Getting Along with Others
Interdependence
Humans are social creatures—it’s simply in our nature. We continually interact with other students and instructors, and we can learn a great deal from these interactions that heighten the learning process. This frequent interaction with others forms a state of interdependence. College students depend on their instructors, but just as importantly, they depend on other students in many ways.
As important as our interactions with others are, we do not automatically possess the skills that help us form good relationships and make the most of our experiences. Consider how these two college students are different:
John often arrives just as class is beginning and leaves immediately afterward. He makes little effort to talk with other students in the classroom, and after class he goes off to study alone or to his part-time job, where he spends most of his time at a computer screen. He is diligent in his studies and generally does well. After two months, he has not gotten to know his roommate very well, and he generally eats alone with a book in hand. He stops by to see his instructors in their offices only if he missed a class due to illness, and on weekends and holidays he often hangs out at his parents’ house or sees old friends.
Kim likes to get to class early and sits near others so they can talk about the reading for class or compare notes about assignments. She enjoys running into other students she knows from her classes and usually stops to chat. Although she is an older working student who lives alone off campus, she often dines in a campus café and asks students she meets in her classes to join her. After two months, with the approach of midterms, she formed a study group with a couple other students. If she feels she doesn’t understand an important lecture topic very well, she gets to her instructor’s office a few minutes ahead of office hours to avoid missing out by having to wait in line. A few weeks into the term, she spent a weekend with a student from another country and learned much about a culture about which she had previously known little.
These students are very different. Which do you think is more fully enjoying the college experience? Which do you think is more likely to do well academically? Most of us fall somewhere between these two extremes, but we can learn to be more like Kim and more actively engage with others.
Recognize the Value of Social Interaction
Building good relationships is important for happiness and a successful college experience. College offers the opportunity to meet many people you would likely not meet otherwise in life. Make the most of this opportunity to gain a number of benefits:
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A growing understanding of diverse other people, how they think, and what they feel that will serve you well throughout your life and in your future career
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A heightened sense of your own identity, especially as you interact with others with different personalities and from different backgrounds
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Emotional comfort from friendship with someone who understands you and with whom you can talk about your problems, joys, hopes, and fears
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An opportunity to grow with wider intellectual and emotional horizons
College often offers an opportunity to be stimulated and excited by new relationships and interactions with people who will challenge your thinking and help you become your best. Still, it can be difficult to get started with new relationships in college.
Making New Friends
Some people just make friends naturally, but many first-year college students are more shy or quiet and may need to actively seek new friends. Here are some starting points:
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Keep all doors open for meeting new people.
If you live in a dorm, literally keep the door open. Try to sit with different people at meals so you can get to know them better. Study in a common area or lounge where you’ll be among others.
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Be open in your interests.
Don’t limit yourself to people who share only certain interests. Meeting people by studying together is an excellent way to get to know people with different interests.
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Don’t try to get involved in everything going on around you.
Committing to too many activities or joining too many social groups will spread your time too thin, and you may not spend enough time with anyone to get to know them.

Let people get to know the things you’re interested in, your real passions. People who really know you are more likely to become good friends.
-
Make an effort to get to know others, too.
Show some interest. Don’t talk just about
your
interests—ask them about theirs. Show others that you’re interested, that you think they’re worth spending time with, and that you really do want to get to know them.
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Once a friendship has started, be a good friend.
Respect your friends for what they are and don’t criticize them or talk about them behind their back. Give emotional support when your friends need it and accept their support as well when you need it.
Are You Shy?
If you’re shy, try meeting and talking to people in situations where you can interact one-to-one, such as talking with another student after class. Start with what you have in common—“How’d you do on the test?”—and let the conversation grow from there. Avoid the emotional trap of thinking everyone but you is making new friends and start some conversations with others who look interesting to you. You’ll soon find other “shy” or quiet people eager to interact with you as well and get to know you.
Shy people may be more likely to feel lonely at times, especially while still feeling new at college. Loneliness is usually a temporary emotional state, however. For tips for how to overcome feelings of loneliness, seek out a counselor or ally on your campus to help you.
Communication Skills
Communication is at the core of almost all social interactions, including those involved in friendships and relationships with your instructors. Communication with others has a huge effect on our lives, what we think and feel, and what and how we learn. Communication is, many would say, what makes us human.
Oral communication involves not only speech and listening, of course, but also nonverbal communication: facial expressions, tone of voice, and many other body language signals that affect the messages sent and received. Many experts think that people pay more attention, often unconsciously, to
how
people say something than to
what
they are saying. When the nonverbal message is inconsistent with the verbal (spoken) message, just as when the verbal message itself is unclear because of poorly chosen words or vague explanations, then miscommunication may occur.
Miscommunication is at the root of many misunderstandings among people and makes it difficult to build relationships.
Figure 1.12
Miscommunication is at the root of many misunderstandings.
The next chapter discusses oral communication skills in general and guidelines for communicating well with your instructors. The same communication skills are important for building and maintaining significant relationships.
Remember that communication is a two-way process. Listening skills are critical for most college students simply because many of us may not have learned how to really listen to another person. Here are some guidelines for how to listen effectively:
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Talk less to listen more.
Most people naturally like to share their thoughts and feelings, and some people almost seem unable to stop talking long enough to ever listen to another person. Try this: next time you’re in a conversation with another student, deliberately try not to speak very much but give the other person a chance to speak fully. You may notice a big difference in how much you gain from the conversation.
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Ask questions.
To keep the conversational ball rolling, show your interest in the other person by asking them about things they are saying. This helps the other person feel that you are interested in them and helps build the relationship.
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Watch and respond to the other person’s body language.
You’ll learn much more about their feelings for what they’re saying than if you listen only to their words.
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Show the other person that you’re really listening and that you care.
Make eye contact and respond appropriately with nods and brief comments like “That’s interesting!” or “I know what you mean” or “Really?” Be friendly, smile when appropriate, and encourage the person to keep speaking.
-
Give the other
person
feedback
.
Show you understand by saying things like “So you’re saying that…” or asking a question that demonstrates you’ve been following what they’re saying and want to know more.
As you learn to improve your listening skills, think also about what you are saying yourself and how. Here are additional guidelines for effective speaking:
-
Be honest, but don’t be critical
. Strongly disagreeing may only put the other person on the defensive—an emotion sure to disrupt the hope for good communication. You can disagree, but be respectful to keep the conversation from becoming emotional. Say “I don’t know, I think that maybe it’s…” instead of “That’s crazy! What’s
really
going on is.…”
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Look for common ground.
Make sure that your side of a conversation relates to what the other person is saying and that it focuses on what you have in common. There’s almost no better way to stop a conversation dead in its tracks than to ignore everything the other person has just said and launch into an unrelated story or idea of your own.
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Avoid sarcasm and irony unless you know the person well.
Sarcasm is easily misunderstood and may be interpreted as an attack on the other person’s ideas or statements.
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Don’t try to talk like the other person,
especially if the person is from a different ethnic or cultural background or speaks with an accent or heavy slang. The other person will feel that you are imitating them and maybe even making fun of them. Be yourself and speak naturally.
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While not imitating the other person, relate to his or her personality and style of thinking.
We do not speak to our parents or instructors the exact same way we speak to our closest friends, nor should we speak to someone we’ve just met the same way. Show your respect for the other person by keeping the conversation on an appropriate level.
-
Remember
that
assertive communication
is
better than passive or aggressive communication.
“Assertive” in this context means you are honest and direct in stating your ideas and thoughts; you are confident and clear and willing to discuss your ideas while still respecting the thoughts and ideas of others. A passive communicator is reluctant to speak up, seems to agree with everything others say, hesitates to say anything that others might disagree with, and therefore seldom communicates much at all. Passive communication simply is not a real exchange in communication. Aggressive communication, at the other extreme, is often highly critical of the thoughts and ideas of others. This communication style may be sarcastic, emotional, and even insulting. Real communication is not occurring because others are not prompted to respond honestly and openly.
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Choose your conversations wisely.
Recognize that you don’t have to engage in all conversations. Make it your goal to form relationships and engage in interactions that help you learn and grow as a person. College life offers plenty of opportunities for making relationships and interacting with others if you keep open to them, so you needn’t try to participate in every social situation around you.
Some students may have difficulty in the opposite direction: their social lives may become so rich or so time consuming that they have problems balancing their social lives with their schoolwork. Online social media, for example, may eat up a lot of time.
Online Social Networking
Most college students know all about Facebook, Twitter, blogging, chat rooms, and other social networking sites. Current studies reveal that over 90 percent of all college students use Facebook or MySpace regularly, although older students use these sites less commonly. The media have often emphasized negative stories involving safety concerns, obsessive behavior, a perceived superficiality of social interaction online, and so on. But more recently, online social networking has been found to have several benefits. Many of those who once criticized Facebook and MySpace are now regularly networking among themselves via LinkedIn, Plaxo, and other “professional” networking sites.
Following are some of the benefits of Facebook—some clear to those using it, others revealed only recently by research in the social sciences:
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Facebook and other forms of online networking makes it easy to stay in touch with friends and family at a geographical distance. College students who have moved away from former friends seem to make the transition more easily when they stay in touch. Maintaining past relationships does not prevent most people from making new friends at college.
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Facebook provides users with increased “social capital,” which is a sum of resources gained through one’s relationships with people. Facebook users gain information, opportunities for participation in activities and groups, greater knowledge about others, some interaction skills, and so forth. Social capital is also associated with self-esteem, success in some endeavors, and general happiness.
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Facebook makes it easier for people who are shy or otherwise slow to initiate or respond to interactions with others to participate socially in a group. Online network sites also offer an outlet for self-expression and sharing.
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For many college students, interactions on Facebook strengthen personal relationships rather than detracting from them.
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Acknowledging that online social networking is a reality for most college students, many college administrators and instructors also use it to stay in better touch with students, to provide information and encouragement, and to help students experience the full richness of the college experience. Your college may have a Facebook page where you can learn much about things happening around campus, and you may even receive tweets about important announcements.
Still, online social networking is not 100 percent beneficial for all college students. Someone who becomes obsessed with constantly updating their profile or attracting a huge number of friends can spend so much time at their computer that they miss out on other important aspects of college life. Hopefully by now everyone knows why you should never post compromising or inappropriate photos or information about yourself anywhere online, even as a joke: many employers, college admissions offices, and others may find this compromising material in the future and deny you the job, internship, graduate program, or other position that you want. It’s important also to protect your identity and privacy on online sites.
Overall, online networking in moderation can help enrich one’s life. When used to build relationships, gain information, and stay in touch with a larger community, it can contribute to success in college. Most college students use Facebook ten to thirty minutes a day. If you’re spending more than that, you might ask yourself if you’re missing out on something else.
Balancing Schoolwork and Social Life
If there’s one thing true of virtually all college students, it’s that you don’t have enough time to do everything you want. Once you’ve developed friendships within the college community and have an active social life, you may feel you don’t have enough time for your studies and other activities such as work. For many students, the numerous social opportunities of college become a distraction, and with less attention to one’s studies, academic performance can drop. Here are some tips for balancing your social life with your studies:
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Keep working on your time management skills,
as you will learn in future chapters, you can’t just “go with the flow” and hope that, after spending time with friends, you have enough time (and energy) left over for studying. Make a study schedule that provides enough time for what you need to do. Study first; socialize after.
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Keep working on your study skills,
apply what you will learn in future chapters covering reading, note taking, studying and test taking, when you have only a limited amount of time for studying, be sure you’re using that time as effectively as possible as you read assignments and prepare for class, organize your notes after class, and prepare for tests.
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If you can’t resist temptations, reduce them.
If you are easily distracted by the opportunity to talk with your roommate, spouse, or family members because you study where you live, then go to the library to study.
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Make studying a social experience.
 You will learn more than you would alone by gaining from the thoughts of others, and you can enjoy interacting with others without falling behind.
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Keep your social life from affecting your studying.
Simply scheduling study time doesn’t mean you’ll use it well. If you stayed up late the night before, you may not today be able to concentrate well as you study for that big test. This is another reason for good time management and scheduling your time well, looking ahead.
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Get help if you need it.
If you’re still having difficulty balancing your study time with other activities, talk with your academic advisor or a counselor. Maybe something else is keeping you from doing your best. Maybe you need some additional study skills or you need to get some extra help from a tutor or campus study center. Remember, your college wants you to succeed and will try to help those who seek help.
Campus Groups
The college social experience also includes organized campus groups and activities. Participating in organized activities requires taking some initiative—you can’t be passive and expect these opportunities to come knocking on your door—but is well worthwhile for fully enriching college interactions. The active pursuit of a stimulating life on campus offers many benefits:
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Organized groups and activities speed your transition into your new life.
New students can be overwhelmed by their studies and every aspect of a new life, and they may be slow to build a new life. Rather than waiting for it to come along on its own, you can immediately begin broadening your social contacts and experiences by joining groups that share your interests.
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Organized groups and activities help you experience a much greater variety of social life than you might otherwise.
New students often tend to interact more with other students their own age and with similar backgrounds—this is just natural. But if you simply go with the flow and don’t actively reach out, you are much less likely to meet and interact with others from the broader campus diversity: students who are older and may have a perspective you may otherwise miss, upper-level students who have much to share from their years on campus, and students of diverse heritage or culture with whom you might otherwise be slow to interact.
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Organized groups and activities help you gain new skills, whether technical, physical, intellectual, or social.
Such skills may find their way into your résumé when you next seek a job or your application for a scholarship or other future educational opportunity. Employers and others like to see well-rounded students with a range of proficiencies and experiences.
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Organized groups and activities are fun and a great way to stay healthy and relieve stress.
Exercise and physical activity are essential for health and well-being, and many organized activities offer a good way to keep moving.
Participating in Groups and Activities
College campuses offer a wide range of clubs, organizations, and other activities open to all students. College administrators view this as a significant benefit and work to promote student involvement in such groups. When you made your decision to attend your college, you likely received printed materials or studied the college’s Web site and saw many opportunities. But you may have been so busy attending to academic matters that you haven’t thought of these groups since. It’s a good time now to check out the possibilities:
Figure 1.14
Check bulletin boards on campus to learn about cultural events.
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Browse the college Web site, where you’re likely to find links to pages for student clubs and organizations.
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Watch for club fairs, open houses, and similar activities on campus. Especially near the beginning of the year, an activity fair may include tables set up by many groups to provide students with information. Talk with the representatives from any group in which you may be interested.
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Look for notices on bulletin boards around campus. Student groups really do want new students to join, so they usually try to post information where you can find it.
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Stop by the appropriate college office, such as the student affairs or student activities office or cultural center.
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If you are looking for a group with very specialized interests, check with the academic offices of departments where many students with that interest may be majoring.
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Consider a wide variety of types of organizations. Some are primarily social; some are political or activist; some are based on hobbies (photography, chess, equestrianism, bird watching, videogaming, computer programming); some involve the arts (instrumental music, choral singing, painting, poetry writing, drama club); some are forms of physical recreation (rock-climbing, ballroom dancing, archery, yoga, table tennis, tai chi, team sports); some focus on volunteerism (tutoring other students, community service projects, food drives); and others are related to academic or intellectual pursuits (nursing club, math club, chess club, engineering club, debate club, student literary magazine).
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Consider other forms of involvement and roles beyond clubs. Gain leadership experience by running for office in student government or applying for a residence hall support position. If you are looking for a job, consider what kinds of people you’ll have the opportunity to interact with. Seek out the job placement specialist on your campus for assistance in landing a great on or off-campus job. These may be associated with your financial aid department on campus.
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If your campus doesn’t have a group focused on a particular activity you enjoy yourself, think about starting a new club. Your college will help you get started; talk with the student activities or affairs office.
Whatever your interests, don’t be shy about checking out a club or organization. Take chances and explore. Attending a meeting or gathering is not a commitment—you’re just going the first time to see what it’s like, and you have no obligation to join. Keep an open mind as you meet and observe other students in the group, especially if you don’t feel at first like you fit in: remember that part of the benefit of the experience is to meet others who are not necessarily just like everyone you already know.
A Note on Greek Life
Fraternities and sororities appeal to many students on many campuses. You meet a lot of people quickly and have a social life provided for you almost automatically, with many events and parties as well as usually an active house life. Many people have formed lasting, even lifelong relationships with their fraternity and sorority friends. On the other hand, this living and social experience may limit the kinds of people you meet and present fewer opportunities to interact with others outside the Greek system. If there are frequent activities, it may be important to learn to say no at time when studying becomes a priority. If you are interested in but not yet committed to this life, it’s worthwhile to find out what the houses at your school are really like, consider what your life would likely be like in a fraternity or sorority, and think about how it may impact your college goals.
Bridging the Generation Gap
Is there still a “generation gap” in our society? Maybe not in the same sense as when that phrase came into being in the 1960s, but it remains generally true that most people naturally gravitate toward others of similar age. Even in the open, accepting environment of most colleges, many students interact primarily with others of similar age—which, sadly, misses a great opportunity for both older and younger students to learn from each other.
Younger, “traditional” students just out of high school usually live in residence halls and immediately meet other students of the same age. New students who are just a few years older, who usually have spent some time in the workforce before returning to their education, are more likely to live in a house or apartment and probably spend less time on campus interacting with other students. Some students may be decades older than both traditional and most untraditional students, returning to college sometimes with the desire to change careers or simply to take classes of special interest; their lives may be so well settled in other respects that they have little interest at all in the social world of college. Students in all of these groups may be slow to initiate interactions with each other.
This is one of the great benefits of organized campus groups and activities, however. Regardless of your age or background, you can attend a meeting of those with similar interests and have the opportunity to meet people you simply would not have crossed paths with otherwise. Age barriers rapidly break down when people share the same interests.
Overcoming Difficulties and Resolving Conflicts
Conflicts among people who are interacting are natural. People have many differences in opinions, ideas, emotions, and behaviors, and these differences sometimes cause conflicts. Here are just a few examples of conflicts that may occur among college students:
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Your roommate is playing loud music in your room, and you need some quiet to study for a test.
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You want to have a nice dinner out, but your spouse wants to save the money to buy new furniture.
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Your instructor gave you a C on a paper because it lacks some of the required elements, but you feel it deserves a better grade because you think it accomplished more important goals.
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Others at your Greek house want to invite only members of other fraternities and sororities to an upcoming party, but you want the party to be more inclusive and to invite more diverse students.

So how can such conflicts be resolved? Two things are necessary for conflict resolution that does not leave one or more of the people involved feeling negative about the outcome: attitude and communication.
A conflict cannot be resolved satisfactorily unless all people involved have the right attitude:
-
Respect the options and behaviors of others.
Accept that people are not all alike and learn to celebrate your differences. Most situations do not involve a single right or wrong answer.
-
Be open minded.
Just because at first you are sure that that you are right, do not close the door to other possibilities. Look at the other’s point of view. Be open to change—even when that means accepting constructive criticism.
-
Calm down.
You can’t work together to resolve a conflict while you’re still feeling strong emotions. Agree with the other to wait until you’re both able to discuss it without strong emotions.
-
Recognize the value of compromise.
Even if you disagree after calmly talking over an issue, accept that as a human reality and understand that a compromise may be necessary in order to get along with others.
With the right attitude, you can then work together to resolve the issue. This process depends on good communication:
-
Listen.
Don’t simply argue for your position, but listen carefully to what the other says. Pay attention to their body language as you try to understand their point of view and ask questions to ensure that you do. Paraphrase what you think you hear to give the other a chance to correct any misunderstanding.
-
Use “I statements” rather than “you statements.”
Explain your point of view about the situation in a way that does not put the other person on the defensive and evoke emotions that make resolution more difficult. Don’t say, “You’re always playing loud music when I’m trying to study.” Instead, say, “I have difficulty studying when you play loud music, and that makes me frustrated and irritable.” Don’t blame the other for the problem—that would just get emotions flowing again.
-
Brainstorm together to find a solution that satisfies both of you.
Some compromise is usually needed, but that is usually not difficult to reach when you’re calm and have the right attitude about working together on a solution. In some cases, you may simply have to accept a result that you still do not agree with, simply in order to move on.
The process of conflict resolution is often uncomfortable at times. In most cases, when the people involved have a good attitude and are open to compromise, conflicts can be resolved successfully.
Yet sometimes there seems to be no resolution. Sometimes the other person may simply be difficult and refuse to even try to work out a solution. Regrettably, not everyone on or off campus is mature enough to be open to other perspectives. With some interpersonal conflicts, you may simply have to decide not to see that person anymore or find other ways to avoid the conflict in the future. But remember, most conflicts can be solved among adults, and it’s seldom a good solution to run away from a problem that will continue to surface and keep you from being happy with your life.
Roommate Issues
At many colleges students just out of high school must live in a campus residence hall. Other students may live in a shared apartment with new roommates. This is the first time many students have had to share a room, suite, or apartment with others who were not family members, and this situation may lead to conflicts and strong feelings that can even affect your academic success.
As in other interactions, the keys to forming a good relationship with a roommate are communication and attitude. From the beginning, you should talk about everyone’s expectations of the other(s) and what matters most to you about where you live. Don’t wait until problems happen before talking. It’s often good to begin with the key practical issues: agreeing on quiet hours for study (limiting not only loud music but also visits from others), time for lights out, neatness and cleaning up, things shared and private things not to touch.
Show respect for the other’s ideas and possessions, respect their privacy, and try to listen more than you talk. Even if your roommate does not become a close friend, you can have a harmonious, successful relationship that makes your residence a good home for both of you. Millions of college students before you have learned to work this out, and if both (or all) of you respect each other and keep communication open and nonconfrontational, you will easily get through the small bumps in the road ahead, too. Follow these guidelines to help ensure you get along well:
-
Anticipate problems before they happen.
Think about things that you consider essential in your living environment and talk with a new roommate about these essentials now.
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Deal with any problem promptly.
Don’t wait until a behavior is well established before speaking up, as if the other person will somehow catch on that it aggravates you. It may be as simple as a roommate using your coffee cup or borrowing your toothpaste without asking, but if you say nothing, trying to be polite, the habit may expand to other things.
-
Be patient, flexible, and willing to compromise.
It may take a while for each of you to get used to each other and to establish a communication pattern of openness so that you can be honest with each other about what really matters.
-
Be warm, use humor, and be sensitive.
Telling someone that they’re doing something bothersome can be very difficult for many people. Think before speaking, looking for the best way to communicate what you feel. Remember, you’ll be spending a lot of time around this person, so do you really want them to think of you as bossy or obsessive-compulsive?
-
Get out more.
Sometimes it helps to spend more time elsewhere on campus, studying in the library or another quiet place. You just might need a certain amount of time a day alone. That’s fine, but don’t expect your roommate to have to leave just to give you that time alone!
But What If You
Really
Have a Roommate Problem?
In some situations and with some people who will not compromise and do not respect you and your needs, a roommate can be a serious problem. In some circumstances, you may able to move to a different room. Room changes usually are not granted simply because you “don’t get along,” but certain circumstances may justify a change. The following are some examples:
-
Your roommate smokes in the room.
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Your roommate uses illegal drugs, drinks alcohol underage, or conducts other illegal activities in the room.
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Your roommate repeatedly refuses to limit activities at any hour to allow you to sleep.
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Your roommate does anything that threatens your physical well-being or safety.
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Your roommate denies you your rights to practice your religion or other basic rights.
If you have a problem like this, first talk with your resident advisor (RA) or other residence hall authority. They will explain the process for a room change, if warranted, or other ways for managing the problem,
Dealing with Harassment
Although college campuses are for the most part safe, secure, and friendly places where social and intellectual interaction is generally mature and responsible, harassment can occur in any setting. Harassment is a general term referring to behavior that is intended to disturb or threaten another person in some way, often psychologically. Typically the person or people doing the harassment target their victim because of a difference in race, ethnicity, religion, nationality, sex, age, sexual orientation, or disability.
Acts of harassment may be verbal, physical, psychological, or any other behavior intended to disturb another person. Bullying behavior, name-calling, belittling, gesturing obscenely, stalking, mobbing—any action intended to torment or deliberately make another person uncomfortable or feel humiliated is harassment. Harassment may also be intended to manipulate a targeted person to act in some specific way.
Sexual harassment is a special term referring to persistent, unwanted sexual behaviors or advances. Sexual harassment may begin with words but progress to unwanted touching and potentially even rape.
Many types of harassment are illegal.  Students who deliberately malign members of another race may be guilty of committing a hate crime. Physically tormenting another student in a hazing may be judged assault and battery. Any discrimination in the workplace based on race, religion, age, sex, and so on is illegal. On a college campus, any harassment of a student by a faculty member or college employee is expressly forbidden, unethical, and also possibly illegal.
Harassment of any type, at any time, of any person, is wrong and unacceptable. You will know it if you are harassed, and you should know also that it is your basic right to be free of harassment and that your college has strict policies against all forms of harassment. Here’s what you should do if you are being harassed:
-
Tell the person to stop the behavior—or if you feel at any risk of harm, get out of the situation immediately.
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Document the incident, particularly with ongoing harassment. Keep notes of the details. Tell someone you trust about the situation.
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Report the harassment to the appropriate college authority. If you are unsure which to go to, go to the dean of students first.
 Any time there is change, issues may arise.
Figure 1.15
A video communication program like Skype makes it easy to stay in touch with friends and family.
theunquietlibrarian – Skyping with Cherished Friends Elisabeth and Michelle! – CC BY-NC 2.0.
As in other relationships, try to understand the other’s perspective. Honesty is particularly important—but with tact and understanding. Here are some tips for getting along:
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Understand that your parents may not change their attitudes toward you as quickly as you yourself may be changing. They may still think of you as a younger person in need of their continued guidance. They will worry about you and fear that you might fall in with the wrong crowd or engage in risky behavior. Be patient.  Let them gradually accept you as a more mature person who can make your own decisions wisely.
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Stay in touch. You may be busier than ever and feel you haven’t time for a phone call or e-mail, but communication is very important for parents—especially if they are now empty nesters without other children at home. Even if they seem to want too much involvement in your life and to make decisions for you, realize that this at least in part is simply a desire to stay in touch with you—and to feel they still matter in your life.
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Use your best listening skills. Understanding what they’re really feeling, which is often simply a concern for you born of their love for you, will usually help you know best to respond.
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Be assured that over time your parents and other family members will get used to your being on your own and will accept your ability to make your own decisions. Time itself often solves issues.
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With your own family, now that you are busier than ever with classes and work, you may need to pay special attention to ensuring you stay active in family relationships. Schedule times for family outings and make room in your days for casual interactions. But remember, it’s not how much time you spend together but the quality of that time, so give your family your full attention when you are together.
Key Takeaways
-
A rich, diverse social life is an important dimension of the college experience that contributes also to academic success.
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Getting along with others involves communication skills and a willingness to interact with different people in a number of different ways.
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Effective listening skills are as important as expressing yourself well verbally and nonverbally.
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Online social networking used in moderation can be beneficial.
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Balancing one’s social life with schoolwork requires time management skills as well as good study skills.
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Because social interactions frequently involve conflicting values, behaviors, or ideas, it’s important to respect others, stay open minded, be open to compromise, and understand how to resolve conflicts.
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Acknowledge that family relationships will likely change after you enter college and work to ease the transition for everyone.
Checkpoint Exercises
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List three or four guidelines for interacting successfully with others.
__________________________________________________________________
__________________________________________________________________
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You are talking after class with another student with whom you’d like to be friends, but you’re distracted by a test you have to study for. If you’re not careful, what nonverbal communication signals might you accidentally send that could make the other person feel you are
not
friendly? Describe two or three nonverbal signals that could give the wrong impression.
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What are the best things
to say
when you’re actively engaged in
listening
to another?
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For each of the following statements about effective communication, circle T for true or F for false:
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You are upset because your roommate (or a family member) always seems to have several friends over just when you need to study most. Write in the space below what you might say to this person to explain the problem, using “I statements” rather than “you statements.”
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If another person is acting very emotionally and is harassing you, what should you
not
do at that moment?
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31.6: Binding Energy
Learning Objectives
By the end of this section, you will be able to:
- Define and discuss binding energy.
- Calculate the binding energy per nucleon of a particle.
The more tightly bound a system is, the stronger the forces that hold it together and the greater the energy required to pull it apart. We can therefore learn about nuclear forces by examining how tightly bound the nuclei are. We define the binding energy (BE) of a nucleus to be the energy required to completely disassemble it into separate protons and neutrons . We can determine the BE of a nucleus from its rest mass. The two are connected through Einstein’s famous relationship \(E = (\Delta m)c^2\).
A bound system has a smaller mass than its separate constituents; the more tightly the nucleons are bound together, the smaller the mass of the nucleus.
Imagine pulling a nuclide apart as illustrated in Figure \(\PageIndex{1}\). Work done to overcome the nuclear forces holding the nucleus together puts energy into the system. By definition, the energy input equals the binding energy BE. The pieces are at rest when separated, and so the energy put into them increases their total rest mass compared with what it was when they were glued together as a nucleus. That mass increase is thus \(\Delta m = BE/c^2\) This difference in mass is known as mass defect . It implies that the mass of the nucleus is less than the sum of the masses of its constituent protons and neutrons. A nuclide \(^AX\) has \(Z\) protons and \(N\) neutrons, so that the difference in mass is
\[ \Delta m = (Zm_p + Nm_n) - M_{tot}.\] Thus,
\[BE = (∆m)c^2 = [(Zm_p + Nm_n) − m_{tot}]c^2.\]
where \(m_{tot}\) is the mass of the nuclide \(^AX\), \(m_p\) is the mass of a proton, and \(m_n\) is the mass of a neutron. Traditionally, we deal with the masses of neutral atoms. To get atomic masses into the last equation, we first add \(Z\) electrons to \(m_{tot}\) which gives \(m(^AX)\), the atomic mass of the nuclide. We then add \(Z\) electrons to the \(Z\) protons, which gives \(Zm(^1H)\), or \(Z\) times the mass of a hydrogen atom. Thus the binding energy of a nuclide \(^AX\) is
\[BE = [(Z_m(^1H) + Nm_n) - m(^AX)]c^2 \label{BE}\]
The atomic masses can be found in Appendix A , most conveniently expressed in unified atomic mass units u \((1 \, u = 931.5 \, MeV/c^2)\). BE is thus calculated from known atomic masses.
Nuclear Decay Helps Explain Earth’s Hot Interior
A puzzle created by radioactive dating of rocks is resolved by radioactive heating of Earth’s interior. This intriguing story is another example of how small-scale physics can explain large-scale phenomena.
Radioactive dating plays a role in determining the approximate age of the Earth. The oldest rocks on Earth solidified about \(3.5 \times 10^9\) years ago—a number determined by uranium-238 dating. These rocks could only have solidified once the surface of the Earth had cooled sufficiently. The temperature of the Earth at formation can be estimated based on gravitational potential energy of the assemblage of pieces being converted to thermal energy. Using heat transfer concepts discussed in Thermodynamics it is then possible to calculate how long it would take for the surface to cool to rock-formation temperatures. The result is about \(10^9\) years. The first rocks formed have been solid for \(3.5 \times 10^9\) years, so that the age of the Earth is approximately \(4.5 \times 10^9\) years. There is a large body of other types of evidence (both Earth-bound and solar system characteristics are used) that supports this age. The puzzle is that, given its age and initial temperature, the center of the Earth should be much cooler than it is today (Figure \(\PageIndex{2}\)).
We know from seismic waves produced by earthquakes that parts of the interior of the Earth are liquid. Shear or transverse waves cannot travel through a liquid and are not transmitted through the Earth’s core. Yet compression or longitudinal waves can pass through a liquid and do go through the core. From this information, the temperature of the interior can be estimated. As noticed, the interior should have cooled more from its initial temperature in the \(4.5 \times 10^9\) years since its formation. In fact, it should have taken no more than about \(10^9\) years to cool to its present temperature. What is keeping it hot? The answer seems to be radioactive decay of primordial elements that were part of the material that formed the Earth (see the blowup in Figure \(\PageIndex{2}\)).
Nuclides such as \(^{238}U\) and \(^{40}K\) have half-lives similar to or longer than the age of the Earth, and their decay still contributes energy to the interior. Some of the primordial radioactive nuclides have unstable decay products that also release energy— \(\ce{^{238}U}\) has a long decay chain of these. Further, there were more of these primordial radioactive nuclides early in the life of the Earth, and thus the activity and energy contributed were greater then (perhaps by an order of magnitude). The amount of power created by these decays per cubic meter is very small. However, since a huge volume of material lies deep below the surface, this relatively small amount of energy cannot escape quickly. The power produced near the surface has much less distance to go to escape and has a negligible effect on surface temperatures.
A final effect of this trapped radiation merits mention. Alpha decay produces helium nuclei, which form helium atoms when they are stopped and capture electrons. Most of the helium on Earth is obtained from wells and is produced in this manner. Any helium in the atmosphere will escape in geologically short times because of its high thermal velocity.
What patterns and insights are gained from an examination of the binding energy of various nuclides? First, we find that BE is approximately proportional to the number of nucleons \(A\) in any nucleus. About twice as much energy is needed to pull apart a nucleus like \(^{24}Mg\) compared with pulling apart \(^{12}C\), for example. To help us look at other effects, we divide BE by \(A\) and consider the binding energy per nucleon , \(BE/A\). The graph of \(BE/A\) in Figure \(\PageIndex{3}\) reveals some very interesting aspects of nuclei. We see that the binding energy per nucleon averages about 8 MeV, but is lower for both the lightest and heaviest nuclei. This overall trend, in which nuclei with \(A\) equal to about 60 have the greatest \(BE/A\) and are thus the most tightly bound, is due to the combined characteristics of the attractive nuclear forces and the repulsive Coulomb force.
It is especially important to note two things—the strong nuclear force is about 100 times stronger than the Coulomb force, and the nuclear forces are shorter in range compared to the Coulomb force. So, for low-mass nuclei, the nuclear attraction dominates and each added nucleon forms bonds with all others, causing progressively heavier nuclei to have progressively greater values of \(BE/A\). This continues up to \(A \approx 60\), roughly corresponding to the mass number of iron. Beyond that, new nucleons added to a nucleus will be too far from some others to feel their nuclear attraction. Added protons, however, feel the repulsion of all other protons, since the Coulomb force is longer in range. Coulomb repulsion grows for progressively heavier nuclei, but nuclear attraction remains about the same, and so \(BE/A\) becomes smaller. This is why stable nuclei heavier than \(A \approx 40\) have more neutrons than protons. Coulomb repulsion is reduced by having more neutrons to keep the protons farther apart (Figure \(\PageIndex{4}\)).
There are some noticeable spikes on the \(BE/A\) graph, which represent particularly tightly bound nuclei. These spikes reveal further details of nuclear forces, such as confirming that closed-shell nuclei (those with magic numbers of protons or neutrons or both) are more tightly bound. The spikes also indicate that some nuclei with even numbers for \(Z\) and \(N\) and with \(Z = N\), are exceptionally tightly bound. This finding can be correlated with some of the cosmic abundances of the elements. The most common elements in the universe, as determined by observations of atomic spectra from outer space, are hydrogen, followed by \(^4He\) with much smaller amounts of \(^{12}C\) and other elements. It should be noted that the heavier elements are created in supernova explosions, while the lighter ones are produced by nuclear fusion during the normal life cycles of stars, as will be discussed in subsequent chapters. The most common elements have the most tightly bound nuclei. It is also no accident that one of the most tightly bound light nuclei is \(^4He\) emitted in \(\alpha\) decay.
Example \(\PageIndex{1}\): What Is \(BE/A\) for an Alpha Particle?
Calculate the binding energy per nucleon of \(^4He\) the \(\alpha\) particle.
Strategy
To find \(BE/A\) we first find BE using the Equation \(BE = [(Z_m(^1H) + Nm_n) - m(^AX)]c^2\) and then divide by \(A\). This is straightforward once we have looked up the appropriate atomic masses in Appendix A .
Solution
The binding energy for a nucleus is given by the equation
\[BE = [(Z_m(^1H) + Nm_n) - m(^AX)]c^2 \nonumber\]
For \(^4He\), we have \(Z = N = 2; thus,
\[BE = [(2m(^1H) + 2Nm_n) - m(^4He)]c^2. \nonumber\]
Appendix A gives these masses as \(m(^4He) = 4.002602 \, u\). \(m(^1H) = 1.007825 \, u\), and \(m_n = 1.008665 \, u\). Thus
\[BE = (0.030378 \, u)c^2. \nonumber\]
Noting that \(1 \, u = 931.5 \, MeV/c^2\), we find
\[BE = (0.030378)(931.5 \, MeV/c^2)c^2 = 28.3 \, MeV. \nonumber\]
Since \(A = 4\), we see that \(BE/A\) is this number divided by 4, or
\[BE/A = 7.07 \, MeV/nucleon. \nonumber\]
Discussion
This is a large binding energy per nucleon compared with those for other low-mass nuclei, which have \(BE/A \approx 3 \, MeV/nucleon\). This indicates that \(^4He\) is tightly bound compared with its neighbors on the chart of the nuclides. You can see the spike representing this value of \(BE/A\) for \(^4He\) on the graph in Figure . This is why \(^4He\) is stable. Since \(^4He\) is tightly bound, it has less mass than other \(A = 4\) nuclei and, therefore, cannot spontaneously decay into them. The large binding energy also helps to explain why some nuclei undergo decay. Smaller mass in the decay products can mean energy release, and such decays can be spontaneous. Further, it can happen that two protons and two neutrons in a nucleus can randomly find themselves together, experience the exceptionally large nuclear force that binds this combination, and act as a \(^4He\) unit within the nucleus, at least for a while. In some cases, the \(^4He\) escapes, and \(\alpha\) decay has then taken place.
There is more to be learned from nuclear binding energies. The general trend in \(BE/A\) is fundamental to energy production in stars, and to fusion and fission energy sources on Earth, for example. This is one of the applications of nuclear physics covered in Medical Applications of Nuclear Physics . The abundance of elements on Earth, in stars, and in the universe as a whole is related to the binding energy of nuclei and has implications for the continued expansion of the universe.
Problem-Solving Strategies: For Reaction, Binding Energies, & Activity Calculations
- Identify exactly what needs to be determined in the problem (identify the unknowns) . This will allow you to decide whether the energy of a decay or nuclear reaction is involved, for example, or whether the problem is primarily concerned with activity (rate of decay).
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
- For reaction and binding-energy problems, we use atomic rather than nuclear masses. Since the masses of neutral atoms are used, you must count the number of electrons involved. If these do not balance (such as in \(\beta^+\) decay), then an energy adjustment of 0.511 MeV per electron must be made. Also note that atomic masses may not be given in a problem; they can be found in tables.
- For problems involving activity, the relationship of activity to half-life, and the number of nuclei given in the equation \(R = \frac{0.693N}{t_{1/2}}\) can be very useful. Owing to the fact that number of nuclei is involved, you will also need to be familiar with moles and Avogadro’s number.
- Perform the desired calculation; keep careful track of plus and minus signs as well as powers of 10.
- Check the answer to see if it is reasonable: Does it make sense? Compare your results with worked examples and other information in the text. (Heeding the advice in Step 5 will also help you to be certain of your result.) You must understand the problem conceptually to be able to determine whether the numerical result is reasonable.
PHET EXPLORATIONS: NUCLEAR FISSION
Download the PhET simulation and start a chain reaction, or introduce non-radioactive isotopes to prevent one. Control energy production in a nuclear reactor!
Summary
The binding energy (BE) of a nucleus is the energy needed to separate it into individual protons and neutrons. In terms of atomic masses, \[BE = [(Zm(^1H) + Nm_n] - m(^AX)]c^2, \nonumber\] where \(m(^1H)\) is the mass of a hydrogen atom, \(m(^AX)\) is the atomic mass of the nuclide, and \(m_n\) is the mass of a neutron. Patterns in the binding energy per nucleon, \(BE/A\), reveal details of the nuclear force. The larger the \(BE/A\), the more stable the nucleus.
Glossary
- binding energy
- the energy needed to separate nucleus into individual protons and neutrons
- binding energy per nucleon
- the binding energy calculated per nucleon; it reveals the details of the nuclear force—larger the \(BE/A\) the more stable the nucleu


5.9: Exercise-2- Exploration
1. This question explores how bidding dishonestly can end up hurting the cheater. Four partners are dividing a million-dollar property using the lone-divider method. Using a map, Danny divides the property into four parcels \(s_1\), \(s_2\), \(s_3\), and \(s_4\). The following table shows the value of the four parcels in the eyes of each partner (in thousands of dollars):
\(\begin{array}{|c|l|l|l|l|}
\hline & \mathrm{s}_{1} & \mathrm{s}_{2} & \mathrm{s}_{3} & \mathrm{s}_{4} \\
\hline \textbf { Danny } & \$ 250 & \$ 250 & \$ 250 & \$ 250 \\
\hline \textbf { Brianna } & \$ 460 & \$ 180 & \$ 200 & \$ 160 \\
\hline \textbf { Carlos } & \$ 260 & \$ 310 & \$ 220 & \$ 210 \\
\hline \textbf { Greedy } & \$ 330 & \$ 300 & \$ 270 & \$ 100 \\
\hline
\end{array}\)
- Assuming all players bid honestly, which piece will Greedy receive?
- Assume Brianna and Carlos bid honestly, but Greedy decides to bid only for s1, figuring that doing so will get him s1. In this case there is a standoff between Brianna and Greedy. Since Danny and Carlos are not part of the standoff, they can receive their fair shares. Suppose Danny gets s3 and Carlos gets s2, and the remaining pieces are put back together and Brianna and Greedy will split them using the basic divider-chooser method. If Greedy gets selected to be the divider, what will be the value of the piece he receives?
- Extension: Create a Sealed Bids scenario that shows that sometimes a player can successfully cheat and increase the value they receive by increasing their bid on an item, but if they increase it too much, they could end up receiving less than their fair share.
- Explain why divider-chooser method with two players will always result in an envy-free division.
- Will the lone divider method always result in an envy-free division? If not, will it ever result in an envy-free division?
- The Selfridge-Conway method is an envy-free division method for three players. Research how the method works and prepare a demonstration for the class.
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Suppose that two people are dividing a $12 pizza that is half pepperoni, half cheese. Steve likes both equally, but Maria likes cheese twice as much as pepperoni. As divider, Steve divides the pizza so that one piece is 1/3 cheese and 2/3 pepperoni, and the second piece is 1/3 pepperoni and 2/3 cheese.
- Describe the value of each piece to each player
- Since the value to each player is not the same, this division is not equitable. Find a division that would be equitable. Is it still envy-free?
- The original division is not Pareto optimal. To show this, find another division that would increase the value to one player without decreasing the value to the other player. Is this division still envy-free?
- Would it be possible with this set of preferences to find a division that is both equitable and Pareto optimal? If so, find it. If not, explain why.
- Is the Sealed Bids method Pareto optimal when used with two players? If not, can you adjust the method to be so?
- Is the Sealed Bids method envy-free when used with two players? If not, can you adjust the method to be so?
- Is the Sealed Bids method equitable when used with two players? If not, can you adjust the method to be so?
- All the problems we have looked at in this chapter have assumed that all participants receive an equal share of what is being divided. Often, this does not occur in real life. Suppose Fred and Maria are going to divide a cake using the divider-chooser method. However, Fred is only entitled to 30% of the cake, and Maria is entitled to 70% of the cake (maybe it was a $10 cake, and Fred put in $3 and Maria put in $7). Adapt the divider-choose method to allow them to divide the cake fairly.
Assume (as we have throughout this chapter) that different parts of the cake may have different values to Fred and Maria, and that they don't communicate their preferences/values with each other. You goal is to come up with a method of fair division, meaning that although the participants may not receive equal shares, they should be guaranteed their fair share. Your method needs to be designed so that each person will always be guaranteed a share that they value as being worth at least as much as they're entitled to.
The last few questions will be based on the Adjusted Winner method, described here:
For discrete division between two players, there is a method called Adjusted Winner that produces an outcome that is always equitable, envy-free, and Pareto optimal. It does, however, require that items can be split or shared. The method works like this:
1) Each player gets 100 points that they assign to the items to be divided based on their relative worth.
2) In the initial allocation, each item is given to the party that assigned it more points. If there were any items with both parties assigned the same number of points, they’d go to the person with the fewest points so far.
3) If the assigned point values are not equal, then begin transferring items from the person with more points to the person with fewer points. Start with the items that have the smallest point ratio, calculated as (higher valuation/lower valuation).
4) If transferring an entire item would move too many points, only transfer a portion of the item.
Example: A couple is attempting to settle a contentious divorce[1]. They assign their 100 points to the issues in contention:
\(\begin{array}{|l|l|l|}
\hline & \textbf { Mike } & \textbf { Carol } \\
\hline \textbf { Custody of children } & 25 & 65 \\
\hline \textbf { Alimony } & 60 & 25 \\
\hline \textbf { House } & 15 & 10 \\
\hline
\end{array}\)
In the initial allocation, Mike gets his way on alimony and house, and Carol gets custody of the children. In the initial allocation, Mike has 75 points and Carol has 65 points. To decide what to transfer, we calculate the point ratios.
\(\begin{array}{|l|l|l|l|}
\hline & \textbf { Mike } & \textbf { Carol } & \textbf { Point ratio } \\
\hline \textbf { Custody of children } & 25 & 65 & 65 / 25=2.6 \\
\hline \textbf { Alimony } & 60 & 25 & 60 / 25=2.4 \\
\hline \textbf { House } & 15 & 10 & 15 / 10=1.5 \\
\hline
\end{array}\)
Since the house has the smallest point ratio, the house will be the item we work with first. Since transferring the entire house would give Carol too many points, we instead need to transfer some fraction, \(p\), of the house to that Carol and Mike will end up with the same point values. If Carol receives a fraction \(p\) of the house, then Mike will give up \((1-p)\) of the house. The value Carol will receive is \(10p\): the fraction \(p\) of the 10 points Carol values the house at. The value Mike will get is \(15(1-p)\). We set their point totals equal to solve for \(p\):
\(\begin{array} {lll} 65+10 p & = & 60+15(1-p) \\ 65+10 p & = & 60+15-15 p \\ 25 p & = & 10 \\ \end{array}\)
Where \(p=10 / 25=0.4=40 \%\). So Carol should receive 40% of the house.
- Apply the Adjusted Winner method to settle a divorce where the couples have assigned the point values below
\(\begin{array}{|l|l|l|}
\hline & \textbf { Sandra } & \textbf { Kenny } \\
\hline \textbf { Home } & 20 & 30 \\
\hline \textbf { Summer home } & 15 & 10 \\
\hline \textbf { Retirement account } & 50 & 40 \\
\hline \textbf { Investments } & 10 & 10 \\
\hline \textbf { Other } & 5 & 10 \\
\hline
\end{array}\)
- In 1974, the United States and Panama negotiated over US involvement and interests in the Panama Canal. Suppose that these were the issues and point values assigned by each side[2]. Apply the Adjusted Winner method.
\(\begin{array}{|l|l|l|}
\hline & \textbf { United States } & \textbf { Panama } \\
\hline \textbf { US defense rights } & 22 & 9 \\
\hline \textbf { Use rights } & 22 & 15 \\
\hline \textbf { Land and water } & 15 & 15 \\
\hline \textbf { Expansion rights } & 14 & 3 \\
\hline \textbf { Duration } & 11 & 15 \\
\hline \textbf { Expansion routes } & 6 & 5 \\
\hline \textbf { Jurisdiction } & 2 & 7 \\
\hline \textbf { US military rights } & 2 & 7 \\
\hline \textbf { Defense role of Panama } & 2 & 13 \\
\hline
\end{array}\)
[1] From Negotiating to Settlement in Divorce , 1987
[2] Taken from The Art and Science of Negotiation , 1982


6: Making Healthy Choices
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Succeeding in the college environment involves more than just one singular trait or skill set. Preparing for academic success is about more than attending class or writing papers. How we care for our bodies and minds can also influence achievement, or lack thereof, in school and work pursuits.
In this module, we explore a variety of factors every individual should consider pertaining to personal health and well-being. We begin by taking a look at ways to promote physical health and well-being, including how to make positive choices regarding food and substances. Next, we present a variety of tools and strategies to protect yourself while on campus or while you’re out and about, through methods of maintaining personal safety. Your learning investigation also includes taking a look at ways to ask for help—something many people struggle with in today’s society.
Through exploring perspectives and skills related to health, safety, and overall well-being, you will gain valuable insight as you prepare for success in your academic path.
Thumbnail: www.pexels.com/photo/woman-sleeping-on-a-wooden-table-3767371/


3.1: A Model of Computing
A modern computer is a tremendously complex system, with many things to understand at the level of the hardware, the operating system, and the programming software (e.g. Python). It is helpful to consider a simplified computing model , which is basically a "cartoon" representation of a computer that omits the unimportant facts about how it operates, and focuses on the most important aspects of what a computer program does.
The standard paradigm of computing that we use today is the Von Neumann architecture , which divides a computer into three inter-connected units: processor, memory, and input/output devices. We'll use this as the basis of our simplified model, with an emphasis on the processor and memory parts.
A computer's memory is essentially a chunk of space where we can store numbers. For now, we won't concern ourselves with how the contents of memory are organized or formatted. The processor can read one or more numbers from any locations (or addresses ) in memory, perform some basic operations on them, and then write the results into any other addresses. We also make three other important assumptions:
- The capacity is effectively infinite; we don't worry about running out.
- The memory is random-access memory (RAM), meaning that the processor can access any addresses in memory, one after another, with the same speed. (In real life, not all types of memory are random-access. Disk drives, for instance, are not, because the scanning head must physically move to different positions in order to read different parts of memory. However, ordinary computer programs can ignore such details, which are left to the operating system to manage.)
- The processor can only do one thing at a time. For example, if you ask it to read two numbers from memory, that takes twice as long as reading a single number from memory. (Again, real computers violate this assumption to some extent; computers now commonly have multiple processors that can performs multiple operations simultaneously. Our present simple model ignores these complications.)
A
program
is a set of instructions for the processor. For example, the following line of code is a program (or part of a program) that tells the processor to add up four numbers that are currently stored in memory, and save the result to another memory address labeled
x
:
x = a + b + c + d
Because the processor can only do one thing at a time, even a simple line of code like this involves several sequential steps. The processor can't simultaneously read all four memory addresses (corresponding to
a
through
d
); it must read them one at a time. The following figure shows how the processor might carry out the above addition program:


15: Adolescence - Social Emotional Development
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learning objectives
After this chapter, you should be able to:
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Compare Erikson and Marcia’s Theories
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Explain Identity and Self-concept
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Summarize the Stages of Ethnic Identity Development
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Explain the Development of Gender Identity
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Summarize Sexuality Identity and Orientation
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Describe Antisocial Behaviors
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Explain the Developmental Stage of Emerging Adulthood
Adolescents continue to refine their sense of self as they relate to others. Adolescent’s main questions are “Who am I?” and “Who do I want to be?” Some adolescents adopt the values and roles that their parents expect of them. Other teens develop identities that align more with the peer groups rather than their parents’ expectations. This is common as adolescents work to form their identities. They pull away from their parents and the peer group becomes very important (Shanahan, McHale, Osgood, & Crouter, 2007). Despite spending less time with their parents, most teens report positive feelings toward them (Moore, Guzman, Hair, Lippman, & Garrett, 2004). Warm and healthy parent-child relationships have been associated with positive outcomes for the adolescent, such as better grades and fewer school behavior problems, in the United States as well as in other countries (Hair et al., 2005). 1
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15.1: Erik Erikson – Theory of Psychosocial Development
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Erikson proposed that each period of life has a unique challenge or crisis that a person must face. This is referred to as a psychosocial development. According to Erikson, successful development involves dealing with and resolving the goals and demands of each of these crises in a positive way. These crises are usually called stages, although that is not the term Erikson used. If a person does not resolve a crisis successfully, it may hinder their ability to deal with later crises.
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15.2: James Marcia – Theory of Identity Development
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One approach to assessing identity development was proposed by James Marcia. In his approach, adolescents are asking questions regarding their exploration of and commitment to issues related to occupation, politics, religion, and sexual behavior. Studies assessing how teens pass through Marcia’s stages show that although most teens eventually succeed in developing a stable identity, the path to it is not always easy and there are many routes that can be taken.
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15.3: Development of Identity and Self Concept - Who am I?
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Developmental psychologists have researched several different areas of identity development for adolescence and some of the main areas include:
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15.4: Development of Gender Identity
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From birth, children are assigned a gender and are socialized to conform to certain gender roles based on their biological sex. “Sex,” refers to physical or physiological differences between males, females, and intersex persons, including both their primary and secondary sex characteristics. “Gender,” on the other hand, refers to social or cultural distinctions associated with a given sex.
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15.5: Socialization Agents During Adolescence
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Adolescence is a crucial period in social development, research shows there are four main types of relationships that influence an adolescent: parents, peers, community, and society.
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15.6: Community, Society, and Culture
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There are certain characteristics of adolescent development that are more rooted in culture than in human biology or cognitive structures. Culture is learned and socially shared, and it affects all aspects of an individual’s life. Social responsibilities, sexual expression, and belief-system development, for instance, are all likely to vary based on culture. Furthermore, many distinguishing characteristics of an individual are all products of culture.
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15.7: Media- Influences on Teens
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Media is another agent of socialization that influences our political views; our tastes in popular culture; our views of women, people of color, and the LGBTQ+ community; and many other beliefs and practices. In an ongoing controversy, the media is often blamed for youth violence and many other of society’s ills. The average child sees thousands of acts of violence on television and in the movies before reaching young adulthood.
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15.8: Development of Sexual Identity
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Adolescence is the developmental period during which romantic relationships typically first emerge. By the end of adolescence, most American teens have dated others and have had at least one romantic relationship. Dating serves many purposes for teens, including having fun, companionship, status, socialization, intimacy, sexual experimentation, and partner selection for those in late adolescence.
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15.9: Antisocial Behavior in Adolescence
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Bullying is defined as unwanted, aggressive behavior among school aged children that involves a real or perceived power imbalance. Further, the aggressive behavior happens more than once or has the potential to be repeated. There are different types of bullying.
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15.10: Antisocial Behaviors, Violence, and Child Abuse
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"Most kids will act up or become disruptive or defiant sometimes. Disruptive and conduct disorders, however, involve much more severe and longer-lasting behaviors than typical, short-lived episodes. Disruptive, impulse-control and conduct disorders refer to a group of disorders that include oppositional defiant disorder, conduct disorder, intermittent explosive disorder, kleptomania and pyromania. These disorders can cause people to behave angrily or aggressively toward people or property."
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15.11: Violence and Abuse
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Violence and abuse are among the most disconcerting of the challenges that today’s families face. Abuse can occur between spouses, between parent and child, as well as between other family members. The frequency of violence among families is difficult to determine because many cases of spousal abuse and child abuse go unreported. In any case, studies have shown that abuse (reported or not) has a major impact on families and society as a whole.
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15.12: Emerging Adulthood - The Bridge Between Adolescence and Adulthood
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The next stage of development is emerging adulthood and is characterized as an in-between time where identity exploration is focused on work and love, which occurs from approximately 18 years of age to the mid to late 20s.
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15.S: Summary
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Thumbnail: pixabay.com/photos/love-coup...young-1716825/


5.13: Statistical Literacy
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Learning Objectives
Example \(\PageIndex{1}\)
The playbill for the Alley Theatre in Houston wants to appeal to advertisers. They reported the mean household income and the median age of theatergoers. What might have guided their choice of the mean or median?
Solution
It is likely that they wanted to emphasize that theatergoers had high income but de-emphasize how old they are. The distributions of income and age of theatergoers probably have positive skew. Therefore the mean is probably higher than the median, which results in higher income and lower age than if the median household income and mean age had been presented.


8.1.11: Suggested Resources
8.1.2 Overview of Entrepreneurial Finance and Accounting Strategies
Small Business Administration lending program: https://www.sba.gov/funding-programs/loans
“The Top Twenty Venture Capitalists Worldwide,” New York Times (March 27, 2017): https://www.nytimes.com/interactive/...ists.html?_r=0
National Public Radio “How I Built This” podcast series (example episodes: Instagram, Airbnb, Spanx): https://www.npr.org/podcasts/510313/how-i-built-this
TED Radio Hour (example episodes: Big Data Revolution, Disruptive Leadership): https://www.npr.org/programs/ted-rad...ate=2019-07-05
8.1.3 Special Funding Strategies
Tax information for charitable organizations from the Internal Revenue Service: https://www.irs.gov/charities-non-pr...-organizations
Information from the National Institutes of Health on grants and the application process: https://grants.nih.gov/grants/about_grants.htm
Small Business Administration, “How to Start a Non-Profit”: www.sba.gov/blogs/how-start-non-profit
Kickstarter: www.kickstarter.com
Indiegogo, “The Essential Guide to Crowdfunding”: https://learn.indiegogo.com/the-esse...o-crowdfunding
Startupfest, “The $100K Investment Prize”: http://www.startupfestival.com/news/...estment-prize/
G-Startup Worldwide’s global startup competition: https://www.f6s.com/g-startupworldwide/
Extending Credit to Your Customers: https://fitsmallbusiness.com/invoici...-to-customers/
7 Ways to Bootstrap Your Business to Success: https://www.entrepreneur.com/article/254217
4 Must-Follow Tips for Moonlighting as an Entrepreneur: https://www.entrepreneur.com/article/298310
5 Ways Nonprofits Can Engage Donors: https://www.ama.org/publications/Mar...ge-donors.aspx
8.1.4 Accounting Basics for Entrepreneurs
Small Business Administration Introduction to Accounting Course: https://www.sba.gov/tools/learning-c...-course/364071
Accounting Coach free Accounting Basics Explanation: www.accountingcoach.com/acco...cs/explanation
Khan Academy Accounting and Financial Statements Course: https://www.khanacademy.org/economic...ancial-stateme
8.1.5 Developing Startup Financial Statements and Projections
U.S. Small Business Administration: www.sba.gov/blogs/3-essentia...small-business
U.S. Securities and Exchange Commission: https://www.sec.gov/reportspubs/inve...tguidehtm.html
Forbes , 5 Financial Reports You Should Be Running: https://www.forbes.com/sites/sagewor.../#183ae98d6548
Medium , 3 Phases of Financial Indicators for Startups: https://medium.com/@DanielleMorrill/...s-b18ddb97f5f0


4.17: Assignment- Research
Retailers are becoming increasingly aware of how charitable contributions to local communities and non-profit organizations can impact customer buy-in and brand loyalty to the company. Your general manager has asked you to research a possible community service opportunity for your workplace. Some examples include reading to children at a school, sponsoring a blood drive with the Red Cross, or a local wildlife volunteer event, or a local volunteer litter clean-up event etc.
Your task for this assignment is to write a three-paragraph (minimum) memo addressed to the general manager and team leads at your store. In your memo, recommend one charitable organization you wish to partner or volunteer with that you found via a web search. This memo will briefly discuss the rationale and benefits of partnering with a non-profit, and propose a date for a meeting for your team and members of the charitable organization.
Part 1: Research
Create a short screen capture demonstrating your online search methods for the questions below (should only be 2–3 minutes long). You will submit this video per the course instructor’s directions.
- How many non-profit or charitable organizations are in your city?
- How many non-profit or charitable organizations are within a 5, 10, 15, and 20 mile radius of the school/business you work for?
- Find the five best rated non-profit or charitable organizations within a 5, 10, 15, and 20 mile radius of the school location/business you work for.
Part 2: Analysis
After searching and collecting the information for the questions in Part 1, you will analyze this data to decide which charitable organizations would be the best match with your team. Use the chart linked below to compare your data from Part 1, Question 3.
You’ll rank the charities, provide their names and answer the following two questions: 1) What is the organization’s mission and what do they do? 2) How does this match the school or company’s goals?
Part 3: Writing
After you decide on one organization that best aligns with the goals and mission of your company write a memo or email message analyzing your findings. Make it clear in this memo which organization is your first recommendation that you wish to partner with and propose a date for a meeting on the organization to work with. This memo should briefly discuss the rationale and benefits of partnering with a non-profit, explain your recommendation for a non-profit foundation found during your research, and propose a date for a meeting with the charitable organization.
Grading Rubric
| Criteria | Not Evident | Developing | Proficient | Exemplary | Points |
|---|---|---|---|---|---|
| Organization and format |
4 pts
Writing lacks logical organization. It may show some coherence but ideas lack unity. Serious errors and generally is an unorganized format and information. |
6 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Some points may be contextually misplaced and/or stray from the topic. Transitions may be evident but not used throughout the essay. Organization and format used may detract from understanding the material presented. |
8 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Transitions between ideas and paragraphs create coherence. Overall unity of ideas is supported by the format and organization of the material presented. |
10 pts
Writing shows high degree of attention to details and presentation of points. Format used enhances understanding of material presented. Unity clearly leads the reader to the writer’s conclusion and the format and information could be used independently. |
10 pts |
| Content |
4 pts
Some but not all required questions are addressed. Content and/or terminology is not properly used or referenced. Little or no original thought is present in the writing. Concepts presented are merely restated from the source, or ideas presented do not follow the logic and reasoning presented throughout the writing. |
6 pts
All required questions are addressed but may not be addressed with thoughtful consideration and/or may not reflect proper use of content terminology or additional original thought. Additional concepts may not be present and/or may not be properly cited sources. |
8 pts
All required questions are addressed with thoughtful consideration reflecting both proper use of content terminology and additional original thought. Some additional concepts may be presented from other properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
10 pts
All required questions are addressed with thoughtful in-depth consideration reflecting both proper use of content terminology and additional original thought. Additional concepts are clearly presented from properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
10 pts |
| Development – Critical Thinking |
8 pts
Shows some thinking and reasoning but most ideas are underdeveloped, unoriginal, and/or do not address the questions asked. Conclusions drawn may be unsupported, illogical or merely the author’s opinion with no supporting evidence presented. |
12 pts
Content indicates thinking and reasoning applied with original thought on a few ideas, but may repeat information provided and/ or does not address all of the questions asked. The author presents no original ideas, or ideas do not follow clear logic and reasoning. The evidence presented may not support conclusions drawn. |
16 pts
Content indicates original thinking, cohesive conclusions, and developed ideas with sufficient and firm evidence. Clearly addresses all of the questions or requirements asked. The evidence presented supports conclusions drawn. |
20 pts
Content indicates synthesis of ideas, in-depth analysis and evidence beyond the questions or requirements asked. Original thought supports the topic, and is clearly a well-constructed response to the questions asked. The evidence presented makes a compelling case for any conclusions drawn. |
20 pts |
| Grammar, Mechanics, Style |
4 pts
Writing contains many spelling, punctuation, and grammatical errors, making it difficult for the reader to follow ideas clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices disrupts the content. Additional information may be presented but in an unsuitable style, detracting from its understanding. |
6 pts
Some spelling, punctuation, and grammatical errors are present, interrupting the reader from following the ideas presented clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices may detract from the content. Additional information may be presented, but in a style of writing that does not support understanding of the content. |
8 pts
Writing is free of most spelling, punctuation, and grammatical errors, allowing the reader to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented in a cohesive style that supports understanding of the content. |
10 pts
Writing is free of all spelling, punctuation, and grammatical errors and written in a style that enhances the reader’s ability to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented to encourage and enhance understanding of the content. |
10 pts |
| Total: | 50 pts |


4.2: The Tangent Function
- How is the tangent function defined in terms of the sine and cosine functions?
- Why is the graph of the tangent function so different from the graphs of the sine and cosine functions?
- What are important applications of the tangent function?
In Activity 4.1.4 , we determined the distance between two points \(A\) and \(B\) on opposite sides of a river by knowing a length along one shore of the river and the angle formed between a point downstream and the point on the opposite shore, as pictured in Figure 4.2.1. By first using the cosine of the angle, we determined the value of \(z\) and from there were able to use the sine of the angle to find \(w\text{,}\) the river's width, which turns out to be
\[ w = 50 \cdot \frac{\sin(56.4)}{\cos(56.4)}\text{.} \nonumber \]
It turns out that we regularly need to evaluate the ratio of the sine and cosine functions at the same angle, so it is convenient to define a new function to be their ratio.
For any real number \(t\) for which \(\cos(t) \ne 0\text{,}\) we define the tangent of \(t\), denoted \(\tan(t)\text{,}\) by
\[ \tan(t) = \frac{\sin(t)}{\cos(t)}\text{.} \nonumber \]
Through the following questions, we work to understand the special values and overall behavior of the tangent function.
- Without using computational device, find the exact value of the \(\tan(t)\) at the following values: \(t = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}\text{.}\)
- Why is \(\tan \left( \frac{\pi}{2} \right)\) not defined? What are three other input values \(x\) for which \(\tan(x)\) is not defined?
- Point your browser to http://gvsu.edu/s/0yO (“zero-y-Oh”) to find a Desmos worksheet with data from the tangent function already input. Click on several of the orange points to compare your exact values in (a) with the decimal values given by Desmos . Add one entry to the table: \(x = \frac{11\pi}{24}\text{,}\) \(y = T(\frac{11\pi}{24})\text{.}\) At about what coordinates does this point lie? What are the respective values of \(\sin(\frac{11\pi}{24})\) and \(\cos(\frac{11\pi}{24})\text{?}\) Why is the value of \(\tan(\frac{11\pi}{24})\) so large?
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At the top of the input lists on the left side of the
Desmos
worksheet, click the circle to highlight the function \(T(x) = \tan(x)\) and thus show its plot along with the data points in orange. Use the plot and your work above to answer the following important questions about the tangent function:
- What is the domain of \(y = \tan(x)\text{?}\)
- What is the period of \(y = \tan(x)\text{?}\)
- What is the range of \(y = \tan(x)\text{?}\)
Two perspectives on the tangent function
Because the tangent function is defined in terms of the two fundamental circular functions by the rule \(\tan(t) = \frac{\sin(t)}{\cos(t)}\text{,}\) we can use our understanding of the sine and cosine functions to make sense of the tangent function. In particular, we can think of the tangent of an angle from two different perspectives: as an angle in standard position in the unit circle, or as an angle in a right triangle.
From the viewpoint of Figure \(\PageIndex{3}\) , as the point corresponding to angle \(t\) traverses the circle and generates the point \((a,b)\text{,}\) we know \(\cos(t) = a\) and \(\sin(t) = b\text{,}\) and therefore the tangent function tracks the ratio of these two quantities, and is given by
From the perspective of any right triangle (not necessarily in the unit circle) with hypotenuse “hyp” and legs “adj” and “opp” that are respectively adjacent and opposite the known angle \(\theta\text{,}\) as seen in Figure 4.2.4 , we know that \(\sin(\theta) = \frac{\text{opp}}{\text{hyp}}\) and \(\cos(\theta) = \frac{\text{adj}}{\text{hyp}}\text{.}\) Substituting these expressions for \(\sin(\theta)\) and \(\cos(\theta)\) in the rule for the tangent function, we see that
We typically use the first perspective of tracking the ratio of the \(y\)-coordinate to the \(x\)-coordinate of a point traversing the unit circle in order to think of the overall behavior and graph of the tangent function, and use the second perspective in a right triangle whenever we are working to determine missing values in a triangle.
Properties of the tangent function
Because the tangent function is defined in terms of the sine and cosine functions, its values and behavior are completely determined by those two functions. To begin, we know the value of \(\tan(t)\) for every special angle \(t\) on the unit circle that we identified for the sine and cosine functions. For instance, we know that
Executing similar computations for every familiar special angle on the unit circle, we find the results shown in Table 4.2.5 and Table 4.2.6. We also note that anywhere \(\cos(t) = 0\text{,}\) the value of \(\tan(t)\) is undefined. We record such instances in the table by writing “u”.
| \(t\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\frac{3\pi}{4}\) | \(\frac{5\pi}{6}\) | \(\pi\) |
| \(\sin(t)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(\0\) |
| \(\cos(t)\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-1\) |
| \(\tan(t)\) | \(0\) | \(\frac{1}{\sqrt{3}}\) |
\(1\) |
\(\frac{3}{\sqrt{3}}\) |
u | \(-\frac{3}{\sqrt{3}}\) | \(-1\) | \(-\frac{1}{\sqrt{3}}\) | \(\0\) |
| \(t\) | \(\frac{7\pi}{6}\) | \(\frac{5\pi}{4}\) | \(\frac{4\pi}{3}\) | \(\frac{3\pi}{2}\) | \(\frac{5\pi}{3}\) | \(\frac{7\pi}{4}\) | \(\frac{11\pi}{6}\) | \(2\pi\) |
| \(\sin(t)\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-1\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) |
| \(\cos(t)\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(0\) |
| \(\tan(t)\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\frac{3}{\sqrt{3}}\) | u | \(-\frac{3}{\sqrt{3}}\) | \(-1\) | \(-\frac{1}{\sqrt{3}}\) | \(0\) |
Table 4.2.6. Additional values of the sine, cosine, and tangent functions at special points on the unit circle.
Table 4.2.5 and Table 4.2.6 helps us identify trends in the tangent function. For instance, we observe that the sign of \(\tan(t)\) is positive in Quadrant I, negative in Quadrant II, positive in Quadrant III, and negative in Quadrant IV. This holds because the sine and cosine functions have the same sign in the first and third quadrants, and opposite signs in the other two quadrants.
In addition, we observe that as \(t\)-values in the first quadrant get closer to \(\frac{\pi}{2}\text{,}\) \(\sin(t)\) gets closer to \(1\text{,}\) while \(\cos(t)\) gets closer to \(0\) (while being always positive). Noting that \(\frac{\pi}{2} \approx 1.57\text{,}\) we observe that
and
Because the ratio of numbers closer and closer to \(1\) divided by numbers closer and closer to \(0\) (but positive) increases without bound, this means that \(\tan(t)\) increases without bound as \(t\) approaches \(\frac{\pi}{2}\) from the left side. Once \(t\) is slightly greater than \(\frac{\pi}{2}\) in Quadrant II, the value of \(\sin(t)\) stays close to \(1\text{,}\) but now the value of \(\cos(t)\) is negative (and close to zero). For instance, \(\cos(1.58) \approx -0.0092\text{.}\) This makes the value of \(\tan(t)\) decrease without bound (negative and getting further away from \(0\)) for \(t\) approaching \(\frac{\pi}{2}\) from the right side, and results in \(h(t) = \tan(t)\) having a vertical asymptote at \(t = \frac{\pi}{2}\text{.}\) The periodicity and sign behaviors of \(\sin(t)\) and \(\cos(t)\) mean this asymptotic behavior of the tangent function will repeat.
Plotting the data in the table along with the expected asymptotes and connecting the points intuitively, we see the graph of the tangent function in Figure 4.2.7.
We see from Table 4.2.5 and Table 4.2.6 as well as from Figure \(\PageIndex{7}\) that the tangent function has period \(P = \pi\) and that the function is increasing on any interval on which it is defined. We summarize our recent work as follows.
For the function \(h(t) = \tan(t)\text{,}\)
- its domain is the set of all real numbers except \(t = \frac{\pi}{2} \pm k\pi\) where \(k\) is any whole number;
- its range is the set of all real numbers;
- its period is \(P = \pi\text{;}\)
- is increasing on any interval on which the function is defined at every point in the interval.
While the tangent function is an interesting mathematical function for its own sake, its most important applications arise in the setting of right triangles, and for the remainder of this section we will focus on that perspective.
Using the tangent function in right triangles
The tangent function offers us an additional choice when working in right triangles with limited information. In the setting where we have a right triangle with one additional known angle, if we know the length of the hypotenuse, we can use either the sine or cosine of the angle to help us easily find the remaining side lengths. But in the setting where we know only the length of one leg, the tangent function now allows us to determine the value of the remaining leg in a similarly straightforward way, and from there the hypotenuse.
Use the tangent function to determine the width, \(w\text{,}\) of the river in Figure \(\PageIndex{9}\) (Note that here we are revisiting the problem in Activity 4.1.4 , which we previously solved without using the tangent function.) What other information can we now easily determine?
- Solution
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Using the perspective that \(\tan(\theta) = \frac{\text{opp}}{\text{adj}}\) in a right triangle, in this context we have
\[ \tan(56.4^\circ) = \frac{w}{50} \nonumber \]and thus \(w = 50\tan(56.4)\) is the exact width of the river. Using a computational device, we find that \(w \approx 75.256\text{.}\)
Once we know the river's width, we can use the Pythagorean theorem or the sine function to determine the distance from \(P\) to \(A\text{,}\) at which point all \(6\) parts of the triangle are known.
The tangent function finds a wide range of applications in finding missing information in right triangles where information about one or more legs of the triangle is known.
The top of a \(225\) foot tower is to be anchored by four cables that each make an angle of \(32.5^{\circ}\) with the ground. How long do the cables have to be and how far from the base of the tower must they be anchored?
Supertall 1 high rises have changed the Manhattan skyline. These skyscrapers are known for their small footprint in proportion to their height, with their ratio of width to height at most \(1:10\text{,}\) and some as extreme as \(1:24\text{.}\) Suppose that a relatively short supertall has been built to a height of \(635\) feet, as pictured in Figure \(\PageIndex{10}\) , and that a second supertall is built nearby. Given the two angles that are computed from the new building, how tall, \(s\text{,}\) is the new building, and how far apart, \(d\text{,}\) are the two towers?
Surveyors are trying to determine the height of a hill relative to sea level. First, they choose a point to take an initial measurement with a sextant that shows the angle of elevation from the ground to the peak of the hill is \(19^\circ\text{.}\) Next, they move \(1000\) feet closer to the hill, staying at the same elevation relative to sea level, and find that the angle of elevation has increased to \(25^\circ\text{,}\) as pictured in Figure \(\PageIndex{11}\) . We let \(h\) represent the height of the hill relative to the two measurements, and \(x\) represent the distance from the second measurement location to the “center” of the hill that lies directly under the peak.
- Using the right triangle with the \(25^\circ\) angle, find an equation that relates \(x\) and \(h\text{.}\)
- Using the right triangle with the \(19^\circ\) angle, find a second equation that relates \(x\) and \(h\text{.}\)
- Our work in (a) and (b) results in a system of two equations in the two unknowns \(x\) and \(h\text{.}\) Solve each of the two equations for \(h\) and then substitute appropriately in order to find a single equation in the variable \(x\text{.}\)
- Solve the equation from (c) to find the exact value of \(x\) and determine an approximate value accurate to \(3\) decimal places.
- Use your preceding work to solve for \(h\) exactly, plus determine an estimate accurate to \(3\) decimal places.
- If the surveyors' initial measurements were taken from an elevation of \(78\) feet above sea level, how high above sea level is the peak of the hill?
Summary
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The tangent function is defined defined to be the ratio of the sine and cosine functions according to the rule
\[ \tan(t) = \frac{\sin(t)}{\cos(t)} \nonumber \]
for all values of \(t\) for which \(\cos(t) \ne 0\text{.}\)
- The graph of the tangent function differs substantially from the graphs of the sine and cosine functions, primarily because near values where \(\cos(t) = 0\text{,}\) the ratio of \(\frac{\sin(t)}{\cos(t)}\) increases or decreases without bound, producing vertical asymptotes. In addition, while the period of the sine and cosine functions is \(P = 2\pi\text{,}\) the period of the tangent function is \(P = \pi\) due to how the sine and cosine functions repeat the same values (with different signs) as a point traverses the unit circle.
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The tangent function finds some of its most important applications in the setting of right triangles where one leg of the triangle is known and one of the non-right angles is known. Computing the tangent of the known angle, say \(\alpha\text{,}\) and using the fact that
\[ \tan(\alpha) = \frac{\text{opp}}{\text{adj}} \nonumber \]
we can then find the missing leg's length in terms of the other and the tangent of the angle.
Exercises
A wheelchair ramp is to be built so that the angle it forms with level ground is \(4^{\circ}\text{.}\) If the ramp is going to rise from a level sidewalk up to a front porch that is \(3\) feet above the ground, how long does the ramp have to be? How far from the front porch will it meet the sidewalk? What is the slope of the ramp?
A person is flying a kite and at the end of a fixed length of string. Assume there is no slack in the string.
At a certain moment, the kite is \(170\) feet off the ground, and the angle of elevation the string makes with the ground is \(40^\circ\text{.}\)
- How far is it from the person flying the kite to another person who is standing directly beneath the kite?
- How much string is out between the person flying the kite and the kite itself?
- With the same amount of string out, the angle of elevation increases to \(50^\circ\text{.}\) How high is the kite at this time?
An airplane is flying at a constant speed along a straight path above a straight road at a constant elevation of \(2400\) feet. A person on the road observes the plane flying directly at them and uses a sextant to measure the angle of elevation from them to the plane. The first measurement they take records an angle of \(36^\circ\text{;}\) a second measurement taken \(2\) seconds later is \(41^\circ\text{.}\)
How far did the plane travel during the two seconds between the two angle measurements? How fast was the plane flying?
<1> See, for instance, this article