7.2: The Communication Paradigm
A paradigm is a pattern of ideas that form the foundation for a body of knowledge. A paradigm for (tele-) communication theory is a pattern of basic building blocks that may be applied to the dual problems of (i) reliably transmitting information from source to receiver at high speed or (ii) reliably storing information from source to memory at high density. High-speed communication permits us to accommodate many low-rate sources (such as audio) or one high-rate source (such as video). High-density storage permits us to store a large amount of information in a small space. For example, a typical 1.2 Mbyte floppy disc stores \(9.6×10^6\) bits of information, whereas a typical CD stores about \(2×10^9\) bits, enough for one hour's worth of high-quality sound.
Figure 1 illustrates the basic building blocks that apply to any problem in the theory of (tele-) communication. The source is an arbitrary source of information. It can be the time-varying voltage at the output of a vibration sensor (such as an integrating accelerometer for measuring motion or a microphone for measuring sound pressure); it can be the charges stored in the CCD array of a solid-state camera; it can be the addresses generated from a sequence of keystrokes at a computer terminal; it can be a sequence of instructions in a computer program. The source coder is a device for turning primitive source outputs into more efficient representations. For example, in a recording studio, the source coder would convert analog voltages into digital approximations using an A/D converter; a fancy source coder would use a fancy A/D converter that finely quantized likely analog values and crudely quantized unlikely values. If the source is a source of discrete symbols like letters and numbers, then a fancy source code would assign short binary sequences to likely symbols (such as \(e\)) and long binary sequences to unlikely symbols (such as \(z\)). The channel coder adds “redundant bits” to the binary output of the source coder so that errors of transmission or storage may be detected and corrected. In the simplest example, a binary string of the form 01001001 would have an extra bit of 1 added to give even parity (an even number of l's) to the string; the string 10110111 would have an extra bit of 0 added to preserve the even parity . If one bit error is introduced in the channel, then the parity is odd and the receiver knows that an error has occurred. The modulator takes outputs of the channel coder, a stream of binary digits, and constructs an analog waveform that represents a block of bits. For example, in a 9600 baud Modem, five bits are used to determine one of \(2^5=32\) phases that are used to modulate the signal Acos(ωt+φ)Acos(ωt+φ). Each possible string of five bits has its own personalized phase, \(\phi\), and this phase can be determined at the receiver. The signal \(A \cos (\omega t+\phi)\). \(A \cos (\omega t+\phi)\) is an analog signal that may be transmitted over a channel (such as a telephone line, a microwave link, or a fiber-optic cable). The channel has a finite bandwidth, meaning that it distorts signals, and it is subject to noise or interference from other electromagnetic radiation. Therefore transmitted information arrives at the demodulator in imperfect form. The demodulator uses filters matched to the modulated signals to demodulate the phase and look up the corresponding bit stream. The channel decoder converts the coded bit stream into the information bit stream, and the source decoder looks up the corresponding symbol. This sequence of steps is illustrated symbolically in Figure 7.3.
In your subsequent courses on communication theory you will study each block of Figure 1 in detail. You will find that every source of information has a characteristic complexity, called entropy , that determines the minimum rate at which bits must be generated in order to represent the source. You will also find that every communication channel has a characteristic tolerance for bits, called channel capacity . This capacity depends on signal-to-noise ratio and bandwidth. When the channel capacity exceeds the source entropy, then you can transmit information reliably; if it does not, then you cannot.


7.3: Compostion and Structure of Planets
By the end of this section, you will be able to:
- Describe the characteristics of the giant planets, terrestrial planets, and small bodies in the solar system
- Explain what influences the temperature of a planet’s surface
- Explain why there is geological activity on some planets and not on others
The fact that there are two distinct kinds of planets—the rocky terrestrial planets and the gas-rich jovian planets—leads us to believe that they formed under different conditions. Certainly their compositions are dominated by different elements. Let us look at each type in more detail.
The Giant Planets
The two largest planets, Jupiter and Saturn , have nearly the same chemical makeup as the Sun; they are composed primarily of the two elements hydrogen and helium, with 75% of their mass being hydrogen and 25% helium. On Earth, both hydrogen and helium are gases, so Jupiter and Saturn are sometimes called gas planets. But, this name is misleading. Jupiter and Saturn are so large that the gas is compressed in their interior until the hydrogen becomes a liquid. Because the bulk of both planets consists of compressed, liquefied hydrogen, we should really call them liquid planets.
Under the force of gravity, the heavier elements sink toward the inner parts of a liquid or gaseous planet. Both Jupiter and Saturn, therefore, have cores composed of heavier rock, metal, and ice, but we cannot see these regions directly. In fact, when we look down from above, all we see is the atmosphere with its swirling clouds (Figure 7.11). We must infer the existence of the denser core inside these planets from studies of each planet’s gravity.
Uranus and Neptune are much smaller than Jupiter and Saturn, but each also has a core of rock, metal, and ice. Uranus and Neptune were less efficient at attracting hydrogen and helium gas, so they have much smaller atmospheres in proportion to their cores.
Chemically, each giant planet is dominated by hydrogen and its many compounds. Nearly all the oxygen present is combined chemically with hydrogen to form water (H 2 O). Chemists call such a hydrogen-dominated composition reduced . Throughout the outer solar system, we find abundant water (mostly in the form of ice) and reducing chemistry.
The Terrestrial Planets
The terrestrial planets are quite different from the giants. In addition to being much smaller, they are composed primarily of rocks and metals. These, in turn, are made of elements that are less common in the universe as a whole. The most abundant rocks, called silicates, are made of silicon and oxygen, and the most common metal is iron. We can tell from their densities (see Table 7.2) that Mercury has the greatest proportion of metals (which are denser) and the Moon has the lowest. Earth , Venus , and Mars all have roughly similar bulk compositions: about one third of their mass consists of iron-nickel or iron-sulfur combinations; two thirds is made of silicates. Because these planets are largely composed of oxygen compounds (such as the silicate minerals of their crusts), their chemistry is said to be oxidized .
When we look at the internal structure of each of the terrestrial planets, we find that the densest metals are in a central core, with the lighter silicates near the surface. If these planets were liquid, like the giant planets, we could understand this effect as the result the sinking of heavier elements due to the pull of gravity. This leads us to conclude that, although the terrestrial planets are solid today, at one time they must have been hot enough to melt.
Differentiation is the process by which gravity helps separate a planet’s interior into layers of different compositions and densities. The heavier metals sink to form a core, while the lightest minerals float to the surface to form a crust. Later, when the planet cools, this layered structure is preserved. In order for a rocky planet to differentiate, it must be heated to the melting point of rocks, which is typically more than 1300 K.
Moons, Asteroids, and Comets
Chemically and structurally, Earth’s Moon is like the terrestrial planets, but most moons are in the outer solar system, and they have compositions similar to the cores of the giant planets around which they orbit. The three largest moons—Ganymede and Callisto in the jovian system, and Titan in the saturnian system—are composed half of frozen water, and half of rocks and metals. Most of these moons differentiated during formation, and today they have cores of rock and metal, with upper layers and crusts of very cold and—thus very hard—ice (Figure 7.12).
Most of the asteroids and comets, as well as the smallest moons, were probably never heated to the melting point. However, some of the largest asteroids, such as Vesta , appear to be differentiated; others are fragments from differentiated bodies. Many of the smaller objects seem to be fragments or rubble piles that are the result of collisions. Because most asteroids and comets retain their original composition, they represent relatively unmodified material dating back to the time of the formation of the solar system. In a sense, they act as chemical fossils, helping us to learn about a time long ago whose traces have been erased on larger worlds.
Temperatures: Going to Extremes
Generally speaking, the farther a planet or moon is from the Sun, the cooler its surface. The planets are heated by the radiant energy of the Sun, which gets weaker with the square of the distance. You know how rapidly the heating effect of a fireplace or an outdoor radiant heater diminishes as you walk away from it; the same effect applies to the Sun. Mercury , the closest planet to the Sun, has a blistering surface temperature that ranges from 280–430 °C on its sunlit side, whereas the surface temperature on Pluto is only about –220 °C, colder than liquid air.
Mathematically, the temperatures decrease approximately in proportion to the square root of the distance from the Sun. Pluto is about 30 AU at its closest to the Sun (or 100 times the distance of Mercury) and about 49 AU at its farthest from the Sun. Thus, Pluto’s temperature is less than that of Mercury by the square root of 100, or a factor of 10: from 500 K to 50 K.
In addition to its distance from the Sun, the surface temperature of a planet can be influenced strongly by its atmosphere. Without our atmospheric insulation (the greenhouse effect, which keeps the heat in), the oceans of Earth would be permanently frozen. Conversely, if Mars once had a larger atmosphere in the past, it could have supported a more temperate climate than it has today. Venus is an even more extreme example, where its thick atmosphere of carbon dioxide acts as insulation, reducing the escape of heat built up at the surface, resulting in temperatures greater than those on Mercury. Today, Earth is the only planet where surface temperatures generally lie between the freezing and boiling points of water. As far as we know, Earth is the only planet to support life.
In the classic film The Wizard of Oz , Dorothy, the heroine, concludes after her many adventures in “alien” environments that “there’s no place like home.” The same can be said of the other worlds in our solar system. There are many fascinating places, large and small, that we might like to visit, but humans could not survive on any without a great deal of artificial assistance.
 Mars, on the other hand, has temperatures generally below freezing, with air (also mostly carbon dioxide) so thin that it resembles that found at an altitude of 30 kilometers (100,000 feet) in Earth’s atmosphere. And the red planet is so dry that it has not had any rain for billions of years.
The outer layers of the jovian planets are neither warm enough nor solid enough for human habitation. Any bases we build in the systems of the giant planets may well have to be in space or one of their moons—none of which is particularly hospitable to a luxury hotel with a swimming pool and palm trees. Perhaps we will find warmer havens deep inside the clouds of Jupiter or in the ocean under the frozen ice of its moon Europa.
All of this suggests that we had better take good care of Earth because it is the only site where life as we know it could survive. Recent human activity may be reducing the habitability of our planet by adding pollutants to the atmosphere, especially the potent greenhouse gas carbon dioxide. Human civilization is changing our planet dramatically, and these changes are not necessarily for the better. In a solar system that seems unready to receive us, making Earth less hospitable to life may be a grave mistake.
Geological Activity
The crusts of all of the terrestrial planets, as well as of the larger moons, and Pluto, have been modified over their histories by both internal and external forces. Externally, each has been battered by a slow rain of projectiles from space, leaving their surfaces pockmarked by impact craters of all sizes (see Figure 7.4). We have good evidence that this bombardment was far greater in the early history of the solar system, but it certainly continues to this day, even if at a lower rate. The collision of more than 20 large pieces of Comet Shoemaker–Levy 9 with Jupiter in the summer of 1994 (see Figure 7.13) is one dramatic example of this process.
Figure 7.14 shows the aftermath of these collisions, when debris clouds larger than Earth could be seen in Jupiter ’s atmosphere.
During the time all the planets have been subject to such impacts, internal forces on the terrestrial planets have buckled and twisted their crusts, built up mountain ranges, erupted as volcanoes, and generally reshaped the surfaces in what we call geological activity. (The prefix geo means “Earth,” so this is a bit of an “Earth-chauvinist” term, but it is so widely used that we bow to tradition.) Among the terrestrial planets, Earth and Venus have experienced the most geological activity over their histories, although some of the moons in the outer solar system are also surprisingly active. In contrast, our own Moon is a dead world where geological activity ceased billions of years ago.
Geological activity on a planet is the result of a hot interior. The forces of volcanism and mountain building are driven by heat escaping from the interiors of planets. As we will see, each of the planets was heated at the time of its birth, and this primordial heat initially powered extensive volcanic activity, even on our Moon. But, small objects such as the Moon soon cooled off. The larger the planet or moon, the longer it retains its internal heat, and therefore the more we expect to see surface evidence of continuing geological activity. The effect is similar to our own experience with a hot baked potato: the larger the potato, the more slowly it cools. If we want a potato to cool quickly, we cut it into small pieces.
For the most part, the history of volcanic activity on the terrestrial planets conforms to the predictions of this simple theory. The Moon, the smallest of these objects, is a geologically dead world. Although we know less about Mercury, it seems likely that this planet, too, ceased most volcanic activity about the same time the Moon did. Mars represents an intermediate case. It has been much more active than the Moon, but less so than Earth. Earth and Venus, the largest terrestrial planets, still have molten interiors even today, some 4.5 billion years after their birth.


12.2: What Is Social Psychology?
Learning Objectives
- Define social psychology
- Describe situational versus dispositional influences on behavior
- Describe the fundamental attribution error
Social psychology examines how people affect one another, and it looks at the power of the situation. Social psychologists assert that an individual’s thoughts, feelings, and behaviors are very much influenced by social situations. Essentially, people will change their behavior to align with the social situation at hand. If we are in a new situation or are unsure how to behave, we will take our cues from other individuals.
The field of social psychology studies topics at both the intra- and interpersonal levels. Intrapersonal topics (those that pertain to the individual) include emotions and attitudes, the self, and social cognition (the ways in which we think about ourselves and others). Interpersonal topics (those that pertain to dyads and groups) include helping behavior (See figure \(\PageIndex{1}\)), aggression, prejudice and discrimination, attraction and close relationships, and group processes and intergroup relationships.
Social psychologists focus on how people construe or interpret situations and how these interpretations influence their thoughts, feelings, and behaviors (Ross & Nisbett, 1991). Thus, social psychology studies individuals in a social context and how situational variables interact to influence behavior. In this chapter, we discuss the intrapersonal processes of self-presentation, cognitive dissonance and attitude change, and the interpersonal processes of conformity and obedience, aggression and altruism, and, finally, love and attraction.
Situational and Dispositional Influences on Behavior
Behavior is a product of both the situation (e.g., cultural influences, social roles, and the presence of bystanders) and of the person (e.g., personality characteristics). Subfields of psychology tend to focus on one influence or behavior over others. Situationism is the view that our behavior and actions are determined by our immediate environment and surroundings. In contrast, dispositionism holds that our behavior is determined by internal factors (Heider, 1958). An internal factor is an attribute of a person and includes personality traits and temperament. Social psychologists have tended to take the situationist perspective, whereas personality psychologists have promoted the dispositionist perspective. Modern approaches to social psychology, however, take both the situation and the individual into account when studying human behavior (Fiske, Gilbert, & Lindzey, 2010). In fact, the field of social-personality psychology has emerged to study the complex interaction of internal and situational factors that affect human behavior (Mischel, 1977; Richard, Bond, & Stokes-Zoota, 2003).
Fundamental Attribution Error
In the United States, the predominant culture tends to favor a dispositional approach in explaining human behavior. Why do you think this is? We tend to think that people are in control of their own behaviors, and, therefore, any behavior change must be due to something internal, such as their personality, habits, or temperament. According to some social psychologists, people tend to overemphasize internal factors as explanations—or attributions—for the behavior of other people. They tend to assume that the behavior of another person is a trait of that person, and to underestimate the power of the situation on the behavior of others. They tend to fail to recognize when the behavior of another is due to situational variables, and thus to the person’s state . This erroneous assumption is called the fundamental attribution error (Ross, 1977; Riggio & Garcia, 2009). To better understand, imagine this scenario: Greg returns home from work, and upon opening the front door his wife happily greets him and inquires about his day. Instead of greeting his wife, Greg yells at her, “Leave me alone!” Why did Greg yell at his wife? How would someone committing the fundamental attribution error explain Greg’s behavior? The most common response is that Greg is a mean, angry, or unfriendly person (his traits). This is an internal or dispositional explanation. However, imagine that Greg was just laid off from his job due to company downsizing. Would your explanation for Greg’s behavior change? Your revised explanation might be that Greg was frustrated and disappointed for losing his job; therefore, he was in a bad mood (his state). This is now an external or situational explanation for Greg’s behavior.
The fundamental attribution error is so powerful that people often overlook obvious situational influences on behavior. A classic example was demonstrated in a series of experiments known as the quizmaster study (Ross, Amabile, & Steinmetz, 1977). Student participants were randomly assigned to play the role of a questioner (the quizmaster) or a contestant in a quiz game. Questioners developed difficult questions to which they knew the answers, and they presented these questions to the contestants. The contestants answered the questions correctly only \(4\) out of \(10\) times (See fig. \(\PageIndex{2}\)). After the task, the questioners and contestants were asked to rate their own general knowledge compared to the average student. Questioners did not rate their general knowledge higher than the contestants, but the contestants rated the questioners’ intelligence higher than their own. In a second study, observers of the interaction also rated the questioner as having more general knowledge than the contestant. The obvious influence on performance is the situation. The questioners wrote the questions, so of course they had an advantage. Both the contestants and observers made an internal attribution for the performance. They concluded that the questioners must be more intelligent than the contestants.
As demonstrated in the example above, the fundamental attribution error is considered a powerful influence in how we explain the behaviors of others. However, it should be noted that some researchers have suggested that the fundamental attribution error may not be as powerful as it is often portrayed. In fact, a recent review of more than \(173\) published studies suggests that several factors (e.g., high levels of idiosyncrasy of the character and how well hypothetical events are explained) play a role in determining just how influential the fundamental attribution error is (Malle, 2006).
Is the Fundamental Attribution Error a Universal Phenomenon?
You may be able to think of examples of the fundamental attribution error in your life. Do people in all cultures commit the fundamental attribution error? Research suggests that they do not. People from an individualistic culture , that is, a culture that focuses on individual achievement and autonomy, have the greatest tendency to commit the fundamental attribution error. Individualistic cultures, which tend to be found in western countries such as the United States, Canada, and the United Kingdom, promote a focus on the individual. Therefore, a person’s disposition is thought to be the primary explanation for her behavior. In contrast, people from a collectivistic culture , that is, a culture that focuses on communal relationships with others, such as family, friends, and community (See fig. \(\PageIndex{3}\)), are less likely to commit the fundamental attribution error (Markus & Kitayama, 1991; Triandis, 2001).
Why do you think this is the case? Collectivistic cultures, which tend to be found in east Asian countries and in Latin American and African countries, focus on the group more than on the individual (Nisbett, Peng, Choi, & Norenzayan, 2001). This focus on others provides a broader perspective that takes into account both situational and cultural influences on behavior; thus, a more nuanced explanation of the causes of others’ behavior becomes more likely. The Table \(\PageIndex{1}\) below summarizes compares individualistic and collectivist cultures.
| Individualistic Culture | Collectivistic Culture |
|---|---|
| Achievement oriented | Relationship oriented |
| Focus on autonomy | Focus on group autonomy |
| Dispositional perspective | Situational perspective |
| Independent | Interdependent |
| Analytic thinking style | Holistic thinking style |
Actor-Observer Bias
Returning to our earlier example, Greg knew that he lost his job, but an observer would not know. So a naïve observer would tend to attribute Greg’s hostile behavior to Greg’s disposition rather than to the true, situational cause. Why do you think we underestimate the influence of the situation on the behaviors of others? One reason is that we often don’t have all the information we need to make a situational explanation for another person’s behavior. The only information we might have is what is observable. Due to this lack of information we have a tendency to assume the behavior is due to a dispositional, or internal, factor. When it comes to explaining our own behaviors, however, we have much more information available to us. If you came home from school or work angry and yelled at your dog or a loved one, what would your explanation be? You might say you were very tired or feeling unwell and needed quiet time—a situational explanation. The actor-observer bias is the phenomenon of attributing other people’s behavior to internal factors (fundamental attribution error) while attributing our own behavior to situational forces (Jones & Nisbett, 1971; Nisbett, Caputo, Legant, & Marecek, 1973; Choi & Nisbett, 1998). As actors of behavior, we have more information available to explain our own behavior. However as observers, we have less information available; therefore, we tend to default to a dispositionist perspective.
One study on the actor-observer bias investigated reasons male participants gave for why they liked their girlfriend (Nisbett et al., 1973). When asked why participants liked their own girlfriend, participants focused on internal, dispositional qualities of their girlfriends (for example, her pleasant personality). The participants’ explanations rarely included causes internal to themselves, such as dispositional traits (for example, “I need companionship.”). In contrast, when speculating why a male friend likes his girlfriend, participants were equally likely to give dispositional and external explanations. This supports the idea that actors tend to provide few internal explanations but many situational explanations for their own behavior. In contrast, observers tend to provide more dispositional explanations for a friend’s behavior (See figure \(\PageIndex{4}\) below).
Self-Serving Bias
Following an outcome, self-serving bias are those attributions that enable us to see ourselves in favorable light (for example, making internal attributions for success and external attributions for failures). When you do well at a task, for example acing an exam, it is in your best interest to make a dispositional attribution for your behavior (“I’m smart,”) instead of a situational one (“The exam was easy,”). The tendency of an individual to take credit by making dispositional or internal attributions for positive outcomes but situational or external attributions for negative outcomes is known as the self-serving bias (Miller & Ross, 1975). This bias serves to protect self-esteem. You can imagine that if people always made situational attributions for their behavior, they would never be able to take credit and feel good about their accomplishments.
We can understand self-serving bias by digging more deeply into attribution , a belief about the cause of a result. One model of attribution proposes three main dimensions: locus of control (internal versus external), stability (stable versus unstable), and controllability (controllable versus uncontrollable). In this context, stability refers the extent to which the circumstances that result in a given outcome are changeable. The circumstances are considered stable if they are unlikely to change. Controllability refers to the extent to which the circumstances that are associated with a given outcome can be controlled. Obviously, those things that we have the power to control would be labeled controllable (Weiner, 1979).
Consider the example of how we explain our favorite sports team’s wins. Research shows that we make internal, stable, and controllable attributions for our team’s victory (See figure \(\PageIndex{5}\)) (Grove, Hanrahan, & McInman, 1991). For example, we might tell ourselves that our team is talented (internal), consistently works hard (stable), and uses effective strategies (controllable). In contrast, we are more likely to make external, unstable, and uncontrollable attributions when our favorite team loses. For example, we might tell ourselves that the other team has more experienced players or that the referees were unfair (external), the other team played at home (unstable), and the cold weather affected our team’s performance (uncontrollable).
Just-World Hypothesis
One consequence of westerners’ tendency to provide dispositional explanations for behavior is victim blame (Jost & Major, 2001). When people experience bad fortune, others tend to assume that they somehow are responsible for their own fate. A common ideology, or worldview, in the United States is the just-world hypothesis. The just-world hypothesis is the belief that people get the outcomes they deserve (Lerner & Miller, 1978). In order to maintain the belief that the world is a fair place, people tend to think that good people experience positive outcomes, and bad people experience negative outcomes (Jost, Banaji, & Nosek, 2004; Jost & Major, 2001). The ability to think of the world as a fair place, where people get what they deserve, allows us to feel that the world is predictable and that we have some control over our life outcomes (Jost et al., 2004; Jost & Major, 2001). For example, if you want to experience positive outcomes, you just need to work hard to get ahead in life.
Can you think of a negative consequence of the just-world hypothesis?  What common explanations are given for why people live in poverty?  What types of explanations are these, dispositional or situational? These dispositional explanations are clear examples of the fundamental attribution error. Blaming poor people for their poverty ignores situational factors that impact them, such as high unemployment rates, recession, poor educational opportunities, and the familial cycle of poverty (See figure \(\PageIndex{6}\)). Other research shows that people who hold just-world beliefs have negative attitudes toward people who are unemployed and people living with AIDS (Sutton & Douglas, 2005). In the United States and other countries, victims of sexual assault may find themselves blamed for their abuse. Victim advocacy groups, such as Domestic Violence Ended (DOVE), attend court in support of victims to ensure that blame is directed at the perpetrators of sexual violence, not the victims.
Summary
Social psychology is the subfield of psychology that studies the power of the situation to influence individuals’ thoughts, feelings, and behaviors. Psychologists categorize the causes of human behavior as those due to internal factors, such as personality, or those due to external factors, such as cultural and other social influences. Behavior is better explained, however, by using both approaches. Lay people tend to over-rely on dispositional explanations for behavior and ignore the power of situational influences, a perspective called the fundamental attribution error. People from individualistic cultures are more likely to display this bias versus people from collectivistic cultures. Our explanations for our own and others behaviors can be biased due to not having enough information about others’ motivations for behaviors and by providing explanations that bolster our self-esteem.
Glossary
- actor-observer bias
- phenomenon of explaining other people’s behaviors are due to internal factors and our own behaviors are due to situational forces
- attribution
- explanation for the behavior of other people
- collectivist culture
- culture that focuses on communal relationships with others such as family, friends, and community
- dispositionism
- describes a perspective common to personality psychologists, which asserts that our behavior is determined by internal factors, such as personality traits and temperament
- fundamental attribution error
- tendency to overemphasize internal factors as attributions for behavior and underestimate the power of the situation
- individualistic culture
- culture that focuses on individual achievement and autonomy
- internal factor
- internal attribute of a person, such as personality traits or temperament
- just-world hypothesis
- ideology common in the United States that people get the outcomes they deserve
- self-serving bias
- tendency for individuals to take credit by making dispositional or internal attributions for positive outcomes and situational or external attributions for negative outcomes
- situationism
- describes a perspective that behavior and actions are determined by the immediate environment and surroundings; a view promoted by social psychologists
- social psychology
- field of psychology that examines how people impact or affect each other, with particular focus on the power of the situation
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


16.17: Regulating Gene Expression in Cell Development - Cell Migration in Multicellular Organisms
- Describe how cells can migrate within an organism
Cell migration
Cell migration is a central process in the development and maintenance of multicellular organisms. Processes such as tissue formation during embryonic development, wound healing, and immune responses, all require the orchestrated movement of cells in particular directions to specific locations. Errors during this process have serious consequences, including intellectual disability, vascular disease, tumor formation and metastasis. An understanding of the mechanism by which cells migrate may lead to the development of novel therapeutic strategies for controlling, for example, invasive tumor cells.
Cells often migrate in response to specific external signals, including chemical signals and mechanical signals. Due to a highly viscous environment, cells need to permanently produce forces in order to move. Cells achieve active movement by very different mechanisms. Many less complex prokaryotic organisms (and sperm cells) use flagella or cilia to propel themselves. Eukaryotic cell migration typically is far more complex and can consist of combinations of different migration mechanisms. It generally involves drastic changes in cell shape which are driven by the cytoskeleton, for instance a series of contractions and expansions due to cytoplasmic displacement. Two very distinct migration scenarios are crawling motion (most commonly studied) and blebbing motility.
The migration of cultured cells attached to a surface is commonly studied using microscopy. As cell movement is very slow (only a few µm/minute), time-lapse microscopy videos are recorded of the migrating cells to speed up the movement. Such videos reveal that the leading cell front is very active with a characteristic behavior of successive contractions and expansions. It is generally accepted that the leading front is the main motor that pulls the cell forward.
Cell Migration : Phase images of BSC 1 cells migrating in a scratch assay in the absence of serum over a period of 15 hours.
Common features of cell migration
The processes underlying mammalian cell migration are believed to be consistent with those of (non-spermatozoic) locomotion. Observations in common include cytoplasmic displacement at the leading front and laminar removal of dorsally-accumulated debris toward trailing end. The latter feature is most easily observed when aggregates of a surface molecule are cross-linked with a fluorescent antibody or when small beads become artificially bound to the front of the cell. Other eukaryotic cells are observed to migrate similarly. The amoeba Dictyostelium discoideum is useful to researchers because they consistently exhibit chemotaxis in response to cyclic AMP; they move more quickly than cultured mammalian cells; and they have a haploid genome that simplifies the process of connecting a particular gene product with its effect on cellular behavior.
Key Points
- The disruption or dysfunction of cell migration processes can lead to formation of various diseases such as metastasis, tumor formation and vascular disease.
- In prokaryotic organisms, and some eukaryotic cells such as sperm cells, cell migration occurs via the use of a cilia or flagella to propel forward.
- In eukaryotic organisms, cell migration is a much more complex process and can include, but is not excluded to, changes in the cytoskeleton, motor proteins, blebbing, and cytoplasmic displacement; it involves both external and internal signals that mediate these processes.
Key Terms
- bleb : an irregular bulge in the plasma membrane of a cell
- chemotaxis : the movement of a cell or an organism in response to a chemical stimulant
- laminar : of fluid motion, smooth and regular, flowing as though in different layers
- metastasis : the transference of a bodily function or disease to another part of the body; specifically the development of a secondary area of disease remote from the original site, as with some cancers


16.9: Waves
Learning Objectives
By the end of this section, you will be able to:
- State the characteristics of a wave.
- Calculate the velocity of wave propagation.
What do we mean when we say something is a wave? The most intuitive and easiest wave to imagine is the familiar water wave. More precisely, a wave is a disturbance that propagates, or moves from the place it was created. For water waves, the disturbance is in the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker. For earthquakes, there are several types of disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves. Visualizing water waves is useful because there is more to it than just a mental image. Water waves exhibit characteristics common to all waves, such as amplitude, period, frequency and energy. All wave characteristics can be described by a small set of underlying principles.
A wave is a disturbance that propagates, or moves from the place it was created. The simplest waves repeat themselves for several cycles and are associated with simple harmonic motion. Let us start by considering the simplified water wave in Figure \(\PageIndex{2}\) . The wave is an up and down disturbance of the water surface. It causes a sea gull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The time for one complete up and down motion is the wave’s period \(T\). The wave’s frequency is \(f = 1/T\), as usual. The wave itself moves to the right in the figure. This movement of the wave is actually the disturbance moving to the right, not the water itself (or the bird would move to the right). We define wave velocity \(v_w\) to be the speed at which the disturbance moves. Wave velocity is sometimes also called the propagation velocity or propagation speed , because the disturbance propagates from one location to another.
MISCONCEPTION ALERT
Many people think that water waves push water from one direction to another. In fact, the particles of water tend to stay in one location, save for moving up and down due to the energy in the wave. The energy moves forward through the water, but the water stays in one place. If you feel yourself pushed in an ocean, what you feel is the energy of the wave, not a rush of water.
The water wave in Figure \(\PageIndex{2}\) also has a length associated with it, called its wavelength \(\lambda\) the distance between adjacent identical parts of a wave. (\(\lambda\) is the distance parallel to the direction of propagation.) The speed of propagation \(v_w\) is the distance the wave travels in a given time, which is one wavelength in the time of one period. In equation form, that is
\[v_w = \dfrac{\lambda}{T} \label{eq1}\]
or
\[v_w = f\lambda. \label{eq2}\]
This fundamental relationship holds for all types of waves. For water waves, \(v_w\) is the speed of a surface wave; for sound, \(v_w\) is the speed of sound; and for visible light, \(v_w\) is the speed of light, for example.
TAKE-HOME EXPERIMENT: WAVES IN A BOWL
Fill a large bowl or basin with water and wait for the water to settle so there are no ripples. Gently drop a cork into the middle of the bowl. Estimate the wavelength and period of oscillation of the water wave that propagates away from the cork. Remove the cork from the bowl and wait for the water to settle again. Gently drop the cork at a height that is different from the first drop. Does the wavelength depend upon how high above the water the cork is dropped?
Example \(\PageIndex{1}\): Calculate the Velocity of Wave Propagation: Gull in the Ocean
Calculate the wave velocity of the ocean wave in Figure \(\PageIndex{2}\) if the distance between wave crests is 10.0 m and the time for a sea gull to bob up and down is 5.00 s.
Strategy
We are asked to find \(v_w\). The given information tells us that \(\lambda = 10.0 \, m\) and \(T = 5.00 \, s\). Therefore, we can use Equation \ref{eq1} to find the wave velocity.
Solution
- Enter the known values into Equation \ref{eq1}: \[\begin{align*} v_w &= \frac{\lambda}{T} \\[5pt] &= \dfrac{10.0 \, m}{5.00 \, s}. \end{align*}\]
- Solve for \(v_w\) to find \(v_w = 2.00 \, m/s.\)
Discussion
This slow speed seems reasonable for an ocean wave. Note that the wave moves to the right in the figure at this speed, not the varying speed at which the sea gull moves up and down.
Transverse and Longitudinal Waves
A simple wave consists of a periodic disturbance that propagates from one place to another. The wave in Figure \(\PageIndex{3}\) propagates in the horizontal direction while the surface is disturbed in the vertical direction. Such a wave is called a transverse wave or shear wave; in such a wave, the disturbance is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation. Figure \(\PageIndex{4}\) shows an example of a longitudinal wave. The size of the disturbance is its amplitude X and is completely independent of the speed of propagation \(v_w\).
Waves may be transverse, longitudinal, or a combination of the two . (Water waves are actually a combination of transverse and longitudinal. The simplified water wave illustrated in Figure \(\PageIndex{2}\)shows no longitudinal motion of the bird.) The waves on the strings of musical instruments are transverse—so are electromagnetic waves, such as visible light.
Sound waves in air and water are longitudinal. Their disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and thus the sound waves in them must be longitudinal or compressional. Sound in solids can be both longitudinal and transverse.
Earthquake waves under Earth’s surface also have both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). These components have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water.
Exercise \(\PageIndex{1}\):Check Your Understanding
Why is it important to differentiate between longitudinal and transverse waves?
- Answer
-
In the different types of waves, energy can propagate in a different direction relative to the motion of the wave. This is important to understand how different types of waves affect the materials around them.
 Wiggle the end of the string and make waves, or adjust the frequency and amplitude of an oscillator. Adjust the damping and tension. The end can be fixed, loose, or open.
Summary
- A wave is a disturbance that moves from the point of creation with a wave velocity \(v_w\).
- A wave has a wavelength \(\lambda\) which is the distance between adjacent identical parts of the wave.
- Wave velocity and wavelength are related to the wave’s frequency and period by \(v_w = \frac{\lambda}{T}\) or \(v_w = f\lambda\).
- A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.
Glossary
- longitudinal wave
- a wave in which the disturbance is parallel to the direction of propagation
- transverse wave
- a wave in which the disturbance is perpendicular to the direction of propagation
- wave velocity
- the speed at which the disturbance moves. Also called the propagation velocity or propagation speed
- wavelength
- the distance between adjacent identical parts of a wave


1.8: Thesis Statements
What this handout is about
This handout describes what a thesis statement is, how thesis statements work in your writing, and how you can discover or refine one for your draft.
Introduction
Writing in college often takes the form of persuasion—convincing others that you have an interesting, logical point of view on the subject you are studying.  You persuade your roommate to clean up, your parents to let you borrow the car, your friend to vote for your favorite candidate or policy. In college, course assignments often ask you to make a persuasive case in writing. You are asked to convince your reader of your point of view. This form of persuasion, often called academic argument, follows a predictable pattern in writing. After a brief introduction of your topic, you state your point of view on the topic directly and often in one sentence. This sentence is the thesis statement, and it serves as a summary of the argument you’ll make in the rest of your paper .
What is a thesis statement?
A thesis statement:
- tells the reader how you will interpret the significance of the subject matter under discussion.
- is a road map for the paper; in other words, it tells the reader what to expect from the rest of the paper.
- directly answers the question asked of you. A thesis is an interpretation of a question or subject, not the subject itself. The subject, or topic, of an essay might be World War II or Moby Dick; a thesis must then offer a way to understand the war or the novel.
- makes a claim that others might dispute.
- is usually a single sentence somewhere in your first paragraph that presents your argument to the reader. The rest of the paper, the body of the essay, gathers and organizes evidence that will persuade the reader of the logic of your interpretation.
If your assignment asks you to take a position or develop a claim about a subject, you may need to convey that position or claim in a thesis statement near the beginning of your draft. The assignment may not explicitly state that you need a thesis statement because your instructor may assume you will include one. When in doubt, ask your instructor if the assignment requires a thesis statement. When an assignment asks you to analyze, to interpret, to compare and contrast, to demonstrate cause and effect, or to take a stand on an issue, it is likely that you are being asked to develop a thesis and to support it persuasively. (Check out our handout on understanding assignments for more information.)
How do I get a thesis?
A thesis is the result of a lengthy thinking process. Formulating a thesis is not the first thing you do after reading an essay assignment. Before you develop an argument on any topic, you have to collect and organize evidence, look for possible relationships between known facts (such as surprising contrasts or similarities), and think about the significance of these relationships. Once you do this thinking, you will probably have a “working thesis,” a basic or main idea, an argument that you think you can support with evidence but that may need adjustment along the way.
Writers use all kinds of techniques to stimulate their thinking and to help them clarify relationships or comprehend the broader significance of a topic and arrive at a thesis statement. For more ideas on how to get started, see our handout on brainstorming .
How do I know if my thesis is strong?
If there’s time, run it by your instructor or make an appointment at the Writing Center to get some feedback. Even if you do not have time to get advice elsewhere, you can do some thesis evaluation of your own. When reviewing your first draft and its working thesis, ask yourself the following:
- Do I answer the question? Re-reading the question prompt after constructing a working thesis can help you fix an argument that misses the focus of the question.
- Have I taken a position that others might challenge or oppose? If your thesis simply states facts that no one would, or even could, disagree with, it’s possible that you are simply providing a summary, rather than making an argument.
- Is my thesis statement specific enough?
Thesis statements that are too vague often do not have a strong argument. If your thesis contains words like “good” or “successful,” see if you could be more specific: why is something “good”; what specifically makes something “successful”?
- Does my thesis pass the “So what?” test? If a reader’s first response is, “So what?” then you need to clarify, to forge a relationship, or to connect to a larger issue.
- Does my essay support my thesis specifically and without wandering? If your thesis and the body of your essay do not seem to go together, one of them has to change. It’s o.k. to change your working thesis to reflect things you have figured out in the course of writing your paper. Remember, always reassess and revise your writing as necessary.
- Does my thesis pass the “how and why?” test? If a reader’s first response is “how?” or “why?” your thesis may be too open-ended and lack guidance for the reader. See what you can add to give the reader a better take on your position right from the beginning.
Examples
Suppose you are taking a course on 19th-century America, and the instructor hands out the following essay assignment: Compare and contrast the reasons why the North and South fought the Civil War. You turn on the computer and type out the following:
The North and South fought the Civil War for many reasons, some of which were the same and some different.
This weak thesis restates the question without providing any additional information. You will expand on this new information in the body of the essay, but it is important that the reader know where you are heading. A reader of this weak thesis might think, “What reasons? How are they the same? How are they different?” Ask yourself these same questions and begin to compare Northern and Southern attitudes (perhaps you first think, “The South believed slavery was right, and the North thought slavery was wrong”). Now, push your comparison toward an interpretation—why did one side think slavery was right and the other side think it was wrong? You look again at the evidence, and you decide that you are going to argue that the North believed slavery was immoral while the South believed it upheld the Southern way of life. You write:
While both sides fought the Civil War over the issue of slavery, the North fought for moral reasons while the South fought to preserve its own institutions.
Now you have a working thesis! Included in this working thesis is a reason for the war and some idea of how the two sides disagreed over this reason. As you write the essay, you will probably begin to characterize these differences more precisely, and your working thesis may start to seem too vague. Maybe you decide that both sides fought for moral reasons, and that they just focused on different moral issues. You end up revising the working thesis into a final thesis that really captures the argument in your paper:
While both Northerners and Southerners believed they fought against tyranny and oppression, Northerners focused on the oppression of slaves while Southerners defended their own right to self-government.
Compare this to the original weak thesis. This final thesis presents a way of interpreting evidence that illuminates the significance of the question. Keep in mind that this is one of many possible interpretations of the Civil War—it is not the one and only right answer to the question . There isn’t one right answer; there are only strong and weak thesis statements and strong and weak uses of evidence.
Let’s look at another example. Suppose your literature professor hands out the following assignment in a class on the American novel: Write an analysis of some aspect of Mark Twain’s novel Huckleberry Finn. “This will be easy,” you think. “I loved Huckleberry Finn!” You grab a pad of paper and write:
Mark Twain’s Huckleberry Finn is a great American novel.
Why is this thesis weak? Think about what the reader would expect from the essay that follows: you will most likely provide a general, appreciative summary of Twain’s novel. The question did not ask you to summarize; it asked you to analyze. Your professor is probably not interested in your opinion of the novel; instead, she wants you to think about why it’s such a great novel— what do Huck’s adventures tell us about life, about America, about coming of age, about race relations, etc.? First, the question asks you to pick an aspect of the novel that you think is important to its structure or meaning—for example, the role of storytelling, the contrasting scenes between the shore and the river, or the relationships between adults and children. Now you write:
In Huckleberry Finn, Mark Twain develops a contrast between life on the river and life on the shore.
Here’s a working thesis with potential: you have highlighted an important aspect of the novel for investigation; however, it’s still not clear what your analysis will reveal. Your reader is intrigued, but is still thinking, “So what? What’s the point of this contrast? What does it signify?” Perhaps you are not sure yet, either. That’s fine—begin to work on comparing scenes from the book and see what you discover. Free write, make lists, jot down Huck’s actions and reactions. Eventually you will be able to clarify for yourself, and then for the reader, why this contrast matters. After examining the evidence and considering your own insights, you write:
Through its contrasting river and shore scenes, Twain’s Huckleberry Finn suggests that to find the true expression of American democratic ideals, one must leave “civilized” society and go back to nature.
This final thesis statement presents an interpretation of a literary work based on an analysis of its content. Of course, for the essay itself to be successful, you must now present evidence from the novel that will convince the reader of your interpretation.
Works consulted
We consulted these works while writing the original version of this handout. This is not a comprehensive list of resources on the handout’s topic, and we encourage you to do your own research to find the latest publications on this topic. Please do not use this list as a model for the format of your own reference list, as it may not match the citation style you are using. For guidance on formatting citations, please see the UNC Libraries citation tutorial .
Anson, Chris M. and Robert A. Schwegler. The Longman Handbook for Writers. 2nd ed. New York: Longman, 2000.
Hairston, Maxine and John J. Ruszkiewicz. The Scott, Foresman Handbook for Writers. 4th ed. New York: HarperCollins, 1996.
Lunsford, Andrea and Robert Connors. The St. Martin’s Handbook. 3rd ed. New York: St. Martin’s, 1995.
Rosen, Leonard J. and Laurence Behrens. The Allyn & Bacon Handbook. 3rd ed. Boston: Allyn & Bacon, 1997.


47.1: The Biodiversity Crisis Last updated Save as PDF Page ID 14238 Boundless Boundless 47.1A: Loss of Biodiversity Human activity is the driving force behind the current biodiversity crisis, which is causing great species loss in a short time period. 47.1B: Types of Biodiversity Genetic diversity, ecosystem diversity, and human-derived diversity are measures of biodiversity that currently define life on earth. 47.1C: Biodiversity Change through Geological Time Biodiversity has been affected by five mass extinction periods, which greatly influenced speciation and extinction rates. 47.1D: The Pleistocene Extinction Biodiversity loss, especially the disappearance of megafauna, during the Pleistocene Extinction has been linked to the arrival of humans. 47.1E: Present-Time Extinctions Human activities probably caused the Holocene mass extinctions; many methods have been employed to estimate these extinction rates.


11.3: Hands-on Activity
Overview
Individual students write concise reports on a recent trend in the sector they have been studying.
Learning Objectives
- Identify emerging trends and demonstrate an understanding of emerging technologies.
- Understand the Internet of Things (IoT) and how it expands the cyber “attack surface.”
- Be able to make educated predictions of what the future might look like for the cybersecurity critical infrastructure framework.
- Discuss ethical issues that can arise in relation to new technology and new defense strategies.
Based on your team’s investigation of your chosen sector and created fictitious organization, research recent trends in cybersecurity relevant to your team’s selected sector. Identify at least 5 references on relevant trends.
Assignment Options
- Write a short paper describing your findings on how these trends will impact your sector.
- Prepare 2–3 presentation slides on your findings on how these trends will impact your sector.
Grading Criteria Rubric
- Content
- Evidence of teamwork
- References
- Use of American Psychological Association (APA) style in writing the assignment
Grade Points: 100


1.4: Diffusion and Osmosis
The cell theory states that all living things are composed of cells and that cells only arise from other cells. Some cells are fairly simple in structure, while others are extremely complex. For example, some organisms are unicellular—they exist as a single cell, while multicellular organisms are composed of many cells that form tissues and organs. In either case, all cells share some common properties: the presence of DNA, intracellular proteins that enable the cell to perform its functions, and a plasma membrane. Some cells, known as eukaryotic cells, also contain membrane-bound organelles that allow a more complex level of functioning.
Homeostasis is defined as the maintenance of a stable internal environment. In order to maintain homeostasis, cells continually transport substances in and out across their cell (plasma) membrane.
The cell membrane serves as a “gatekeeper” and is the cellular structure that regulates the transport of materials into and out of the cell. The phospholipid bilayer architecture of the cell membrane allows certain molecules to pass through while keeping others out, therefore the cell membrane is selectively permeable (or semipermeable) . Things that need to enter a cell for it to function properly include ions, nucleotides, sugars, oxygen, amino acids, water, vitamins, and some hormones. Cells also allow certain molecules like water, ions, and secreted proteins to leave. Additionally, cells must eliminate waste products like urea and carbon dioxide
In the following exercises, you will examine the semipermeable nature of the cell membrane. You will also explore the concept of tonicity , which refers to the solute concentration of a solution, and its inherent ability to influence the rate and direction of osmosis.
PART 1: DIFFUSION & OSMOSIS
Diffusion is the movement of molecules from an area in which they are high in concentration to an area in which they are low in concentration. Molecules move down a concentration gradient until they are equally distributed, or equilibrium is reached (Fig. 1). At equilibrium, there is no concentration gradient. Molecules still move once equilibrium is reached, but there is no net movement in any one direction.
Osmosis is a specific type of diffusion : the diffusion of water molecules across a semipermeable membrane. Like other molecules, water molecules diffuse down a concentration gradient, from an area of higher free water concentration to an area of lower free water concentration. This means that water will move across a semipermeable membrane, like the cell membrane, in the direction of the higher solute concentration. (In solution, high solute concentration = low free water concentration; conversely, low solute concentration = high free water concentration.) You will observe this concept in Part 2: Osmosis & Tonicity .
In living organisms, most substances are transported as solutes , dissolved in water, a solvent . For example, if we dissolve salt (\(\ce{NaCl}\)) in a beaker of water, salt (\(\ce{NaCl}\)) is the solute and water is the solvent. Examples of solutes in the human body include glucose, small proteins, and electrolytes like calcium and sodium ions. Waste products, such as \(\ce{CO2}\) and urea are also transported as solutes. Solutes are carried by body fluids, such as blood plasma, and pass into and out of cells through passive and active transport . In either case, the cell membrane will either inhibit or facilitate the process of diffusion: some molecules can easily diffuse across a plasma membrane and some cannot. For example, small, nonpolar molecules (such as \(\ce{CO2}\) and \(\ce{O2}\)) can cross a membrane by simple diffusion. Large molecules or polar molecules, however, cannot easily diffuse across a membrane. Cells must have specialized membrane-bound proteins that function to transport such substances across the membrane.
In Part 1: Diffusion & Osmosis , you will learn about diffusion and osmosis using dialysis membrane, a selectively permeable sheet of cellulose that permits the passage of water and small solutes, but does not allow larger molecules to diffuse across. This is because the membrane has microscopic pores that only allow small molecules through; anything larger than the size of the pores is prevented from crossing. Some of the solutes in this experiment, sucrose (\(\ce{C12H22O11}\)) and starch (\(\ce{(C6H10O5})n}\) are too large to pass through the pores of the dialysis tubing, but the solvent molecules (\(\ce{H2O}\)) and glucose (\(\ce{C12H22O11}\)), are small enough to pass easily.
Exercise 1: Molecular Weight and Diffusion Rate
Molecular weight is an indication of the mass and size of a molecule. The purpose of this experiment is to determine the relationship between molecular weight and the rate of diffusion through a semisolid gel. You will investigate two dyes, methylene blue and potassium permanganate.
| Molecule | Molecular Weight | Color |
|---|---|---|
| Methylene blue | 300 grams/mole | blue |
| Potassium permanganate | 150 grams/mole | purple |
Employing Steps in the Scientific Method:
- Record the Question that is being investigated in this experiment. ________________________________________________________________
- Record a Hypothesis for the question stated above. ________________________________________________________________
- Predict the results of the experiment based on your hypothesis (if/then). ________________________________________________________________
- Perform the experiment below and collect your data.
Materials:
- Petri dish of agar semi-solid gel (Mueller Hinton agar plates, 150 x 15 mm) - make sure the agar has been allowed to come to room temperature
- Methylene blue solution (0.2% in 25% EtOH)
- Potassium permanganate solution (0.1% KMnO4)
- Small straws
- Small plastic metric ruler
Procedure:
1. Obtain a petri dish of agar
2. Take the plastic straw and gently stick it down into one side of the agar. Lift up the straw, withdrawing a small plug of agar. Repeat on the other side of the dish.
3. Using a 1mL transfer pipet, place a single drop of each dye into the appropriate agar well. (Fig. 2).
4. After 20 minutes, place a small, clear metric ruler underneath the Petri dish to measure the distance (diameter) that the dye has moved. Enter the data in Table 1.
5. Repeat step 4 at 40, 60, and 80 minutes.
|
Molecular Weight (g/mole) |
Diameter after 20 min. (mm) |
Diameter after 40 min. (mm) |
Diameter after 60 min. (mm) |
Diameter after 80 min. (mm) |
|
|---|---|---|---|---|---|
|
Methylene blue |
|||||
|
Potassium permanganate |
Questions:
- What is the relationship between molecular weight and the rate of diffusion? Explain. ________________________________________________________________
- Go back and look at your initial hypothesis. Does your data support this hypothesis? Explain. _______________________________________________________________
Extension Activity: (Optional)
The results of this experiment can be presented graphically. The presentation of your data in a graph will assist you in interpreting your results. Based on your results, you can complete the final step of scientific investigation, in which you must be able to propose a logical argument that either allows you to support or reject your initial hypothesis.
- Graph your results using the data from Table 1.
- What is the dependent variable? Which axis is used to graph this data? ______________________________________________________________________
- What is your independent variable? Which axis is used to graph this data? ______________________________________________________________________
Exercise 2: Diffusion Across a Membrane
Employing Steps in the Scientific Method:
- Record the Question that is being investigated in this experiment. ________________________________________________________________
- Record a Hypothesis for the question stated above. ________________________________________________________________
- Predict the results of the experiment based on your hypothesis (if/then). ________________________________________________________________
- Perform the experiment below and collect your data.
Materials:
- Dialysis tubing
- Plastic clips or string
- 5 x 400 mL beakers (or plastic cups)
- Electronic balance
- Weigh boats
- 15% Sucrose solution (MW sucrose = 342 g/mol)
- 30% Sucrose solution (MW sucrose = 342 g/mol)
- 30% Glucose solution (MW glucose = 180 g/mol)
- Graduated cylinders (10 mL and 100 mL)
- Wax pencil or sharpie
- 15% starch solution (MW = variable)
- Iodine solution (MW = 166 g/mol)
- Benedict’s reagent
- Hot plate or heat block
Procedure:
- Cut 5 pieces of dialysis membrane approximately 10 cm long. Soak the pieces in tap water until they are soft and pliable (3-5 minutes). *This step may be done for you; check with your instructor.
- Obtain 5 beakers (plastic cups) and label them #1 - 5. Fill each beaker with 150 mL of a solution as follows:
- Beaker #1 – H2O
- Beaker #2 – H2O
- Beaker #3 – H2O
- Beaker #4 – 30% sucrose solution
- Beaker #5 - H2O and 1mL Iodine solution
- Set beakers aside.
- Remove one piece of dialysis membrane from the soaking water and open it, forming a tube. Close one end of the tube with a plastic clip, a piece of string, or simply tie it with a knot (Fig. 3)
- Fill the bag with 10 mL of H2O. Remove excess air, and close the other end of the bag with a plastic clip, a piece of string, or tie with a knot. Set aside on a paper towel.
- Repeat steps 4 and 5 for the 4 remaining dialysis tubes, filling them with 10 mL of a solution as follows:
• Bag #2 – 15% sucrose
• Bag #3 – 30% sucrose
• Bag #4 – H2O
• Bag #5 - 5 mL 30% glucose solution and 5 mL 15% starch solution
- Rinse off the outside of the bags with water and carefully blot dry.
- Weigh bags #1 - 4 to the nearest 0.5g. Record the weights in Table 1 below, in the column labeled “0 min.”
- Place each bag in the corresponding beaker (Bag #1 in Beaker #1, etc.). Make sure each bag is fully submerged in the solution.
- Set a timer for 5 minutes.
- At the end of 5 minutes, remove bags 1 - 4 from their beakers, blot excess fluid, and record the mass (in grams) in Table 1.
- Return the bags to the appropriate beaker, and wait another 5 minutes.
- Repeat steps 11 - 13 every 5 minutes and record the weights in Table 1.
|
Time (min.) |
0 |
5 |
10 |
15 |
20 |
|
|---|---|---|---|---|---|---|
|
Beaker #1 (water) |
Bag #1 (water) |
|||||
|
Beaker #2 (water) |
Bag #2 (15% sucrose) |
|||||
|
Beaker #3 (water) |
Bag #3 (30% sucrose) |
|||||
|
Beaker #4 (30% sucrose) |
Bag #4 (water) |
- Calculate the total weight change (weight change = final weight – initial weight) for each bag. Record the values in Table 2. Calculate the rate (g/min) of osmosis for each bag by dividing the weight change by the time change. Since all 4 bags were recorded for a total of 20 minutes, the time change for all 4 bags is 20 minutes. Record the rate of osmosis for all 4 bags in Table 2.
|
Weight Change (g) |
Time (min) |
Rate (g/min) |
|
|---|---|---|---|
|
Bag #1 (water) |
|||
|
Bag #2 (15% sucrose) |
|||
|
Bag #3 (30% sucrose) |
|||
|
Bag #4 (water) |
- Make observations about bag #5 and beaker #5 in Table 3.
- Remove several mL of liquid from bag #5 and beaker #5 and add each to separate test tubes.
- Add several drops of Benedict's solution to each of the two test tubes and heat to 100 degrees Celsius in a boiling water bath or heat block for 2 - 5 minutes. Record the test results in Table 3.
|
Appearance of liquid after 20 minutes |
Appearance of liquid after heating |
|
|---|---|---|
|
Bag #5 (5 mL 30% glucose solution and 5 mL 15% starch solution) |
||
|
Beaker #5 (H2O and 1mL Iodine solution) |
Questions
- Did the weight of each bag (#1 - #4) change significantly over 20 minutes? Explain.
- In which bag(s) was there a net movement of water?
- Explain what is meant by “net movement”.
- Which carbohydrate molecules (glucose, sucrose, starch) were not able to move across the membrane? Explain.
- In terms of solvent (water) concentration, water moved from the area of _______________ concentration to the area of __________________ concentration across a selectively permeable membrane, which is defined as ________________________.
- What can you conclude about the movement of Iodine, glucose, and starch across the dialysis membrane based on your results in Table 3? Support your answers for each with the observation from bag #5 and beaker #5.
- Iodine -
- Glucose -
- Starch -
- We used the dialysis tubing to simulate a cell membrane. How is the dialysis tubing functionally the same as a cell membrane?
- We used the dialysis tubing to simulate a cell membrane. How is the dialysis tubing functionally different from a cell membrane?
Extension Activity: (Optional)
The results of this experiment can be presented graphically. The presentation of your data in a graph will assist you in interpreting your results. Based on your results, you can complete the final step of scientific investigation, in which you must be able to propose a logical argument that either allows you to support or reject your initial hypothesis.
- Prepare a line graph using the data from Table 2.
- What is the dependent variable? Which axis is used to graph this data? ______________________________________________________________________
- What is your independent variable? Which axis is used to graph this data? ______________________________________________________________________
PART 2: OSMOSIS & TONICITY
Tonicity is the relative concentration of solute (particles), and therefore also a solvent (water), outside the cell compared with inside the cell.
• An isotonic solution has the same concentration of solute (and therefore of free water) as the cell. When cells are placed in an isotonic solution, there is no net movement of water.
• A hypertonic solution has a higher solute (therefore, lower free water) concentration than the cell. When cells are placed in a hypertonic solution, water moves out of the cell into the lower free water solution.
• A hypotonic solution has a lower solute (therefore, higher free water) concentration than the cell. When cells are placed in a hypotonic solution, water moves into the cell from the higher free water solution.
IMPORTANT NOTE: Notice that all of the above definitions have ‘solution’ as the noun. Sometimes the noun will refer to the cell instead of the solution. For example, a hypotonic cell will experience a net movement of water out of the cell. What this means is that if the ‘solution’ is hypotonic, the cell is hypertonic and vice versa.
Exercise 1: Observing Osmosis in Potato Strips
Employing Steps in the Scientific Method:
- Record the Question that is being investigated in this experiment. ________________________________________________________________
- Record a Hypothesis for the question stated above. ________________________________________________________________
- Predict the results of the experiment based on your hypothesis (if/then). ________________________________________________________________
- Perform the experiment below and collect your data.
Materials:
- Potatoes
- Cork borer
- Petri dish
- Wax pencil or Sharpie
- Metric ruler
- 10% NaCl solution
- 0.9% NaCl solution
- Forceps and scalpel
Procedure:
- Obtain a potato and use a cork borer to prepare 3 cylinders of potato. Push the borer through the length of the potato. When the borer is filled, use the flat end of a wooden skewer to gently push out the potato cylinder into the petri dish. Use the scalpel to cut each potato cylinder into a length of 5 cm.
- With a wax pencil or Sharpie, label 3 test tubes (#1, #2, #3).
- Using the metric ruler, mark each tube at the 10 cm mark level from the bottom of the tube.
-
Set up three test tubes with 10 cm of solutions as follows:
- Tube #1 - distilled water
- Tube #2 - 10% sodium chloride (NaCl)
- Tube #3 - 0.9% NaCl
- Place one potato cylinder into each test tube and allow them to soak for about 15 minutes in the solutions.
- You can now move on to Exercises 2 and 3 while your potatoes soak.
- After the elapsed time, observe each strip for limpness (water loss, flaccid) or stiffness (water gain,turgor).
Questions
- Which tube contained the limp (flaccid) potato strip? Explain.
- Which tube contained the stiff (turgid) potato strip? Explain.
- Which solution is isotonic to the inside of the potato cell?
- What happened to the potato strip in the isotonic solution?
Osmoregulation in Living Cells
Some organisms, known as osmoregulators , have special adaptations to keep tight control over their internal osmotic conditions while still others, known as osmoconformers , are able to live in a variety of osmotic conditions. Most living cells, however, are often at the mercy of their surrounding osmotic environment. Many freshwater plants live in an isotonic or hypotonic environment, so they have no adaptations to protect them from a hypertonic environment. Likewise, mammalian red blood cells live in the isotonic plasma inside your circulatory system so they have no protection from either a hypertonic nor a hypotonic environment.
A plant cell is surrounded by a rigid cell wall, so when the cell is placed in a hypotonic environment, the net flow of water is from the surrounding medium into the cell, and it simply expands to the cell wall and becomes turgid. When the same plant cell is placed in a hypertonic environment, water leaves the central vacuole and the cytoplasm shrinks. This causes the cell membrane to pull away from the cell wall. In this situation, the plant cell will undergo plasmolysis and die. Animal cells have no cell wall so when they are in a hypotonic environment they will expand and fill with water until they burst in a condition known as lysis. Figures 4 and 5 demonstrate these conditions.
Exercise 2: Observing Osmosis in Elodea Cells
Employing Steps in the Scientific Method:
- Record the Question that is being investigated in this experiment. ________________________________________________________________
- Record a Hypothesis for the question stated above. ________________________________________________________________
- Predict the results of the experiment based on your hypothesis (if/then). ________________________________________________________________
- Perform the experiment below and collect your data.
Materials
- Microscope
- Blank slides and coverslips
- DI water
- 10% salt solution
- Forceps
- Elodea leaf
Procedure
- Place a drop of water onto a clean microscope slide.
- Using the forceps, gently tear off a small leaf from the Elodea plant.
- Prepare a wet mount by placing the Elodea leaf into the drop of water on your slide.
- Place a coverslip onto the slide.
- Use the scanning (4x) objective to bring the Elodea cells into focus. You may not be able to observe individual cells at this power.
- Switch to the low power (10x) objective. The Elodea cell walls should be visible. They will look like dark green grid lines. Use the fine focus adjustment to focus the specimen.
- Once you think you have located an Elodea cell, switch to the high power (40x) objective and refocus using the fine focus adjustment.
- Next, add several drops of 10% salt (NaCl) solution to the edge of the coverslip to allow the salt to diffuse under the coverslip. Observe what happens to the cells (this may require you to search around along the edges of the leaf). Look for cells that have been visibly altered.
- Record your observations in the following table. The cells in distilled water should look similar to the figure below.
|
Solution |
Appearance of Elodea Cells |
|
Distilled water (0% NaCl) |
|
|
10% NaCl |
Questions
- Which solution is hypertonic to an Elodea cell? Use your observations to support your answer.
- Would you expect pond water to be isotonic, hypotonic, or hypertonic to Elodea cells? Explain your answer.
- Explain what happens to a plant cell that undergoes plasmolysis.
Exercise 3: Observing Osmosis in Red Blood Cells (Erythrocytes)
Employing Steps in the Scientific Method:
- Record the Question that is being investigated in this experiment. ________________________________________________________________
- Record a Hypothesis for the question stated above. ________________________________________________________________
- Predict the results of the experiment based on your hypothesis (if/then). ________________________________________________________________
- Perform the experiment below and collect your data.
Materials
- Microscope
- Wax pencil or Sharpie
- Blank slides and coverslips
- DI water
- 10% NaCl solution
- 0.9% NaCl solution
- Sheep red blood cells
Procedure
- Obtain 3 test tubes and label them (#1, #2, #3) with a wax pencil or Sharpie.
-
Set up the three test tubes with 5 mL of solutions as follows:
- Tube #1 - Distilled water
- Tube #2 - 10% NaCl
- Tube #3 - 0.9% NaCl
- Using a new transfer pipette, add 2 drops of sheep blood to each tube and swirl gently to mix the contents.
- Hold each test tube up to a sheet of paper with printed text. Attempt to read the print through each tube and record your results in the following table.
|
Test Tube / Solution |
Appearance of Solution |
Can you read print? |
|---|---|---|
|
#1 - Distilled water |
||
|
#2 - 10% NaCl |
||
|
#3 - 0.9% NaCl |
- Label 3 microscope slides (#1, #2, #3) with a wax pencil or Sharpie.
- Prepare wet mounts of each tube by placing a drop of the solution in each tube (#1 - 3) on the appropriate microscope slide (#1 -3). Add a coverslip to each slide.
- View slide #1 through the microscope using the scanning (4x) objective first. Focus the image using the coarse adjustment. Then view the blood cells under low power and then high power. Only use the fine focus adjustment to focus the specimen.
- Observe slide #2 and slide #3 in the same manner.
- Record the appearance of the red blood cells in each solution in the following table.
|
Solution |
Appearance of RBCs |
|---|---|
|
#1 - Distilled water |
|
|
#2 - 10% NaCl |
|
|
#3 - 0.9% NaCl |
Questions
- Which solution allowed you to read print through the solution? Explain.
- Which solution is hypertonic to the RBCs? Use your observations to support your answer.
- Which solution is hypotonic to the RBCs? Use your observations to support your answer.
- Which solution is isotonic to the RBCs? Use your observations to support your answer.
Questions for Review
- Define diffusion. What is the energy source for diffusion? Is diffusion considered an active or passive process? Explain.
- Name a molecule that diffused through the artificial membrane (dialysis tubing) that we used in the laboratory. Can diffusion occur without a membrane? Give an example to support your answer.
- What is osmosis? Is it an active or a passive process? Explain.
-
Fill in the blank in the following statements.
- A solution that has a lower solute concentration than another solution is said to be ______________________ when compared with the second solution.
- A solution that has the same solute concentration as another solution is said to be ______________________ when compared with the second solution.
- A solution that has a higher solute concentration than another solution is said to be ______________________ when compared with the second solution.
- What does it mean when a membrane is selectively permeable?
Practical Challenge
-
Briefly explain what happens to a red blood cell when placed into the following solutions.
- Isotonic solution –
- Hypertonic solution –
- Hypotonic solution -
- The concentration of glucose inside Elodea cells is 5 mM. What is the solution in moles (M)? What will happen to an Elodea cell if it is placed in a 1 M glucose solution? Explain.
- Apply what you learned in the lab to explain why it is said that marine organisms, which live in saline environments, literally live in a desert environment.


4.2: Overcounting
In this section, we will explore a common fallacy that occurs while counting. Allow us to consider (and critique!) the following solution:
Poker dice is played by simultaneously rolling 5 fair six-sided dice. Find the probability that we obtain one pair.
Note: By one pair , we mean that the dice have denominations \(a, a, b, c, d\) where \(a, b, c\) and \(d\) are all distinct. For instance, the following are examples of one pair: \( (2, 1, 2, 3, 5 ) \) and \( ( 6, 2, 1, 5, 5 ) \). Meanwhile, the outcomes \( (2, 1, 2, 3, 2 ) \) and \( (2, 2, 1, 1, 1) \) do not meet the conditions of being one pair.
Solution
As usual, we apply our counting framework. Let \( A = \{ \text{we obtain one pair} \} \) and let \(S\) be the set of all outcomes where an outcome is an ordered 5-tuple. That is, an outcome looks like \( ( \_\_, \_\_, \_\_, \_\_, \_\_ ) \). Notice there is no reason to believe that a certain ordered 5-tuple is more likely that another ordered 5-tuple and so \(S\) is indeed a simple sample space. Clearly, \( |S| = 6^5 \). All that remains is to compute \( |E| \). To do this, we proceed in steps:
- We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so.
- We now must select a value of \(b\) that is different from \(a\). There are \( \binom{5}{1} \) ways to do so.
- We now must select a value of \(c\) that is different from \(a\) and \(b\). There are \( \binom{4}{1} \) ways to do so.
- We now must select a value of \(d\) that is different from \(a, b\) and \(c\). There are \( \binom{3}{1} \) ways to do so.
- Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1~ 1!} \) ways to do so.
Hence \( |A| = \binom{6}{1} \times \binom{5}{1} \times \binom{4}{1} \times \binom{3}{1} \times \frac{5!}{2! ~ 1! ~ 1!~ 1!} \).
Putting everything together, we obtain
\[ P(A) = \frac{|A|}{|S|} = \frac{ \binom{6}{1} \times \binom{5}{1} \times \binom{4}{1} \times \binom{3}{1} \times \frac{5!}{2! ~ 1! ~ 1!~ 1!} }{6^5} \nonumber \]
Does the above solution seem correct? If not, then can you explain what is wrong?
- Answer
-
It turns out the answer above is incorrect! How do we know this? Observe that \[ P(A) = \frac{|A|}{|S|} = \frac{ \binom{6}{1} \times \binom{5}{1} \times \binom{4}{1} \times \binom{3}{1} \times \frac{5!}{2! ~ 1! ~ 1!~ 1!} }{6^5} \approx 2.78 \nonumber \] and so clearly something is wrong since for any event \(A\) of sample space \(S\), we proved that \( P(A) \leq 1 \).
We will now explore what is wrong in our solution and why.
An overcount is when any distinct outcome has been incorrectly counted as multiple distinct outcomes.
The above example and definition raises two questions:
- How can we recognize if an overcount has occurred?
- If an overcount has occurred, then is there anything we can do to fix our answer?
We first answer question 1 by providing a recipe to check for overcounting.
- After writing down the steps for computing the cardinality of some event, go through your steps and fix an actual outcome which we will denote by \(s_1\).
- Now go through the steps again and change your choice on at least one step. Call the resulting outcome \(s_2\).
- If for some change in choice, \(s_1 = s_2 \), then you have overcounted. If for all possible changes in your selection for \(s_2\), \(s_1 \neq s_2 \), then we have not overcounted.
Allow us to apply the recipe to our poker dice question discussed above. We do so by creating a chart:
| Steps | \(s_1\) | \(s_2\) |
|---|---|---|
| 1) We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so. | ||
| 2) We now must select a value of \(b\) that is different from \(a\). There are \( \binom{5}{1} \) ways to do so. | ||
| 3) We now must select a value of \(c\) that is different from \(a\) and \(b\). There are \( \binom{4}{1} \) ways to do so. | ||
| 4) We now must select a value of \(d\) that is different from \(a, b\) and \(c\). There are \( \binom{3}{1} \) ways to do so. | ||
| 5) Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1!~ 1!} \) ways to do so. |
We will now go through our steps and fix an outcome for \(s_1\). Let us suppose we come up with the following choices:
| Steps | \(s_1\) | \(s_2\) |
|---|---|---|
| 1) We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so. | 4, 4 | |
| 2) We now must select a value of \(b\) that is different from \(a\). There are \( \binom{5}{1} \) ways to do so. | 3 | |
| 3) We now must select a value of \(c\) that is different from \(a\) and \(b\). There are \( \binom{4}{1} \) ways to do so. | 1 | |
| 4) We now must select a value of \(d\) that is different from \(a, b\) and \(c\). There are \( \binom{3}{1} \) ways to do so. | 2 | |
| 5) Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1!~ 1!} \) ways to do so. | (4, 2, 1, 4, 3) |
Here, \(s_1 = (4, 2, 1, 4, 3) \). To complete our recipe, we check if there is a way to change our choice on at least one step that will result in generating \(s_1 \) again. If there is a way to do so, then we have overcounted. Let us change our choice on steps 2, 3, and 4.
| Steps | \(s_1\) | \(s_2\) |
|---|---|---|
| 1) We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so. | 4, 4 | 4,4 |
| 2) We now must select a value of \(b\) that is different from \(a\). There are \( \binom{5}{1} \) ways to do so. | 3 | 1 |
| 3) We now must select a value of \(c\) that is different from \(a\) and \(b\). There are \( \binom{4}{1} \) ways to do so. | 1 | 2 |
| 4) We now must select a value of \(d\) that is different from \(a, b\) and \(c\). There are \( \binom{3}{1} \) ways to do so. | 2 | 3 |
| 5) Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1!~ 1!} \) ways to do so. | (4, 2, 1, 4, 3) | (4, 2, 1, 4, 3) |
Here \(s_2 = (4, 2, 1, 4, 3) \). Since \(s_1 = s_2 \), then we have overcounted. This leads to our second question. How do we fix an overcount?
To fix an overcount, you may simply divide by the number of ways you have overcounted.
To use Method 1, we have to understand how did we overcount? Looking back at our steps, we see that the order in which we select \(b, c\) and \(d\) does not matter since their arrangement can yield the same outcome. How many ways can we arrange\(b, c\) and \(d\)? There are 3! ways to do so. Hence, all we have to do is divide \( |A| \) by 3! to obtain the correct answer.
\[ P(A) = \frac{|A|}{|S|} = \frac{ \Bigg[ \binom{6}{1} \binom{5}{1} \binom{4}{1} \binom{3}{1} \frac{5!}{2! ~ 1! ~ 1~ 1!} \Bigg] / 3! }{6^5} \approx 0.4630 \nonumber \]
To fix an overcount, redo the counting process and condense the steps so that the overcount no longer occurs in the first place.
Consider the follow revision in our counting process:
| Old Steps | New Steps |
|---|---|
| 1) We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so. | 1) We first choose a value of \(a\). That is, we must select the denomination to be repeated. There are \( \binom{6}{1} \) ways to do so. |
| 2) We now must select a value of \(b\) that is different from \(a\). There are \( \binom{5}{1} \) ways to do so. | 2) We now select the values of \(b, c\) and \(d\). There are \( \binom{5}{3} \) ways to do so. |
| 3) We now must select a value of \(c\) that is different from \(a\) and \(b\). There are \( \binom{4}{1} \) ways to do so. | |
| 4) We now must select a value of \(d\) that is different from \(a, b\) and \(c\). There are \( \binom{3}{1} \) ways to do so. | |
| 5) Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1!~ 1!} \) ways to do so. | 3) Finally, we must impose an ordering on \(a, a, b, c, d \). There are \( \frac{5!}{2! ~ 1! ~ 1!~ 1!} \) ways to do so. |
\[ P(A) = \frac{|A|}{|S|} = \frac{ \binom{6}{1} \binom{5}{3} \frac{5!}{2! ~ 1! ~ 1!~ 1!} }{6^5} \approx 0.4630 \nonumber \]
Notice how steps 2, 3 and 4 under the old method have now been collapsed into one step in the new method.


11.23: A Few Thoughts about Mortgages
There are a few key points regarding mortgages, which require summary and notice.
- Interest payments are tax deductible. Tax deductibility is important simply because it reduces the after-tax cost of the mortgage. Funds that would otherwise have gone to pay tax instead go to debt service. This benefit is reduced as the interest portion of the mortgage payment is reduced with time.
- Partway through the loan, half the loan will have been paid off. A mortgage has a kind of “half-life.” In the foregoing example, half the loan will have been paid off after six years. This half-life will be greater than half the length of the mortgage because initially the annuity payments are used mainly to pay interest on the loan!
- Total interest paid may be much greater than and in longer-term cases, a multiple of the principal – this can be calculated by evaluating the total payments minus the principal.
In the foregoing example, over the ten years the borrower will have paid a total of 10 annual payments of $15,576.32 for a total of $155,763.20. If you subtract from this figure the loan principal of $100,000, you are left with a figure of $55,763.20, which represents the total amount of interest paid over the life of the mortgage (unadjusted for time value). In other words, interest paid represents an additional 55% approximately of the principal borrowed. (We take note again that most mortgages require monthly payments and that, in today’s marketplace, most mortgages are 15 t o 30 years. )
Let us compare 15- and 30-year mortgages in terms of the ratio of interest payments made relative to t he principal. We remind ourselves that in the above instance (10 years and 9%) we had 55% interest payments relative to the loan principal. Let us use $100,000 of principal again, and, this time, 6% in interest. We shall again employ the formula:
Principal ÷ PV Annuity Factor = Periodic Payment
15 years : ($100,000) ÷ (9.7122) = $10,296.33
Over 15 years total payments will equal (15) ($10,296.33) = $154,444.95. In this case, interest payments will equal 54.44% of the principal.
30 years : ($100,000) ÷ (13.7648) = $7,264.91
Over 30 years, total payments will equal (30) ($7,264.91) = $217,947.30. In this case, interest payments will equal 117.95% of the principal.
Even though the annual payments are less in the 30-year case, we see that interest payments multiply enormously over time.


4.2E: Modeling with Linear Functions (Exercises)
24. Find the area of a triangle bounded by the \(y\) axis, the line \(f(x)=10-2 x,\) and the line perpendicular to \(f\) that passes through the origin.
25. A town's population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.
26. The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004,875 people were inflicted. Find the linear function that models the number of people afflicted with the common cold \(C\) as a function of the year, \(t\). When will no one be afflicted?
For the following exercises, use the graph in Figure 1 showing the profit, \(y\), in thousands of dollars, of a company in a given year, \(x\), where \(x\) represents years since \(1980 .\)
Figure 1
27. Find the linear function \(y,\) where \(y\) depends on \(x,\) the number of years since \(1980 .\)
28. Find and interpret the \(y\)-intercept.
For the following exercise, consider this scenario: In \(2004,\) a school population was \(1,700 .\) By 2012 the population had grown to 2,500
-
Assume the population is changing linearly.
- How much did the population grow between the year 2004 and \(2012 ?\)
- What is the average population growth per year?
- Find an equation for the population, \(P\), of the school \(t\) years after 2004 .
For the following exercises, consider this scenario: In \(2000,\) the moose population in a park was measured to be \(6,500 .\) By \(2010,\) the population was measured to be \(12,500 .\) Assume the population continues to change linearly.
30. Find a formula for the moose population, \(P\).
31. What does your model predict the moose population to be in \(2020 ?\)
For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in Table \(1 .\) Assume that the house values are changing linearly.
| Year | Pima Central | East Valley |
|---|---|---|
| 1970 | 32,000 | 120,250 |
| 2010 | 85,000 | 150,000 |
32. In which subdivision have home values increased at a higher rate?
33. If these trends were to continue, what would be the median home value in Pima Central in \(2015 ?\)