3.8: Inverses and Radical Functions
In this section, we will explore the inverses of polynomial and rational functions, and in particular the radical functions that arise in the process.
Example \(\PageIndex{1}\)
A water runoff collector is built in the shape of a parabolic trough as shown below. Find the surface area of the water in the trough as a function of the depth of the water.
Solution
Since it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system at the cross section, with \(x\) measured horizontally and \(y\) measured vertically, with the origin at the vertex of the parabola.
From this we find an equation for the parabolic shape. Since we placed the origin at the vertex of the parabola, we know the equation will have form \(y(x)=ax^{2}\). Our equation will need to pass through the point (6,18), from which we can solve for the stretch factor \(a\):
\[\begin{array}{l} {18=a6^{2} } \\ {a=\dfrac{18}{36} =\dfrac{1}{2} } \end{array}\nonumber \]
Our parabolic cross section has equation \(y(x)=\dfrac{1}{2} x^{2}\)
Since we are interested in the surface area of the water, we are interested in determining the width at the top of the water as a function of the water depth. For any depth \(y\) the width will be given by \(2x\), so we need to solve the equation above for \(x\). However notice that the original function is not one-to-one, and indeed given any output there are two inputs that produce the same output, one positive and one negative.
To find an inverse, we can restrict our original function to a limited domain on which it is one-to-one. In this case, it makes sense to restrict ourselves to positive x values. On this domain, we can find an inverse by solving for the input variable:
\[\begin{array}{l} {y=\dfrac{1}{2} x^{2} } \\ {2y=x^{2} } \end{array}\quad x=\pm \sqrt{2y}\nonumber \]
This is not a function as written. Since we are limiting ourselves to positive \(x\) values, we eliminate the negative solution, giving us the inverse function we’re looking for
\[x(y)=\sqrt{2y}\nonumber \]
Since \(x\) measures from the center out, the entire width of the water at the top will be \(2x\). Since the trough is 3 feet (36 inches) long, the surface area will then be 36(2x), or in terms of \(y\):
\[\text{Area}=72x=72\sqrt{2y}\nonumber\]
The previous example illustrated two important things:
- When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
- The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
Functions involving roots are often called radical functions .
Example \(\PageIndex{2}\)
Find the inverse of \(f(x)=(x-2)^{2} -3=x^{2} -4x+1\)
Solution
From the transformation form of the function, we can see this is a transformed quadratic with vertex at (2,-3) that opens upwards. Since the graph will be decreasing on one side of the vertex, and increasing on the other side, we can restrict this function to a domain on which it will be one-to-one by limiting the domain to\(x \ge 2\).
To find the inverse, we will use the vertex form of the quadratic. We start by replacing the \(f(x)\) with a simple variable \(y\), then solve for \(x\).
\[y=(x-2)^{2} -3\nonumber\] Add 3 to both sides
\[y+3=(x-2)^{2}\nonumber\] Take the square root
\[\pm \sqrt{y+3} =x-2\nonumber\] Add 2 to both sides
\[2\pm \sqrt{y+3} =x\nonumber \]
Of course, as written this is not a function. Since we restricted our original function to a domain of \(x\ge 2\), the outputs of the inverse should be the same, telling us to utilize the positive case:
\[x=f^{-1} (y)=2+\sqrt{y+3}\nonumber\]
If the quadratic had not been given in vertex form, rewriting it into vertex form is probably the best approach. Alternatively, we could have taken the standard equation and rewritten it equal to zero:
\[0=x^{2} -4x+1-y\nonumber\]
We would then be able to use the quadratic formula with \(a=1\), \(b=-4\), and \(c=(1-y)\), resulting in the same solutions we found above:
\[x=\dfrac{-(-4)\pm \sqrt{(-4)^{2} -4(1)(1-y)} }{2} =2\pm \dfrac{\sqrt{12+4y} }{2} =2\pm \sqrt{3+y}\nonumber \]
Exercise \(\PageIndex{1}\)
Find the inverse of the function \(f(x)=x^{2} +1\), on the domain \(x \ge 0\).
- Answer
-
\[y=x^{2} +1\nonumber\]
\[y-1=x^{2}\nonumber\]
\[x=f^{-1} (y)=\sqrt{y-1}\nonumber\]
While it is not possible to find an inverse of most polynomial functions, some other basic polynomials are invertible.
Example \(\PageIndex{3}\)
Find the inverse of the function \(f(x)=5x^{3} +1\).
Solution
This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for \(x\)
\[\begin{array}{l} {y=5x^{3} +1} \\ {y-1=5x^{3} } \\ {\dfrac{y-1}{5} =x^{3} } \\ {x=f^{-1} (y)=\sqrt[{3}]{\dfrac{y-1}{5} } } \end{array}\nonumber \]
Notice that this inverse is also a transformation of a power function with a fractional power, \(x^{1/3}\).
Exercise \(\PageIndex{2}\)
Which toolkit functions have inverse functions without restricting their domain?
- Answer
-
identity, cubic, square root, cube root
Besides being important as an inverse function, radical functions are common in important physical models.
Example \(\PageIndex{4}\)
The velocity, \(v\) in feet per second, of a car that slammed on its brakes can be determined based on the length of skid marks that the tires left on the ground. This relationship is given by
\[v(d)=\sqrt{2gfd} \nonumber\]
In this formula, \(g\) represents acceleration due to gravity (32 ft/sec\({}^{2}\)), d is the length of the skid marks in feet, and \(f\) is a constant representing the friction of the surface. A car lost control on wet asphalt, with a friction coefficient of 0.5, leaving 200 foot skid marks. How fast was the car travelling when it lost control?
Solution
Using the given values of \(f\) = 0.5 and \(d\) = 200, we can evaluate the given formula:
\(v(200)=\sqrt{2(32)(0.5)(200)} =80\) ft/sec, which is about 54.5 miles per hour.
When radical functions are composed with other functions, determining domain can become more complicated.
Example \(\PageIndex{5}\)
Find the domain of the function \(f(x)=\sqrt{\dfrac{(x+2)(x-3)}{(x-1)} }\).
Solution
Since a square root is only defined when the quantity under the radical is non-negative, we need to determine where \(\dfrac{(x+2)(x-3)}{(x-1)} \ge 0\). A rational function can change signs (change from positive to negative or vice versa) at horizontal intercepts and at vertical asymptotes. For this equation, the graph could change signs at \(x\) = -2, 1, and 3.
To determine on which intervals the rational expression is positive, we could evaluate the expression at test values, or sketch a graph. While both approaches work equally well, for this example we will use a graph.
This function has two horizontal intercepts, both of which exhibit linear behavior, where the graph will pass through the intercept. There is one vertical asymptote, corresponding to a linear factor, leading to a behavior similar to the basic reciprocal toolkit function. There is a vertical intercept at (0, 6). This graph does not have a horizontal asymptote, since the degree of the numerator is larger than the degree of the denominator.
From the vertical intercept and horizontal intercept at \(x\) = -2, we can sketch the left side of the graph. From the behavior at the asymptote, we can sketch the right side of the graph.
From the graph, we can now tell on which intervals this expression will be non-negative, so the original function \(f(x)\) will be defined.
\(f(x)\) has domain \(-2\le x<1\; or\; x\ge 3\), or in interval notation, \([-2,1)\bigcup [3,\infty )\).
Like with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function, particularly of rational functions that are the ratio of linear functions, such as our concentration examples.
Example \(\PageIndex{6}\)
The function \(C(n)=\dfrac{20+0.4n}{100+n}\) was used in the previous section to represent the concentration of an acid solution after \(n\) mL of 40% solution has been added to 100 mL of a 20% solution. We might want to be able to determine instead how much 40% solution has been added based on the current concentration of the mixture.
Solution
To do this, we would want the inverse of this function:
\[C=\dfrac{20+0.4n}{100+n}\nonumber\] multiply both sides by the denominator
\[C(100+n)=20+0.4n\nonumber\] distribute
\[100C+Cn=20+0.4n\nonumber\] group everything with \(n\) on one side
\[100C-20=0.4n-Cn\nonumber\] factor out \(n\)
\[100C-20=(0.4-C)n\nonumber\] divide to find the inverse
\[n(C)=\dfrac{100C-20}{0.4-C}\nonumber\]
If, for example, we wanted to know how many mL of 40% solution need to be added to obtain a concentration of 35%, we can simply evaluate the inverse rather than solving an equation involving the original function:
\(n(0.35)=\dfrac{100(0.35)-20}{0.4-0.35} =\dfrac{15}{0.05} =300\) mL of 40% solution would need to be added.
Exercise \(\PageIndex{3}\)
Find the inverse of the function \(f(x)=\dfrac{x+3}{x-2}\).
- Answer
-
\[y=\dfrac{x+3}{x-2}\nonumber\]
\[y(x-2)=x+3\nonumber\]
\[yx-2y=x+3\nonumber\]
\[yx-x=2y+3\nonumber\]
\[x(y-1)=2y+3\nonumber\]
\[f^{-1} (y)=\dfrac{2y+3}{y-1}\nonumber\]
Important Topics of this Section
- Imposing a coordinate system
- Finding an inverse function
- Restricting the domain
- Invertible toolkit functions
- Radical Functions
- Inverses of rational functions


6.4: What is a Planet?
The International Astronomical Union (IAU) ruled in August 2006 that to be a planet an object must meet three criteria:
- It must have enough mass and gravity to gather itself into a sphere
- It must orbit the sun
- It must reign supreme in its own orbit, having “cleared the neighborhood” of other competing bodies
Is there a controversy?
Yes. This ruling demoted Pluto from a planet to a dwarf planet. According to the new rule, Pluto has not ‘reigned supreme in its own orbit, ‘clearing the neighborhood’ of other competing bodies.
The International Astronomical Union (IAU) has sanction over astronomical definitions and rules. Most of these are minor decisions, like adding leap seconds to a year. The decision – stating what a planet is – is a major decision. The Pluto Planet debate had been brewing for several years.
Yet this was not the IAU’s first vote on the subject. One week prior to the “final” planet definition vote, the IAU’s Planet Definition Committee endorsed the following:
- It must have enough mass and gravity to gather itself into a sphere
- It must orbit the sun
What were the implications of this first definition?
Pluto was still a planet, so was its moon Charon , the asteroid Ceres , and the recently discovered Eris. For about a week, the solar system would have had twelve planets instead of nine. The IAU rejected this definition and the now-infamous definition accepted.
What are the implications of the new planet definition?
There are now eight planets in our solar system, along with a new classification of objects called Dwarf Planets . Currently there are three Dwarf Planets: Pluto, Ceres (previously classified as an asteroid), and Eris. A subclass of Dwarf planets is called the Plutoids ; these ice dwarfs are trans-Neptunian (past the planet Neptune) planets that orbit the Sun and are large enough to be somewhat spherical in shape. It’s important to note that these rulings only apply to our solar system, not to other stellar systems with planets .
This decision is still hotly contested some ten years later. With the arrival of the New Horizons spacecraft at Pluto and its five known moons in 2015, astronomers will take a closer look at this icy world.


9.2: SIFT
The process of evaluating your resources may seem overwhelming, but it doesn’t have to be. Academic books and journals usually present themselves as scholarly and may be easily assessed for their quality of information. But others, especially websites, social media, newspapers, and magazines are not the same and sorting truth from fiction can be very challenging.
Our solution gives you a list of things to do when looking at a source, and hooks each of those things to one or two highly effective evaluation techniques. We call the “things to do” moves and there are four of them S top, I nvestigate, F ind, and T rack.
Licenses and Attributions: SIFT (The Four Moves) Authored by Mike Caulfield. Located at: https://hapgood.us/2019/06/19/sift-the-four-moves
License: CC BY 4.0 Adaptions: Revised text and one move.


2.1: Introduction
-
- Last updated
- Save as PDF
- Michael Curtin and Kevin Sanson
- University of California Press
The prevailing media credo, in domains that matter both a lot (popular, capitalist, and state discourse and action) and a little (communication, cultural, and media studies), is upheaval. The litany goes something like this: Corporate power is challenged. State authority is compromised. Avant-garde art and politics are centered. The young are masters, not victims. Technologies represent freedom, not domination. Revolutions are fomented by Twitter, not theory; by memes, not memos; by Facebook, not Foucault; by phone, not protest.
Political participation is just a click away. Tweets are the new streets and online friends the new vanguard, as 140ism displaces Maoism. Cadres are created and destroyed via BlackBerry. Teens tease technocrats. Hackers undermine hierarchy. Leakers dowse the fire of spies and illuminate the shady world of diplomats.
The endless iterations offered by digital reproduction and the immediate exchanges promised by the Internet have turned the world on its head. We are advised that the media in particular are being transformed. Tradition is rent asunder. Newspapers are metaphorically tossed aside. What was once their fate in a literal sense (when we dispensed with print in poubelles ) is now a figure of speech that refers to their financial decline. Journalists are recycled as public relations people, and readers become the new journalists. Cinema is irrelevant, TV is on the way out, gaming is the future, telephony is timeless, and the entire panoply of scholarship on the political economy of ownership and control is of archaeological interest at best.
This technophilic vision of old and middle-aged media being shunted aside by new media is espoused by a wide variety of actors. The corporate world is signed up: Netflix proudly proclaims that “Internet TV is replacing linear TV. Apps are replacing channels, remote controls are disappearing, and screens are proliferating.” 1 IBM disparages “Massive Passives . . . in the living room . . . a ‘lean back’ mode in which consumers do little more than flip on the remote and scan programming.” By contrast, it valorizes and desires “Gadgetiers and Kool Kids” who “force radical change” because they demand “anywhere, anytime content.” 2 I wish someone would pay me to come up with lines like those.
The state loves this new world too, despite the risks allegedly posed to its own essence. Let’s drop in on a Pentagon web site to see it share the joy: “Take the world’s most powerful sea, air and land force with you wherever you go with the new America’s Navy iPhone app. Read the latest articles. See the newest pics and videos. And learn more about the Navy—from its vessels and weapons to its global activities. You can do it all right on your iPhone—and then share what you like with friends via your favorite social media venues.” 3
Civil society is also excited. The wonderfully named Progress & Freedom Foundation’s “Magna Carta for the Information Age” proposes that the political-economic gains made through democratic action since the thirteenth century have been eclipsed by technological ones: “The central event of the 20th century is the overthrow of matter. In technology, economics, and the politics of nations, wealth—in the form of physical resources—has been losing value and significance. The powers of mind are everywhere ascendant over the brute force of things.” 4
The foundation has closed its doors, no doubt overtaken by pesky progress, but its discourse of liberty still rings loudly in our ears. Meanwhile, a prominent international environmental organization surveys me about its methods and appeal, asking whether I am prepared to sign petitions and embark on actions under its direction that might lead to my arrest. I prefer cozily comfortable middle-aged clicking to infantile attention-seeking incarceration, but either way, twinning the two is a telling sign of the times—as is doing so via corporate marketing techniques.
Even the bourgeois media take a certain pride in pronouncing their end of days. On the liberal left, the Guardian is prey to this beguiling magic: someone called “You” heads its 2013 list of the hundred most important folks in the media, with unknowns like Rupert Murdoch lagging far behind. 5 Time magazine exemplified just such love of a seemingly immaterial world when it chose “You” as “Person of the Year” for 2006 because “You control the Information Age. Welcome to your world.” 6 For its part, the New Statesman , a progressive British weekly, heralds the new epoch in a nationalistic way: “Our economic and political clout wanes,” but “when it comes to culture, we remain a superpower” because popular culture provides “critical tools through which Britain can market itself and its ideas to the world.” 7
Many academics love this new age too, not least because it’s avowedly green: the Australian Council for the Humanities, Arts and Social Sciences informs the country’s Productivity Commission that we dwell in a “post-smokestack era” 8 —a blessed world for workers, consumers, and residents, with residues of code rather than carbon. 9
The illustrations gathered above—arbitrarily selected but emblematic of profound tendencies across theories, industries, and places—amount to a touching but maddening mythology: cybertarianism, the belief that new media technologies are obliterating geography, sovereignty, and hierarchy in an alchemy of truth and beauty. Cybertarianism promises libertarian ideals and forms of life made real and whole thanks to the innately individualistic and iconoclastic nature of the newer media. 10
In this cybertarian world, corporate and governmental cultural gatekeepers and hegemons are allegedly undermined by innovative possibilities of creation and distribution. The comparatively cheap and easy access to making and circulating meaning afforded by Internet media and genres is thought to have eroded the one-way hold on culture that saw a small segment of the world as producers and the larger one as consumers, even as it makes for a cleaner economy that glides into an ever-greener postindustrialism. Cybertarians celebrate their belief that new technologies allow us all to become simultaneously cultural consumers and producers—no more factory conditions, no more factory emissions. 11
Crucial to these fantasies is the idea of the prosumer. This concept was invented by Alvin Toffler, a lapsed leftist and Reaganite signatory to the Progress & Freedom Foundation’s “Magna Carta.” Toffler was one of a merry band of futurists who emerged in the 1970s. He coined the term prosumer in 1980 to describe the vanguard class of a technologized future. (Toffler had a nifty knack for knee-jerk neologisms, as we will see.) 12
Rather than being entirely new, the prosumer partially represented a return to subsistence, to the period prior to the Industrial Revolution’s division of labor—a time when we ate what we grew, built our own shelters, and gave birth without medicine. The specialization of agriculture and manufacturing and the rise of cities put an end to such autarky: the emergence of capitalism distinguished production from consumption via markets. But Toffler discerned a paradoxical latter-day blend of the two seemingly opposed eras, symbolized by the French invention and marketing of home pregnancy tests in the 1970s. These kits relied on the formal knowledge, manufacture, and distribution that typified modern life but permitted customers to make their own diagnoses, cutting out the role of doctors as expert gatekeepers between applied science and the self.
Toffler called this “production for self-use.” He saw it at play elsewhere as well: in the vast array of civil society organizations that emerged at the time, the craze for “self-help,” the popularity of self-serve gas stations as franchises struggled to survive after the 1973–74 oil crisis, and the proliferation of automatic teller machines as banks sought to reduce their retail labor force.
The argument Toffler made thirty-five years ago—that we are simultaneously cultural consumers and producers, that is, prosumers—is an idea whose time has come, as his fellow reactionary Victor Hugo almost put it. 13 Readers become authors. Listeners transform into speakers. Viewers emerge as stars. Fans are academics. Zine writers are screenwriters. Bloggers are copywriters. Children are columnists. Bus riders are journalists. Coca-Cola hires African Americans to drive through the inner city selling soda and playing hip-hop. AT&T pays San Francisco buskers to mention the company in their songs. Urban performance poets rhyme about Nissan cars for cash, simultaneously hawking, entertaining, and researching. Subway’s sandwich commercials are marketed as made by teenagers. Cultural studies majors turn into designers. Graduate students in New York and Los Angeles read scripts for producers, then pronounce on whether they tap into the zeitgeist. Internally divided—but happily so—each person is, as Foucault put it forty years ago, “a consumer on the one hand, but . . . also a producer.” 14
Along the way, all that seemed scholarly has melted into the air. Bitcoin and Baudrillard, creativity and carnival, heteroglossia and heterotopia—they’re all present but simultaneously theorized and realized by screen-based activists rather than academics. Vapid victims of ideology are now credible creators of meaning, and active audiences are neither active nor audiences—their uses and gratifications come from sitting back and enjoying the career of their own content, not from viewing others’. They resist authority not via aberrant decoding of texts that have been generated by professionals, but by ignoring such things in favor of making and watching their own.
Whether scholars like to attach electrodes to peoples’ naughty bits to establish whether porn turns them on, or interview afternoon TV viewers to discern progressive political tendencies in their interpretation of courtroom shows, they’re yesterday’s people. It doesn’t matter if they purvey rats and stats and are consummate quantoids, or eschew that in favor of populist authenticity as acafans and credulous qualtoids. Their day has passed. “Media effects” describes what people do to the media, not the other way round.
People in all spheres of scholarship say “my children” enjoy this, that, or the other by way of media use. These choices are held up as predictions of the future. No one says the same about, for example, their children’s food preferences, as if abjuring vegetables at age seven will be a lifetime activity. But when it comes to the media, children are mini-Tofflers, forecasters of a world they are also bringing into being.
Like Toffler all those decades ago, cybertarian discourse buys into individualistic fantasies of reader, audience, consumer, and player autonomy—the neoliberal intellectual’s wet dream of music, movies, television, and everything else converging under the sign of empowered and creative fans. The New Right of communication and cultural studies invests with unparalleled gusto in Schumpeterian entrepreneurs, evolutionary economics, and creative industries. It’s never seen an “app” it didn’t like or a socialist idea it did. Faith in devolved media-making amounts to a secular religion, offering transcendence in the here and now via a “literature of the eighth day, the day after Genesis.” 15 This is narcissography at work, with the critic’s persona a guarantor of assumed audience revelry and Dionysian joy. Welcome to “Readers’ Liberation Movement” media studies. 16
So strong a utopian line about digital technologies and the Internet is appealing in its totality, its tonality, its claims, its cadres, its populism, its popularity, its happiness, and its hopefulness. But such utopianism has seen a comprehensive turn away from addressing unequal infrastructural and cultural exchange, toward an extended dalliance with new technology and its supposedly innate capacity to endow users with transcendence. 17 In 2011, the cost of broadband in the Global South was 40.3 percent of the average individual gross national income (GNI). Across the Global North, by comparison, the price was less than 5 percent of GNI per capita. 18 Within Latin America, for example, there are major disparities in pricing. One megabit a second in Mexico costs US$9, or 1 percent of average monthly income; in Bolivia, it is US$63, or 31 percent. Access is also structured unequally in terms of race, occupation, and region: indigenous people represent a third of rural workers in Latin America, and over half in some countries are essentially disconnected. The digital divide between indigenous people and the rest of the population in Mexico is 0.3, in Panama 0.7, and Venezuela 0.6. 19 Rather than seeing new communications technologies as magical agents that can produce market equilibrium and hence individual and collective happiness, we should note their continued exclusivity.
It is also worth noting that there are anticybertarian skeptics aplenty in both public intellectual and cloistered worlds and the third sector. They offer ways of thinking that differ from the dominant ones. Consider Evgeny Morozov’s striking journalistic critiques, which have resonated powerfully in their refusal of technocentric claims for social change. 20 On more scholarly tracks, many authors have done ethnographic and political-economic work on the labor conditions experienced by people in the prosumer world as well as policy explorations of digital capitalism and the state. 21 Case studies of WikiLeaks, for instance, show the ambivalent and ambiguous sides to a phenomenon that has been uncritically welcomed by cybertarians, while we now know the extent of corporate surveillance enabled by their embrace of Facebook and friends. 22 Beyond the Global North, thick descriptions of technocentric, cybertarian exploitation and mystification proliferate as the reality of successive liberatory “springs” supposedly unleashed by social media networks is exposed. 23 And nongovernment organizations raise the flag against crass celebrations of new media technologies that damage workers and the environment. 24 This array of work provides a sturdy counterdiscourse to the admittedly still dominant cybertarian position.


Unit IV: Evolutionary Processes
Evolution is change in the heritable traits of biological populations over successive generations. Evolutionary processes give rise to diversity at every level of biological organization, including the levels of species, individual organisms, and molecules. In Unit 4, the core concepts of evolution are discussed in this unit with examples illustrating evolutionary processes. Additionally, the evolutionary basis of biology reappears throughout the textbook in general discussion and is reinforced through special call-out features highlighting specific evolution-based topics.
Thumbnail: A silhouette of human evolution. (CC BY-SA 3.0; Tkgd2007 via Wikimedia Commons ).
- Contributors
-
Connie Rye (East Mississippi Community College), Robert Wise (University of Wisconsin, Oshkosh), Vladimir Jurukovski (Suffolk County Community College), Jean DeSaix (University of North Carolina at Chapel Hill), Jung Choi (Georgia Institute of Technology), Yael Avissar (Rhode Island College) among other contributing authors. Original content by OpenStax (CC BY 4.0; Download for free at<EMAIL_ADDRESS>).


9.7: Review Questions
-
- Last updated
- Save as PDF
1.
A public health nurse is reviewing data that shows that families living closer to an industrial plant have higher rates of cancer. Which of the following determinants should the nurse consider in this situation?
-
Epigenetics
-
Sociocultural environment
-
Biological processes
-
Physical environment
2.
A nurse researcher has found that clients living in rural areas tend to have higher rates of hospitalization for chronic lung disease than those living in affluent urban areas. Which of the following terms describes what the nurse researcher has identified?
-
Health disparities
-
Health determinants
-
Intersectionality
-
Socially disadvantaged populations
3.
A community health nurse educates a group of families on healthy eating habits to reduce their risk of diabetes. The nurse encourages them to share healthy meal ideas with each other. Which domain and level of influence does this intervention address?
-
Behavioral and community
-
Behavioral and individual
-
Sociocultural environment and interpersonal
-
Sociocultural environment and societal
4.
A community health nurse is working with a group of gay Black clients who are describing some of the negative social interactions they have experienced. This group is likely affected by which of the following:
-
Intersectionality
-
Entitlements
-
Gender disparities
-
Racial disparities
5.
A home health nurse notices that, among clients who were recently released from the hospital, female clients were more likely than male clients to report experiencing uncontrolled pain while hospitalized. What contributing factor should the nurse consider in this situation?
-
Gender roles
-
Gender biases
-
Biological differences
-
Environmental differences
6.
A public health nurse determines that a non-English-speaking Hispanic client needs additional medical evaluation and treatment for cancer. Which of the following statements best describes the nurse’s assessment of this situation?
-
This client is insured and should be able to receive treatment accompanied by a medical interpreter.
-
This client has a primary care provider who should be able to help the client navigate the system.
-
This client’s access to health care may be limited by the ability of the health care system to provide services when they are needed.
-
This client’s access to health care may be limited by the availability of culturally competent providers.
7.
Which of the following statements by a nursing student would indicate the need for further education?
-
“A difference in health status between older adults and the young may indicate a health disparity.”
-
“Race is not a determinant of health.”
-
“Individuals who live in high-poverty areas may need assistance accessing health care services.”
-
“If I don’t have a high school diploma, my risk for chronic disease rises.”
8.
A nurse studying the percentage of the population that reports having a regular source of health care notices that some states have significantly higher rates of care than other states. Which of the following factors is most likely the reason for the difference?
-
Differences in health literacy rates
-
The geographic distribution of health care providers
-
The income requirements for Medicaid eligibility
-
Availability of public transportation
9.
A nurse educating a group of men on the importance of performing testicular self-exams recognizes that which of the following factors influences the likelihood that they will adhere to this recommendation?
-
Low health literacy levels
-
Poor social support
-
Gender bias related to health care access
-
Gender norms related to masculinity
10.
A nurse working in a rural public health clinic recognizes that which of the following risk factors for poor health outcomes is prevalent in the community?
-
An increased likelihood of engaging in risky behaviors
-
A decreased risk of experiencing chronic lung disease
-
An increased likelihood of wearing seat belts
-
A decreased risk of cancer


12.16: The Third Crusade
The Crusader States had endured from 1099 to 1187 because the Muslim Middle East was politically fragmented. Once Saladin had overthrown Egypt’s Fatimid Caliphate and united Egypt to Muslim-controlled Syria and northern Iraq (see Chapter Eight), he was able to turn his resources to destroying the Crusader States. Eventually, at the 1187 Battle of Hattin, his forces met the combined forces of the Kingdom of Jerusalem. The result was a complete victory for Saladin. With the manpower of most of the Kingdom of Jerusalem killed or captured—knights and noble prisoners would be held for a hefty ransom, while lower-ranked soldiers would go to slave markets—he was easily able to capture most of the castles and cities that made up the kingdom, to include the city of Jerusalem itself.
The result shocked the Christian world, and Pope Gregory VIII quickly issued the bull (that is, an official papal pronouncement) Audita tremendi , which called on the Christian world to retake Jerusalem. The kings of England and France, Richard I (known as Lionheart, r. 1189 – 1199) and Philip Augustus (r. 1180 – 1223), respectively, took vows to launch a crusade, as did Holy Roman Emperor Frederick Barbarossa. (As usual, the Christians of Iberia took little part in crusades in the Levant, as their efforts focused on the Reconquista .)
Although Frederick Barbarossa died en route (he drowned in a stream in the mountains of Anatolia), both Richard I and Philip Augustus eventually arrived in the Levant by sea. Although Philip soon returned to France, King Richard battled Saladin over the course of two years, to results that were mostly inconclusive. The crusading army seized most of the castles and cities on the coast, and these became the center for a restored, but smaller Kingdom of Jerusalem, but the Crusaders ultimately failed to take Jerusalem itself. The Crusade finally ended in a truce in which both parties ratified this state of affairs, with Christian pilgrims allowed to visit the city of Jerusalem, even though it remained under Muslim rule.


2.1: Linear Functions
As you hop into a taxicab in Las Vegas, the meter will immediately read $3.50; this is the “drop” charge made when the taximeter is activated. After that initial fee, the taximeter will add $2.76 for each mile the taxi drives (Nevada Taxicab Authority, retrieved May 8, 2017. There is also a waiting fee assessed when the taxi is waiting at red lights, but we’ll ignore that in this discussion. This chapter is part of Precalculus: An Investigation of Functions © Lippman & Rasmussen 2017. This material is licensed under a Creative Commons CC-BY-SA license.). In this scenario, the total taxi fare depends upon the number of miles ridden in the taxi, and we can ask whether it is possible to model this type of scenario with a function. Using descriptive variables, we choose \(m\) for miles and \(C\) for Cost in dollars as a function of miles: \(C(m)\).
We know for certain that \(C(0)=3.50\), since the $3.50 drop charge is assessed regardless of how many miles are driven. Since $2.67 is added for each mile driven, then
\[C(1)=3.50+2.67=6.17\nonumber \]
If we then drove a second mile, another $2.67 would be added to the cost:
\[C(2)=3.50+2.67+2.67=3.50+2.67(2)=8.84\nonumber \]
If we drove a third mile, another $2.67 would be added to the cost:
\[C(3)=3.50+2.67+2.67+2.67=3.50+2.67(3)=11.51\nonumber \]
From this we might observe the pattern, and conclude that if \(m\) miles are driven,
\(C(m)=3.50+2.67m\) because we start with a $3.50 drop fee and then for each mile increase we add $2.67.
It is good to verify that the units make sense in this equation. The $3.50 drop charge is measured in dollars; the $2.67 charge is measured in dollars per mile.
\[C(m)=3.50\text{ dollars}+\left(2.67\dfrac{\text{dollars}}{\text{mile}} \right)\left(m\; \text{ miles}\right)\nonumber \]
When dollars per mile are multiplied by a number of miles, the result is a number of dollars, matching the units on the 3.50, and matching the desired units for the C function.
Notice this equation \(C(m)=3.50+2.67m\) consisted of two quantities. The first is the fixed $3.50 charge which does not change based on the value of the input. The second is the $2.67 dollars per mile value, which is a rate of change . In the equation, this rate of change is multiplied by the input value.
Looking at this same problem in table format we can also see the cost changes by $2.67 for every 1 mile increase.
| \(m\) | 0 | 1 | 2 | 3 |
| \(C(m)\) | 3.50 | 6.17 | 8.84 | 11.51 |
It is important here to note that in this equation, the rate of change is constant ; over any interval, the rate of change is the same.
Graphing this equation, \(C(m)=3.50+2.67m\) we see the shape is a line, which is how these functions get their name: linear functions .
When the number of miles is zero the cost is $3.50, giving the point (0, 3.50) on the graph. This is the vertical or \(C(m)\) intercept. The graph is increasing in a straight line from left to right because for each mile the cost goes up by $2.67; this rate remains consistent.
In this example, you have seen the taxicab cost modeled in words, an equation, a table and in graphical form. Whenever possible, ensure that you can link these four representations together to continually build your skills. It is important to note that you will not always be able to find all 4 representations for a problem and so being able to work with all 4 forms is very important.
A linear function is a function whose graph produces a line. Linear functions can always be written in the form
\(f(x)=b+mx\) or \(f(x)=mx+b\); they’re equivalent
where
- \(b\) is the initial or starting value of the function (when input, x = 0), and
- \(m\) is the constant rate of change of the function
Many people like to write linear functions in the form \(f(x)=b+mx\) because it corresponds to the way we tend to speak: “The output starts at \(b\) and increases at a rate of \(m\).”
For this reason alone we will use the \(f(x)=b+mx\) form for many of the examples, but remember they are equivalent and can be written correctly both ways.
\(m\) is the constant rate of change of the function (also called slope ). Slope The slope determines if the function is an increasing function or a decreasing function.
\(f(x)=b+mx\) is an increasing function if \(m>0\)
\(f(x)=b+mx\) is a decreasing function if \(m<0\)
If \(m=0\), the rate of change zero, and the function \(f(x)=b+0x=b\) is just a horizontal line passing through the point (0, \(b\)), neither increasing nor decreasing.
Marcus currently owns 200 songs in his iTunes collection. Every month, he adds 15 new songs. Write a formula for the number of songs, \(N\), in his iTunes collection as a function of the number of months, \(m\). How many songs will he own in a year?
Solution
The initial value for this function is 200, since he currently owns 200 songs, so \(N(0)=200\). The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. With this information, we can write the formula:
\[N(m)=200+15m\nonumber \]
\(N(m)\) is an increasing linear function. With this formula we can predict how many songs he will have in 1 year (12 months):
\[N(12)=200+15(12)=200+180=380\nonumber \] Marcus will have 380 songs in 12 months.
If you earn $30,000 per year and you spend $29,000 per year write an equation for the amount of money you save after y years, if you start with nothing.
“The most important thing, spend less than you earn!( http://www.thesimpledollar.com/2009/...than-you-earn/ ) ”
- Answer
-
\(S(y)=30,000y - 29,000y = 1000y\) $1000 is saved each year.
Given two values for the input,\(x_{1} {\rm \; and\; }x_{2}\), and two corresponding values for the output, \(y_{1} {\rm \; and\; }y_{2}\), or a set of points, \((x_{1} {\rm ,\; \; }y_{1} )\) and\((x_{2} {\rm ,\; \; }y_{2} )\), if we wish to find a linear function that contains both points we can calculate the rate of change, m :
\[m=\dfrac{\rm change\; in\; output}{\rm change\; in\; input} =\dfrac{\Delta y}{\Delta x} =\dfrac{y_{2} -y_{1} }{x_{2} -x_{1} }\]
Rate of change of a linear function is also called the slope of the line.
Note in function notation,\(y_{1} =f(x_{1} )\) and \(y_{2} =f(x_{2} )\), so we could equivalently write
\[m=\dfrac{f\left(x_{2} \right)-f\left(x_{1} \right)}{x_{2} -x_{1} }\]
The population of a city increased from 23,400 to 27,800 between 2002 and 2006. Find the rate of change of the population during this time span.
Solution
The rate of change will relate the change in population to the change in time. The population increased by \(27800-23400=4400\)people over the 4 year time interval. To find the rate of change, the number of people per year the population changed by:
\[\dfrac{4400\text{ people}}{4\text{ years}} =1100\dfrac{\text{people}}{\text{year}}= 1100\text{ people per year}\nonumber\]
Notice that we knew the population was increasing, so we would expect our value for \(m\) to be positive. This is a quick way to check to see if your value is reasonable.
The pressure, \(P\), in pounds per square inch (PSI) on a diver depends upon their depth below the water surface, \(d\), in feet, following the equation \(P(d)=14.696+0.434d\). Interpret the components of this function.
Solution
The rate of change, or slope, 0.434 would have units \(\dfrac{\text{output}}{\text{input}} = \dfrac{\text{pressure}}{\text{depth}} = \dfrac{\text{PSI}}{\text{ft}}\). This tells us the pressure on the diver increases by 0.434 PSI for each foot their depth increases.
The initial value, 14.696, will have the same units as the output, so this tells us that at a depth of 0 feet, the pressure on the diver will be 14.696 PSI.
If \(f(x)\) is a linear function, \(f(3)=-2\), and \(f(8)=1\), find the rate of change.
Solution
\(f(3)=-2\) tells us that the input 3 corresponds with the output -2, and \(f(8)=1\) tells us that the input 8 corresponds with the output 1. To find the rate of change, we divide the change in output by the change in input:
\[m=\dfrac{\text{change in output}}{\text{change in input}} =\dfrac{1-(-2)}{8-3} =\dfrac{3}{5}\nonumber\] If desired we could also write this as \(m = 0.6\)
Note that it is not important which pair of values comes first in the subtractions so long as the first output value used corresponds with the first input value used.
Given the two points (2, 3) and (0, 4), find the rate of change. Is this function increasing or decreasing?
- Answer
-
\(m=\dfrac{4-3}{0-2} =\dfrac{1}{-2} =-\dfrac{1}{2}\) ; Decreasing because \(m < 0\)
We can now find the rate of change given two input-output pairs, and can write an equation for a linear function once we have the rate of change and initial value. If we have two input-output pairs and they do not include the initial value of the function, then we will have to solve for it.
Write an equation for the linear function graphed to the right.
Solution
Looking at the graph, we might notice that it passes through the points (0, 7) and (4, 4). From the first value, we know the initial value of the function is \(b = 7\), so in this case we will only need to calculate the rate of change:
\[m=\dfrac{4-7}{4-0} =\dfrac{-3}{4}\nonumber \]
This allows us to write the equation:
\[f(x)=7-\dfrac{3}{4} x\nonumber \]
If \(f(x)\)is a linear function, \(f(3)=-2\), and \(f(8)=1\), find an equation for the function.
Solution
In example 3, we computed the rate of change to be \(m=\dfrac{3}{5}\). In this case, we do not know the initial value\(f(0)\), so we will have to solve for it. Using the rate of change, we know the equation will have the form \(f(x)=b+\dfrac{3}{5} x\). Since we know the value of the function when \(x = 3\), we can evaluate the function at 3.
\[f(3)=b+\dfrac{3}{5} (3)\nonumber \] Since we know that\(f(3)=-2\), we can substitute on the left side
\[-2=b+\dfrac{3}{5} (3)\nonumber \] This leaves us with an equation we can solve for the initial value
\[b=-2-\dfrac{9}{5} =\dfrac{-19}{5}\nonumber \]
Combining this with the value for the rate of change, we can now write a formula for this function:
\[f(x)=\dfrac{-19}{5} +\dfrac{3}{5} x\nonumber \]
Working as an insurance salesperson, Ilya earns a base salary and a commission on each new policy, so Ilya’s weekly income, \(I\), depends on the number of new policies, \(n\), he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies, and earned $920. Find an equation for \(I(n)\), and interpret the meaning of the components of the equation.
Solution
The given information gives us two input-output pairs: (3,760) and (5,920). We start by finding the rate of change.
\[m=\dfrac{920-760}{5-3} =\dfrac{160}{2} =80\nonumber \]
Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy; Ilya earns a commission of $80 for each policy sold during the week.
We can then solve for the initial value
\[I(n)=b+80n\nonumber\] then when \(n = 3\), \(I(3)=760\), giving
\[760=b+80(3)\nonumber \] this allows us to solve for \(b\)
\[b=760-80(3)=520\nonumber \]
This value is the starting value for the function. This is Ilya’s income when \(n = 0\), which means no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold.
Writing the final equation:
\[I(n)=520+80n\nonumber \]
Our final interpretation is: Ilya’s base salary is $520 per week and he earns an additional $80 commission for each policy sold each week.
Looking at Example 7:
Determine the independent and dependent variables.
What is a reasonable domain and range?
Is this function one-to-one?
- Answer
-
\(n\) (number of policies sold) is the independent variable
\(I(n)\) (weekly income as a function of policies sold) is the dependent variable.
A reasonable domain is (0, 15)\({}^{*}\)
A reasonable range is ($540, $1740)\({}^{*}\)
\({}^{*}\)answers may vary given reasoning is stated; 15 is an arbitrary upper limit based on selling 3 policies per day in a 5 day work week and $1740 corresponds with the domain.
Yes this function is one-to-one
The balance in your college payment account, \(C\), is a function of the number of quarters, \(q\), you attend. Interpret the function \(C(a) = 20000 – 4000q\) in words. How many quarters of college can you pay for until this account is empty?
- Answer
-
Your College account starts with $20,000 in it and you withdraw $4,000 each quarter (or your account contains $20,000 and decreases by $4000 each quarter.) Solving \(C(a) = 0\) gives \(a = 5\). You can pay for 5 quarters before the money in this account is gone.
Given the table below write a linear equation that represents the table values
| \(w\), number of weeks | 0 | 2 | 4 | 6 |
| \(P(w)\), number of rats | 1000 | 1080 | 1160 | 1240 |
Solution
We can see from the table that the initial value of rats is 1000 so in the linear format
\[P(w)=b+mw,\: b = 1000\nonumber\]
Rather than solving for \(m\), we can notice from the table that the population goes up by 80 for every 2 weeks that pass. This rate is consistent from week 0, to week 2, 4, and 6. The rate of change is 80 rats per 2 weeks. This can be simplified to 40 rats per week and we can write
\[P(w)=b+mw\text{ as }P(w)=1000+40w\nonumber \]
If you didn’t notice this from the table you could still solve for the slope using any two points from the table. For example, using (2, 1080) and (6, 1240),
\[m=\dfrac{1240-1080}{6-2} =\dfrac{160}{4} =40\text{ rats per week}\nonumber\]
Important Topics of this Section
- Definition of Modeling
- Definition of a linear function
- Structure of a linear function
- Increasing & Decreasing functions
- Finding the vertical intercept (0, b )
- Finding the slope/rate of change, m
- Interpreting linear functions


1.10: Navigating The College Website
selected template will load here
This action is not available.
“A journey of a thousand sites begins with a single click.”
– Author Unknown
A college website can be like a cookbook, full of great information and ideas, and can be completely overwhelming. Where do you start? Looking at the pictures? Scrolling down through the information? Can you taste the food in your mind just by looking at a picture? Can you imagine a food just from the list of ingredients? Is the number of ingredients needed to make the recipe exciting or paralyzing? How experienced a cook you are can impact your reaction to the cookbook.
As you begin navigating new information, remember, world-famous chef and author, Julia Child wasn’t always a great chef! In fact, when she got married, she could barely cook. Experience isn’t everything. An open mind is paramount. Let the joy of discovering be your guide.
“Take risks and you’ll get pay offs. Learn from your mistakes until you succeed. It’s that simple.”
– Bobby Flay, Master Chef
The role of a college website has changed substantially over the past few years. Student expectations for easy, accessible information drives colleges to get as much information online as possible. It also can lead to a battle for what information makes it onto the home page and how many clicks it will take to find what the student is looking for.
Student services are increasingly utilizing the college website to communicate with students and expecting that students will be proficient in navigating the college website. Students expect to easily locate information; this is helped when the college uses logical organization to the information architecture and design of the website. College websites can be very frustrating to new users, especially if the new user is a first-time college student and is unfamiliar with the underlying structure of the college system. The people creating the college web site may be very familiar with the way their system works and not see the structure as confusing.
Adding to potential confusion can be the lack of ability to view the entire home page of the college depending on the size of monitor or mobile device the student is accessing the website from. Students are increasingly using smart phones and tablets as their primary viewing device for the Internet. Sometimes key information a student needs may be just out of view on the screen. The experienced user knows to make adjustments, but new users may not. Knowing where and how to get started may not be as easy as the “start here” button.
Logical arrangement of information for the college’s needs may not be a logical progression of information for the student’s needs. From the college perspective, students come in different groups/classifications. Here are some examples:
Each of these groups can have variations on what their first steps should be. Students aren’t necessarily used to thinking of themselves in terms of these classifications/groups. It can be difficult for first time students, who may fit into more than one of these groups, to decide which one is the place to start.
Most college websites have a “Getting Started” type button on the home page. After clicking that button, a student begins to make a decision about what category of student he or she is. To an experienced user, this is not an obstacle, but to the first-time college student it may be a barrier. For example, what if a student falls into more than one group or classification? Where does the person fit? The answer may vary from college to college.
In addition, some college websites may not be mobile friendly so that students who are trying to use smartphones or tablets may face additional obstacles. Despite the potential difficulties, today’s college students need to become savvy users of the college website and recognize the role it will play in the communication process.
Pick 2 different colleges and examine their websites. Try to find the following information on each of the websites.
Website Challenge Reflection
The college website will be part of your communication system with the college you attend.
As you watch John Green’s Ted Talk, think about what information applies to college websites.
A TED element has been excluded from this version of the text. You can view it online here: https://press.rebus.community/blueprint2/?p=33
CC licensed content, Previously shared:
A Different Road To College: A Guide For Transitioning To College For Non-traditional Students. Authored by: Alise Lamoreaux. Located at: https://openoregon.pressbooks.pub/co...ter/chapter-6/ License: CC BY: Attribution.
Adaptions: Reformatted, some content adapted or deleted for broader audience. Edits made based on peer review suggestions.
John Green: The Nerd’s Guide to Learning Everything Online. Authored by TED.com
Located at: https://www.ted.com/talks/john_green...rything_online
License: CC BY – NC – ND 4.0 International.


1.14.21: Donnan Membrane Equilibria
A given experimental system comprises two compartments, I and II, separated by a membrane. The two compartments contain aqueous solutions at common temperature and pressure. The experimental system is set up by placing in compartment I an aqueous salt solution; e.g. \(\mathrm{NaCl}(\mathrm{aq})\) having concentration \(\mathrm{c}_{1} \mathrm{~mol dm}^{-3}\). However compartment II contains a salt, \(\mathrm{R}^{+}\mathrm{Cl}^{-} (\mathrm{aq})\), concentration \(\mathrm{c}_{2} \mathrm{~mol dm}^{-3}\). The membrane is permeable to both \(\mathrm{Na}^{+} and \(\mathrm{Cl}^{-}\) ions but not to \(\mathrm{R}^{+}\) cations. The sodium and chloride ions spontaneously diffuse across the membrane until the two solutions are in thermodynamic equilibrium. We represent the equilibrium system as follows where | | represents the membrane.
[I]
\[ \mathrm{Na}^{+}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{cq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{1}-\alpha\right)^{\mathrm{eq}}||[\mathrm{II}] \mathrm{Na}^{+}(\alpha)^{\mathrm{eq}} \mathrm{R}^{+}\left(\mathrm{c}_{2}\right)^{\mathrm{eq}} \mathrm{Cl}^{-}\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}} \nonumber \]
The solutions on both sides are electrically neutral. A thermodynamic analysis is somewhat complicated if account is taken of the role of ion-ion interactions. However the essential features of the argument are revealed if we identify the activities of the ions as equal to their concentrations.
Hence at equilibrium at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
\[\begin{aligned}
&{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Na}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}} \\
&+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{I}}= \\
&{\left[\mu^{0}\left(\mathrm{Na}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\alpha_{\mathrm{Na+}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}}} \\
&+\left[\mu^{0}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\left(\mathrm{c}_{2}+\alpha\right)_{\mathrm{Cl-}}^{\mathrm{eq}} / \mathrm{c}_{\mathrm{r}}\right\}\right]_{\mathrm{II}}
\end{aligned} \nonumber \]
But
\[\left(c_{1}-\alpha\right)_{\mathrm{Na+}}^{\mathrm{eq}}=\left(\mathrm{c}_{1}-\alpha\right)_{\mathrm{Cl}-}^{\mathrm{eq}} \nonumber \]
Or,
\[\left[\left(c_{1}-\alpha\right)_{\mathrm{Na}+}^{\mathrm{eq}}\right]^{2}=\alpha_{\mathrm{Na}+}^{\mathrm{eq}} \,\left(\mathrm{c}_{2}+\alpha\right)^{\mathrm{eq}} \mathrm{CI}^{-} \nonumber \]
Then [1],
\[\frac{\alpha^{\mathrm{eq}}}{\mathrm{c}_{1}}=\frac{\mathrm{c}_{1}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}} \nonumber \]
The latter is Donnan’s Equation. The ratio \(\left(\alpha^{\mathrm{eq}} / \mathrm{c}_{1}\right)\) tends to be smaller the larger is \(\mathrm{c}_{2}\). This conclusion is confirmed by experiment.
We have simplified the algebra by writing \(\mathrm{R}^{+} \mathrm{Cl}^{-}\) as the salt in compartment II. In practice the Donnan equilibrium finds major application where salt \(\mathrm{RCl}\) is a macromolecule [2-4].
Footnotes
[1] \(\mathrm{c}_{1}^{2}-2 \, \alpha \, \mathrm{c}_{1}+\alpha^{2}=\alpha \, \mathrm{c}_{2}+\alpha^{2}\)
Then, \(c_{1}^{2}-2 \, \alpha \, c_{1}=\alpha \, c_{2}\)
Or, \(\alpha=\frac{\mathrm{c}_{1}^{2}}{2 \, \mathrm{c}_{1}+\mathrm{c}_{2}}\)
[2] F. G. Donnan et al, J. Chem. Soc.,1911, 1554; 1914,1941.
[3] F. G. Donnan, Chem. Rev.,1924, 1 ,73.
[4] F. G. Donnan and E. A. Guggenheim, Z. Physik Chem. A, 1932, 162 ,346.


1.14: Auger Electron Spectroscopy
Basic Principles
Auger electron spectroscopy (AES) is one of the most commonly employed surface analysis techniques. It uses the energy of emitted electrons to identify the elements present in a sample, similar to X-ray photoelectron spectroscopy (XPS). The main difference is that XPS uses an X-ray beam to eject an electron while AES uses an electron beam to eject an electron. In AES, the sample depth is dependent on the escape energy of the electrons. It is not a function of the excitation source as in XPS. In AES, the collection depth is limited to 1-5 nm due to the small escape depth of electrons, which permits analysis of the first 2 - 10 atomic layers. In addition, a typical analysis spot size is roughly 10 nm. A representative AES spectrum illustrating the number of emitted electrons, N, as a function of kinetic energy, E, in direct form (red) and in differentiated form (black) is shown in Figure \(\PageIndex{1}\).
Like XPS, AES measures the kinetic energy (E k ) of an electron to determine its binding energy (E b ). The binding energy is inversely proportional to the kinetic energy and can be found from \ref{1}, where hν is the energy of the incident photon and ΔΦ is the difference in work function between the sample and the detector material.
\[ E_{b}\ =\ h\nu \ -\ E_{k}\ +\ \Delta \Phi \label{1} \]
Since the E b is dependent on the element and the electronic environment of the nucleus, AES can be used to distinguish elements and their oxidation states. For instance, the energy required to remove an electron from Fe 3 + is more than in Fe 0 . Therefore, the Fe 3 + peak will have a lower E k than the Fe 0 peak, effectively distinguishing the oxidation states.
Auger Process
An Auger electron comes from a cascade of events. First, an electron beam comes in with sufficient energy to eject a core electron creating a vacancy (see Figure \(\PageIndex{2}\)a). Typical energies of the primary electrons range from 3 - 30 keV. A secondary electron (imaging electron) of higher energy drops down to fill the vacancy (see Figure \(\PageIndex{2}\) b) and emits sufficient energy to eject a tertiary electron (Auger electron) from a higher shell (see Figure \(\PageIndex{2}\) c).
The shells from which the electrons move from lowest to highest energy are described as the K shell, L shell, and M shell. This nomenclature is related to quantum numbers. Explicitly, the K shell represents the 1s orbital, the L shell represents the 2s and 2p orbitals, and the M shell represents the 3s, 3p, and 3d orbitals. The cascade of events typically begins with the ionization of a K shell electron, followed by the movement of an L shell electron into the K shell vacancy. Then, either an L shell electron or M shell electron is ejected. It depends on the element, which peak is prevalent but often both peaks will be present. The peak seen in the spectrum is labeled according to the shells involved in the movement of the electrons. For example, an electron ejected from a gold atom could be labeled as Au KLL or Au KLM.
The intensity of the peak depends on the amount of material present, while the peak position is element dependent. Auger transitions characteristic of each elements can be found in the literature. Auger transitions of the first forty detectable elements are listed in Table \(\PageIndex{1}\).
| Atomic Number | Element | AES transition | Kinetic Energy of Transition (eV) |
|---|---|---|---|
| 3 | Li | KLL | 43 |
| 4 | Be | KLL | 104 |
| 5 | B | KLL | 179 |
| 6 | C | KLL | 272 |
| 7 | N | KLL | 379 |
| 8 | O | KLL | 508 |
| 9 | F | KLL | 647 |
| 11 | Na | KLL | 990 |
| 12 | Mg | KLL | 1186 |
| 13 | Al | LMM | 68 |
| 14 | Si | LMM | 92 |
| 15 | P | LMM | 120 |
| 16 | S | LMM | 152 |
| 17 | Cl | LMM | 181 |
| 19 | K | KLL | 252 |
| 20 | Ca | LMM | 291 |
| 21 | Sc | LMM | 340 |
| 22 | Ti | LMM | 418 |
| 23 | V | LMM | 473 |
| 24 | Cr | LMM | 529 |
| 25 | Mn | LMM | 589 |
| 26 | Fe | LMM | 703 |
| 27 | Co | LMM | 775 |
| 28 | Ni | LMM | 848 |
| 29 | Cu | LMM | 920 |
| 30 | Zn | LMM | 994 |
| 31 | Ga | LMM | 1070 |
| 32 | Ge | LMM | 1147 |
| 33 | As | LMM | 1228 |
| 34 | Se | LMM | 1315 |
| 35 | Br | LMM | 1376 |
| 39 | Y | MNN | 127 |
| 40 | Zr | MNN | 147 |
| 41 | Nb | MNN | 167 |
| 42 | Mo | MNN | 186 |
Instrumentation
Important elements of an Auger spectrometer include a vacuum system, an electron source, and a detector. AES must be performed at pressures less than 10 -3 pascal (Pa) to keep residual gases from adsorbing to the sample surface. This can be achieved using an ultra-high-vacuum system with pressures from 10 -8 to 10 -9 Pa. Typical electron sources include tungsten filaments with an electron beam diameter of 3 - 5 μm, LaB 6 electron sources with a beam diameter of less than 40 nm, and Schottky barrier filaments with a 20 nm beam diameter and high beam current density. Two common detectors are the cylindrical mirror analyzer and the concentric hemispherical analyzer discussed below. Notably, concentric hemispherical analyzers typically have better energy resolution.
Cylindrical Mirror Analyzer (CMA)
A CMA is composed of an electron gun, two cylinders, and an electron detector (Figure \(\PageIndex{2}\)). The operation of a CMA involves an electron gun being directed at the sample. An ejected electron then enters the space between the inner and outer cylinders (IC and OC). The inner cylinder is at ground potential, while the outer cylinder’s potential is proportional to the kinetic energy of the electron. Due to its negative potential, the outer cylinder deflects the electron towards the electron detector. Only electrons within the solid angle cone are detected. The resulting signal is proportional to the number of electrons detected as a function of kinetic energy.
Concentric Hemispherical Analyzer (CHA)
A CHA contains three parts (Figure \(\PageIndex{4}\)):
- A retarding and focusing input lens assembly
- An inner and outer hemisphere (IH and OH)
- An electron detector
Electrons ejected from the surface enter the input lens, which focuses the electrons and retards their energy for better resolution. Electrons then enter the hemispheres through an entrance slit. A potential difference is applied on the hemispheres so that only electrons with a small range of energy differences reach the exit. Finally, an electron detector analyzes the electrons.
Applications
AES has widespread use owing to its ability to analyze small spot sizes with diameters from 5 μm down to 10 nm depending on the electron gun. For instance, AES is commonly employed to study film growth and surface-chemical composition, as well as grain boundaries in metals and ceramics. It is also used for quality control surface analyses in integrated circuit production lines due to short acquisition times. Moreover, AES is used for areas that require high spatial resolution, which XPS cannot achieve. AES can also be used in conjunction with transmission electron microscopy (TEM) and scanning electron microscopy (SEM) to obtain a comprehensive understanding of microscale materials, both chemically and structurally. As an example of combining techniques to investigate microscale materials, Figure \(\PageIndex{5}\) shows the characterization of a single wire from a Sn-Nb multi-wire alloy. Figure \(\PageIndex{5}\) a is a SEM image of the singular wire and Figure \(\PageIndex{5}\) b is a schematic depicting the distribution of Nb and Sn within the wire. Point analysis was performed along the length of the wire to determine the percent concentrations of Nb and Sn.
AES is widely used for depth profiling. Depth profiling allows the elemental distributions of layered samples 0.2 – 1 μm thick to be characterized beyond the escape depth limit of an electron. Varying the incident and collection angles, and the primary beam energy controls the analysis depth. In general, the depth resolution decreases with the square root of the sample thickness. Notably, in AES, it is possible to simultaneously sputter and collect Auger data for depth profiling. The sputtering time indicates the depth and the intensity indicates elemental concentrations. Since, the sputtering process does not affect the ejection of the Auger electron, helium or argon ions can be used to sputter the surface and create the trench, while collecting Auger data at the same time. The depth profile does not have the problem of diffusion of hydrocarbons into the trenches. Thus, AES is better for depth profiles of reactive metals (e.g., gold or any metal or semiconductor). Yet, care should be taken because sputtering can mix up different elements, changing the sample composition.
Limitations
While AES is a very valuable surface analysis technique, there are limitations. Because AES is a three-electron process, elements with less than three electrons cannot be analyzed. Therefore, hydrogen and helium cannot be detected. Nonetheless, detection is better for lighter elements with fewer transitions. The numerous transition peaks in heavier elements can cause peak overlap, as can the increased peak width of higher energy transitions. Detection limits of AES include 0.1 – 1% of a monolayer, 10 -16 – 10 -15 g of material, and 10 12 – 10 13 atoms/cm 2 .
Another limitation is sample destruction. Although focusing of the electron beam can improve resolution; the high-energy electrons can destroy the sample. To limit destruction, beam current densities of greater than 1 mA/cm 2 should be used. Furthermore, charging of the electron beam on insulating samples can deteriorate the sample and result in high-energy peak shifts or the appearance of large peaks.


8.1.3A: Class Structure in the U.S.
-
- Last updated
- Save as PDF
- Anonymous
- LibreTexts
American society is stratified into social classes based on wealth, income, educational attainment, occupation, and social networks.
Learning Objectives
- Discuss America’s class structure and its relation to the concept of the “American Dream”
Key Points
- There are competing models for thinking about social classes in the U.S. — most Americans recognize a three-tier structure that includes the upper, middle, and lower classes, but variations delineate an upper-middle class and a working class.
- High income earners likely are substantially educated, have high- status occupations, and maintain powerful social networks.
- According to the “American Dream,” American society is meritocratic and class is achievement-based. In other words, one’s membership in a particular social class is based on educational and career accomplishments.
Key Terms
- social network : The web of a person’s social, family, and business contacts, who provide material and social resources and opportunities.
- The American Dream : The belief that with hard work, courage, and determination, anyone can prosper and achieve success.
- Corporate Elite : A class of high-salaried stockholders, such as corporate CEOs, who do not necessarily have inherited privilege but have achieved high status through their careers.
Most social scientists in the U.S. agree that society is stratified into social classes. Social classes are hierarchical groupings of individuals that are usually based on wealth, educational attainment, occupation, income, or membership in a subculture or social network. Social class in the United States is a controversial issue, having many competing definitions, models, and even disagreements over its very existence. Many Americans recognize a simple three-tier model that includes the upper class, the middle class, and the lower or working class. Some social scientists have proposed more complex models that may include as many as a dozen class levels. Meanwhile, some scholars deny the very existence of discrete social classes in American society. In spite of debate, most social scientists do agree that in the U.S. people are hierarchically ranked in a social class structure.
Models of U.S. Social Classes
A team of sociologists recently posited that there are six social classes in America. In this model, the upper class (3% of the population ) is divided into upper-upper class (1% of the U.S. population, earning hundreds of millions to billions per year) and the lower-upper class (2%, earning millions per year). The middle class (40%) is divided into upper-middle class (14%, earning $76,000 or more per year) and the lower-middle class (26%, earning $46,000 to $75,000 per year). The working class (30%) earns $19,000 to $45,000 per year. The lower class (27%) is divided into working poor (13%, earning $9000 to 18,000 per year) and underclass (14%, earning under $9000 per year). This model has gained traction as a tool for thinking about social classes in America, but it does not fully account for variations in status based on non-economic factors, such as education and occupational prestige. This critique is somewhat mitigated by the fact that income is often closely aligned with other indicators of status; for example, those with high incomes likely have substantial education, high status occupations, and powerful social networks.
A commonly used model for thinking about social classes in the U.S. attributes the following general characteristics to each tier: the upper class has vast accumulated wealth and significant control over corporations and political institutions, and their privilege is usually inherited; the corporate elite consists of high-salaried stockholders, such as corporate CEOs, who did not necessarily inherit privilege but have achieved high status through their careers; the upper-middle class consists of highly educated salaried professionals whose occupations are held in high esteem, such as lawyers, engineers, and professors; the middle class (the most vaguely defined and largest social class) is generally thought to include people in mid-level managerial positions or relatively low status professional positions, such as high school teachers and small business owners; the working class generally refers to those without college degrees who do low level service work, such as working as a sales clerk or housekeeper, and includes most people whose incomes fall below the poverty line. In the above outline of social class, status clearly depends not only on income, but also occupational prestige and educational attainment.
Debates over the Existence and Significance of U.S. Social Classes
According to the “American Dream,” American society is meritocratic and class is achievement-based. In other words, membership in a particular social class is based on educational and career accomplishments. Many sociologists dispute the existence of such class mobility and point to the ways in which social class is inherited. For example, a son or daughter of a wealthy individual may carry a higher status and different cultural connotations than a member of the nouveau riche (“new money”). Likewise, being born into a particular social class may confer advantages or disadvantages that increase the likelihood that an adult will remain in the social class into which they were born.
Social theorists who dispute the existence of social classes in the U.S. tend to argue that society is stratified along a continuous gradation, rather than into delineated categories. In other words, there is inequality in America, with some people attaining higher status and higher standards of living than others. But there is no clear place to draw a line separating one status group from the next. Whether one ascribes to the view that classes are discrete groups or levels along a continuum, it is important to remember that all social classes in the United States, except the upper class, consist of tens of millions of people. Thus social classes form social groups so large that they feature considerable internal diversity and any statement regarding a given social class’ culture should be seen as a broad generalization.


2.6: The Wanderers
The word planet is from the Greek
planete
, meaning
“wanderer”
. All of the known planets moved against the background stars; the historically-known planets included Mercury, Venus, Mars, Jupiter, and Saturn (and Earth, of course). Mercury and Venus stayed close to the Sun. Mars, Jupiter, and Saturn could be tracked as they moved across the sky. But occasionally the planets were observed to move backwards against the stars. This phenomenon is called
retrograde
. At the time, the thinking was that the Earth was the center of the Solar System and even the Universe. So the Sun, Moon, and known planets revolved around Earth – called the
Geocentric Solar System
; Geo means Earth.
How could this retrograde motion occur if Earth was the center of the Solar System and even the Universe? Thus, the planets presented problems.


2.19: Histograms (2 of 4)
Learning Objectives
- Describe the distribution of quantitative data using a histogram.
We have discussed two types of graphs that summarize a distribution of a quantitative variable: dotplots and histograms.
From a dotplot, we also described the pattern in the data with statements about shape, center, and spread. We have to be more cautious making similar statements using a histogram because our perception of shape, center, and spread can be affected by how the bins are defined. We investigate this important point in the next example.
Example
We used the same set of data to construct these three histograms of student scores. Are you surprised by how different the distribution looks in each histogram?
The histogram on the left has a bin width of 20. The first bin starts at 40. To create the middle histogram, we changed the bin width to 10 but kept the first bin starting at 40. To create the last histogram, we kept the bin width at 10 but started the first bin at 45.
These changes affect our description of the shape, center, and spread of this set of data. For example, in the histogram on the left, the distribution looks symmetric with a central peak. In the histogram on the right, the distribution looks slightly skewed to the right. Based on the middle histogram, we might estimate that most students scored between 70 and 80. But the histogram on the right suggests that typical students scored between 65 and 75.
Why does changing the bin size and the starting point of the first bin change the histogram so drastically?
When we change the bins, the data gets grouped differently. The different grouping affects the appearance of the histogram.
To illustrate this point, we highlighted the five students who scored in the 70s in each histogram.
- In the histogram on the left, these five students are grouped in the middle bin with other students who scored between 60 and 80.
- In the histogram in the middle, these five students form a bin of their own, since no other students scored between 70 and 80.
- In the histogram on the right, these five students are in separate bins.
Which histogram gives the most helpful summary of the distribution?
For this situation, the middle histogram is probably the most useful summary because the intervals correspond to letter grades.
Our general advice is as follows:
- Avoid histograms with large bin widths that group data into only a few bins. A histogram constructed with large bin widths will show the distribution as a “skyscraper.” This does not give good information about variability in the distribution.
- Avoid histograms with small bin widths that group data into lots of bins. A histogram constructed with small bin widths will show the distribution as a “pancake.” This does not help us see the pattern in the data.
Use the simulation below to answer the questions in the next Try It.
Click here to open this simulation in its own window.
These next exercises focus on recognizing the shape of a distribution using a histogram. We know that changes in the bin width can change the appearance of the distribution. But a histogram with an appropriate bin width can give good information about the shape of the distribution.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


7.2.3: Triangles with 3 Common Measures
Lesson
Let's contrast triangles.
Exercise \(\PageIndex{1}\): 3 Sides; 3 Angles
Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different?
Set 1:
Set 2:
Exercise \(\PageIndex{2}\): 2 Sides and 1 Angle
Examine this set of triangles.
- What is the same about the triangles in the set? What is different?
- How many different triangles are there? Explain or show your reasoning.
Exercise \(\PageIndex{3}\): 2 Angles and 1 Side
Examine this set of triangles.
- What is the same about the triangles in the set? What is different?
- How many different triangles are there? Explain or show your reasoning.
Summary
Both of these quadrilaterals have a right angle and side lengths 4 and 5.
However, in one case, the right angle is between the two given side lengths; in the other, it is not.
If we create two triangles with three equal measures, but these measures are not next to each other in the same order, that usually means the triangles are different. Here is an example:
Practice
Exercise \(\PageIndex{4}\)
Are these two triangles identical? Explain how you know.
Exercise \(\PageIndex{5}\)
Are these triangles identical? Explain your reasoning.
Exercise \(\PageIndex{6}\)
Tyler claims that if two triangles each have a side length of 11 units and a side length of 8 units, and also an angle measuring \(100^{\circ}\), they must be identical to each other. Do you agree? Explain your reasoning.
Exercise \(\PageIndex{7}\)
The markings on the number line are equally spaced. Label the other markings on the number line.
(From Unit 5.3.1)
Exercise \(\PageIndex{8}\)
A passenger on a ship dropped his camera into the ocean. If it is descending at a rate of -4.2 meters per second, how long until it hits the bottom of the ocean, which is at -1,875 meters?
(From Unit 5.3.2)
Exercise \(\PageIndex{9}\)
Apples cost $1.99 per pound.
- How much do \(3\frac{1}{4}\) pounds of apples cost?
- How much do \(x\) pounds of apples cost?
- Clare spent $5.17 on apples. How many pounds of apples did Clare buy?
(From Unit 4.1.3)
Exercise \(\PageIndex{10}\)
Diego has a glue stick with a diameter of 0.7 inches. He sets it down 3.5 inches away from the edge of the table, but it rolls onto the floor. How many rotations did the glue stick make before it fell off of the table?
(From Unit 3.1.5)


9.1: Muscles of the Rotator Cuff
Information
The rotator cuff is the name given to the group of four muscles that are largely responsible for the ability to rotate the arm. Three of the four rotator cuff muscles are deep to the deltoid and trapezius muscles and cannot be seen unless those muscles are first removed and one is on the anterior side of the scapula bone and cannot be seen from the surface.
On the anterior side of scapula bone is a single muscle, the subscapularis . It is triangular in shape and covers the entire bone. Its origin is along the fossa that makes up most of the “wing” of the scapula and it inserts on the lesser tubercle of the humerus bone. The subscapularis muscle is shown in Figure \(\PageIndex{1}\).
On the posterior side of the scapula bone are the other three muscles of the rotator cuff. All three insert on the greater tubercle of the humerus, allowing them, in combination with the subscapularis, to control rotation of the arm. The supraspinatus muscle is above the spine of the scapula. The infraspinatus muscle is below the spine of the scapula. The relatively thin teres minor muscle is the most inferior of the rotator cuff muscles. The three posteriorly- positioned muscles of the rotator cuff are shown in Figure \(\PageIndex{2}\).
The teres major muscle has its origin on the scapula, like the rotator cuff muscles, but is not involved in rotating the arm. It inserts lower on the humerus than the rotator cuff muscles and is involved in adducting the arm (bringing it closer to the midline of the body.)
LAB 9 EXERCISES \(\PageIndex{1}\)
1. Using the full-scale arm model, locate and identify all four muscles of the rotator cuff, as well as the deltoid muscle and the teres major muscle.
2. The following are muscles of arm rotation and adduction . For each, give its origin(s) and insertion(s) and whether or not it is part of the rotator cuff.
|
Muscle |
Origin(s) |
Insertion(s) |
Part of rotator cuff? |
|
Subscapularis |
|
|
|
|
Supraspinatus |
|
|
|
|
Infraspinatus |
|
|
|
|
Teres major |
|
|
|
|
Teres minor |
|
|
|
|
Deltoid |
|
|
|
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, ORIGINAL
A&P Labs. Authored by : Ross Whitwam. Provided by : Mississippi University for Women. Located at : http://www.muw.edu . License : CC BY-SA: Attribution-ShareAlike
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
Figure \(\PageIndex{1}\). The subscapularis muscle of the rotator cuff, in red, anterior view.. Authored by : Was a bee. Provided by : Images in Figure \(\PageIndex{1}\) were made out of, or made from, content published in a BodyParts3D/Anatomography web site. The content of their website is published under the Creative Commons Attribution 2.1 Japan license. The author and licenser of the contents is http://lifesciencedb.jp/bp3d/?lng=en.. Located at : https://commons.wikimedia.org/wiki/F...le_frontal.png . License : CC BY-SA: Attribution-ShareAlike
Figure \(\PageIndex{2}\). The muscles of the rotator cuff and arm, posterior view.. Authored by : OpenStax College. Located at : http://cnx.org/resources/e5ba9b5bb7343a347f55336ebd7a61f3b35b0cdc/1119_Muscles_that_Move_the_Humerus.jpg . Lice nse : CC BY-SA: Attribution-ShareAlike


8.1.5: More Estimating Probabilities
Lesson
Let's estimate some probabilities.
Exercise \(\PageIndex{1}\): Is it Likely?
- If the weather forecast calls for a 20% chance of light rain tomorrow, would you say that it is likely to rain tomorrow?
- If the probability of a tornado today is \(\frac{1}{10}\), would you say that there will likely be a tornado today?
- If the probability of snow this week is 0.85, would you say that it is likely to snow this week?
Exercise \(\PageIndex{2}\): Making My Head Spin
Work with your group to decide which person will use each spinner. Make sure each person selects a different spinner.
Answer the first set of questions on your own.
Spinner A
Spinner B
Spinner C
Spinner D
- Spin your spinner 10 times and record your outcomes.
- Did you get all of the different possible outcomes in your 10 spins?
-
What fraction of your 10 spins landed on 3?
Work with your group to answer the next set of questions. -
Share your outcomes with your group and record their outcomes.
- Outcomes for spinner A:
- Outcomes for spinner B:
- Outcomes for spinner C:
- Outcomes for spinner D:
- Do any of the spinners have the same sample space? If so, do they have the same probabilities for each number to result?
- For each spinner, what is the probability that it ends on the number 3? Explain or show your reasoning.
- For each spinner, what is the probability that it lands on something other than the number 3? Explain or show your reasoning.
- Noah put spinner D on top of his closed binder and spun it 10 times. It never landed on the number 1. How might you explain why this happened?
- Han put spinner C on the floor and spun it 10 times. It never landed on the number 3, so he says that the probability of getting a 3 is 0. How might you explain why this happened?
Are you ready for more?
Design a spinner that has a \(\frac{2}{3}\) probability of landing on the number 3. Explain how you could precisely draw this spinner.
Exercise \(\PageIndex{3}\): How much Green?
 Do not look into the bag or take out more than 1 block at a time. Repeat these steps until everyone in your group has had 4 turns.
- Take one block out of the bag and record whether or not it is green.
- Put the block back into the bag, and shake the bag to mix up the blocks.
- Pass the bag to the next person in the group.
- What do you think is the probability of taking out a green block from this bag? Explain or show your reasoning.
- How could you get a better estimate without opening the bag?
Summary
Suppose a bag contains 5 blocks. If we select a block at random from the bag, then the probability of getting any one of the blocks is \(\frac{1}{5}\).
Now suppose a bag contains 5 blocks. Some of the blocks have a star, and some have a moon.  The probability of getting a star block depends on how many there are in the bag.
In this example, the probability of selecting a star block at random from the first bag is \(\frac{1}{5}\), because it contains only 1 star block. (The probability of getting a moon block is \(\frac{4}{5}\).) The probability of selecting a star block at random from the second bag is \(\frac{3}{5}\), because it contains 3 star blocks. (The probability of getting a moon block from this bag is \(\frac{2}{5}\).)
This shows that two experiments can have the same sample space, but different probabilities for each outcome.
Glossary Entries
Definition: Chance Experiment
A chance experiment is something you can do over and over again, and you don’t know what will happen each time.
For example, each time you spin the spinner, it could land on red, yellow, blue, or green.
Definition: Event
An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes.
Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”
Definition: Outcome
An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails.
Definition: Probability
The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen.
For example, the probability of selecting a moon block at random from this bag is \(\frac{4}{5}\).
Definition: Random
Outcomes of a chance experiment are random if they are all equally likely to happen.
Definition: Sample Space
The sample space is the list of every possible outcome for a chance experiment.
For example, the sample space for tossing two coins is:
| heads-heads | tails-heads |
| heads-tails | tails-tails |
Practice
Exercise \(\PageIndex{4}\)
What is the same about these two experiments? What is different?
- Selecting a letter at random from the word “ALABAMA”
- Selecting a letter at random from the word “LAMB”
Exercise \(\PageIndex{5}\)
Andre picks a block out of a bag 60 times and notes that 43 of them were green.
- What should Andre estimate for the probability of picking out a green block from this bag?
- Mai looks in the bag and sees that there are 6 blocks in the bag. Should Andre change his estimate based on this information? If so, what should the new estimate be? If not, explain your reasoning.
Exercise \(\PageIndex{6}\)
Han has a number cube that he suspects is not so standard.
- Han rolls the cube 100 times, and it lands on a six 40 times.
- Kiran rolls the cube 50 times, and it lands on a six 21 times.
- Lin rolls the cube 30 times, and it lands on a six 11 times.
Based on these results, is there evidence to help prove that this cube is not a standard number cube? Explain your reasoning.
Exercise \(\PageIndex{7}\)
A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages.
- What is the sample space for this experiment?
- What is the probability that you turn to page 45?
- What is the probability that you turn to an even numbered page?
- If you repeat this experiment 50 times, about how many times do you expect you will turn to an even numbered page?
(From Unit 8.1.3)
Exercise \(\PageIndex{8}\)
A rectangular prism is cut along a diagonal on each face to create two triangular prisms. The distance between \(A\) and \(B\) is 5 inches.
What is the surface area of the original rectangular prism? What is the total surface area of the two triangular prisms together?
(From Unit 7.3.5)