massachusetts. The Los Angeles Harbor area of San Pedro has a sizeable Croatian American population. Small Amish/Mennonite colonies exist in an area bordered by the towns Oakdale, Riverbank and Ripon near Modesto and in Reedley, Sanger and Orange Cove near Fresno in the San Joaquin Valley.
Brief History
The first Europeans to arrive in California were the Spanish, and the area was under Spanish and Mexican rule for many years. The population was diverse, with people of White, Mestizo, African, and Indigenous ancestry. By 1846, more White Americans had begun to enter California, and they made up 10% of the non-Native population.
The California Gold Rush of 1848-1855 brought a massive influx of people to California, including many White Americans. By 1850, the non-Native population of California had grown to 92,597, and two-thirds of these new arrivals were Americans.
While Northern California became predominantly white by the mid-19th century, Southern California remained mainly Mexican until the 1880s. This changed after the Southern Pacific railroad reached Los Angeles, and Southern California's population grew rapidly. By 1920, the population of Southern California had reached 1.3 million, and white Americans made up most of the population.
In the 1930s, the Dust Bowl in the American Midwest caused many people to leave their homes and seek new opportunities in California. These migrants, known as Okies, were mostly white, and they helped to further diversify California's population.
By 1940, 90% of California's population identified as white. However, in recent decades, the state has become more diverse, and by 1990, the white non-Hispanic population had decreased to 43%. This is due to increased immigration from other countries, as well as higher birth rates among non-white Californians.


1.2: Organization of the Elements - The Periodic Table
The number of protons in the nucleus of an element is called the atomic number of that element. Chemists typically place elements in order of increasing atomic numbers in a special arrangement that is called the periodic table.
The periodic table is not simply a grid of elements arranged numerically. In the periodic table, the elements are arranged in horizontal rows called periods (numbered in blue) and vertically into columns called groups. These groups are numbered by two, somewhat conflicting, schemes. In the simplest presentation, favored by the International Union of Pure and Applied Chemistry ( IUPAC ), the groups are simply numbered 1-18. The convention in much of the world, however, is to number the first two groups 1A and 2A, the last six groups 3A-8A; the middle ten groups are then numbered 1B-8B (but not in that order!). While the IUPAC numbering appears much simpler, in this text we will use the current USA nomenclature (1A-8A). The reason for this choice will become more apparent in Chapter 3 when we discuss “valence” and electron configuration in more detail. The actual layout of the periodic table is based on the grouping of the elements according to chemical properties. For example, elements in each Group of the periodic table (each vertical column) will share many of the same chemical properties. As we discuss the properties of elements and the ways they combine with other elements, the reasons for this particular arrangement of the periodic table will become more obvious.
As you can see, each element in the periodic table is represented by a box containing the chemical symbol, the atomic number (the number of protons in the nucleus) and the atomic mass of the element. Remember that the atomic mass is the weighted average of the masses of all of the natural isotopes of the particular element.
Periodic tables are often colored, or shaded, to distinguish groups of elements that have similar properties or chemical reactivity. The broadest classification is into metals, metalloids (or semi-metals) and nonmetals. The elements in Groups 1A – 8A are called the representative elements and the elements in Groups 3 - 15 are called the transition metals. the metallic elements are shown in purple. Metals are solids (except for mercury), can conduct electricity and are usually malleable (can be rolled into sheets) and are ductile (can be drawn into wires). Metals are usually separated into the main group metals (the elements colored purple in Groups 1A - 5A) and the transition metals (in Groups 3 – 15). Nonmetals (yellow in the Figure) do not conduct electricity (with the exception of carbon in the form of graphite) and have a variety of physical states (some are solids, some liquids and some gasses). Two important subclasses of nonmetals are the halogens (Group 7A) and the inert gasses (or noble gasses; Group 8A). At the border between metals and nonmetals lie the elements boron, silicon, germanium, astatine, antimony and tellurium. These elements share physical properties of metals and nonmetals and are called metalloids, or semi-metals. The common semiconductors silicon and germanium are in this group and it is their unique electrical properties that make transistors and other solid-state devices possible. Later in this book we will see that the position of elements in the periodic table also correlates with their chemical reactivity.


7.7: Solve Rational Inequalities
- Solve rational inequalities
- Solve an inequality with rational functions
Before you get started, take this readiness quiz.
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Find the value of \(x-5\) when ⓐ \(x=6\) ⓑ \(x=-3\) ⓒ \(x=5\)
If you missed this problem, review Example 1.2.16 . -
Solve: \(8-2 x<12\)
If you missed this problem, review Example 2.6.13 . -
Write in interval notation: \(-3 \leq x<5 \)
If you missed this problem, review Example 2.6.4 .
Solve Rational Inequalities
We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.
Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.
A rational inequality is an inequality that contains a rational expression.
Inequalities such as\(\quad \dfrac{3}{2 x}>1, \quad \dfrac{2 x}{x-3}<4, \quad \dfrac{2 x-3}{x-6} \geq x,\quad\) and \(\quad \dfrac{1}{4}-\dfrac{2}{x^{2}} \leq \dfrac{3}{x}\quad \) are rational inequalities as they each contain a rational expression.
When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.
Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.
When we solve an equation and the result is \(x=3\), we know there is one solution, which is 3.
When we solve an inequality and the result is \(x>3\), we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.
To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.
Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.
We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.
We write the solution in interval notation being careful to determine whether the endpoints are included.
Solve and write the solution in interval notation: \(\dfrac{x-1}{x+3} \geq 0\)
Solution
Step 1 . Write the inequality as one quotient on the left and zero on the right.
Our inequality is in this form.\[\dfrac{x-1}{x+3} \geq 0 \nonumber \]
Step 2 . Determine the critical points—the points where the rational expression will be zero or undefined.
The rational expression will be zero when the numerator is zero. Since \(x-1=0\) when \(x=1\), then 1 is a critical point.
The rational expression will be undefined when the denominator is zero. Since \(x+3=0\) when \(x=-3\), then -3 is a critical point.
The critical points are 1 and -3.
Step 3 . Use the critical points to divide the number line into intervals.
The number line is divided into three intervals:
\[(-\infty,-3) \quad (-3,1) \quad (1,\infty) \nonumber \]
Step 4 . Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.
To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.
\[\text { Interval }(-\infty,-3) \nonumber \]
The number -4 is in the interval \((-\infty,-3)\). Test \(x=-4\) in the expression in the numerator and the denominator.
The numerator:
\[\begin{array}{l} {x-1} \\ {-4-1} \\ {-5} \\ {\text {Negative}} \end{array} \nonumber \]
The denominator:
\[\begin{array}{l} {x+3} \\ {-4+3} \\ {-1} \\ {\text {Negative}} \end{array} \nonumber \]
Above the number line, mark the factor \(x-1\) negative and mark the factor \(x+3\) negative.
Since a negative divided by a negative is positive, mark the quotient positive in the interval \((-\infty,-3)\)
\[\text {Interval } (-3,1) \nonumber \]
The number 0 is in the interval \((-3,1)\). Test \(x=0\).
The numerator:
\[\begin{array}{l} {x-1} \\ {0-1} \\ {-1} \\ {\text {Negative}} \end{array} \nonumber \]
The denominator:
\[\begin{array}{l} {x+3} \\ {0+3} \\ {3} \\ {\text {Positive}} \end{array} \nonumber \]
Above the number line, mark the factor \(x-1\) negative and mark \(x+3\) positive.
Since a negative divided by a positive is negative, the quotient is marked negative in the interval \((-3,1)\).
\[\text {Interval }(1, \infty) \nonumber \]
The number 2 is in the interval \((1, \infty)\). Test \(x=2\).
The numerator:
\[\begin{array}{l} {x-1} \\ {2-1} \\ {1} \\ {\text {Positive}} \end{array} \nonumber \]
The denominator:
\[\begin{array}{l} {x+3} \\ {2+3} \\ {5} \\ {\text {Positive}} \end{array} \nonumber \]
Above the number line, mark the factor \(x-1\) positive and mark \(x+3\) positive.
Since a positive divided by a positive is positive, mark the quotient positive in the interval \((1, \infty)\).
Step 5 . Determine the intervals where the inequality is correct. Write the solution in interval notation.
We want the quotient to be greater than or equal to zero, so the numbers in the intervals \((-\infty,-3)\) and \((1, \infty) \)are solutions.
But what about the critical points?
The critical point \(x=-3\) makes the denominator 0, so it must be excluded from the solution and we mark it with a parenthesis.
The critical point \(x=1\) makes the whole rational expression 0. The inequality requires that the rational expression be greater than or equal to. So, 1 is part of the solution and we will mark it with a bracket.
Recall that when we have a solution made up of more than one interval we use the union symbol, \(\cup \), to connect the two intervals. The solution in interval notation is \((-\infty,-3) \cup[1, \infty)\).
Solve and write the solution in interval notation: \(\dfrac{x-2}{x+4} \geq 0\)
- Answer
-
\((-\infty,-4) \cup[2, \infty)\)
Solve and write the solution in interval notation: \(\dfrac{x+2}{x-4} \geq 0\)
- Answer
-
\((-\infty,-2] \cup(4, \infty)\)
We summarize the steps for easy reference.
Step 1. Write the inequality as one quotient on the left and zero on the right.
Step 2. Determine the critical points–the points where the rational expression will be zero or undefined.
Step 3. Use the critical points to divide the number line into intervals.
Step 4. Test a value in each interval. Above the number line show the sign of each factor of the numerator and denominator in each interval. Below the number line show the sign of the quotient.
Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
The next example requires that we first get the rational inequality into the correct form.
Solve and write the solution in interval notation: \(\dfrac{4 x}{x-6}<1\)
Solution
\[\dfrac{4 x}{x-6}<1 \nonumber \]
Subtract 1 to get zero on the right.
\[\dfrac{4 x}{x-6}-1<0 \nonumber \]
Rewrite 1 as a fraction using the LCD.
\[\dfrac{4 x}{x-6}-\frac{x-6}{x-6}<0 \nonumber \]
Subtract the numerators and place the difference over the common denominator.
\[\dfrac{4 x-(x-6)}{x-6}<0 \nonumber \]
Simplify.
\[\dfrac{3 x+6}{x-6}<0 \nonumber \]
Factor the numerator to show all factors.
\[\dfrac{3(x+2)}{x-6}<0 \nonumber \]
Find the critical points.
The quotient will be zero when the numerator is zero. The quotient is undefined when the denominator is zero.
\[\begin{array}{rlrl} {x+2} & {=0} & {x-6} & {=0} \\ {x} & {=-2} & {x} & {=6} \end{array} \nonumber \]
Use the critical points to divide the number line into intervals.
Test a value in each interval.
| \((-\infty,-2)\) | \((-2,6)\) | \((6, \infty)\) | |
|---|---|---|---|
| \(x+2)\) |
x+2 -3+2 -1 - |
x+2 0+2 2 + |
x+2 7+2 9 + |
| \(x-6\) |
x-6 -3-6 -9 - |
x-6 0-6 -6 - |
x-6 7-6 1 + |
Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.
Determine the intervals where the inequality is correct. We want the quotient to be negative, so the solution includes the points between −2 and 6. Since the inequality is strictly less than, the endpoints are not included.
We write the solution in interval notation as (−2, 6).
Solve and write the solution in interval notation: \(\dfrac{3 x}{x-3}<1\).
- Answer
-
\(\left(-\dfrac{3}{2}, 3\right)\)
Solve and write the solution in interval notation: \(\dfrac{3 x}{x-4}<2\).
- Answer
-
\((-8,4)\)
In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.
Solve and write the solution in interval notation: \(\dfrac{5}{x^{2}-2 x-15}>0\).
Solution
The inequality is in the correct form.
\[\dfrac{5}{x^{2}-2 x-15}>0 \nonumber \]
Factor the denominator.
\[\dfrac{5}{(x+3)(x-5)}>0 \nonumber \]
Find the critical points. The quotient is 0 when the numerator is 0. Since the numerator is always 5, the quotient cannot be 0.
The quotient will be undefined when the denominator is zero.
\[\begin{aligned} &(x+3)(x-5)=0\\ &x=-3, x=5 \end{aligned} \nonumber \]
Use the critical points to divide the number line into intervals.
Test values in each interval. Above the number line show the sign of each factor of the denominator in each interval. Below the number line, show the sign of the quotient.
Write the solution in interval notation.
\[(-\infty,-3) \cup(5, \infty) \nonumber \]
Solve and write the solution in interval notation: \(\dfrac{1}{x^{2}+2 x-8}>0\).
- Answer
-
\((-\infty,-4) \cup(2, \infty)\)
Solve and write the solution in interval notation: \(\dfrac{3}{x^{2}+x-12}>0 \).
- Answer
-
\((-\infty,-4) \cup(3, \infty)\)
The next example requires some work to get it into the needed form.
Solve and write the solution in interval notation: \(\dfrac{1}{3}-\dfrac{2}{x^{2}}<\dfrac{5}{3 x}\).
Solution
\[\dfrac{1}{3}-\dfrac{2}{x^{2}}<\dfrac{5}{3 x} \nonumber \]
Subtract \(\dfrac{5}{3 x}\) to get zero on the right.
\[\dfrac{1}{3}-\dfrac{2}{x^{2}}-\dfrac{5}{3 x}<0 \nonumber \]
Rewrite to get each fraction with the LCD
\[\dfrac{1 \cdot x^{2}}{3 \cdot x^{2}}-\dfrac{2 \cdot 3}{x^{2} \cdot 3}-\dfrac{5 \cdot x}{3 x-x}<0 \nonumber \]
Simplify.
\[\dfrac{x^{2}}{3 x^{2}}-\dfrac{6}{3 x^{2}}-\dfrac{5 x}{3 x^{2}}<0 \nonumber \]
Subtract the numerators and place the difference over the common denominator.
\[\dfrac{x^{2}-5 x-6}{3 x^{2}}<0 \nonumber \]
Factor the numerator.
\[\dfrac{(x-6)(x+1)}{3 x^{2}}<0 \nonumber \]
Find the critical points.
\[\begin{array}{rlrl} {3 x^{2}=0} && {x-6=0} && {x+1=0} \\ {x=0} && {x=6} && {x=-1} \end{array} \nonumber \]
Use the critical points to divide the number line into intervals.
Above the number line show the sign of each factor in each interval. Below the number line, show the sign of the quotient.
Since, 0 is excluded, the solution is the two \((-1,0) \cup(0,6)\) intervals, \((-1,0)\) and \((0,6)\).
Solve and write the solution in interval notation: \(\dfrac{1}{2}+\dfrac{4}{x^{2}}<\dfrac{3}{x}\).
- Answer
-
\((2,4)\)
Solve and write the solution in interval notation: \(\dfrac{1}{3}+\dfrac{6}{x^{2}}<\dfrac{3}{x}\).
- Answer
-
\((3,6)\)
Solve an Inequality with Rational Functions
When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.
Given the function \(R(x)=\dfrac{x+3}{x-5}\), find the values of x that make the function less than or equal to 0.
Solution
We want the function to be less than or equal to 0.
\[R(x) \leq 0 \nonumber \]
Substitute the rational expression for \(R(x)\).
\[\dfrac{x+3}{x-5} \leq 0 \quad x \neq 5 \nonumber \]
Find the critical points.
\[\begin{array}{rlrl} {x+3=0} && {x-5=0} \\ {x=-3} && {x=5} \end{array} \nonumber \]
Use the critical points to divide the number line into intervals.
Test values in each interval. Above the number line, show the sign of each factor in each interval. Below the number line, show the sign of the quotient. Write the solution in interval notation. Since 5 is excluded we, do not include it in the interval.
\[[-3,5) \nonumber \]
Given the function \(R(x)=\dfrac{x-2}{x+4}\), find the values of \(x\) that make the function less than or equal to 0.
- Answer
-
\((-4,2]\)
Given the function \(R(x)=\dfrac{x+1}{x-4}\), find the values of \(x\) that make the function less than or equal to 0.
- Answer
-
\([-1,4)\)
In economics, the function \(C(x)\) is used to represent the cost of producing \(x\) units of a commodity. The average cost per unit can be found by dividing \(C(x)\) by the number of items \(x\). Then, the average cost per unit is \(c(x)=\dfrac{C(x)}{x}).
The function\(C(x)=10 x+3000\) represents the cost to produce \(x\), number of items. Find:
- The average cost function, \(c(x)\)
- How many items should be produced so that the average cost is less than $40.
Solution
- \[C(x)=10 x+3000 \nonumber \]
The average cost function is \(c(x)=\dfrac{C(x)}{x})\). To find the average cost function, divide the cost function by \(x\).
\[\begin{aligned} &c(x)=\dfrac{C(x)}{x}\\ &c(x)=\dfrac{10 x+3000}{x} \end{aligned} \nonumber \]
The average cost function is \(c(x)=\dfrac{10 x+3000}{x} \)
- We want the function \(c(x)\) to be less than 40.
\[c(x)<40 \nonumber \]
Substitute the rational expression forc(x).
\[\dfrac{10 x+3000}{x}<40, \quad x \neq 0 \nonumber \]
Subtract 40 to get 0 on the right.
\[\dfrac{10 x+3000}{x}-40<0 \nonumber \]
Rewrite the left side as one quotient by finding the LCD and performing the subtraction.
\[\begin{aligned} \dfrac{10 x+3000}{x}-40\left(\dfrac{x}{x}\right) &<0\\ \dfrac{10 x+3000}{x}-\dfrac{40 x}{x} &<0\\ \dfrac{10 x+3000-40 x}{x} &<0 \\ \dfrac{-30 x+3000}{x} &<0 \end{aligned} \nonumber \]
Factor the numerator to show all factors.
\[\begin{array}{ll} {\dfrac{-30(x-100)}{x}<0} \\ {-30(x-100)=0} && {x=0} \end{array} \nonumber \]
Find the critical points.
\[\begin{array}{rl} {-30 \neq 0} & {x-100=0} \\ &{x=100} \end{array} \nonumber \]
More than 100 items must be produced to keep the average cost below $40 per item.
The function\(C(x)=20 x+6000\) represents the cost to produce \(x\), number of items. Find:
- How many items should be produced so that the average cost is less than $60.
- Answer
-
- \(c(x)=\dfrac{20 x+6000}{x}\)
- More than 150 items must be produced to keep the average cost below $60 per item.
The function\(C(x)=5 x+900\) represents the cost to produce \(x\), number of items. Find:
- How many items should be produced so that the average cost is less than $20.
- Answer
-
- \(c(x)=\dfrac{5 x+900}{x}\)
- More than 60 items must be produced to keep the average cost below $20 per item.


2.7: Native American (1600 – 1800s)
Introduction
What is Native American art? Unfortunately, the past stereotypes still prevail; a warrior with his feathered headdress or a woman in a beaded dress, etc., images perpetuated today by media practices, the educational systems, and individual perceptions. Native groups lived across North America and were as diverse in languages, traditions, rituals, and artwork as any population of people covering a large geographic area. With any populace in the world, their artwork was created using materials found in the people's location and culture. The art of Native Americans was deemed "primitive," and cultural artifacts as viewed through the lens of the invading populations. Art, sacred objects, and human remains were taken and moved to unknown places and displayed as curiosities. The rightful status of the artwork was not acknowledged; much of the sacred meanings created by people for their own cultures were lost.
This book uses the name Native American to discuss the artwork. Other terms might include Indigenous, First Nations, First Peoples, American Indian, all of which are generic names and only identify a location as North America, not individual cultures. From the 1600s – 1800s, the cultures were generally grouped into geographic designations (2.7.1); Arctic and sub-Arctic, the Northeast Woodlands, Southeast, Plains, Southwest, Great Basin, California, Plateau, and the Northwest Coast. Patterns of art in a unique section were similar because of materials available in the area; for example, the particular type of clay in a given area defined the style of pottery, or the type of indigenous animals determined how clothing was made and its ornamentation. After the 1600s, artwork in every area was also influenced by the effects of European invasion and colonization.
In many other cultures, an artist's concept was based on the wealth and support of the elite, royalty, and religious leaders who employed artists to produce art for religious or commemorative reasons. Although records do not exist for individual artists in early Native American cultures, surviving art demonstrates a person's talent who made a beautiful, well-formed carving versus a poorly made version. Generally, decorative elements of their design were based on natural forms or geometric shapes, and the style of a particular artist might be found if more than one version of a piece of art survived.
Arctic
The Arctic region is flat with trackless tundra, cold and dark with little vegetation. The population of Inuit and Aleut people was small because of the inhospitable land, living in scattered groups as nomads or small villages along with the ocean areas. Weather ranged from the dark, frigid winters, the rainy seasons to the hot summers of long days and forced the people to be resourceful, using limited land resources. However, the oceans brought abundant food sources; shellfish, whales, salmon, seals, other sea life, and birds of all varieties. Hunting was risky, the ice and seas unforgiving. The materials available for art were based on the sea and animals; using driftwood, bone, skin, or fur, they made smaller objects portable for use in everyday life or ceremonies.
Clothing for winter was made from the fur and skin of the local animals; however, the most effective material for protection from the weather was the gut garment (2.7.2). They were made from the intestines of the large sea mammals or the bear and decorated with feathers, fur, or other found materials. The intestinal tubes were cleaned with scraping tools; the gut was blown up, stretched into a straight line, and left to dry. If the gut was dried in the winter when it was cold and dark, it became white and known as the 'winter gut.' A gut dried in the summer with warmer days had a yellow hue called the 'summer gut." When the gut tubes were dried, they were cut lengthwise into long strips, ready to be sewn together. Sinew was the typical material used for sewing, and then the garment was embellished or left plain. "In all traditions, the making of a gut garment is considered a gift to the animal whose flesh served to provide such an excellent protection from the elements. Each parka is constructed with care and honor from the hunter and for the animals he has received and will gather for his family." [1]
The Aleut were known for their carvings, passing the long winter nights making weapons for hunting, images of animals and people, or sewing needles and jewelry. Their houses were partially subterranean, small dwellings of driftwood with sod roofs, the dark interior lit by the fat from sea mammals burned in small stone lamps. Ivory from the massive walrus was the favorite material to use, and driftwood washed up on the beaches for carving with chipped stone tools. The carving of the small (7.8 centimeters) seated female (2.7.3) with her knees folded up to her chest was made from the walrus' tusk. The miniature objects were easy to transport and generally used as fetish figures or storytelling.
Sub-Arctic
Mostly Algonquin-speaking people (Cree, Ojibwa, Naskapi) inhabited the Sub-arctic region and followed a nomadic life into the 19 th century. Archaeologists believe they had the longest uniformity of cultures of the native peoples and, until their initial contact with Europeans, followed their traditional lifestyles for over 7,000 years. [2] The land stretched from the Atlantic to the Pacific across mountains, tundra, and forests. The weather was too extreme to grow crops, and they remained dependent on the bountiful natural flora and fauna for all aspects of their lives. The people generally lived in small groups; however, travel was possible across flat tundra and the multiple rivers and lakes. A variety of artwork was developed in different areas based on natural resources available and later materials brought by Europeans. Initially, they may have used moose hair to sew and embroider designs on clothing or porcupine quills for baskets. The traders who came brought glass beads, and women artists wove the beads into their designs.
Bandolier bags were based initially on the bags the soldiers from Europe used to carry their ammunition, a concept adopted by multiple Native American groups across the Sub-arctic and Northeast regions. Men wore them with the wide strap across the chest and the bag section resting on the hip, not intended for battle, rather as decorative elements and part of ceremonial clothing. Women made the bags, and the early ones (2.7.4) were generally made from native-tanned leather for the base and decorated with porcupine quills which were softened, dyed, and then bent into the required shape. White quills form the zigzag pattern across the bag and the stripes on the handle. Deer hair, yarn, and natural fibers were added to complete the bag.
The later bandolier bag (2.7.5) demonstrates the influence of trade with the Europeans and the acquisition of glass beads, yarn, and ribbons. The women artists changed from porcupine quills to glass beads, adopting new ideas to decorate their clothing and bags with different colors and techniques. The patterns were intricate, the beads reflecting light and the ribbons moving in the breeze.
The Naskapi coat (2.7.6)was based on different styles of European coats and the design and needs of the Naskapi artist. They used colors from natural products, which were ground into powder and mixed with a binder. The proteins and lipids from salmon eggs made a suitable binder with the pigment. The fish egg acted much like the egg tempera used for the frescoes of civilizations along the Mediterranean, a durable process.
Together, the female artist and the hunter who would wear this coat conceived the geometric, colorful design based on the man's dreams of the motifs that would give him power and ensure his success in the hunt. Coats such as this example were worn for only one caribou hunting season, as the Naskapis (Innus) believed they lost their protective power over time. [3]
Northeast
Although prehistoric art existed in the region for thousands of years, early contact with European explorers and settlers from the 1600s onward changed the concepts of indigenous art. Different explorers, missionaries, and settlers wrote about, sketched, and unfortunately "collected" art and arbitrarily defined cultural characteristics. "But this emphasis has resulted in a frozen time perspective and an erroneously narrow view of the great historical depth, diversity, and richness" [4] of the art history existing in pre-contact Native peoples.
The Iroquois-speaking people (Oneida, Seneca, Erie, etc.) inhabited the northern part of the region and were most affected by the invasions of people from Europe. Algonquian-speaking people (Pequot, Shawnee, etc.) lived in the southern sections. They had farms and permanent villages and based their artwork on their social institutions. The European colonists forced the different groups to fight against each other, taking sides in the European battles to control the territories. By the early 1800s, most native peoples migrated to the west or were forced to leave and be resettled in other areas. After the 1600s, the contacts from the Europeans changed the economy of the Native Americans, who had established trading partnerships. As the European incursions expanded and territory was seized, the local economy was severely disrupted. By 1800 the native populations were restricted to reservations, and their lifestyles changed forever. The wooden comb (2.7.7) was made from a moose antler, a common material since the moose shed their antlers yearly in early winter and grow a new set in the spring. The carved comb depicts two Europeans dressed in coats and hats and may have been made for trading.
Both male and female Iroquois wore moccasins (2.7.8) made from one piece of deer or elk skin, which was smoked, creating long-lasting and sturdy leathers. The moccasins had a seam in the back for the heel and one in the front across the top of the foot. The bottom was flat, comfortable, and durable. Around the ankle, the sides are folded down a few inches for the cuff and decorated. Initially, the decorations were made from porcupine quills, and after the Europeans came, they used glass beads. Sometimes the beadwork was sewn on a separate piece and then attached, and when the moccasins wore out, the beadwork could be removed and used on a new pair.
The False face mask (2.7.9) was used during ceremonial and healing rituals. Different types of masks were made for each of the ceremonies based on the type of illness a person was suffering from and which spirits needed to be driven away using the appropriate mask with an exaggerated nose, no eye sockets, or a distorted mouth. The mask creator searched through the woods until he came to a tree representing a spirit and carved the wooden part of the mask from that tree. Masks were painted red if they were created during the morning or black when they started in the afternoon. Horsetail hairs were used for the masks, and before the Europeans came, they used corn husks.
Southeast
The Southeast culture was based on agriculture in the fertile region, growing maize, tobacco, and beans. Living in villages, their houses, burial sites, religious sites, and storage facilities were frequently constructed in the shape of mounds. They each developed hierarchical societies with powerful chiefs as rulers and those in leadership positions. From this area came names familiar today; Cherokee, Chickasaw, or Seminole, and others. Large numbers of people perished from diseases introduced by the European settlers, and by 1830, only about 100,000 Native Americans lived in the territory. However, the European colonists wanted to expand their farms to grow cotton. The Indian Removal Act forced the relocation of most of the native people into the plains areas, a journey called the Trail of Tears, reflecting on the human toll and misery of the relocation.
The lidded basket (2.7.10) was made from the canes growing in thick forests along the rivers. The river canes were gathered, and the hard-exterior parts split into strips, ready to be woven. The lidded basket has diagonal lines along the base, darker triangles around the sides, and rectangles decorating the top. Some patterns were based on the natural colors of the canes, and other strips were dyed with walnuts or berries to produce specific colors. Baskets were an important part of life both in mythology and practical uses in life.
Tobacco was native to South America and grown throughout the Americas, an important ceremonial plant for most native peoples. Tobacco grew well in this territory; however, if tobacco did not grow in a region, it was obtained from other people through trades. Tobacco was an integral part of ceremonies where smoke carried prayers to the heavens. It was also used as a medicine for multiple ailments, included in medicine bundles, and was smoked in social situations by both men and women. Depending on its purpose, the pipes used for smoking might be considered sacred during rituals or an everyday item. The pipe (2.7.11) was made of steatite or black pipestone. Originally a light gray in its natural state, steatite turns deep black when carved and polished with wax or oil. This stone was a common material used by Native Americans for their pipes, soft enough to cut and shape and hard enough to support intense heat.
Plains
The Plains area was a vast space across the country from the Rocky Mountains to the Mississippi, filled with mountains, plains, tablelands, and river valleys. Originally, the people were hunters or farmed in small communities, the regions filled with wildlife of all kinds. Many people moved throughout an area to follow the weather, water, fields they could cultivate, and animal food sources. When the Spanish came, they imported horses, and the local groups (Crow, Comanche, Arapaho, Sioux, etc.) used that resource to pursue the bison, adopting a more nomadic lifestyle. They made tipis from the hide of buffalos around lodgepoles, easy to assemble and move to follow buffalo or avoid danger. White traders' excursions into the area brought guns and disease to the people, and sport hunters exterminated most of the herds of bison until the people were finally forced onto government-run reservations.
Bison, deer, sheep, and antelope were abundant in numbers on the plains and the most important part for the life of the people, providing food and hides for their shelter, clothing, and other necessities. The thick, heavy buffalo skins were valuable as blankets in the cold winters, while the lighter weight and thinner deer and antelope hides were used for clothing. To make anything from the hides was difficult and complex. The men were the hunters, following the herds and bringing back the animals, and the women prepared the hides.
The women work to prepare the hides by removing the hair and tanning the skin. It is messy, hard work and requires muscles and brains—the muscles of the women, and the brains of the animal! In one tanning method, the hide is first soaked in water mixed with ashes for several days.  Next, cooked brains of the animal are applied to the hide in order to soften it. The hide is then rinsed and stretched and pulled until—as one dressmaker says—your arms are "so tired, they feel like they will fall off!" This process is referred to as "brain-tanning" and is still done today to prepare hides that will be made into dresses, shirts, jackets, purses, and other items. [6]
The people of the plains regions have a history of decorating their clothing and other articles. Before the Europeans brought glass beads, they used paints, natural minerals, and fat from the plentiful bison. They found the hip bone in the bison very porous and able to hold paint, a useful tool along with grasses. Porcupine quills could be dyed for color, and other animal bones, teeth, or claws made attractive decorations or supported ritual meanings.
Made from native-tanned leather, the Sioux dress (2.7.12) was probably made for ceremonial purposes. The elaborate glass beading forms a bright blue background with meaningful symbols in green, yellow, red, and black beads across the top. The fringe was cut in small strips from leather as part of the sleeves and bottom and added to the dress, giving the woman another movement.
After the hide was cleaned and tanned, women adorned the clothing with sacred designs or symbols of life using paint or beads and porcupine quills. The robe (2.7.13) was tanned and painted with a traditional Arapaho design, a box-like structure in the middle filled with precisely painted triangular shapes. The outside edge is bordered by white and red lines reflecting Arapaho's use of dynamic colors.
All types and styles of bags were made and used throughout the different regions. The parfleche bag (2.7.14), made from hide, was shaped like an envelope used by men and women to carry pemmican or other dried foods. Each bag was decorated in an abstract style to symbolize the natural landscape of the plains, rivers, or hills or a family symbol. A hide was cleaned, tanned, and stretched into shape, then soaked and dried over a smoky fire to make the hide very hard. To make the design, the hide was incised, painted, and fringed.
The tobacco bag (2.7.115) was also made from hide and was used by males to transport pipes and tobacco for ritual ceremonies. The highly decorated bag was painted, then the hanging strips were covered with beads and horsehair.
Great Basin and Plateau
The people of the Great Basin (Shoshone, Paiute, Ute, etc.) lived in frequently barren areas and moved from place to place to forage. Their settlements were small, and housing was made from trees and brush, easily made and destroyed to move again. Horses from the Europeans gave them more mobility for a while; however, the discovery of gold and silver forced the people onto reservations.
The Plateau region was smaller, less populated, and people (Klamath, Nez Perce, Flathead, etc.) lived in villages by rivers and streams. Natural resources provided food, and they did not have to travel far to survive. When horses came to the area, they expanded their territory; however, when the Lewis and Clark expedition arrived in 1805, other white settlers followed along with diseases. After a gold rush, the government took over six million acres of land. It moved the Nez Perce to a reservation triggering further battles to retake their land, only to be pushed and shoved into smaller areas, losing their homeland. At the end of the 1800s, the people were removed from their lands onto small reservations.
As with other native people in the plains and plateau regions, the Nez Perce who lived in the plateaus fashioned their lives after the same types of natural resources as those from the plains. The invention and use of the blanket strip occurred during this period. The changes started on the plains and moved to the plateau, demonstrating the creativeness of the women. The giant bison was heavy and hard to manipulate when the women tried to cut the skins from the carcass that lay on one side. They learned to cut the skin along the backbone and remove the one side before rolling it over and removing the hide from the other side. When they were ready to tan the leather, they sewed the two pieces together to make a robe. However, this left them with a seam that showed, so they designed a blanket strip (2.7.16) to cover the seam. The blanket strip was decorated and was so popular; other ornamented strips were added even if they didn't cover a seam. The designs on the strips were symbolic, generally holding a sacred significance.
Southwest
Some of the native populations in the area were farmers (Hopi, Zuni, Navajo, and others) who lived in permanent settlements and farmed crops of corn and squash. They used the resources of the landscape and built multistory pueblos and ceremonial centers. Others like the Apache were nomadic, moving around the region, hunting and invading their neighbors' crops. The Spanish were the major European power in the region and exterminated many people in wars or enslaved them. Eventually, most of the remaining population was resettled onto reservations.
Native Americans in the Southwest have a long tradition of art, masters of weaving and pottery. The Pueblo people grew cotton and began weaving with a backstrap loom around 700 CE. The people stayed in the area, developing their skills based on the different natural resources available. Historians believe Hopi and Navajo weavers learned the skills from the Pueblo people around 1650, and each of them perfected their patterns, colors, and styles. Pottery was another art form highly developed in the area created on long traditions of techniques passed through the generations. Clay was abundant, and natural materials for paint were made by grinding rocks or using a wide variety of plants.
Kachinas were a specific form made by Hopi for use in religious practices, icons made from the indigenous root of the cottonwood tree. The Kachinas, used for generations, were carefully painted to resemble figures representing different aspects of mythology and religion. Kachina figures (2.7.17) are small. They resemble the shape of a human, carved, rubbed with sandstone to smooth the figures, then painted with pigments made from natural materials, ready to be used in ceremonial activities.
In the southwestern region, cotton was grown and an integral part of weaving to make clothing. When the Spanish came with their sheep, the Navajo adopted the techniques of producing wool, incorporating it as an important source of materials. Weaving traditions and patterns were reflected in the horizontal, diamond, and zigzag themes using vibrant colors. A significant item of clothing was the blanket. The striped wearing blanket (2.7.18) was woven with undyed wool, fleece dyed with indigo, and unusual raveled bayeta. During the Navajo trade with Mexico, they obtained a red flannel material named bayeta, a fabric getting its red color from a specific insect species. The Navajo stripped each of the fibers from the bayeta and re-twisted it into new yarn to use the material.
Later, blankets (2.7.19) were made with vividly colored yarns and patterns of diamonds, zigzags, and stripes, a contrasting single strand of yarn used to highlight the design.
Zuni women generally made the pottery, gathering clay, sifting it to remove impurities, and mixing the powdered clay with water. They used the coil method to shape the pots (2.7.20) and smoothed the sides with a handmade tool. The designs were based on clan designations and events in their life, painted with natural dyes and brushes created from the yucca plant, then fired in kilns. Firing the pottery followed a ritual, using respectful, soft voices or even silence; the women carefully placed the pottery on the fire, allowing the voice of the clay to emerge.
The Apaches moved throughout their region, not building permanent housing in the area, requiring lightweight and portable methods to store food and other items. They perfected basket making; a tradition passed from mothers to their daughters. Regional natural materials in trees, grasses, and roots were common to use in a basket. The large jug-shaped storage basket (2.7.21) could be waterproofed with resin from pine trees to hold liquids, or it might be used to store food or firewood. The basket was made from the shoots of the willow tree, root of the yucca, and a plant called the devil's claw, forming the intricate pattern based on symbolic beliefs.
Northwest Coast
The climate of the Northwest coastal region was mild, and natural resources were abundant, especially from the ocean. Although they were hunter-gathers, they did not have to travel from place to place to survive; water sources provided a constant supply of fish, seals, and shellfish. They built permanent structures forming large villages and broad social networks. Europeans did not inhabit this region in large numbers until the late 1800s, so the native populations (Tlingit, Penutian Chinook, etc.) maintained their traditions until then. With the influx of Europeans, new diseases, and suppression of their social structures, the societies became fragmented.
The native people from the Northwest are known for their artistic traditions, design, and artistry. Many practices are formalized, and the "formline" became the primary design element that defines outlines and colors. The colors of formline painting were based on black lines, red lines, and additional blue and green tones. The different color pigments were mixed with dried salmon eggs as a fixative, and their brushes were usually made from porcupine hairs.
Artwork frequently defined kinship, spirit powers, or territories. Potlatches were ceremonial events to display objects of art and value. The crest had significant meaning, similar to heraldic art of England, portraying real and imaginary animals. The crests were added to totem poles, house fronts, or ceremonial clothing, a legacy from ancestors to descendants; the crest was a sacred object.
Most of the art was made by men who formed in groups and were supported by patrons. They made the crests, totem poles from the abundant trees; carved from whalebone, ivory from the walrus, or wood from the trees, natural materials were available everywhere for the artist. Natural forms of whales, eagles, bears, or other animals and spiritual or legendary creatures were used as patterns for their designs.
The grip of the Tlingit dagger (2.7.22) appears as an animal, staring with one eye made from a stone. The wooden carved face has deep ridges reflecting the early form line style. The techniques to forge iron are not documented; however, historians believe native smiths developed metalworking techniques well before the Europeans arrived. This dagger was generated from a whole piece of iron using the hot chasing process. Holes along the top of the head were pierced, and the eyes and spaces were between some teeth. Many historians believe the dagger was made by the female metal smith Saayina.aat because of its proportions and signature design. Most of the daggers had long blades, however, the blade of this one is broad and short, with two ridges down the sides, reflecting her style. [7]
One of the unique styles for weaving blankets was done by the Salish, who lived along the coast of British Columbia. Robes were made with a process called twining. In the usual weaving process, the single thread of the crossing or weaving weft stays perpendicular to the installed warp. Twining has a pair of weft threads that switch back and forth as they move along the warp threads, twisting the warp and creating the diagonal slant. The warp thread had to be sturdy to survive the twisting, and the Salish used nettles or the bark inside the cedar tree. The Salish robe (2.7.23) is made with natural dyed material of reds, blues, black and brown along with undyed yarns, all made with patterns of checkerboards, zigzags, geometric shapes, and lines.
The Tsimshian lived in British Columbia and Alaska, a matrilineal society with a clan system based on four different totems; the eagle, a wolf, a raven/frog, or the orca. A person belonged to their mother's clan, and the ceremonial chief could be male or female. The raven rattle (2.7.24) was a standard instrument for shamanic rituals, so the beak pointed towards the ground to guide the shaman. The figure has the face of the wolf, perhaps identifying the owner of the rattle. The rattle was carved in two different pieces, filled with seeds or pebbles, joined with wooden pegs, and then painted in traditional colors. An artist might adorn the rattle with feathers, beads, or other natural decorations.
California
In the late sixteenth century, it is estimated California had over 300,000 people with many different groups (Miwok, Pomo, Serrano, etc.), all speaking different dialects. The population was one of "the largest and most diverse in the Western hemisphere is exhibited by the no fewer than sixty-four distinct languages they spoke…Before white contact, California had more linguistic variety than all of Europe." [8] Despite the diversity of languages, most native Californians practiced similar live styles, organized around small groups as hunter-gathers who traded with each other. As the Spanish colonized California, they brought disease, forced labor, brutal religious practices, and nearly exterminated the native populations.
Basketry was the primary art form of the Native Californians, each varied according to the local group with different designs and uses. Baskets were used for "gathering food, sifting acorn meal, storing tobacco, gambling, cooking, and even transporting water. Native Californians could weave so tightly that their baskets held boiling water—heated by hot rocks—without leaking." [9]
The Chumash, who lived in the southern part of California, made the tray (2.7.25) of sumac shoots and juncus stems, forming a set of concentric circles of geometric shapes, alternating with a triangle pattern and parallel bars. The Miwok extended from the valley to the mountains through the state's middle and had many types of natural materials for their baskets. They could split small sapling trees into strips or use grasses like the carex barbarae root for the lighter colors.
Baskets were so well made and so familiar; making pottery was very rare.
A basket (2.7.26) incorporated dark brown or black using ferns or dyeing material with ashes, pulverized manzanita, or the bark of the black oak. The primary tool was an awl made from bone to pierce the material while making the tightly coiled basket.
The Pomo basket (2.7.27) from the northern part of California followed the tradition of including multiple patterns in the basket. A substantial portion of the design is the dau or spirit door to allow good spirits into the basket, made by leaving a small opening or turn in the stitching. The Pomo also had many natural materials available, and willow shoots, river canes, and sedge roots were the most common. The materials were cleaned and dried and, if needed, split or dyed. Males constructed baskets for fish and bird traps, while women made the baskets for food storage and religious rites.
By the end of the 1800s, most of the tribal lands were taken, and the Native Americans who lived on those lands for centuries were moved to reservations, splitting up tribal groups and customs.
Chief Joseph said in the famous statement attributed to him;
…My people, some of them, have run away to the hills and have no blankets, no food. No one knows where they are—perhaps freezing to death. I want to have time to look for my children and see how many of them I can find. Maybe I shall find them among the dead. Hear me, my chiefs! I am tired.  From where the sun not stands, I will fight no more forever. [10]
The traditions of Native American art changed, and their art became a commodity, sold to outsiders and tourists, divorced from the people and its meanings. Many works by Native people were meant for sacred ceremonies, not displayed in museums. Today, in both the United States and Canada, a few museums are starting to honor these beliefs and remove sacred items or grave goods from their displays. There is still a market for different groups to recreate the art of the past for sale. However, today there is a resurgence in their artistic traditions by Native Americans in North America, bringing their cultural backgrounds into their modern styles.
[1] Reed, Fran, "Embellishments of the Alaska Native Gut Parka" (2008). Textile Society of America Symposium Proceedings. 127. Retrieved from http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1127&context=tsaconf . 29 January 2019.
[2] Kopper, P. (1986). North American Indians, Smithsonian Books, Washington D.C. p.113.
[3] Retrieved from https://www.metmuseum.org/art/collec...p=20&pos=1
[4] Vastokas, J., History of Indigenous Art in Canada (2018). In The Canadian Encyclopedia . Retrieved from https://www.thecanadianencyclopedia....-art-in-canada
[5] https://books.google.com/books?id=uh...ensils&f=false
[6] National Museum of the American Indian, A Life in Beads: The Stories a Plains Dress Can Tell,
Retrieved from https://americanindian.si.edu/sites/1/files/pdf/education/NMAI_lifeinbeads.pdf
28 January 2019
[7] Torrence, G., (2018). Art of Native America: The Charles and Valerie Diker Collection, Metropolitan Museum of Art.
[8] Retrieved from https://www.parks.ca.gov/?page_id=23548
[9] Retrieved from https://calisphere.org/exhibitions/e...pre-columbian/
[10] https://www.pbs.org/weta/thewest/peo...hiefjoseph.htm
[11] Retrieved from https://www.metmuseum.org/toah/hd/plai/hd_plai.htm ) (20 April 2020)


1.1: Introduction
Notes to Teachers and Students
When we teach complex numbers to beginning engineering students, we encourage a geometrical picture supported by an algebraic structure. Every algebraic manipulation carried out in a lecture is accompanied by a care-fully drawn picture in order to fix the idea that geometry and algebra go hand-in-glove to complete our understanding of complex numbers. We assign essentially every problem for homework.
We use the MATLAB programs in this chapter to illustrate the theory of complex numbers and to develop skill with the MATLAB language. The numerical experiment introduces students to the basic quadratic equation of electrical and computer engineering and shows how the roots of this quadratic equation depend on the coefficients of the equation.
“Representing Complex Numbers in a Vector Space,” is a little demanding for freshmen but easily accessible to sophomores. It may be covered for additional insight, skipped without consequence, or covered after Chapter 4. “An Electric Field Computation,” is well beyond most freshmen, and it is demanding for sophomores. Nonetheless, an expert in electromagnetics might want to cover the section "An electric Field Computation" for the insight it brings to the use of complex numbers for representing two-dimensional real quantities.
Introduction
It is hard to overestimate the value of complex numbers. They first arose in the study of roots for quadratic equations. But, as with so many other great discoveries, complex numbers have found widespread application well outside their original domain of discovery. They are now used throughout mathematics, applied science, and engineering to represent the harmonic nature of vibrating systems and oscillating fields. For example, complex numbers may be used to study
- traveling waves on a sea surface
- standing waves on a violin string
- the pure tone of a Kurzweil piano
- the acoustic field in a concert hall
- the light of a He-Ne laser
- the electromagnetic field in a light show
- the vibrations in a robot arm
- the oscillations of a suspension system
- the carrier signal used to transmit AM or FM radio
- the carrier signal used to transmit digital data over telephone lines
- the 60 Hz signal used to deliver power to a home
In this chapter we develop the algebra and geometry of complex numbers. In Chapter 3 we will show how complex numbers are used to build phasor representations of power and communication signals


2.8: Cultural Notes and Extra/Optional Online Materials
Cultural Notes
| Topics | Explanation |
| Namaste नमस्ते | Namaste is used by Hello and bye at anytime of a day by Hindi speakers. It is culturally odd to shake hands so when you say namaste then just make the hand gesture. However, in business setting and bigger metropolitan cities, you will notice that people shake hands because of the effect of western world. |
| Personal Pronoun “You” |
Unlike English, Hindi has three different second persona, personal pronoun “you.”
तू tu : It is used with younger, intimate relation such as mother, and God, and some one who is lower in socio-economic status. It is a very informal way to say “You” in Hindi. तुम tum : It is used with friends, and with someone who is of equal status. It is somewhat informal. It should be used carefully only after confirming with your the person you are speaking to. आप aap : It is used with older and respected people of the community. It is also used for someone who is of higher socio-economic status. It is a very formal way to say “You” in Hindi. |
| Introduction |
Typically, in Hindi speaking culture, you typically do not have to go on introducing yourself to everyone. Culturally, you will be introduced to other people by your host, a friend, and/or someone who already knows you.
However, you should expect some very personal questions which may be culturally odd for you in your first meeting with someone, such as: are you married? how much money do you earn? etc. Please know that the sense of privacy is very different in different cultures. |
| Yes/No Question |
You learned in this chapter that you can simply add क्या
kyaa
, at the beginning of a simple sentence to form a yes/no question.
For example: क्या आप अमेरिकी हैं? Are you an American? However, it is very common for the native speakers to pose yes/no question by using simple sentence in raised tone. It is also very common in English. For example: आप अमेरिकी हैं? You are an American? |
| Gender | In this chapter, you learned that all Hindi nouns have grammatical gender. However, you may hear the native speakers to use Hindi words with incorrect gender during the conversation. It is because of two major reasons: (a) Not every one knows the gender of the noun hundred percent, (b) there are many dialects of Hindi which are mutually intelligible but do not have gender feature. So, they claim to be the native speaker of Hindi but their L1/mother tongue is not modern standard Hindi. |
Extra/Optional Online Materials
Self-introduction:
धन्यवाद !


13.3: Equality
The notion of equality is fundamental in mathematics, and especially in algebra and algebraic thinking. The symbol “=”’ expresses a relationship . It is not an operation in the way that + and are × operations. It should not be read left-to-right, and it definitely does not mean “… and the answer is …”.
For your work to be clear and easily understood by others, it is essential that you use the symbol = appropriately. And for your future students to understand the meaning of the = symbol and use it correctly, it is essential that you are clear and precise in your use of it.
Let’s start by working on some problems.
Akira went to visit his grandmother, and she gave him $1.50 to buy a treat.
He went to the store and bought a book for $3.20. After that, he had $2.30 left.
How much money did Akira have before he visited his grandmother?
Examine the following equations. Decide: Is the statement always true, sometimes true, or never true? Justify your answers.
- \(5+3=8 \ldotp\)
- \(\frac{2}{3} + \frac{1}{2} = \frac{3}{5} \ldotp\)
- \(5 + 3 = y \ldotp\)
- \(\frac{a}{5} = \frac{5}{a} \ldotp\)
- \(n + 3 = m \ldotp\)
- \(3x = 2x + x \ldotp\)
- \(5k = 5k + 1 \ldotp\)
Consider the equation
\[18 - 7 = \_\_\_ \ldotp \nonumber \]
- Fill in the blank with something that makes the equation always true .
- Fill in the blank with something that makes the equation always false .
- Fill in the blank with something that makes the equation sometimes true and sometimes false .
If someone asked you to solve the equations in Problem 8, what would you do in each case and why?
Kim solved Problem 7 this way this way:
Let’s see:
\[2.30 + 3.20 = 5.50 - 1.50 = 4, \nonumber \]
so the answer is 4.
What do you think about Kim’s solution? Did she get the correct answer? Is her solution clear? How could it be better?
Although Kim found the correct numerical answer, her calculation really doesn’t make any sense. It is true that
\[2.30 + 3.20 = 5.50 \ldotp \nonumber \]
But it is definitely not true that
\[2.30 + 3.20 = 5.50 - 1.40 \ldotp \nonumber \]
She is incorrectly using the symbol “=”, and that makes her calculation hard to understand.
- Can you write a good definition of the symbol “=”? What does it mean and what does it represent?
- Give some examples: When should the symbol “=”’ be used, and when should it not be used?
- Do these two equations express the same relationships or different relationships? Explain your answer. \(x^{2} - 1 = (x + 1)(x - 1) \ldotp\) \((x+1)(x-1) = x^{2} -1 \ldotp\)
This picture shows a (very simplistic) two-pan balance scale. Such a scale allows you to compare the weight of two objects. Place one object in each pan. If one side is lower than the other, then that side holds heavier objects. If the two sides are balanced, then the objects on each side weigh the same.
In the pictures below:
- The orange triangles all weigh the same.
- The green circles all weigh the same.
- The purple squares all weigh the same.
- The silver stars all weigh the same.
- The scale is balanced.
1. In the picture below, what do you know about the weights of the triangles and the circles? How do you know it?
2. In the picture below, what do you know about the weights of the circles and the stars? How do you know it?
3. In the picture below, what do you know about the weights of the stars and the squares? How do you know it?
In the pictures below:
- The orange triangles all weigh the same.
- The green circles all weigh the same.
- The purple squares all weigh the same.
- The scale is balanced.
How many purple squares will balance with one circle? Justify your answer.
In the pictures below:
- The orange triangles all weigh the same.
- The green circles all weigh the same.
- The purple squares all weigh the same.
- The silver stars all weigh the same.
- The scale is balanced.
How many purple squares will balance the scale in each case? Justify your answers.
In the pictures below:
- The orange triangles all weigh the same.
- The green circles all weigh the same.
- The purple squares all weigh the same.
- The scale is balanced.
What will balance the last scale? Can you find more than one answer?
In the pictures below:
- The orange triangles all weigh the same.
- The green circles all weigh the same.
- The purple squares all weigh the same.
- The scale is balanced.
- Which shape weighs the most: the square, the triangle, or the circle? Which shape weighs the least? Justify your answers.
- Which of the two scales is holding the most total weight? How do you know you’re right?
What do Problems 11–14 above have to do with the “=” symbol?


11.5: Pascal’s Principle
Learning Objectives
By the end of this section, you will be able to:
- Define pressure.
- State Pascal’s principle.
- Understand applications of Pascal’s principle.
- Derive relationships between forces in a hydraulic system.
Pressure is defined as force per unit area. Can pressure be increased in a fluid by pushing directly on the fluid? Yes, but it is much easier if the fluid is enclosed. The heart, for example, increases blood pressure by pushing directly on the blood in an enclosed system (valves closed in a chamber). If you try to push on a fluid in an open system, such as a river, the fluid flows away. An enclosed fluid cannot flow away, and so pressure is more easily increased by an applied force.
What happens to a pressure in an enclosed fluid? Since atoms in a fluid are free to move about, they transmit the pressure to all parts of the fluid and to the walls of the container. Remarkably, the pressure is transmitted undiminished . This phenomenon is called Pascal’s principle , because it was first clearly stated by the French philosopher and scientist Blaise Pascal (1623–1662): A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
Definition: Pascal’s Principle
A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
Pascal’s principle, an experimentally verified fact, is what makes pressure so important in fluids. Since a change in pressure is transmitted undiminished in an enclosed fluid, we often know more about pressure than other physical quantities in fluids. Moreover, Pascal’s principle implies that the total pressure in a fluid is the sum of the pressures from different sources . We shall find this fact—that pressures add—very useful.
Blaise Pascal had an interesting life in that he was home-schooled by his father who removed all of the mathematics textbooks from his house and forbade him to study mathematics until the age of 15. This, of course, raised the boy’s curiosity, and by the age of 12, he started to teach himself geometry. Despite this early deprivation, Pascal went on to make major contributions in the mathematical fields of probability theory, number theory, and geometry. He is also well known for being the inventor of the first mechanical digital calculator, in addition to his contributions in the field of fluid statics.
Application of Pascal’s Principle
One of the most important technological applications of Pascal’s principle is found in a hydraulic system , which is an enclosed fluid system used to exert forces. The most common hydraulic systems are those that operate car brakes. Let us first consider the simple hydraulic system shown in Figure \(\PageIndex{1}\).
Relationship Between Forces in a Hydraulic System
We can derive a relationship between the forces in the simple hydraulic system shown in Figure \(\PageIndex{1}\) by applying Pascal’s principle. Note first that the two pistons in the system are at the same height, and so there will be no difference in pressure due to a difference in depth. Now the pressure due to \(F_1\) acting on area \(A_1\) is simply \(P_1 = \frac{F_1}{A_1}\), as defined by \(P = \frac{F}{A}\). According to Pascal’s principle, this pressure is transmitted undiminished throughout the fluid and to all walls of the container. Thus, a pressure \(P_2\) is felt at the other piston that is equal to \(P_1\). That is
\[P_1 = P_2\]
But since \(P_2 - \frac{F_2}{A_2}\), we see that
\[\dfrac{F_1}{A_1} = \frac{F_2}{A_2}. \label{eq20}\]
This equation relates the ratios of force to area in any hydraulic system, providing the pistons are at the same vertical height and that friction in the system is negligible. Hydraulic systems can increase or decrease the force applied to them. To make the force larger, the pressure is applied to a larger area. For example, if a 100-N force is applied to the left cylinder in Figure \(\PageIndex{1}\) and the right one has an area five times greater, then the force out is 500 N. Hydraulic systems are analogous to simple levers, but they have the advantage that pressure can be sent through tortuously curved lines to several places at once.
Example \(\PageIndex{1}\): Calculating Force of Slave Cylinders: Pascal Puts on the Brakes
Consider the automobile hydraulic system shown in Figure \(\PageIndex{2}\).
A force of 100 N is applied to the brake pedal, which acts on the cylinder—called the master—through a lever. A force of 500 N is exerted on the master cylinder. (The reader can verify that the force is 500 N using techniques of statics from Applications of Statics, Including Problem-Solving Strategies .) Pressure created in the master cylinder is transmitted to four so-called slave cylinders. The master cylinder has a diameter of 0.500 cm, and each slave cylinder has a diameter of 2.50 cm. Calculate the force \(F_2\) created at each of the slave cylinders.
Strategy
We are given the force \(F_1\) that is applied to the master cylinder. The cross-sectional areas \(A_1\) and \(A_2\) can be calculated from their given diameters. Then Equation \ref{eq20} can be used to find the force \(F_2\). Manipulate this algebraically to get \(F_2\)
on one side and substitute known values:
Solution
Pascal’s principle applied to hydraulic systems is given by Equation \ref{eq20}:
\[\begin{align*} F_2 &= \dfrac{A_2}{A_1}F_1 \\[5pt] &= \dfrac{\pi r_2^2}{\pi r_1^2}F_1 \\[5pt] &= \dfrac{(1.25 \, cm)^2}{(0.250 \, cm)^2} \times 500 \, N \\[5pt] &= 1.25 \times 10^4 \, N. \end{align*} \]
Discussion
This value is the force exerted by each of the four slave cylinders. Note that we can add as many slave cylinders as we wish. If each has a 2.50-cm diameter, each will exert \(1.25 \times 10^4 \, N.\)
A simple hydraulic system, such as a simple machine, can increase force but cannot do more work than done on it. Work is force times distance moved, and the slave cylinder moves through a smaller distance than the master cylinder. Furthermore, the more slaves added, the smaller the distance each moves. Many hydraulic systems—such as power brakes and those in bulldozers—have a motorized pump that actually does most of the work in the system. The movement of the legs of a spider is achieved partly by hydraulics. Using hydraulics, a jumping spider can create a force that makes it capable of jumping 25 times its length!
Definition: Conservation of Energy
Conservation of energy applied to a hydraulic system tells us that the system cannot do more work than is done on it. Work transfers energy, and so the work output cannot exceed the work input. Power brakes and other similar hydraulic systems use pumps to supply extra energy when needed.
Summary
- Pressure is force per unit area.
- A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
- A hydraulic system is an enclosed fluid system used to exert forces.
Glossary
- Pascal’s Principle
- a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container