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3232Complete the Description of the Piecewise Function Graphed Below
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Wed, 07 Sep 2022 17:11:45 +0000https://estatename.com/?p=1003EstateName.com – Complete the Description of the Piecewise Function Graphed Below Piecewise functions are functions that have a different expression for different intervals in their domain. These functions can be thought of as functions composed of different pieces of two or more functions. In this article, we will see a more detailed definition of these …

]]>EstateName.com – Complete the Description of the Piecewise Function Graphed Below

Piecewise functions are functions that have a different expression for different intervals in their domain. These functions can be thought of as functions composed of different pieces of two or more functions.

In this article, we will see a more detailed definition of these functions and we will learn how to obtain their graphs.

ALGEBRA

Relevant for…

Learning about piecewise functions and their graphs.

See definition

ALGEBRA

Relevant for…

Learning about piecewise functions and their graphs.

See definition

What is a piecewise function?

Piecewise functions are functions that are defined by different formulas or functions for each interval. As the name implies, these functions are defined by chunks of functions for each part of the domain.

In the function above, we can see thatf(x) is a piecewise function since it is defined differently for the three intervalsx> 0,x = 0, andx <0. We can interpret piecewise functions by looking at the different given intervals. The functionf (x) given above can be read as:

Whenx> 0,f(x) is equal to 3x.

Whenx = 0,f(x) is equal to 2,

Whenx <0,f(x) equals -3x.

These changes can be clearly observed in the graph of the function:

How to solve piecewise defined functions?

Now that we have learned a bit about these functions, we need to learn how to solve piecewise functions.

To solve piecewise functions, we have to take into account the following:

Check carefully where thex lies in the given interval.

Evaluate the value using the corresponding function.

For example, let’s say we want to findf(5) in the following function:

Since 5 is greater than 0, the function with which we will use to evaluatef (5) is f(x) = 3x . Therefore, we havef(5) = 3 (5) = 15.

This also means that we havef(0) = 2 and alsof(-1) = -3 (-1) = 3.

To graph piecewise functions, we have to consider that each interval will have a different graph since the function is different in each interval.

We can take into account the following recommendations when graphing piecewise functions:

We can think about how each function will look individually.

For inclusive intervals (such as x≥0), we include the end points using filled points.

For exclusive intervals (such as x> 0), we exclude the end points with empty points.

Some of the more common functions we can expect are the following:

Constant functions likef(x) = 4.

Linear functions likef(x) = 3x+1.

Quadratic functions likef(x) = 2x²+x-3.

You can explore more types of functions and their graphs in our article ontypes of functions. This will help you get an idea of what we can expect each individual graph to look like.

Now, let’s graph the following function as an example:

When we havex>0, andx<0, the function returns a linear expression. We can graph these linear parts simply by using two points that satisfy these expressions and drawing a line through the points taking into account thatf(x) = 2x+1 only corresponds to values ofx greater than 0 and thatf(x) = –x-3, only corresponds to values ofx less than 0. Since both are exclusive inequalities, we have an empty point at their endpoints:

Now, we only have to complete the condition whenx = 0. Since the value is constant atf(x) = 1, we can graph the point (0, 1):

This is the final graph of this piecewise function. In the graph, we can see that the function has a domain of (-∞, ∞) and a range of (-3, ∞).

See also

Interested in learning more about functions? Take a look at these pages:

Finding Asymptotes of a Function

Floor and Ceiling Functions

Learn mathematics with our additional resources in different topics

LEARN MORE

Complete the Description of the Piecewise Function Graphed Below

]]>Which of the Following Shows the Diameter of a Circle
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Wed, 07 Sep 2022 14:58:25 +0000https://estatename.com/?p=997EstateName.com – Which of the Following Shows the Diameter of a Circle Related Pages Circles Tangent Of A Circle Chords Of A Circle The following figures show the different parts of a circle: tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, major sector. Scroll down the page for more …

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Related Pages

Circles
Tangent Of A Circle
Chords Of A Circle

The following figures show the different parts of a circle: tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, major sector. Scroll down the page for more examples and explanations.

Circle

In geometry, a circle is a closed curve formed by a set of points on a plane that are the same distance from its center O. That distance is known as the radius of the circle.

Diameter

The diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle. All the diameters of the same circle have the same length.

Chord

A chord
is a line segment with both endpoints on the circle. The diameter is a special chord that passes through the center of the circle. The diameter would be the longest chord in the circle.

Radius

The radius of the circle is a line segment from the center of the circle to a point on the circle. The plural of radius is radii.

In the above diagram, O
is the center of the circle and

and
are radii of the circle. The radii of a circle are all the same length. The radius is half the length of the diameter.

Arc

An arc is a part of a circle.

In the diagram above, the part of the circle from B to C forms an arc.

An arc can be measured in degrees.

In the circle above, arc BC
is equal to the ∠BOC
that is 45°.

Tangent

A tangent is a line that touches a circle at only one point. A tangent is perpendicular to the radius at the point of contact. The point of tangency is where a tangent line touches the circle.

In the above diagram, the line containing the points B and C is a tangent to the circle.

It touches the circle at point B and is perpendicular to the radius
. Point B is called the point of tangency.

is perpendicular to

i.e.

Parts Of A Circle

The following video gives the definitions of a circle, a radius, a chord, a diameter, secant, secant line, tangent, congruent circles, concentric circles, and intersecting circles.

A secant line
intersects the circle in two points.

A tangent
is a line that intersects the circle at one point.

A point of tangency
is where a tangent line touches or intersects the circle.

Congruent circles
are circles that have the same radius but different centers.

Concentric circles
are two circles that have the same center, but a different radii.

Intersecting Circles: Two circles may intersect at two points or at one point. If they intersect at one point then they can either be externally tangent or internally tangent.

Two circles that do not intersect can either have a common external tangent or common internal tangent.
In the common external tangent, the tangent does not cross between the two circles.
In the common internal tangent, the tangent crosses between the two circles.

Show Video Lesson

Parts Of A Circle, Including Radius, Chord, Diameter, Central Angle, Arc, And Sector

Show Video Lesson

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Which of the Following Shows the Diameter of a Circle

]]>Which of the Following Statements is True of Imagined Risks
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Wed, 07 Sep 2022 12:45:05 +0000https://estatename.com/?p=991EstateName.com – Which of the Following Statements is True of Imagined Risks Having trouble coping? Stressed out? Feeling overwhelmed? If your answer is YES, you are not alone. Everyone feels stressed from time to time. Some people, though, say that they feel very stressed most of the time. In fact, 21% of Canadians aged 12 …

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Having trouble coping?

Stressed out?

Feeling overwhelmed?

If your answer is YES, you are not alone.

Everyone feels stressed from time to time. Some people, though, say that they feel very stressed most of the time. In fact, 21% of Canadians aged 12 and older rated their life stress as quite a bit or extremely stressful. While stress isn’t always bad and can even be helpful, too much stress can affect your well-being negatively.

Top

What causes stress?

Stress comes up when you feel like the demands of a situation or event are too much to manage. It can come up with everyday situations, such as:

Managing demands at work or school

Managing relationships

Managing finances

Dealing with unfair treatment

Managing long-term health problems

Stress can also come up in response to a specific event or situation. Both positive and negative life events can be stressful, especially those that involve major changes to your regular routines. Here are some examples:

Changes in a relationship

Changes in housing arrangements

The death of a family member or friend

Changes in your job or other source of income

As stress is based on the way you see and react to a situation or event, the events or situations that cause stress are different for different people.

How you feel when issues come up can also affect the way you experience stress. If you feel well and confident in your ability to manage challenges, a problem may not seem very stressful. However, if you already feel stressed or overwhelmed, the same problem may add to your existing stress and feel very overwhelming.

Are you experiencing signs of stress?

Stress can affect your body, your behaviours, your feelings and your thoughts. Here are common signs of stress:

Changes in your body

Tense muscles

Rapid breathing and heart rate

Headaches

Difficulties sleeping well

Fatigue

Changes in sex drive

Weaker immune system

Changes in your behaviours

Withdrawing from others

Fidgeting, feeling restless

Smoking, drinking or using more drugs than usual

Avoiding situations that you think are stressful

Changes in your feelings

Feeling worried or confused

Feeling angry or irritable

Feeling overwhelmed or helpless

Feeling like you can’t cope

Changes in your thoughts

Struggling to concentrate, remember or make decisions

Losing your self-confidence

Having a negative attitude towards yourself and your life

Top

Why does stress make me feel so awful?

Your body is designed to react to stress in ways that protect you from threats, such as predators. Although facing life-threatening predators is not common today, you do have many daily demands, such as paying bills, working, and taking care of family. Your body treats these demands as threats and the fight-flight-freeze response is activated in your body. At times, stress can have a negative effect on the basic dimensions of mental health (your thoughts, emotions, behaviours and body reactions). Stress may affect your health if you use unhealthy behaviours to cope. The negative effects of stress on your wellness can become a source of stress in and of themselves. Just talk to anyone who isn’t sleeping well due to stress! For more information about the dimensions of mental health that can be influenced by stress, see Wellness Module 1: Mental Health Matters at www.heretohelp.bc.ca.

Top

Can stress be a good thing?

Sometimes, stress can have a positive effect on your thoughts, emotions, behaviours and body reactions. Stress is more likely to result in positive outcomes if you see it as a challenge or something you can strive to overcome. It can help motivate you to work hard. Stress can also lead to positive effects if you respond with healthy behaviours that improve your situation. You can experience the positive effects of stress even in the face of some negative effects.

Top

Stress and illness

Your stress levels and your coping skills can also influence your physical health. Higher levels of stress can increase the risk of illness and disease. For example, you’re more likely to catch a cold or the flu when you’re coping with high levels of stress. There is also evidence that stress can aggravate disorders such as rheumatoid arthritis, insulin-dependent diabetes, multiple sclerosis and more. Chronic stress also has a negative impact on your physical health. Some of the connections between stress and illness are determined by the ways you cope with stress.

Top

Coping with stress

There is no right or wrong way to cope with stress. What works for one person may not work for another, and what works in one situation may not work in another situation. Below, you will find common ways to cope with stress and maintain wellness.

Focus on what you can do

There is usually something you can do to manage stress in most situations.

Resist the urge to give up or run away from problems—these coping choices may feel good in the short-term, but often make stress worse in the long run

Manage your emotions

Feelings of sadness, anger or fear are common when coping with stress.

Try expressing your feelings by talking or writing them down. Bottling up your emotions makes it harder to cope with stress

Try not to lash out at other people. Yelling or swearing usually pushes people away when you need them the most

Many of the coping strategies listed below are useful ways of managing your emotions

Seek out support

Seeking social support from other people is helpful—especially when you feel you can’t cope on your own. Family, friends, co-workers and health professionals can all be important sources of support.

Ask someone for their opinion or advice on how to handle the situation

Get more information to help make decisions

Accept help with daily tasks and responsibilities, such as chores or child care

Get emotional support from someone you trust who understands you and cares about you

Focus on helpful and realistic thoughts

This is one of the hardest things to do when coping with stress. At times, it can seem impossible. But, dwelling on the negatives often adds to your stress and takes away your motivation to make things better.

Focus on strengths rather than weaknesses—remind yourself that no one is perfect; think of times where you have been able to overcome challenges in the past

Look for the challenges in a situation by asking, “What can I learn from this?” or, “How can I grow as a person?”

Try to keep things in perspective—is it a hassle or a horror?

Try to keep a sense of humour

Remind yourself you are doing the best you can given the circumstances

Make a plan of action

Problem-solving around aspects of a situation that you can control is one of the most effective ways to lower your stress.

Try breaking a stressful problem into manageable chunks.

Think about the best way to approach the problem. You may decide to put other tasks on hold to concentrate on the main problem, or you may decide to wait for the right time and place to act.

Identify and define the problem

Determine your goal

Brainstorm possible solutions

Consider the pros and cons of each possible solution

Choose the best solution for you—the perfect solution rarely exists

Put your plan into action

Evaluate your efforts and choose another strategy, if needed

See our Wellness Module on problem-solving at www.heretohelp.bc.ca

Self-care

Taking good care of yourself can be difficult during stressful times, but self-care can help you cope with problems more effectively. The trick to self-care is to look for little things you can do everyday to help yourself feel well.

Here are some self-care activities to try. Try to think of other activities that might help!

Eat healthy foods and drink lots of water throughout the day to maintain your energy

Try to exercise or do something active on a regular basis

Try to avoid using alcohol or drugs as a way to cope

Explore relaxation techniques like deep breathing, meditation or yoga

Spend time with family and friends

Spend time on things you enjoy, such as hobbies or other activities

Get a good night’s sleep

Take care of your relationships

Family, friends and co-workers can be affected by your stress—and they can also be part of the problem.

Be assertive about your needs rather than aggressive or passive. Being assertive means expressing your needs in a respectful way, which allows you to keep your feelings and needs, as well as the feelings and needs of others, in mind.

Try to discuss your concerns with others in a firm and calm voice

Consider the other person’s point of view—if needed, take some time before responding

Accept responsibility, apologize or try to put things right when appropriate

Talk to others who are involved and keep them informed about your decisions

Spirituality

Spirituality takes many forms and means different things to different people. It can vary from culture to culture, with religion being one way that people experience or express spirituality. People who engage in a spiritual practice often experience lower levels of distress. If community is part of a spiritual practice, it may also offer helpful social support.

Consider spiritual practices that fit with your beliefs, such as prayer, mediation, tai chi, enjoying nature, or creating art

If you have a formal place of worship, spend time there, or get together with others who share your beliefs

Talk with a respected member or leader of your spiritual community

Acceptance

There may be times when you can’t change something. This can be the most challenging aspect of coping with stress. Acceptance means allowing unpleasant feelings and sensations to surface and come and go without trying to resist or fight them. It allows people to recognize and come to terms with what is out of their control while focusing on the actions they can take to improve their lives. Sometimes, all you can do is manage your distress or grief.

Acceptance is a process that takes time. You may need to remind yourself to be patient

Denying that the problem exists may prolong your suffering and interfere with your ability to take action

Death, illness, major losses or major life changes can be particularly difficult to accept

Engage in the present moment—try not to get caught up in wishful thinking or dwell on what could have been, but focus on what you are experiencing in the here-and-now

Distraction

Distraction can be helpful when coping with short-term stress you can’t control, such as reading a magazine while getting dental work done. Distraction strategies can help you to tolerate distress until it is a more appropriate time to resolve the issue.

Distraction can be harmful if it stops you from taking action on things you can control, such as watching TV when you have school or work deadlines to meet.

Distraction by using drugs, alcohol or over-eating often leads to more stress and problems in the long term.

Distraction by overworking at school or on the job can easily lead to burnout or other problems, like family resentment.

You can do many things to take your mind off problems, such as:

Daydreaming

Going for a drive or walk

Doing something creative

Leisure activities, exercise, hobbies

Housework, yard work or gardening

Watching TV or movies

Playing video games

Spending time with friends or family

Spending time with pets

Connecting with others on social media

Sleeping or taking a short nap

When used for short periods of time, many of these forms of distraction create opportunities to take a break and refuel—an important part of self-care.

Top

If you feel like you can’t cope, try these options

Talk to someone that cares about you. They may be able to provide help and support.

Seek professional help as early as possible. Prevention strategies can strengthen protective factors and improve mental health. Talk to your family doctor or mental health care provider, or visit a drop-in clinic or the hospital emergency room.

Top

Try these helpful numbers

Crisis Line: Call 310-6789 (no area code) to connect to a crisis line in BC

Kids’ Help Phone: Call 1-800-668-6868 (free call anywhere in Canada)

Top

Stress survey: What types of stress are you coping with?

Problem solving the controllable aspects of a stressful situation is one of the most effective ways to lower our stress. Identifying the problem and breaking it down into manageable chunks is the first step in creating a plan of action.

You can use this survey to help identify the different sources of stress in your life, and to track your progress in coping with them in a healthy way. Check the boxes beside the sentences you feel apply to you, then brainstorm strategies for coping with or solving each problem. See the Coping with Stress section of this wellness module for more information and tips on how to ensure your coping choices lead to reductions in stress and a healthier, more fulfilling life for you and your loved ones. Try taking the survey once a month to track patterns in your behaviour—and the positive and negative ways you manage stress.

Adapted from: Holmes and Rahe, 1967; Wheaton, 1997.

About the authors

Canadian Mental Health Association BC Division helps people access the community resources they need to maintain and improve mental health, build resilience, and support recovery from mental illness. CMHA BC has served BC for over 60 years.

Anxiety Canada promotes awareness of anxiety disorders and increases access to proven resources. Visit www.anxietycanada.com.

]]>0.9 0.72
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Wed, 07 Sep 2022 10:31:45 +0000https://estatename.com/?p=983EstateName.com – 0.9 0.72 simmental 1st Enter an EPD Value 2nd Enter an EPD Value 3rd Enter an EPD Value SEARCH * SEARCH RESULTS ARE SORTED BY YOUR FIRST CHOICE New Bull Recently Updated * Black * Red BREED AVERAGES: 10.6 1.8 74.8 110.7 23.7 32.2 .87 .05 -.45 127.74 74.65 9.1 2 73.5 97.6 17.1 19 …

]]>0.003 is 1/10 of What Decimal
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Wed, 07 Sep 2022 08:18:25 +0000https://estatename.com/?p=975EstateName.com – 0.003 is 1/10 of What Decimal NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health. Woloshin S, Schwartz LM, Welch HG. Know Your Chances: Understanding Health Statistics. Berkeley (CA): University of California Press; 2008. Know Your Chances: Understanding Health Statistics. Show details Contents Number Converter and Risk Charts …

NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health.

Woloshin S, Schwartz LM, Welch HG. Know Your Chances: Understanding Health Statistics. Berkeley (CA): University of California Press; 2008.

Know Your Chances: Understanding Health Statistics.

Show details

Contents

Number Converter and Risk Charts

Number Converter

View in own window

1 in __

Decimal

Percent

__ out of 1,000

1 in 1

1.00

100%

1,000 out of 1,000

1 in 2

0.50

50%

500 out of 1,000

1 in 3

0.33

33%

333 out of 1,000

1 in 4

0.25

25%

250 out of 1,000

1 in 5

0.20

20%

200 out of 1,000

1 in 6

0.17

17%

167 out of 1,000

1 in 7

0.14

14%

143 out of 1,000

1 in 8

0.13

13%

125 out of 1,000

1 in 9

0.11

11%

111 out of 1,000

1 in 10

0.10

10%

100 out of 1,000

1 in 20

0.05

5.0%

50 out of 1,000

1 in 25

0.04

4.0%

40 out of 1,000

1 in 50

0.02

2.0%

20 out of 1,000

1 in 100

0.01

1.0%

10 out of 1,000

1 in 200

0.0050

0.50%

5 out of 1,000

1 in 250

0.0040

0.40%

4 out of 1,000

1 in 300

0.0033

0.33%

3.3 out of 1,000

1 in 400

0.0025

0.25%

2.5 out of 1,000

1 in 500

0.0020

0.20%

2.0 out of 1,000

1 in 600

0.0017

0.17%

1.7 out of 1,000

1 in 700

0.0014

0.14%

1.4 out of 1,000

1 in 800

0.0013

0.13%

1.3 out of 1,000

1 in 900

0.0011

0.11%

1.1 out of 1,000

1 in 1,000

0.0010

0.10%

1.0 out of 1,000

1 in 2,000

0.00050

0.050%

0.50 out of 1,000

1 in 3,000

0.00033

0.033%

0.33 out of 1,000

1 in 4,000

0.00025

0.025%

0.25 out of 1,000

1 in 5,000

0.00020

0.020%

0.20 out of 1,000

1 in 10,000

0.00010

0.010%

0.10 out of 1,000

1 in 25,000

0.00004

0.004%

0.040 out of 1,000

1 in 50,000

0.00002

0.002%

0.020 out of 1,000

1 in 100,000

0.00001

0.001%

0.010 out of 1,000

1 in 1,000,000

0.000001

0.0001%

0.001 out of 1,000

Note: For numbers less than 1 out of 1,000 (such as 0.50 out of 1,000), it is clearer to recast them as “___ out of 10,000” (“5 out of 10,000,” for instance), because it allows you to use a whole number rather than a decimal.

Risk Chart for Men

Find the line closest to your age and smoking status. The numbers in that row tell you how many out of 1,000 men in that group will die in the next 10 years from . . .

View in own window

Age

Smoking Status

Vascular Disease

Cancer

Infection

Lung Disease

Accidents

All Causes Combined

Heart Attack

Stroke

Lung

Colon

Prostate

Pneumonia

Flu

AIDS

COPD

35

Never smoked

1

1

2

5

15

Smoker

7

1

1

2

5

42

40

Never smoked

3

1

1

1

2

6

24

Smoker

14

2

4

1

2

1

6

62

45

Never smoked

6

1

1

1

2

6

35

Smoker

21

3

8

1

1

2

2

6

91

50

Never smoked

11

1

1

2

1

1

1

5

49

Smoker

29

5

18

2

1

1

1

3

5

128

55

Never smoked

19

3

1

3

2

1

1

1

5

74

Smoker

41

7

34

3

1

2

1

7

4

178

60

Never smoked

32

5

2

5

3

2

1

1

5

115

Smoker

56

11

59

5

3

3

1

16

4

256

65

Never smoked

52

9

4

8

6

3

3

6

176

Smoker

74

16

89

7

6

5

26

5

365

70

Never smoked

87

18

6

10

12

6

5

7

291

Smoker

100

26

113

9

10

9

45

6

511

75

Never smoked

137

32

8

13

19

12

6

11

449

Smoker

140

39

109

11

15

16

60

9

667

Source: Steven Woloshin, Lisa Schwartz, and H. Gilbert Welch, “The Risk of Death by Age, Sex, and Smoking Status in the United States: Putting Health Risks in Context,” Journal of the National Cancer Institute
100 (2008): 845–853.

Note: Shaded portions mean that the chance is less than 1 out of 1,000. People who have never smoked are defined as those who do not smoke now and who have smoked fewer than 100 cigarettes in their lifetime. Smokers are defined as people who have smoked at least 100 cigarettes in their lifetime and who currently smoke (any amount). COPD is chronic obstructive pulmonary disease, which includes emphysema and chronic bronchitis. The numbers in the All Causes Combined column do not represent row totals because they include many other causes of death in addition to the ones listed in the chart.

Risk Chart for Women

Find the line closest to your age and smoking status. The numbers in that row tell you how many out of 1,000 women in that group will die in the next 10 years from . . .

View in own window

Age

Smoking Status

Vascular Disease

Cancer

Infection

Lung Disease

Accidents

All Causes Combined

Heart Attack

Stroke

Lung

Breast

Colon

Ovarian

Cervical

Pneumonia

Flu

AIDS

COPD

35

Never smoked

1

1

1

2

14

Smoker

1

1

1

1

1

2

14

40

Never smoked

1

2

1

1

2

19

Smoker

4

2

4

2

1

1

2

27

45

Never smoked

2

1

1

3

1

1

1

2

25

Smoker

9

3

7

3

1

1

1

1

2

2

45

50

Never smoked

4

1

1

4

1

1

2

37

Smoker

13

5

14

4

1

1

1

4

2

69

55

Never smoked

8

2

2

6

2

2

1

1

1

2

55

Smoker

20

6

26

5

2

2

1

1

9

2

110

60

Never smoked

14

4

3

7

3

3

1

1

2

2

84

Smoker

31

8

41

6

3

3

1

2

18

2

167

65

Never smoked

25

7

5

8

5

4

1

2

3

3

131

Smoker

45

15

55

7

5

3

1

4

31

3

241

70

Never smoked

46

14

7

9

7

4

1

4

5

4

207

Smoker

66

25

61

8

6

4

1

7

44

4

335

75

Never smoked

86

30

7

10

10

5

1

8

6

7

335

Smoker

99

34

58

10

9

4

14

61

7

463

Source: Steven Woloshin, Lisa Schwartz, and H. Gilbert Welch, “The Risk of Death by Age, Sex, and Smoking Status in the United States: Putting Health Risks in Context,” Journal of the National Cancer Institute
100 (2008): 845–853.

Note: Shaded portions mean that the chance is less than 1 out of 1,000. People who have never smoked are defined as those who do not smoke now and who have smoked fewer than 100 cigarettes in their lifetime. Smokers are defined as people who have smoked at least 100 cigarettes in their lifetime and who currently smoke (any amount). COPD is chronic obstructive pulmonary disease, which includes emphysema and chronic bronchitis. The numbers in the All Causes Combined column do not represent row totals because they include many other causes of death in addition to the ones listed in the chart.

Know Your Chances: Understanding Health Statistics
is hereby licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license, which permits copying, distribution, and transmission of the work, provided the original work is properly cited, not used for commercial purposes, nor is altered or transformed.

]]>Shakespeare’s Plays Often Contain Puns Which Are
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Wed, 07 Sep 2022 06:05:05 +0000https://estatename.com/?p=968EstateName.com – Shakespeare’s Plays Often Contain Puns Which Are Want to know more about Shakespeare comedies? Then read on… Traditionally Shakespeare play types are categorised as C omedy, History, Roman and Tragedy, with some additional categories proposed over the years. Shakespeare comedies (or rather the plays of Shakespeare that are usually categorised as comedies) are …

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Want to know more about Shakespeare comedies? Then read on…

Traditionally Shakespeare play types are categorised as C omedy,

History, Roman and

Tragedy, with some additional categories proposed over the years. Shakespeare comedies (or rather the plays of Shakespeare that are usually categorised as comedies) are generally identifiable as plays full of fun, irony and dazzling wordplay. They also abound in disguises and mistaken identities, with very convoluted plots that are difficult to follow with very contrived endings.

Any attempt at describing Shakespeare’s comedy plays as a cohesive group can’t go beyond that superficial outline. The highly contrived endings of most Shakespeare comedies are the clue to what these plays – all very different – are about.

Take The Merchant of Venice
for example – it has the love and relationship element. As is often the case, there are two couples. One of the women is disguised as a man through most of the text – typical of Shakespearean comedy – but the other is in a very unpleasant situation – a young Jewess seduced away from her father by a shallow, rather dull young Christian. The play ends with the lovers all together, as usual, celebrating their love and the way things have turned out well for their group. That resolution has come about by completely destroying a man’s life.

The Jew, Shylock is a man who has made a mistake and been forced to pay dearly for it by losing everything he values, including his religious freedom. It is almost like two plays – a comic structure with a personal tragedy embedded in it. The ‘comedy’ is a frame to heighten the effect of the tragic elements, which creates something very deep and dark.

Twelfth Night is similar – the humiliation of a man the in-group doesn’t like. As in The Merchant of Venice, his suffering is simply shrugged off in the highly contrived comic ending.

Not one of Shakespearean comedy, no matter how full of life and love and laughter and joy, it may be, is without a darkness at its heart. Much Ado About Nothing , likeAntony and Cleopatra
(a ‘tragedy’ with a comic structure), is a miracle of creative writing. Shakespeare seamlessly joins an ancient mythological love story and a modern invented one, weaving them together into a very funny drama in which light and dark chase each other around like clouds and sunshine on a windy day, and the play threatens to fall into an abyss at any moment and emerges from that danger in a highly contrived ending once again.

Like the ‘tragedies’ Shakespeare comedies defy categorisation. They all draw our attention to a range of human experience with all its sadness, joy, poignancy, tragedy, comedy, darkness and lightness. Below are all of the plays generally regarded as Shakespeare comedy plays.

Origins of Comedy Plays

Early Greek comedy was in sharp contrast to the dignity and seriousness of tragedy. Aristophanes, the towering giant of comedy, used every kind of humour from the slapstick through sexual jokes to satire and literary parody. Unlike tragedy, the plots didn’t originate in traditional myth and legend but were the product of the writer’s creative imagination. The main theme was political and social satire. Over the centuries comedy moved away from those themes to focus on family matters, notably a concentration on relationships and the complications of love. Such a universal theme was bound to survive and, indeed, it has travelled well, from Greece through Roman civilization and, with the Renaissance preoccupation with things classical, into Renaissance Europe, to England and the Elizabethans, and into the modern world of the twentieth and twenty-first centuries, where we see Greek comedy alive and well in films and television.

Shylock in Shakespeare comedy play The Merchant of Venice

]]>Which Cube Root Function is Always Decreasing as X Increases
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Wed, 07 Sep 2022 03:51:45 +0000https://estatename.com/?p=960EstateName.com – Which Cube Root Function is Always Decreasing as X Increases 1.3 The Behavior of Functions Intro If you’ve ever ridden a roller coaster you know how it feels to crawl up a hill and rocket down the other side. You also know how your stomach floats at the top and sinks at the …

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1.3 The Behavior of Functions

Intro

If you’ve ever ridden a roller coaster you know how it feels to crawl up a hill and rocket down the other side. You also know how your stomach floats at the top and sinks at the bottom. It would be a very boring ride if the track never changed direction. The twists and turns, peaks and valleys make a roller coaster exciting and give it character.

The same can be said for functions: they wouldn’t be interesting (or useful) if they didn’t change. The way a function changes is often called its “behavior”.

In this section you will learn to recognize function behaviors by looking at a graph, in much the same way that you could pick out the loops and dips of a roller coaster from a photograph.

Increasing, Decreasing and Constant

When we read the shape of a function’s graph, we always read from left to right. For instance, a function increases
if its graph goes up as we move from left to right and decreases
if it goes down. These behaviors are also referred to as growth and decay, respectively. If the graph is flat then the function is constant
on that interval; it does not grow or decay.

Consider the following graph.

This function is increasing on the interval $(a,b)$, constant on $(b,c)$, decreasing over the interval $(c,d)$ and increasing again on $(d,e)$.

A function is not increasing, decreasing or constant at the endpoints of an interval so we always write those intervals with parenthesis $( \thinspace\thinspace)$, never with brackets $[ \thinspace\thinspace ]$

QUICK CHECK

Determine the intervals on which the graph below is increasing, decreasing or constant.

Answer

The graph is constant on $(-3, -1)$, increasing on $(-1, 0)$, and decreasing on the interval $(0, 3)$.

Maximums and Minimums

Any point where a function changes from increasing to decreasing is known as a local maximum. On a graph it looks something like this.

If $f$ has a local maximum at $x=a$, then $f(a)$ is greater than any other function value in its immediate neighborhood, to the left or right.

If it turns out that $f(a)$ is the largest value of the function over its entire domain, then $f(a)$ is called a global maximum

Likewise, the point where a function changes from decreasing to increasing is called a local minimum.

If $f$ has a local minimum at $x=b$, then $f(b)$ is less than any other function values in the immediate neighborhood.

If it happens that $f(b)$ is the lowest value over the entire domain of the function, then $f(b)$ is called a global minimum.

Maximum and minimum points are important in many real life applications. For instance, we might want to minimize cost, or maximize fuel efficiency, or minimize waste.

The process of finding exact maximum and minimum values is called optimization
and is properly studied in calculus. For our needs, however, a visual inspection of the graph will usually be accurate enough. When more precision is needed, the maximum
and minimum
commands found on most graphing utilities can be used.

QUICK CHECK

Use the graph below to answer the questions that follow.

At which $x$ value does the function obtain a local maximum? What is value of the local maximum?

Answer

The local maximum occurs at $x=0$. The value of the local maximum is $f(0)= 2$.

At which $x$ value does the function obtain a local minimum? What is value of the local minimum?

Answer

The local minimum occurs at $x=2$. The value of the local minimum is $f(2)= 0$.

At which $x$ value does the function obtain a global minimum? What is value of the global minimum?

Answer

The global minimum occurs at $x=-3$. The value of the global minimum is $f(-3)= -4$.

Intercepts

In addition to maximum and minimum points, we should also look for points where the graph intersects the axes. The point where the graph crosses the y-axis is called the y-intercept. An x-intercept
is a point where the graph intersects the x-axis.

Both of these concepts should be familiar to you from earlier algebra courses. The only new thing to be aware of is that x-intercepts will also be known as real zeros.

QUICK CHECK

Decide if the following statements are True or False.

True or False: The x-coordinate of a y-intercept is $0$.

Answer

True.

True or False: The y-coordinate of an x-intercept is $0$.

Answer

True.

True or False: A function can have several y-intercepts.

Answer

False. A graph that has multiple y-intercepts, like this example,

will not pass the vertical line test and cannot be a function.

True or False: A function can have several x-intercepts.

Answer

True. A function can have many x-intercepts. This function, for instance, has four x-intercepts.

True or False: A “real zero” is the same thing as an x-intercept.

Answer

True. That is a new vocabulary term.

Asymptotes

Sometimes a function increases (or decreases) without bound toward infinity as it approaches a particular $x$ value. When this happens we say the function has a vertical asymptote
at that $x$ value. In this example there is a vertical asymptote at $x=-1$.

Formally, the vertical line $x=c$ is a vertical asymptote of a function if $f(x)$ approaches either positive or negative infinity as $x$ gets near to $c$,. This is sometimes written $f(x)\rightarrow\pm \infty$ as $x \rightarrow c$.

We will use dotted vertical lines to indicate vertical asymptotes. Very few graphing utilities will draw asymptotes. You will have to infer their locations from the shape of the graph.

It is also possible for a function to eventually level off to the left or to the right. When this happens we say the function has a horizontal asymptote. In the example below, the function has a horizontal asymptote at $y=3$.

Technically speaking, the horizontal line $y=k$ is a horizontal asymptote of a function if $f(x)\rightarrow k$ as $x \rightarrow \pm \infty$.

Horizontal asymptotes describe a type of end behavior of a function. The end behavior
of a function is how the function changes as $x$ approaches positive infinity or negative infinity. When looking for end behaviors, it doesn’t matter how the function behaves in any local neighborhood, it only matters what happens far out at the ends. We will see functions with various combinations of increasing, decreasing and leveling off end behaviors.

Since a function can only have two end behaviors (one as $x \rightarrow \infty$ and another as $x \rightarrow -\infty$), a function can only have, at most, two horizontal asymptotes. There is no such limit on the number of vertical asymptotes.

QUICK CHECK

Try to identify each of the following behaviors. Click the arrows to check your answers.

y-intercept

This function has a y-intercept at $y=2$.

x-intercepts

There are three x-intercepts: $x=-1$, $x=3$, and $x=5$

Maximums

There is a global maximum at $(2, 4)$ and a local maximum at $(6, 2)$.

Minimums

There is a local minimum at $(4, -2)$, but the function does not have a global minimum.

Vertical Asymptotes

There appears to be a vertical asymptote at $x=-2$

Horizontal Asymptotes

The function seems to level off around $y=1$ as $x \rightarrow \infty$, so that is a horizontal asymptote.

Another important feature of a graph is its curvature, also known as its concavity. If a graph bends up, as if to form the side of a cup, then we say it is concave up
on that interval. If the graph bends down, like a frown, then it is concave down. If the graph is straight then it does not bend and does not have any concavity.

Some graphs change their concavity on different intervals. A point where a graph switches from being concave up to concave down, or vice versa, is called an inflection point. Locating inflection points exactly is a chore that is best left to calculus, though their locations can be approximated from a detailed graph.

Note that concavity does not tell us if a graph is increasing or decreasing. Instead, it gives us information about how quickly, or slowly, a function is changing.

For instance, a function that is both increasing and concave down will see its rate of growth slow down. But the growth rate of an increasing, concave up function will speed up. Rates of change will be studied in detail later in this chapter.

Complete Graphs

A graph that indicates all the important features of a function is called a complete graph. If a function has any intercepts, asymptotes, maximums and/or minimums, then those should be visible on a complete graph.

Creating a complete graph can be a difficult adventure. Very few, if any, graphing programs will automatically give you a complete graph. However, they all have commands for zooming in and out and changing the viewing window.

As you learn more about different types of functions, you will be able to reduce the amount of guess work by recognizing how elements of the equation affect the shape of the graph. For now, however, you will want to experiment with the different zoom and window commands available.

Determine Domain and Range from a Graph

A complete graph can help us identify the domain and range of a function. Since each point on the graph has an $x$ and a $y=f(x)$ coordinate, the domain and range are the spread of the $x$ and $y$ values, respectively.

For instance, a complete graph of $f(x)=x^{2}-3$ is shown below.

Since the graph continues to expand in the x-direction, its domain

is the set of all real numbers $\color{red}{ \lbrace x \thinspace \vert \thinspace x \text{ is in } \mathbb{R} \rbrace }$ or the interval $\color{red}{(\infty, \infty)}$ or as ausing set-builder notation . The $\color{blue}{y}$ values start at $\color{blue}{y=-3}$ and continue upward, so the range

is the set $\color{blue}{\lbrace y \thinspace \vert \thinspace y \geq -3 \rbrace}$ or the interval $\color{blue}{[-3, \infty)}$.

QUICK CHECK

Determine the domain and range of the function shown below. Assume the graph is a complete graph. Answer

The domain is all real numbers. The range is $(-\infty, 5]$, also written as $\lbrace y \thinspace \vert \thinspace y \leq 5 \rbrace$.

Let’s look at another example. Suppose the following is a complete graph of a function.

You might recall that solid dots are always considered part of the graph. Hollow dots, on the other hand, are used to indicate points that are not part of the graph. Here the domain

is the interval $\color{red}{(-2, 2 ]}$ and the range

is $\color{blue}{(-2, 4 ]}$.

In set notation the domain

is $\color{red}{ \lbrace x \thinspace \vert \thinspace -2 < x \leq 2 \rbrace }$ and the range

is $\color{blue}{\lbrace y \thinspace \vert \thinspace -2 < y \leq 4 \rbrace}$

QUICK CHECK

Determine the domain and range of the function shown below. Assume the graph is a complete graph. Answer

The domain is $(-3, 2]$, or $\lbrace x \thinspace \vert \thinspace -3 < x \leq 2 \rbrace$.

The range is $[-3, 5]$, or $\lbrace y \thinspace \vert \thinspace -3 \leq y \leq 5 \rbrace$.

If the graph has asymptotes then the domain and range might require the union of several intervals. In this case, the vertical asymptotes indicate that the values $x=-1$ and $x=1$ are not in the domain.

The domain

is split into three intervals that we union together: $\color{red}{(-\infty, -1) \bigcup (-1, 1) \bigcup (1, \infty)}$. The horizontal asymptote doesn’t impact the range, since the center section rises above the asymptote, so our range

is $\color{blue}{(-\infty, 4]}$.

Graphs of Basic Functions

Every time we encounter a new function we will use its graph to help us identify the domain and range and any other important features, such as intercepts, asymptotes, concavity, etc.. The equation will also be an essential tool, but lots of information can be gleaned from a quick look at the graph.

Below you will be introduced to six basic functions that you should learn by heart. Use the provided graphs to look for important behaviors. Click the link below each graph to check your answers or refer to this downloadable summary.

The Identity Function

The identity function $f(x)=x$ is special because the output is always the same as the input, for example $f(5)=5$ or $f(87)=87$. Its graph is the diagonal line through the origin with a slope of $1$.

Identity Function Properties

Domain and Range:
Both the domain and range include all real numbers.

Intercepts:
Since this function crosses at the origin, both the y-intercept and the x-intercept are at the point $(0, 0)$.

Increasing, Decreasing, Constant:
The identity function is increasing over its entire domain. It is never decreases or becomes constant.

Maximums and Minimums:
There are no maximums or minimums.

Concavity:
This function does not have any concavity.

Asymptotes:
There are no vertical or horizontal asymptotes.

The Square Function

The square function $f(x)=x^2$ returns the square of every input, for example $f(3)=3^2=9$. Its graph is a parabola touching the origin.

With negative inputs it is best to include parenthesis like this $f(-2)=(-2)^2=4$ to get the correct values.

Square Function Properties

The graph of the square function is a *parabola*.

Domain and Range:
The domain is all real numbers. The range is the interval $[0, \infty)$.

Intercepts:
They y-intercept is $y=0$. The x-intercept is $x=0$. Both occur at the same point, $(0, 0)$.

Increasing, Decreasing, Constant:
The function is decreasing on $(-\infty, 0)$ and increasing over the interval $(0, \infty)$. It is not constant anywhere.

Maximums and Minimums:
There are no maximums, but there is a global minimum at $(0, 0)$.

Concavity:
The square function is concave up over its entire domain.

Asymptotes:
There are no vertical asymptotes and no horizontal asymptotes.

The Square Root Function

The principal square root function $f(x)=\sqrt{x}$ evaluates the square root of non-negative inputs only, it is not defined for negative values.

Most calculators have a dedicated button for computing square roots. Its graph starts at the origin.

Square Root Function Properties

Domain and Range:
The domain is ${x : x \geq 0 }$. The range is ${y : y \geq 0 }$.

Intercepts:
Since this function starts at the origin, the y-intercept and the x-intercept are at the same point, $(0, 0)$.

Increasing, Decreasing, Constant:
The function is increasing over its entire domain. It is never decreasing or constant.

Maximums and Minimums:
There are no maximums, but $(0, 0)$ is a global minimum.

Concavity:
The square root function is concave down over its entire domain.

Asymptotes:
There are no vertical or horizontal asymptotes.

The Cube Function

The cube function $f(x)=x^3$ produces the cube of any input, which is simply that number multiplied by itself three times. For instance, $f(2)=2^3=8$ since $2 \times 2 \times 2=8$.

Most calculator do not have a dedicated button for the cube function, instead you must enter something like 2^3

to cube $2$.

Cube Function Properties

Domain and Range:
Both the domain and range include all real numbers.

Intercepts:
This function crosses at the origin, so both the y-intercept and the x-intercept are at the point $(0, 0)$.

Increasing, Decreasing, Constant:
The function is increasing over its entire domain. It is never decreasing or constant.

Maximums and Minimums:
There are no maximums or minimums.

Concavity:
The cube function is concave down on the interval $(-\infty, 0)$ and concave up on $(0, \infty)$.

Asymptotes:
There are no vertical or horizontal asymptotes.

The Cube Root Function

Many calculators have a command somewhere that will evaluate the cube root function $f(x)=\sqrt[3]{x}$ but it might be hard to find.

It is often easier to use the rule of exponents $\sqrt[3]{x}=x^{1/3}$ to evaluate cube roots. For example 125^(1/3)

would give the cube root of $125$.

Cube Root Function Properties

Domain and Range:
Both the domain and range include all real numbers.

Intercepts:
Since this function crosses at the origin, the y-intercept and the x-intercept are both $0$.

Increasing, Decreasing, Constant:
The function is increasing over its entire domain. It is never decreasing or constant.

Maximums and Minimums:
There are no maximums or minimums.

Concavity:
The cube function is concave up on the interval $(-\infty, 0)$ and concave down on $(0, \infty)$.

Asymptotes:
There are no vertical or horizontal asymptotes.

The Reciprocal Function

The reciprocal function $f(x)=\frac{1}{x}$ takes any number (except $0$) as an input and returns the reciprocal of that number. The easiest way to remember what a reciprocal is, is to see a few examples.

The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$.

The reciprocal of $5$ is $\frac{1}{5}$.

The reciprocal of $\frac{-6}{7}$ is $\frac{-7}{6}$.

You might recall that $\frac{1}{x}=x^{-1}$ is a rule of exponents. Because of that, many calculators have a button labeled $x^{-1}$ which will compute the reciprocal of a number.

Reciprocal Function Properties

Domain and Range:
The domain is all real numbers except $0$, since division by $0$ is undefined. The range is also all real numbers except for $0$.

Intercepts:
This function does not have any intercepts.

Increasing, Decreasing, Constant:
The function is increasing on the interval $(-\infty, 0)$ and decreasing on $(0, \infty)$. It is not constant anywhere.

Maximums and Minimums:
There are no maximums or minimums.

Concavity:
The cube function is concave down on the interval $(-\infty, 0)$ and concave up on $(0, \infty)$.

Asymptotes:
The y-axis is a vertical asymptote. The x-axis is a horizontal asymptote.

Looking Ahead

These six basic functions form a library that we will use to explore several different topics. For instance, in Section 1.4 you will learn which of these functions can be reversed. In Section 1.5 we discuss how to move their graphs to new locations and alter their shape. Section 1.6 gives numerical methods to distinguish one function from another, so that we don’t have to rely entirely on visual identification. And in Section 1.7 you will match patterns in data with the shapes of the basic function in order to create mathematical models.

Since these functions are the building blocks for more complicated ideas, you will be expected to memorize their basic properties. This downloadable summary will help your study.

Which Cube Root Function is Always Decreasing as X Increases

]]>Which Element’s Atoms Have the Greatest Average Number of Neutrons
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Wed, 07 Sep 2022 01:38:25 +0000https://estatename.com/?p=952EstateName.com – Which Element’s Atoms Have the Greatest Average Number of Neutrons 4.8: Isotopes – When the Number of Neutrons Varies Last updated Save as PDF Page ID 47477 Learning Objectives Explain what isotopes are and how an isotope affects an element’s atomic mass. Determine the number of protons, electrons, and neutrons of an element …

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4.8: Isotopes – When the Number of Neutrons Varies

Last updated

Save as PDF

Page ID

47477

Learning Objectives

Explain what isotopes are and how an isotope affects an element’s atomic mass.

Determine the number of protons, electrons, and neutrons of an element with a given mass number.

All atoms of the same element have the same number of protons, but some may have different numbers of neutrons. For example, all carbon atoms have six protons, and most have six neutrons as well. But some carbon atoms have seven or eight neutrons instead of the usual six. Atoms of the same element that differ in their numbers of neutrons are called isotopes. Many isotopes occur naturally. Usually one or two isotopes of an element are the most stable and common. Different isotopes of an element generally have the same physical and chemical properties because they have the same numbers of protons and electrons.

An Example: Hydrogen Isotopes

Hydrogen is an example of an element that has isotopes. Three isotopes of hydrogen are modeled in Figure \(\PageIndex{1}\). Most hydrogen atoms have just one proton, one electron, and lack a neutron. These atoms are just called hydrogen. Some hydrogen atoms have one neutron as well. These atoms are the isotope named deuterium. Other hydrogen atoms have two neutrons. These atoms are the isotope named tritium.

For most elements other than hydrogen, isotopes are named for their mass number. For example, carbon atoms with the usual 6 neutrons have a mass number of 12 (6 protons + 6 neutrons = 12), so they are called carbon-12. Carbon atoms with 7 neutrons have an atomic mass of 13 (6 protons + 7 neutrons = 13). These atoms are the isotope called carbon-13.

Example \(\PageIndex{1}\): Lithium Isotopes

What is the atomic number and the mass number of an isotope of lithium containing 3 neutrons?

What is the atomic number and the mass number of an isotope of lithium containing 4 neutrons?

Solution

A lithium atom contains 3 protons in its nucleus irrespective of the number of neutrons or electrons.

Notice that because the lithium atom always has 3 protons, the atomic number for lithium is always 3. The mass number, however, is 6 in the isotope with 3 neutrons, and 7 in the isotope with 4 neutrons. In nature, only certain isotopes exist. For instance, lithium exists as an isotope with 3 neutrons, and as an isotope with 4 neutrons, but it doesn’t exist as an isotope with 2 neutrons or as an isotope with 5 neutrons.

Stability of Isotopes

Atoms need a certain ratio of neutrons to protons to have a stable nucleus. Having too many or too few neutrons relative to protons results in an unstable, or radioactive, nucleus that will sooner or later break down to a more stable form. This process is called radioactive decay. Many isotopes have radioactive nuclei, and these isotopes are referred to as radioisotopes. When they decay, they release particles that may be harmful. This is why radioactive isotopes are dangerous and why working with them requires special suits for protection. The isotope of carbon known as carbon-14 is an example of a radioisotope. In contrast, the carbon isotopes called carbon-12 and carbon-13 are stable.

This whole discussion of isotopes brings us back to Dalton’s Atomic Theory. According to Dalton, atoms of a given element are identical. But if atoms of a given element can have different numbers of neutrons, then they can have different masses as well! How did Dalton miss this? It turns out that elements found in nature exist as constant uniform mixtures of their naturally occurring isotopes. In other words, a piece of lithium always contains both types of naturally occurring lithium (the type with 3 neutrons and the type with 4 neutrons). Moreover, it always contains the two in the same relative amounts (or “relative abundance”). In a chunk of lithium, \(93\%\) will always be lithium with 4 neutrons, while the remaining \(7\%\) will always be lithium with 3 neutrons.

Dalton always experimented with large chunks of an element—chunks that contained all of the naturally occurring isotopes of that element. As a result, when he performed his measurements, he was actually observing the averaged properties of all the different isotopes in the sample. For most of our purposes in chemistry, we will do the same thing and deal with the average mass of the atoms. Luckily, aside from having different masses, most other properties of different isotopes are similar.

There are two main ways in which scientists frequently show the mass number of an atom they are interested in. It is important to note that the mass number is not
given on the periodic table. These two ways include writing a nuclear symbol or by giving the name of the element with the mass number written.

To write a nuclear symbol, the mass number is placed at the upper left (superscript) of the chemical symbol and the atomic number is placed at the lower left (subscript) of the symbol. The complete nuclear symbol for helium-4 is drawn below:

The following nuclear symbols are for a nickel nucleus with 31 neutrons and a uranium nucleus with 146 neutrons.

\[\ce{^{59}_{28}Ni} \nonumber \]

\[ \ce{ ^{238}_{92}U} \nonumber \]

In the nickel nucleus represented above, the atomic number 28 indicates that the nucleus contains 28 protons, and therefore, it must contain 31 neutrons in order to have a mass number of 59. The uranium nucleus has 92 protons, as all uranium nuclei do; and this particular uranium nucleus has 146 neutrons.

Another way of representing isotopes is by adding a hyphen and the mass number to the chemical name or symbol. Thus the two nuclei would be Nickel-59 or Ni-59 and Uranium-238 or U-238, where 59 and 238 are the mass numbers of the two atoms, respectively. Note that the mass numbers (not the number of neutrons) are given to the side of the name.

Example \(\PageIndex{2}\): Potassium-40

How many protons, electrons, and neutrons are in an atom of \(^{40}_{19}\ce{K}\)?

How many protons, electrons, and neutrons are in each atom?

\(^{60}_{27}\ce{Co}\)

Na-24

\(^{45}_{20}\ce{Ca}\)

Sr-90

Answer a:

27 protons, 27 electrons, 33 neutrons

Answer b:

11 protons, 11 electrons, 13 neutrons

Answer c:

20 protons, 20 electrons, 25 neutrons

Answer d:

38 protons, 38 electrons, 52 neutrons

Summary

The number of protons is always the same in atoms of the same element.

The number of neutrons can be different, even in atoms of the same element.

Atoms of the same element that contain the same number of protons, but different numbers of neutrons, are known as isotopes.

Isotopes of any given element all contain the same number of protons, so they have the same atomic number (for example, the atomic number of helium is always 2).

Isotopes of a given element contain different numbers of neutrons, therefore, different isotopes have different mass numbers.

Which Element’s Atoms Have the Greatest Average Number of Neutrons

]]>In an Automobile Collision a 44 Kilogram Passenger
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Two cars collide with each other. Before the collision, one car (m = 1300 kg) is going north at 30 m/s and the other car (m = 900 kg) is going south at 15 m/s. What is the momentum of the system made up of the two cars after the collision?

Best Answer

#6

+33113

For interest’s sake then, let’s suppose the two cars collide elastically, conserving both energy and momentum.

Momentum before: 1300*30 – 900*15 kg.m/s =

Momentum after: 1300v1 + 900v2

so 1300v1 + 900v2 = 25500 …(1)

Kinetic energy before: 1300*30^{2}/2 + 900*15^{2}/2 = 686250 J

Kinetic energy after: 1300*v1^{2}/2 + 900*v2^{2}/2

so 1300*v1^{2}/2 + 900*v2^{2}/2 = 686250 …(2)

Equations (1) and (2) can now be solved for v1 and v2. (v1 = -75/11 m/s ≈ -6.8 m/s; v2 = 420/11 m/s ≈ 38.2 m/s).

.

#1

+26320

Two cars collide with each other. Before the collision, one car (m = 1300 kg) is going north at 30 m/s and the other car (m = 900 kg) is going south at 15 m/s. What is the momentum of the system made up of the two cars after the collision ?

Momentum is a vector quantity. In the absence of any losses, the momentum of the system after the collision is the same as that before the collision. So, taking north as positive the total momentum of the system is:

1300*30 – 900*15 = 25500 kg.m/s

.

After the collision the velocity of the two cars, assuming they are stuck together, is

v = 25500/(1300 + 900)

or v = 11.59 m/s (positive, so going north)

.

.

#3

+394

A question for Alan or Heureka

Why does this equation set fail? Would this work if this were an elastic collision?

Kinetic energy autoN = (0.5(1300)*(30^{2})) = 585,000J

Kinetic energy autoS = (0.5(900)*(15^{2})) = 101,250J

585,000J + (- 101,250J) = 483,750J

Total mass now as a unit = (1300kg) + (900kg) = 2200kg

Solve for velocity: Sqr(483750J / 0.5(2200kg)) = 20.97m/s in north direction.

_7UP_

#4

+33113

1. Energy is a scalar, not a vector quantity, so you would add not subtract the energies.

2. If it were an elastic collision they wouldn’t stick together.

3. It would be unusual not to lose a lot of energy (dissipated through frictional heat loss, for example) in a head-on car crash!

.

#5

+394

Thank you Alan. I usually understand these intuitively, first, then learn the maths. In this case, it is the other way around. I need to develop a mental picture from the scalar and vector dimensions via the maths.

_7UP_

#6

+33113

Best Answer

For interest’s sake then, let’s suppose the two cars collide elastically, conserving both energy and momentum.

Momentum before: 1300*30 – 900*15 kg.m/s =

Momentum after: 1300v1 + 900v2

so 1300v1 + 900v2 = 25500 …(1)

Kinetic energy before: 1300*30^{2}/2 + 900*15^{2}/2 = 686250 J

Kinetic energy after: 1300*v1^{2}/2 + 900*v2^{2}/2

so 1300*v1^{2}/2 + 900*v2^{2}/2 = 686250 …(2)

Equations (1) and (2) can now be solved for v1 and v2. (v1 = -75/11 m/s ≈ -6.8 m/s; v2 = 420/11 m/s ≈ 38.2 m/s).

.

#7

+394

Thank you, Alan. This is like finally seeing the three dimensional picture in a blended color dot image. The two velocities relate to the solution via the harmonic mean. Not unusual considering they are rates. It sure is easy to see this when they are the right values.

2/((38.2)^{-1}+(6.8)^{-1}) = 11.54

I think the most interesting thing is how the lower energy auto dramatically slows the higher one. I saw similar energy-shedding patterns in high-mass low-velocity vs low-mass high-velocity comparisons.

All of this is very cool. Thank you again.

_7UP_

In an Automobile Collision a 44 Kilogram Passenger

]]>In the Figure Pq is Parallel to Rs
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Get premium membership and access questions with answers, video lessons as well as revision papers.

In the figure below PQ is parallel to RS. PS and QR intersect at A. Given that PQ=9cm, RS=3cm and AS=4cm, calculate the length of…

In the figure below PQ is parallel to RS. PS and QR intersect at A. Given that PQ=9cm, RS=3cm and AS=4cm, calculate the length of PS.

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