8.2: Arizona v. Hicks
PETITIONER RESPONDENT
Arizona Hicks
LOCATION
Apartment of Hicks
DOCKET NO. DECIDED BY
85-1027 Rehnquist Court
LOWER COURT
Sate Appellate Court
CITATION
480 US 321 (1987)
ARGUED
Dec 8, 1986
DECIDED
Mar 3, 1987
ADVOCATES
John W. Rood, III By appointment of the Court, argued the cause for the respondent
Linda A. Akers Argued the cause for the petitioner
John William Rood for respondent
Facts of the case
A bullet was fired through the floor of Hicks's apartment which injured a man in the apartment below. To investigate the shooting, police officers entered Hicks's apartment and found three weapons along with a stocking mask. During the search, which was done without a warrant, an officer noticed some expensive stereo equipment which he suspected had been stolen. The officer moved some of the components, recorded their serial numbers, and seized them upon learning from police headquarters that his suspicions were correct.
Question
Was the search of the stereo equipment (a search beyond the exigencies of the original entry) reasonable under the Fourth and Fourteenth Amendments?
Conclusion
6–3 Decision
Majority Opinion by Antonin Scalia
| FOR | AGAINST |
|---|---|
|
Blackmun White Scalia Stevens Brennan Marshall |
Powell Rehnquist O’Connor |
No. The Court found, that the search and seizure of the stereo equipment violated the Fourth and Fourteenth Amendments. Citing the Court's holding in Coolidge v. New Hampshire (1971) , Justice Scalia upheld the "plain view" doctrine which allows police officers under some circumstances to seize evidence in plain view without a warrant. However, critical to this doctrine, argued Scalia, is the requirement that warrantless seizures which rely on no "special operational necessities" be done with probable cause. Since the officer who seized the stereo equipment had only a "reasonable suspicion" and not a "probable cause" to believe that the equipment was stolen, the officer's actions were not reconcilable with the Constitution.


2.9E: Optimization of Functions of Several Variables (Exercises)
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13.8: Optimization of Functions of Several Variables
Finding Critical Points
In exercises 1 - 5, find all critical points.
1) \( f(x,y)=1+x^2+y^2\)
- Answer:
- \( (0,0)\)
2) \( f(x, y) = 1 - (x -2)^2 + (y+3)^2\)
3) \( f(x,y)=(3x−2)^2+(y−4)^2\)
- Answer:
- \( \left(\frac{2}{3},4\right)\)
4) \( f(x,y)=x^4+y^4−16xy\)
- Answer:
- \( (0,0), \quad (-2,-2), \quad (2,2)\)
5) \( f(x,y)=15x^3−3xy+15y^3\)
- Answer:
- \( (0,0), \quad \left(\frac{1}{15},\frac{1}{15}\right)\)
Finding Extrema & the Second Partials Test
In exercises 6 - 9, find the critical points of the function and test for extrema or saddle points by using algebraic techniques (completing the square) or by examining the form of the equation. Where possible, verify your results using the Second Partials Test.
6) \( f(x,y)=-\sqrt{x^2+y^2}\)
- Answer:
-
Crit. pts.: \( (0, 0) \)
Extrema: \( f\) has a relative maximum of \(0\) at \( (0, 0)\).
To justify this, consider the fact that the square root function cannot give a negative value, so this function cannot possibly return a positive value. Since it's value is \(0\) at the critical point \( (0, 0)\), we know it must be the function's absolute maximum value.
7) \( f(x,y)=−x^2−5y^2+8x−10y−13\)
- Answer:
-
Crit. pts.: \( (4, -1) \)
Extrema: \( f\) has a relative maximum of \(8\) at \( (4,−1)\).
To justify this, we complete the square on this function, being careful to factor out the coefficient of the squared terms before completing the square.
\[\begin{align*} f(x, y) &= −x^2−5y^2+8x−10y−13 \\ &= −(x^2-8x\quad\quad)−5(y^2+2y\quad\quad)−13 \\ &= −(x^2-8x+16)−5(y^2+2y+1)−13+16+5 \\ &= -(x-4)^2 -5(y+1)^2+8\end{align*}\]
Note that this quadratic polynomial function takes the form \( z = -(x^2 + y^2)\), so we can see that it will have a relative (and, in fact, absolute) maximum at its vertex (the critical point \( (4, -1) \)). We can also argue that since we are subtracting squared terms from 8, we cannot possibly obtain a function value larger than 8, and since we do get a value of 8 at the critical point \( (4, -1) \), we know it will be the absolute maximum value of this function.
8) \( f(x,y)=x^2+y^2+2x−6y+6\)
9) \( f(x,y)=\sqrt{x^2+y^2}+1\)
- Answer:
-
Crit. Pts.: \( (0, 0) \)
Extrema: \( f\) has a relative minimum of \(1\) at \( (0,0)\).
To justify this, consider the fact that the square root function cannot give a negative value, so this function cannot possibly return a value less than \(1\). Since it's value is \(1\) at the critical point \( (0, 0)\), we know \(1\) must be the function's absolute minimum value.
In exercises 10 - 34, identify any critical points and use the Second Partials Test to determine the behavior of the function at each critical point, whether there is a maximum, minimum, saddle point, or none of these. If the Second Partials Test fails, determine the function's behavior at the point using another method and justify your answer clearly.
10) \( f(x,y)=−x^3+4xy−2y^2+1\)
11) \( f(x,y)=x^2y^2\)
- Answer:
-
Crit. pts.: All points on the lines \( x = 0 \) and \( y = 0\) are critical points of this function.
Extrema: The Second Partials Test fails.
Since \( x^2y^2>0\) for all \( x\) and \( y\) different from zero, and \( x^2y^2=0\) when either \( x\) or \( y\) equals zero (or both), then the absolute minimum of \(0\) occurs at all points on the \(x\)- or \(y\)-axes, that is, for all points on the lines \( x = 0 \) and \( y = 0\).
12) \( f(x,y)=x^2−6x+y^2+4y−8\)
13) \( f(x,y)=2xy+3x+4y\)
- Answer:
-
Crit. pts.: \( \left(−2,−\frac{3}{2}\right) \)
Exrema: \(f\) has a saddle point at \( \left(−2,−\frac{3}{2},−6\right) \).
14) \( f(x,y)=8xy(x+y)+7\)
15) \( f(x,y)=x^2+4xy+y^2\)
- Answer:
-
Crit. pts.: \( (0,0) \)
Exrema: \(f\) has a saddle point at \( (0,0,0)\).
16) \( f(x,y)=x^3+y^3−300x−75y−3\)
17) \( f(x,y)=9−x^4y^4\)
- Answer:
-
Crit. pts.: All points on the lines \( x = 0 \) and \( y = 0\) are critical points of this function.
Extrema: The Second Partials Test fails.
Since the term \( -x^4y^4<0\) for all \( x\) and \( y\) different from zero, and \( -x^4y^4=0\) when either \( x\) or \( y\) equals zero (or both), then this function cannot attain a value greater than \(9\) anywhere, but is \(9\) at the critical points. Thus \(f\) has an absolute maximum of \(9\) at all points on the \(x\)- or \(y\)-axes, that is, for all points on the lines \( x = 0 \) and \( y = 0\).
18) \( f(x,y)=x^2+10xy+y^2\)
- Answer:
-
Crit. pts.: \( (0,0) \)
Extrema: \(f\) has a saddle point at \( (0,0,0)\).
19) \(f(x,y) = x^4 + y^2 + 2xy + 3\)
- Answer:
-
Crit. pts.: \( (0,0), \quad \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), \quad \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \)
Extrema: \(f\) has a saddle point at \( (0, 0, 3) \),
\(f\) has a local minimum of \( 2.75\) at the point \( \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \).
\(f\) has a local minimum of \( 2.75\) at the point \( \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \).
20) \( f(x,y)=7x^2y+9xy^2\)
21) \( f(x,y)=3x^2−2xy+y^2−8y\)
- Answer:
-
Crit. pts.: \( (2,6) \)
Extrema: \(f\) has a relative minimum of \( -24\) located at \( (2,6)\).
22) \( f(x,y)=3x^2+2xy+y^2\)
23) \( f(x,y)=y^2+xy+3y+2x+3\)
- Answer:
-
Crit. pts.: \( (1,−2) \)
Extrema: \(f\) has a saddle point at \( (1,−2,1)\).
24) \( f(x,y)=x^2+xy+y^2−3x\)
25) \( f(x,y)=x^2+2y^2−x^2y\)
- Answer:
-
Crit. pts.: \( (0,0), \quad (-2,1), \quad (2,1)\)
Extrema: \(f\) has a relative minimum of \(0\) at \( (0,0) \) and saddle points at \( (2,1,2)\) and \( (−2,1,2)\).
26) \( f(x,y)=x^2+y−e^y\)
27) \( f(x,y)=e^{−(x^2+y^2+2x)}\)
- Answer:
-
Crit. pts.: \( (-1,0) \)
Extrema: \(f\) has a relative maximum of \( e \) located at \( (-1,0)\).
See this problem illustrated in CalcPlot3D .
28) \( f(x,y)=x^2+xy+y^2−x−y+1\)
29) \(f(x,y) = x^2y(9 - x + y)\)
- Answer:
-
Crit. pts.: \( \left(\frac{9}{2},-\frac{9}{4}\right), \quad (9,0)\), and all points on the line \(x = 0\)
Extrema: \(f\) has a saddle point at \( (9,0,0)\) and a relative minimum of \(-102.515625\) at \( \left(\frac{9}{2},-\frac{9}{4}\right)\).
At the critical points on the line \(x = 0\), \(f\) has neither relative extrema nor saddle points, but they do represent a sort of trough on the surface.
30) \( f(x,y)=−x^2−5y^2+10x−30y−62\)
31) \( f(x,y)=120x+120y−xy−x^2−y^2\)
- Answer:
-
Crit. pts.: \( (40,40) \)
Extrema: \(f\) has a relative maximum of \( 4800 \) located at \( (40,40)\).
32) \( f(x,y)=2x^2+2xy+y^2+2x−3\)
33) \( f(x,y)=x^2+x−3xy+y^3−5\)
- Answer:
-
Crit. pts.: \( \left(\frac{1}{4},\frac{1}{2}\right)\) and \((1, 1) \)
Extrema: \(f\) has a saddle point at \( \left(\frac{1}{4},\frac{1}{2}, -\frac{79}{16}\right)\) and a relative minimum of \( -5 \) at \( (1,1)\).
34) \( f(x,y)=2xye^{−x^2−y^2}\)
In exercises 35 - 37, determine the extreme values and the saddle points. Use a CAS to graph the function.
35) [T] \( f(x,y)=ye^x−e^y\)
- Answer:
-
A saddle point is located at \( (0,0,-1).\)
36) [T] \( f(x,y)=x\sin(y)\)
37) [T] \( f(x,y)=\sin(x)\sin(y),\quad x∈(0,2π),\quad y∈(0,2π)\)
- Answer:
-
There is a saddle point at \( (π,π),\) local maxima at \( \left(\frac{π}{2},\frac{π}{2}\right)\) and \( \left(\frac{3π}{2},\frac{3π}{2}\right)\), and local minima at \( \left(\frac{π}{2},\frac{3π}{2}\right)\) and \( \left(\frac{3π}{2},\frac{π}{2}\right)\).
Contributors
-
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
- Paul Seeburger (Monroe Community College) created problems 19 and 29, and added dynamic figures for problems 27 and 35.


17.3: Applying the Nursing Process to Grief
Grieving a loss is a normal process that has implications for both patient and family well-being. NANDA formally recognizes the dimensions of grief with the nursing diagnoses of Grieving and Complicated Grieving . Recall that grief can be experienced due to many types of loss, in addition to death. For example, when patients receive a diagnosis of breast cancer, they may demonstrate signs of various stages of grief, such as denial, anger, bargaining, depression, and acceptance. When undergoing mastectomy and chemotherapy, the patient may grieve over the loss of prior body image.
Communities can also experience grief. For example, when a town experiences a significant tragedy, such as a devastating flood or a tornado, there can be widespread community grief as families grieve the loss of life, property, or a previous way of life. In these situations, nurses are cognizant of the multiple factors that may impact an individual’s health and grieving process. Identifying these factors can help ensure that appropriate resources are mobilized to facilitate coping and progression through the grief process.
Assessment
Grief assessment includes the patient, family members, and significant others. It begins when a patient is diagnosed with an acute, chronic, or terminal illness and/or when the patient is admitted to a hospital, nursing facility, or assisted living facility. It continues throughout the course of a terminal illness for the patient, family members, and significant others and then continues through the bereavement period for the survivors. During the bereavement period, the nurse monitors for symptoms of complicated grief. [1]
Grief can be manifested by physical, emotional, and cognitive symptoms. Physical symptoms can occur, such as feeling ill, headaches, tremors, muscle aches, exhaustion, insomnia, loss of appetite, or weight loss or gain. Cognitive symptoms may occur, such as lack of concentration, confusion, and hallucinations. Emotional symptoms, such as anxiety, guilt, anger, fear, sadness, helplessness, or feelings of relief may occur. These symptoms of grief and loss can be manifested in many different ways and can vary from day to day. Manifestations of grief are unique to the individual and may be influenced by one’s age, culture, resources, and previous experiences with loss. Additionally, as patients cope with grief and loss, it is important for the nurse to recognize that support is often needed by their family members. [2]
Any behavior that may endanger the patient or family should be reported to the health care provider, such as symptoms of depression, suicidal ideation, or symptoms lasting greater than six months.
Diagnoses
Consult a nursing care planning resource when selecting nursing diagnoses for patients and their family members experiencing grief. See Table 17.3 for definitions and selected defining characteristics of the NANDA-I diagnoses Grieving and Complicated Grieving while also keeping in mind the previous discussion in this chapter regarding stages and tasks of normal grief.
Table 17.3 NANDA-I Nursing Diagnoses Related to Grieving [3]
| NANDA-I Diagnosis | Definition | Selected Defining Characteristics |
|---|---|---|
| Grieving | A normal, complex process that includes emotional, physical, spiritual, social, and intellectual responses and behaviors by which individuals, families, and communities incorporate an actual, anticipated, or perceived loss into their daily lives. |
|
| Complicated Grieving | A disorder that occurs after the death of a significant other, in which the experience of distress accompanying bereavement fails to follow normative expectations and manifests in functional impairment. |
|
Examples
See the following for examples of PES statements related to Grieving and Complicated Grieving:
- A patient diagnosed with metastatic cancer is advised they have less than six months to live. They begin to move through the stages of grief as they assimilate this information. A sample NANDA-I diagnosis in current PES format is: “Grieving related to anticipatory loss as evidenced by detachment, disorganization, and alteration in activity level.” The nurse would plan and implement interventions to enhance coping for this patient.
- A patient’s husband died two years ago, and she continues to be preoccupied with thoughts about her husband. Her grown children live several hours away, and she becomes isolated and unable to complete daily activities, such as cleaning the house and grocery shopping. A sample PES statement is: “Complicated Grieving related to insufficient social support as evidenced by avoidance of decreased functioning and preoccupation with thoughts about her deceased husband.” The nurse would plan interventions to facilitate grief work while also arranging for assistance with ADLs and IADLs in the patient’s home.
Outcome Identification
Goal setting and outcome identification for patients and family members experiencing grief are customized to the specific situation and focus on grief resolution. Grief resolution is evidenced by the following indicators:
- Resolves feelings about the loss
- Verbalizes reality and acceptance of loss
- Maintains living environment
- Seeks social support [4]
For the nursing diagnosis of Grieving and Complicated Grieving , a sample goal is, “The p atient will experience grief resolution. ”
A sample SMART outcome is, “ The patient will discuss the meaning of the loss to their life in the next 2 weeks .” [5]
Planning and Implementing Interventions
Nurses are in the ideal position to assist patients with identifying and expressing their feelings related to loss. The most important intervention that nurses can provide is active listening and offering a supportive presence. Actively listening to the bereaved helps them express their feelings and relate the emotions and feelings related to the loss. Interventions to facilitate grief resolution focus on coping enhancement, anticipatory grieving interventions, and grief work facilitation.
Coping Enhancement
Interventions to enhance coping can be implemented for patients and families experiencing any type of actual, anticipated, or perceived loss. Sample interventions include the following: [6]
- Assist the patient in identifying short- and long-term goals.
- Assist the patient in examining available resources to meet the goals.
- Assist the patient in breaking down complex steps into small, manageable steps.
- Encourage relationships with others who have common interests and goals.
- Assist the patient to solve problems in a constructive manner.
- Appraise the effect of a patient’s life situation on roles and relationships.
- Appraise and discuss alternative responses to the situation.
- Use a calm, reassuring approach.
- Provide an atmosphere of acceptance.
- Help the patient identify information they are most interested in obtaining.
- Provide factual information regarding medical diagnosis, treatment, and prognosis.
- Provide the patient with realistic choices about certain aspects of care.
- Encourage an attitude of realistic hope as a way of dealing with hopelessness.
- Seek to understand the patient’s perspective of a stressful situation.
- Discourage decision-making when the patient is under severe stress.
- Acknowledge the patient’s cultural and spiritual background and encourage use of spiritual resources, if desired.
- Encourage verbalization of feelings, perceptions, and fears.
- Encourage family involvement, as appropriate.
- Assist the patient to identify positive strategies to deal with limitations and manage needed lifestyle or role changes.
- Instruct the patient on the use of relaxation techniques.
See Figure 17.18 [7] for an image of a nurse enhancing a patient’s ability to cope with their illness through active listening and touch.
Anticipatory Grieving Interventions
Anticipatory grieving refers to a grief reaction that occurs in anticipation of an impending loss. Recall that anticipatory grieving can be related to impending death of oneself or a loved one, but it can also occur in anticipation of other losses, such as the loss of a body part due to scheduled surgery or the loss of one’s home due to a move to a long-term care facility. Interventions to facilitate resolution of anticipatory grieving include the following: [8]
- Develop a trusting relationship with the patient and family members by using presence and other therapeutic communication techniques.
- Keep the patient and family members apprised of the patient’s ongoing condition as much as possible.
- Keep the family informed of the patient’s needs for physical care and support in symptom control, and inform them about health care options at the end of life, including palliative care, hospice care, and home care.
- Actively listen as the patient grieves for their own death or loss. Normalize the patient’s expressions of grief.
- Discuss the patient’s preferred place of death and document their wishes.
- Ask family members about having adequate resources to care for themselves and the critically ill family member.
- Recognize caregiver role strain in family members providing long-term care at home.
- Listen to the family member’s story.
- Encourage family members to show their caring feelings and talk with the family members.
- Recognize and respect different feelings and wishes from the patient and their family members.
- Refer the patient and family members to counselors or chaplains for spiritual care as appropriate.
Grief Work Facilitation
Grief work facilitation assists patients and family members in resolution of a significant loss. Sample interventions include the following: [9]
- Identify the loss.
- Assist the patient to identify the initial reaction to the loss.
- Listen to expressions of grief.
- Encourage discussion of previous loss experiences.
- Encourage the verbalization of memories of the loss.
- Make empathetic statements about grief.
- Encourage identification of greatest fears concerning the loss.
- Educate about stages and tasks of the grieving process, as appropriate.
- Support progression through personal grieving stages.
- Assist in identifying personal coping strategies.
- Encourage implementation of cultural, religious, and social customs associated with the loss.
- Answer children’s questions about the loss and encourage discussion of feelings.
- Identify sources of community support.
- Reinforce progress made in the grieving process.
- Assist in identifying modifications needed in lifestyle.
Community Resources
Bereavement follow-up with families is a component of hospice programs and includes formal activities and events to promote closure and acceptance. Many hospices have nondenominational memorial services to honor patients. Family members and staff are invited to participate, which can be effective at helping individuals find closure. Other formal types of support can include organized support groups to facilitate discussion and coping. Individual, group counseling, or psychotherapy are other methods that can assist the bereaved in coping with their loss. See additional resources for family members in the following box.
Patients and family members experiencing depression or anxiety related to the grieving process may be prescribed antianxiety medications or antidepressants. See the “ Central Nervous System ” chapter in Open RN Nursing Pharmacology for additional information about these medications.
Evaluation
It is always important to evaluate the effectiveness of interventions implemented. Nurses assess the effectiveness of interventions in helping individuals cope and work through the grief process based on the customized outcome criteria established for their situation.
- This work is a derivative of Nursing Care at the End of Life by Lowey and is licensed under CC BY-NC-SA 4.0 ↵
- This work is a derivative of Nursing Care at the End of Life by Lowey and is licensed under CC BY-NC-SA 4.0 ↵
- Herdman, T., & Kamitsuru, S. (2017). NANDA international nursing diagnoses: Definitions & classification 2018-2020 (11th ed.). Thieme Publishers, pp. 335, 339-341. ↵
- Ackley, B., Ladwig, G., Makic, M. B., Martinez-Kratz, M., & Zanotti, M. (2020). Nursing diagnosis handbook: An evidence-based guide to planning care (12th ed.). Elsevier, pp. 144-147, 434-444. ↵
- Ackley, B., Ladwig, G., Makic, M. B., Martinez-Kratz, M., & Zanotti, M. (2020). Nursing diagnosis handbook: An evidence-based guide to planning care (12th ed.). Elsevier, pp. 144-147, 434-444. ↵
- National Cancer Institute. (n.d.). Grief, bereavement, and coping with loss (PDQ) - Health professional version. U.S. Department of Health and Human Services. https://www.cancer.gov/about-cancer/advanced-cancer/caregivers/planning/bereavement-hp-pdq#section/all ↵
- "423588130-huge.jpg" by Jacob Lund is used under license from Shutterstock.com ↵
- Ackley, B., Ladwig, G., Makic, M. B., Martinez-Kratz, M., & Zanotti, M. (2020). Nursing diagnosis handbook: An evidence-based guide to planning care (12th ed.). Elsevier, pp. 144-147, 434-444. ↵
- National Cancer Institute. (n.d.). Grief, bereavement, and coping with loss (PDQ) - Health professional version. U.S. Department of Health and Human Services. https://www.cancer.gov/about-cancer/advanced-cancer/caregivers/planning/bereavement-hp-pdq#section/all ↵


6: Functionalism -- Introduction to Philosophy- Philosophy of Mind
Jason Newman
Introduction: Two Monsters We Must Avoid
While passing through the Strait of Messina, between mainland Italy and the isle of Sicily, Homer has Odysseus come upon two monsters, Scylla and Charybdis, one on either side of the strait. If Odysseus is to pass through the strait, he must choose between two very unhappy options; for if he averts one along the way, he will move in the other’s monstrous reach. On the one side is roaring Charybdis, who would surely blot out—as if by colossal whirlpool—Odysseus’s entire ship. (Have you ever been faced with an option so bad that you cannot believe you have to seriously consider it? Well, this is Odysseus’s bleak situation.) On the other side of the strait, things fare little better for Odysseus and his war-weary crew: we have vicious Scylla, who only by comparison to Charybdis, looks like the right choice. The ship makes it through, Homer tells us, minus those who were snatched from the ship’s deck and eaten alive. Six are taken, we are told, one for each of Scylla’s heads. By comparison only, indeed.
In this chapter we consider the theory of mind known as functionalism, the view that minds are really functional systems like the computing systems we rely on every day, only much more complex. The functionalist claims to sail a middle path between materialism (discussed in Chapter 2), or the joint thesis that minds are brains and mental states are brain states, and behaviorism (also discussed in Chapter 2), or the thesis that mental states are behavioural states or dispositions to behave in certain ways.
Avoiding Materialism
One the one side we have materialism, which we must avoid because there appears to be no strict identity between mental states and brain states. Even though human Freya is different than a wild rabbit in many interesting ways, we think they can both be in physical pain. Suppose that while restringing her guitar, Freya lodges a rogue metal splinter off the D string in the top of her ring finger. She winces in pain. Physiologically and neurologically, a lot happened—from the tissue damage caused by the metal splinter, to Freya’s finally wincing from the sensation. But it only took milliseconds.
Now suppose that while out foraging and hopping about, the wild rabbit mishops on the prickly side of a pinecone. The rabbit cries out a bit, winks hard, and hops off fast. A very similar physiological and neurological chain of events no doubt transpired from the mishop on the pinecone to hopping off fast in pain. But as interestingly similar as the wild rabbit’s brain is to human Freya’s, it is not plausible to think that both Freya and the wild rabbit entered into the same brain state. We do want to say they entered into the same mental state, however. That is, they were both in pain. Since the same pain state can be realized in multiple kinds of brains, we can say that mental states like pain are multiply realizable.
Avoiding Behaviorism
Now we look bleary-eyed in the direction of behaviorism. But here, too, we find a suspicious identity claim—this time between mental states, like Freya’s belief that her house is gray, and behavioral states or dispositions to behave in certain circumstances. For example, if Freya were asked what color her house is, she would be disposed to answer, “Gray.” But just as with mental states and brain states, Freya’s believing that her Colonial-period house is painted the original gray from when the house was first built and painted in 1810, and her dispositions to behave accordingly, come apart, showing that they could not be identical.
Suppose Freya wants to throw a housewarming party for herself and includes a colorful direction in the invitation that hers is the “only big gray Colonial on Jones St. Can’t miss it.” We say that Freya would not sincerely include such a thing if she did not believe it to be true. And we have no reason to suspect she is lying. We can go further. We want to say that it is her belief that her Colonial is big, is gray, and the only one like it on Jones Street that causes her, at least in part, to include that direction in the invitation. But if it is her mental state (her belief) that caused her behavior, then the mental state and the behavioral state (her including the colorful direction in the invitation) cannot be strictly identical.
Freya might very well have been disposed to give just such a colorful direction to her home, given her beliefs, as the behaviorist would predict; and this disposition might even come with believing the things Freya does. But if we want to refer to Freya’s beliefs in our explanation of her behavior—and this is the sort of thing we do when we say our beliefs and other mental states cause our behavior—then we must hold that they are distinct, since otherwise our causal explanation would be viciously circular.
It would be circular because the thing to be explained, her Colonial-describing behavior, is the same thing as the thing that is supposed to causally explain it, her Colonial-descriptive beliefs; and the circle would be vicious because nothing would ever really get explained. So the behaviorist, like the materialist, seems to see an identity where there is none.
No Turning Back: The Mind is Natural
The goal is to formulate an alternative to the above two theories of mind that nevertheless both make a promise worth making: to treat the mind as something wholly a part of the natural world. From the failures of materialism and behaviorism, we must not turn back to a problematic Cartesian dualist view of mind and matter (discussed in Chapter 1), where it again would become utterly mysterious how Freya’s beliefs about how her Colonial looks could possibly influence her physical behavior, since her beliefs and physical behavior exist on different planes of existence, as it were. But there is a third way to view beliefs like Freya’s.
Functionalism as the Middle Path
Our way between the two monsters is to take seriously the perhaps dangerous idea that minds really are computing machines. In England, Alan Turing (1912-1954) laid the groundwork for such an idea with his monumental work on the nature of computing machines and intelligence (1936, 230-265; 1950, 433-460). Turing was able to conceive of a computing machine so powerful that it could successfully perform any computable function a human being could be said to carry out, whether consciously, as in the math classroom, or at the subconscious level, as in the many computations involved in navigating from one side to the other of one’s room.
A Turing machine, as it came to be called, is an abstract computer model designed with the purpose of illustrating the limits of computability. Thinking creatures like human beings, of course, are not abstract things. Turing machines are not themselves thinking machines, but insofar as thinking states can be coherently understood as computational states, a Turing machine or Turing machine-inspired model should provide an illuminating account of the mind.
Turing’s ideas were developed in the United States by philosopher Hilary Putnam (1926-2016). Functionalism treats minds as natural phenomena contra Cartesian dualism; mental states, like pain, as multiply realizable, contra materialism; and mental states as causes of behavior, contra behaviorism. In its simple form, it is the joint thesis that the mind is a functional system, kind of like an operating system of a computer, and mental states like beliefs, desires, and perceptual experiences are really just functional states, kind of like inputs and outputs in that operating system. Indeed, often this simple version of functionalism is known as “machine” or “input-output functionalism” to highlight just those mechanical features of the theory.
Nothing’s Shocking: The Functionalist Mind is a Natural Mind
The functionalist says if we conceive of mental stuff in this way—namely, as fundamentally inputs and outputs in a complex, but wholly natural system—then we get to observe the reality of the mind, and the reality of our mental lives. We get to avoid any genuine worries about mental stuff being too spooky, or about how it could possibly interact with material stuff, as one might genuinely worry on a Cartesian dualist theory of mind, where we are asked to construe mental stuff and material stuff as fundamentally two kinds of substances. With functionalism, the how-possible question about interaction between the mental and material simply does not arise, no more than it would for the software and hardware interaction in computers, respectively. So, on the functionalist picture of the mind, the mysterious fog is lifted, and the way is clear.
Multiple Realizability
Let us use a thought experiment of our own to illustrate the functionalist’s theory of mind. Imagine Freya cooks a warm Sunday breakfast for herself and sits on a patio table in the spring sun to enjoy it. Freya’s belief that “my tofu scramble is on the table before me” is to be understood roughly like this: as the OUTPUT of one mental state, her seeing her breakfast on the table before her, and as the INPUT for others, including other beliefs Freya might have or come to have by deductive inference (“something is on the table before me,” and so on and so forth) and behaviors (e.g., sticking a fork into that tofu scramble and scarfing it down). Note well: we have not mentioned anything here about the work Freya’s sensory cortex or thalamus or the role the rods and cones in her retina are playing in getting her to believe what she does; her belief is identified only by its functional or causal role. This seems to imply that Freya’s breakfast belief is multiply realizable, like pain is.
Recall our earlier discussion of the important difference between rabbit-brain stuff and human-brain stuff. Nevertheless, we wanted to say that both Freya and the wild rabbit could be in pain. We said pain, then, is multiply realizable. This is another way of saying that being in pain does not require any specific realization means, just some or other adequate means of realization. The point also strongly implies that the means of realization for Freya’s breakfast belief, no less than her pain, need not be a brain state at all. This signals a major worry for the materialist. Since our beliefs, desires, and perceptual experiences are identified by their functional or causal role, the functionalist has no problem accounting for the multiple realizability of mental states.
Real Cause: The Functionalist Mind Causes Behavior
Finally, we saw that our mental states cannot be counted as the causes of our behavior on a behaviorist view, since on that view of mind, mental states are nothing over and above our behavior (or, dispositions to behave in certain ways in certain circumstances). In an effort to disenchant the mind in general and individual minds in particular, and move mental states like beliefs and pain into scientific view, the behaviorist recoiled too far from spooky Cartesian dualism, leaving nothing in us to be the causes of our own behavior. The functionalist understands, like the behaviorist, that there is a close connection between our beliefs, desires, and pains, on the one hand, and our behavior, on the other. It is just that the connection is a functional, or causal, one, not one of identity. Since mental states (like Freya’s belief that “my tofu scramble is on the table before me”) are identified with their functional or causal role in the larger functional system of inputs and outputs, other mental states and behavioral states, the functionalist has no problem accounting for mental states playing a causal role in the explanations we give of our own behavior. On the functionalist theory of mind, mental states are real causes of behavior.
Objections to Functionalism
Now that we have seen some of the major points in favor of the theory, let us have a look at some of the worries that have been raised against functionalism.
The Chinese Room
John Searle argues against a version of functionalism he calls “strong” artificial intelligence, or “strong AI” In “Minds, Brains and Programs,” Searle develops a thought experiment designed to show that having the right inputs and outputs is not sufficient for having mental states, as the functionalist claims (1980). The specific issue concerns what is required to understand Chinese.
Imagine someone who does not understand Chinese is put in a room and tasked with sorting Chinese symbols in response to other Chinese symbols, according to purely formal rules given in an English-language manual. So, for example, one person can write some Chinese symbols on a card, place it in a basket on a conveyor belt which leads into and out from the little room you are in. Once you receive it, you look at the shape of the symbol, find it in the manual, and read which Chinese symbols to find in the other basket to send back out. Imagine further that you get very good at this manipulation of symbols, so good in fact that you can fool fluent Chinese speakers with the responses you give. To them, you function every bit like you understand Chinese. It appears, however, you have no true understanding at all. Therefore, Searle concludes, functioning in the right way is not sufficient for having mental states.
The functionalist has replied that, of course, as the thought experiment is described, the person in the room does not understand Chinese. But also as the case is described, the person in the room is just a piece of the whole functional system. Indeed, it is the system that functions to understand Chinese, not just one part. So it is the whole system, in this case, the whole room, including the person manipulating the symbols and the instruction manual (the “program”), that understands Chinese.
The Problem of Qualia
The splinter Freya picked up from her D string caused her a bit of pain, and perhaps more so for the behaviorist, as we saw earlier. One major worry for the functionalist is that there seems to be more to Freya’s pain than its just being the putative cause of some pain-related behavior, where this cause is understood to be another mental state, presumably, not identified with pain at all. (Remember, the functionalist wishes to avoid the vicious circularity that plagued the behaviorist’s explanations of behavior.)
There is an undeniable sensation to pain: it is something you feel. In fact, some might argue that at the conscious level, that is all there is to pain. Sure, there is the detection of tissue damage and the host physiological and neurological events transpiring, and yes, there is the pain-related behavior, too. However, we must not leave out of our explanation of pain the feel of pain.  Other qualitative mental states might include experiences of colored objects, such as those a person with normal color vision has every day.
In seeing a Granny Smith apple in the basket on a dining room table, she has a visual experience as of a green object. But the functionalist can only talk about the experience in terms of the function or causal role it plays. So, for example, the functionalist can speak to Freya’s green experience as being the cause of her belief that she sees a green apple in the basket. But the functionalist cannot speak to the feeling Freya (or any of us) has in seeing a ripe green Granny Smith. We think there is a corresponding feeling to color experiences like Freya’s over and above whatever beliefs they might go on to cause us to have. Since mental states like pain and color experiences are identified solely by their functional role, the functionalist seems without the resources to account for these qualitative mental states.
The functionalist might reply by offering a treatment of qualia in terms of what such aspects of experience function to do for us. The vivid, ripe greenness of the Granny Smith functions to inform Freya about a source of food in a way that pulls her visual attention to it. Freya’s color experiences allow her to form accurate beliefs about the objects in her immediate environment. It is certainly true that ordinary visual experience provide us with beautiful moments in our lives. However, they likely function to do much more besides. Likewise, it is more likely that there is a function for the qualitative or feeling aspects of some mental states, and that these aspects can be understood in terms of their functions, than it is that these aspects are free-floating above the causal order of things. So, the functionalist who wishes to try to account for qualia need not remain silent on the issue.
Conclusion
We have not considered all the possible objections to functionalism, nor have we considered more sophisticated versions of functionalism that aim to get around the more pernicious objections we have considered. The idea that minds really are kinds of computing machines is still very much alive and as controversial as ever. Taking that idea seriously means having to wrestle with a host of questions at the intersection of philosophy of mind, philosophy of action, and personal identity.
In what sense is Freya truly an agent of her own actions, if we merely cite a cold input to explain some behavior of hers? That is to say, how does Freya avow her own beliefs on a merely functionalist view? If minds are kinds of computers, then what does that make thinking creatures like Freya? Kinds of robots, albeit sophisticated ones? These and other difficult questions will need to be answered satisfactorily before many philosophers will be content with a functionalist theory of mind. For other philosophers, a start down the right path, away from Cartesian dualism and between the two terrors of materialism and behaviorism, has already been made.
References
Putnam, Hilary. (1960) 1975. “Minds and Machines.” Reprinted in Mind, Language, and Reality , 362-385. Cambridge: Cambridge University Press.
Searle, John. 1980. “Minds, Brains, and Programs.” Behavioral and Brain Sciences 3(3): 417-457.
Turing, Alan, M. 1936. “On Computable Numbers, with an Application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society 42 (1): 230-265.
Turing, Alan, M. 1950. “Computing Machinery and Intelligence.” Mind 49: 433-460.
Further Reading
Block, Ned. 1980a. Readings in the Philosophy of Psychology, Volumes 1 and 2 . Cambridge, MA: Harvard University Press.
Block, Ned. 1980b. “Troubles With Functionalism.” In Block 1980a, 268-305.
Gendler, Tamar. 2008. “Belief and Alief.” Journal of Philosophy 105(10): 634-663.
Jackson, Frank. 1982. “Epiphenomenal Qualia.” Philosophical Quarterly 32: 127-136.
Lewis, David. 1972. “Psychophysical and Theoretical Identifications.” In Block 1980a, 207-215.
Lewis, David. 1980. “Mad Pain and Martian Pain.” In Block 1980, 216-222.
Nagel, Thomas. 1974. “What Is It Like To Be a Bat?” Philosophical Review 83: 435-450.
Putnam, Hilary. 1963. “Brains and Behavior.” Reprinted in Putnam 1975b, 325-341.
Putnam, Hilary. 1967. “The Nature of Mental States.” Reprinted in Putnam 1975b, 429-440.
Putnam, Hilary. 1973. “Philosophy and our Mental Life.” Reprinted in Putnam 1975b, 291-303.
Putnam, Hilary. 1975a. “The Meaning of ‘Meaning.’” Reprinted in Putnam 1975b, 215-271.
Putnam, Hilary. 1975b. Mind, Language, and Reality . Cambridge: Cambridge University Press.
Shoemaker, Sydney. 1984. Identity, Cause, and Mind . Cambridge: Cambridge University Press.
Shoemaker, Sydney. 1996. The First-Person Perspective and Other Essays . Cambridge: Cambridge University Press.


8.6: Glossary
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Assessment center Consists of a series of standardized evaluations of behavior based on multiple inputs.
Behavioral observation scale Identifies observable behaviors as they relate to performance and is less demanding of the evaluator.
Behaviorally anchored rating scale A system that requires considerable work prior to evaluation but, if the work is carefully done, can lead to highly accurate ratings with high inter-rater reliability.
Central tendency error The failure to recognize either very good or very poor performers.
Critical incident technique A technique where supervisors record incidents, or examples, of each subordinate’s behavior that led to either unusual success or unusual failure on some aspect of the job.
Distributive justice Where employees receive (at least a portion of) their rewards as a function of their level of contribution to the organization.
Extrinsic rewards Rewards that are external to the work itself.
Flexible benefits system A rewards program where employees are allowed some discretion in the determination of their own packages and can make trade-offs, within certain limits.
Gain sharing An incentive plan in which employees or customers receive benefits directly as a result of cost-saving measures that they initiate or participate in.
Graphic rating scale A performance appraisal technique where the supervisor or rater is typically presented with a printed or online form that contains both the employee’s name and several evaluation dimensions (quantity of work, quality of work, knowledge of job, attendance). The rater is then asked to rate the employee by assigning a number or rating on each of the dimensions.
Halo effect Results in a supervisor assigning the same rating to each factor being evaluated for an individual.
Intrinsic motivation The desire to do a task because you enjoy it.
Intrinsic rewards Rewards that are external to the work itself.
Leniency error Fails to distinguish adequately between good and bad performers and instead relegates almost everyone to the same or related categories.
Lump-sum pay increase A technique that allows employees to decide how (that is, in what amounts) they wish to receive their pay raises for the coming year.
Management by objectives Closely related to the goal-setting theory of motivation.
Participative pay decisions Involving employees in pay raise decisions.
Performance appraisals A system that provides a means of systematically evaluating employees across various performance dimensions to ensure that organizations are getting what they pay for.
Recency error Occurs when, in an evaluation, a supervisor may give undue emphasis to performance during the past months—or even weeks—and ignore performance levels prior to this.
Reliability The extent to which the instrument consistently yields the same results each time it is used.
Skills-based incentives Rewards employees on the basis of the skills they possess and not just the skills they are allowed to use at work.
Strictness error Fails to distinguish adequately between good and bad performers and instead relegates almost everyone to the same or related categories.
Validity The extent to which an instrument actually measures what it intends to measure.


9.2E: Hyperbolas (Exercises)
section 9.2 exercises
Section 9.2 Exercises
In problems 1–4, match each graph to equations A–D.
A. \(\dfrac{x^2}{4} - \dfrac{y^2}{9} = 1\)
B. \(\dfrac{x^2}{9} - \dfrac{y^2}{4} = 1\)
C. \(y^2 - \dfrac{x^2}{9} = 1\)
D. \(\dfrac{y^2}{9} - x^2 = 1\)
1. 2. 3. 4.
In problems 5–14, find the vertices, length of the transverse axis, and equations of the asymptotes. Sketch the graph. Check using a graphing utility.
5. \(\dfrac{x^2}{4} - \dfrac{y^2}{25} = 1\)
6. \(\dfrac{y^2}{16} - \dfrac{x^2}{9} = 1\)
7. \(y^2 - \dfrac{x^2}{4} = 1\)
8. \(x^2 - \dfrac{y^2}{25} = 1\)
9. \(x^2 - 9y^2 = 9\)
10. \(y^2 - 4x^2 = 4\)
11. \(9y^2 - 16x^2 = 144\)
12. \(16x^2 - 25y^2 = 400\)
13. \(9x^2 - y^2 = 18\)
14. \(4y^2 - x^2 = 12\)
In problems 15–16, write an equation for the graph.
15. 16.
In problems 17–22, find the standard form of the equation for a hyperbola satisfying the given conditions.
17. Vertices at (0, 4) and (0, -4); asymptote \(y = \dfrac{1}{2}x\)
18. Vertices at (-6, 0) and (6, 0); asymptote \(y = 3x\)
19. Vertices at (-3, 0) and (3, 0); passes through (5, 8)
20. Vertices at (0, 4) and (0, -4); passes through (6, 5)
21. Asymptote \(y = x\); passes through (5, 3)
22. Asymptote \(y = x\); passes through (12, 13)
In problems 23–30, match each graph to equations A–H.
A. \(\dfrac{\left( x - 1 \right)^2}{9} - \dfrac{\left( y - 2 \right)^2}{4} = 1\)
B. \(\dfrac{\left( x + 1 \right)^2}{9} - \dfrac{\left( y + 2 \right)^2}{4} = 1\)
C. \(\dfrac{\left( x + 1 \right)^2}{9} - \dfrac{\left( y + 2 \right)^2}{16} = 1\)
D. \(\dfrac{\left( x - 1 \right)^2}{9} - \dfrac{\left( y - 2 \right)^2}{16} = 1\)
E. \(\dfrac{\left( y - 2 \right)^2}{4} - \dfrac{\left( x - 1 \right)^2}{9} = 1\)
F. \(\dfrac{\left( y + 2 \right)^2}{4} - \dfrac{\left( x + 1 \right)^2}{9} = 1\)
G. \(\dfrac{\left( y + 2 \right)^2}{4} - \dfrac{\left( x + 1 \right)^2}{16} = 1\)
H. \(\dfrac{\left( y - 2 \right)^2}{4} - \dfrac{\left( x - 1 \right)^2}{16} = 1\)
23. 24. 25. 26.
27.
28.
29.
30.
In problems 31–40, find the center, vertices, length of the transverse axis, and equations of the asymptotes. Sketch the graph. Check using a graphing utility.
31. \(\dfrac{\left( x - 1 \right)^2}{25} - \dfrac{\left( y + 2 \right)^2}{4} = 1\)
32. \(\dfrac{\left( y - 3 \right)^2}{16} - \dfrac{\left( x + 5 \right)^2}{36} = 1\)
33. \(\dfrac{\left( y - 1 \right)^2}{9} - \left( x + 2 \right)^2 = 1\)
34. \(\dfrac{\left( x - 1 \right)^2}{25} - \left( y - 6 \right)^2 = 1\)
35. \(4x^2 - 8x - y^2 = 12\)
36. \(4y^2 + 16y - 9x^2 = 20\)
37. \(4y^2 - 16y - x^2 - 2x = 1\)
38. \(4x^2 - 16x - y^2 + 6y = 29\)
39. \(9x^2 + 36x - 4y^2 + 8y = 4\)
40. \(9y^2 + 36y - 16x^2 - 96x = - 36\)
In problems 41–42, write an equation for the graph.
41. 42.
In problems 43–44, find the standard form of the equation for a hyperbola satisfying the given conditions.
43. Vertices (-1,-2) and (-1,6); asymptote \(y - 2 = 2\left( {x + 1} \right)\)
44. Vertices (-3,-3) and (5,-3); asymptote \(\;y + 3 = \dfrac{1}{2}\left( {x - 1} \right)\)
In problems 45–48, find the center, vertices, length of the transverse axis, and equations of the asymptotes. Sketch the graph. Check using a graphing utility.
45. \(y = \pm 4\sqrt {9x^2 - 1} \)
46. \(y = \pm \dfrac{1}{4}\sqrt {9x^2 + 1} \)
47. \(y = 1 \pm \dfrac{1}{2}\sqrt {9x^2 + 18x + 10} \)
48. \( = - 1 \pm 2\sqrt {9x^2 - 18x + 8} \)
In problems 49–54, find the foci.
49. \(\dfrac{y^2}{6} - \dfrac{x^2}{19} = 1\)
50. \(x^2 - \dfrac{y^2}{35} = 1\)
51. \(\dfrac{\left( x - 1 \right)^2}{15} - \left( y - 6 \right)^2 = 1\)
52. \(\dfrac{\left( y - 3 \right)^2}{47} - \dfrac{\left( x + 5 \right)^2}{2} = 1\)
53. \(y = 1 \pm \dfrac{4}{3}\sqrt {x^2 + 8x + 25} \)
54. \(y = - 3 \pm \dfrac{12}{5}\sqrt {x^2 - 4x - 21} \)
In problems 55–66, find the standard form of the equation for a hyperbola satisfying the given conditions.
55. Foci (5, 0) and (-5, 0), vertices (4, 0) and (4, 0)
56. Foci (0, 26) and (0, -26), vertices (0, 10) and (0, -10)
57. Focus (0, 13), vertex (0, 12), center (0, 0)
58. Focus (15, 0), vertex (12, 0), center (0, 0)
59. Focus (17, 0) and (-17, 0), asymptotes \(y = \dfrac{8}{15}x\) and \(y = - \dfrac{8}{15}x\)
60. Focus (0, 25) and (0, 25), asymptotes \(y = \dfrac{24}{7}x\) and \(y = - \dfrac{24}{7}x\)
61. Focus (10, 0) and (-10, 0), transverse axis length 16
62. Focus (0, 34) and (0, -34), transverse axis length 32
63. Foci (1, 7) and (1, -3), vertices (1, 6) and (1,-2)
64. Foci (4, -2) and (-6, -2), vertices (2, -2) and (-4, -2)
65. Focus (12, 3), vertex (4, 3), center (-1, 3)
66. Focus (-3, 15), vertex (-3, 13), center (-3, -2)
67. LORAN Stations A and B are 100 kilometers apart and send a simultaneous radio signal to a ship. The signal from A arrives 0.0002 seconds before the signal from B. If the signal travels 300,000 kilometers per second, find an equation of the hyperbola on which the ship is positioned if the foci are located at A and B.
68. Thunder and Lightning Anita and Samir are standing 3050 feet apart when they see a bolt of light strike the ground. Anita hears the thunder 0.5 seconds before Samir does. Sound travels at 1100 feet per second. Find an equation of the hyperbola on which the lighting strike is positioned if Anita and Samir are located at the foci.
69. Cooling Tower The cooling tower for a power plant has sides in the shape of a hyperbola. The tower stands 179.6 meters tall. The diameter at the top is 72 meters. At their closest, the sides of the tower are 60 meters apart. Find an equation that models the sides of the cooling tower.
70. Calibration A seismologist positions two recording devices 340 feet apart at points A and B. To check the calibration, an explosive is detonated between the devices 90 feet from point A. The time the explosions register on the devices is noted and the difference calculated. A second explosion will be detonated east of point A. How far east should the second explosion be positioned so that the measured time difference is the same as for the first explosion?
71. Target Practice A gun at point A and a target at point B are 200 feet apart. A person at point C hears the gun fire and hit the target at exactly the same time. Find an equation of the hyperbola on which the person is standing if the foci are located at A and B. A fired bullet has a velocity of 2000 feet per second. The speed of sound is 1100 feet per second.
72. Comet Trajectories A comet passes through the solar system following a hyperbolic trajectory with the sun as a focus. The closest it gets to the sun is 3×108 miles. The figure shows the trajectory of the comet, whose path of entry is at a right angle to its path of departure. Find an equation for the comet’s trajectory. Round to two decimal places
.
73. The conjugate of the hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\) is \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = - 1\). Show that \(5y^2 - x^2 + 25 = 0\) is the conjugate of \(x^2 - 5y^2 + 25 = 0\).
74. The eccentricity \(e\) of a hyperbola is the ratio \(\dfrac{c}{a}\), where \(c\) is the distance of a focus from the center and \(a\) is the distance of a vertex from the center. Find the eccentricity of \(\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1\).
75. An equilateral hyperbola is one for which a = b. Find the eccentricity of an equilateral hyperbola.
76. The latus rectum of a hyperbola is a line segment with endpoints on the hyperbola that passes through a focus and is perpendicular to the transverse axis. Show that \(\dfrac{2b^2}{a}\)is the length of the latus rectum of \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\).
77. Confocal hyperbolas have the same foci. Show that, for \(0 < k < 6\), all hyperbolas of the form \(\dfrac{x^2}{k} - \dfrac{y^2}{6 - k} = 1\) are confocal.
- Answer
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1. B
3. D
5. Vertices \((\pm 2, 0)\), transverse length = 4, asymptotes \(y = \pm 5/2x\),
7. Vertices \((0, \pm 1)\), transverse length = 2, asymptotes \(y = \pm 1/2x\),
9. Vertices \((\pm 3,0)\) transverse length = 6, asymptotes \(y =\pm 1/3x\),
11. Vertices \((0, \pm 4)\), transverse length = 8, asymptotes \(y = \pm 4/3x\)
13. Vertices \((\pm \sqrt{2}, 0)\), transverse length = \(2\sqrt{2}\), asymptotes \(y = \pm 3x\),
15. \(\dfrac{y^2}{4} - \dfrac{x^2}{9} = 1\)
17. \(\dfrac{y^2}{16} - \dfrac{x^2}{64} = 1\)
19. \(\dfrac{y^2}{9} - \dfrac{x^2}{36} = 1\)
21. \(\dfrac{y^2}{16} - \dfrac{x^2}{16} = 1\)
23. C
25. H
27. B
29. A
31. Center (1, -2), vertices (6, -2) and (-4, -2), transverse length = 10, asymptotes \(y = \pm 2/5(x - 1) - 2\)
33. Center (-2, 1), vertices (-2, 4) and (-2, -2), transverse length = 6, asymptotes \(y = \pm 3(x + 2) + 1\)
35. Center (1, 0), vertices (3, 0) and (-1, 0), transverse length = 4, asymptotes \(y =\pm 2(x-1)\)
37. Center (-1, 2), vertices (-1, 4) and (-1, 0), transverse length = 4, asymptotes \(y = \pm 1/2(x + 1) + 2\)
39. Center (-2, 1), vertices (0, 1) and (-4, 1), transverse length = 4, asymptotes \(y = \pm 3/2(x + 2) + 1\)
41. \(\dfrac{(y + 1)^2}{9} - \dfrac{(x - 4)^2}{4} = 1\)
43. \(\dfrac{(y - 2)^2}{16} - \dfrac{(x + 1)^2}{4} = 1\)
45. Center (0,0), vertices \((\pm 1/3,0)\), transverse length = 2/3, asymptotes \(y = \pm 12 x\)
46. Center (-1, 1), vertices (-1, 3/2) and (-1, 1/2), transverse length = 1, asymptotes \(y = \pm 3/2 (x + 1) + 1\)
49. Foci \((0, \pm 5)\)
51. Foci (5, 6) and (-3, 6)
53. Foci (-4, 6) and (-4, -4)
55. \(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\)
57. \(\dfrac{y^2}{144} - \dfrac{x^2}{25} = 1\)
59. \(\dfrac{x^2}{225} - \dfrac{y^2}{64} = 1\)
61. \(\dfrac{x^2}{64} - \dfrac{y^2}{36} = 1\)
63. \(\dfrac{(y - 2)^2}{16} - \dfrac{(x - 1)^2}{9} = 1\)
65. \(\dfrac{(x + 1)^2}{25} - \dfrac{(y - 3)^2}{144} = 1\)
67. \(\dfrac{x^2}{900} - \dfrac{y^2}{1600} = 1\)
69. \(\dfrac{x^2}{900} - \dfrac{y^2}{14400.3636} = 1\)
71. \(\dfrac{x^2}{3025} - \dfrac{y^2}{6975} = 1\)
73. \(5y^2 - x^2 + 25 = 0\) can be put in the form \(\dfrac{y^2}{5} - \dfrac{x^2}{25} = -1\). \(x^2 - 5y^2 + 25 = 0\) can be put in the form \(\dfrac{y^2}{5} - \dfrac{x^2}{25} = 1\) showing they are conjugate.
75. \(\sqrt{2}\)
77. No matter the value of \(k\), the foci are at \((\pm \sqrt{6}, 0)\)


3.7: Shoes
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Shoes
Norwegian-American Decorated by Olga Edseth, 1998 Recycled shoes, paint Gift of Olga Edseth MHAHS 2003.056.0001 |
Clothing often expresses ethnic identity, and these hot pink pumps with two-inch heels make a size eight cultural fashion statement. In 1998, Olga Edseth purchased this pair of leather shoes at a Dodgeville garage sale for fifty cents. Norwegian folk painting has a rich tradition of rosemaling a variety of surfaces. The decoration of these particular functional objects not only draws on this practice, but gains new life and symbolic meaning through Norwegian folk painting. These shoes become heritage “on the go.” In addition to signing and dating her work on the shoes’ soles, Olga lists the price she paid and notes that these are, “The first pair of shoes I ever rosemaled.”
Regardless of where we come from, clothing functions as a highly visible marker of individual and ethnic identity in both the celebratory and everyday realms of life. Articles of clothing worn for special occasions, whether festive or somber, annual or as rites of passage, are commonly referred to as “folk costumes.” In Wisconsin and Dane County, examples that may come to mind range from the Norwegian bunad , German lederhosen , to Hmong paj ntaub . While each folk costume may carry different symbolic significance stretching across generations and sometimes oceans, the wearer becomes, even temporarily, a bearer of traditions and a performer of heritage.
By 1998, Olga Edseth was an active rosemaler in the Mount Horeb community and took this notion of wearing one’s ethnicity to a street-level understanding. That year, while at a garage sale in Dodgeville, Mount Horeb, Wisconsin she came across a hot pink pair of size eight leather pumps. She bought them for fifty cents. Having practiced rosemaling since 1946, Olga was self-taught and practiced for over sixty years, applying her craft not only to her home, where she collected and displayed a massive assortment rosemalled object, but around Mount Horeb. Whether it was a street light or a banner, Olga was happy to apply her craft. Although featuring vibrant floral motifs, such as roses, geometric shapes, and tendrils, rosemaling is characterized by its two-dimensionality. Originally this painting style was applied to wooden surfaces which had been carved or on the interior walls of churches. Over time it came to represent art in the home and was easily expanded to be applied to any wood surface, regardless of various size and purpose. It is utilized in print form for various commercial or decorative usage. These heels must have struck Olga as a new challenge towards which she could apply her craft.
Through applying classic rosemaling features, these pumps became a form of heritage “on the go.” The toe of each shoe becomes the central area of focus through its use of a blue and turquoise Rocco C stem motif with a golden center. The sides feature lively acanthus leafs of gold, green, blue and turquoise, with black outlines and accents which meet at the heel. No section is left untouched as the two inch heels are similarly painted with turquoise tendrils. On the sole of the right shoe not only did Olga sign her artwork, but left two notes. The first, at the bridge, reads, “from a garage sale 50 ¢,” and, as if in celebratory laugher, she wrote, “The first pair of shoes I ever rosemaled” along the right edge.
Wherever we go, we carry with us the stories of countless generations, and how we perform our ethnic identity speaks volumes to the continuation of the traditions of not only our ancestors, but our community and the role we play in their narrative. For Olga Edseth, her artwork demonstrates that rosemaling and Norwegian-American ethnicity are far from static traditions, but dynamic expressions of identity one can wear with pride.
References
Gilmore, Janet. “Mount Horeb’s Oljanna Venden Cunneen. A Norwegian-American Rosemaler ‘on the Edge.’” ARV Nordic Year of Folklore , 2009(65): 25-48.


9.3: Filtration Rate
Filtration is the last stage in the treatment process for removing particles from the water. The filtration media can vary from sand to gravel to anthracite coal to garnet or a variety of materials layered to trap the remaining suspended solids in the water. Filtration rate is the measure of how much water passes through a certain sized filter over a specific time. Typically filtration rates are expressed in gallons per minute per square foot of filter area (gpm/ft\(^2\)). It is the flow of water through the surface area of a filter.
In the picture below a filter is 5 ft by 5 ft and divided into 1 ft by 1 ft sections. This is done to demonstrate how filtration rates are expressed (in 1 ft\(^2\) sections).
As flow enters a filter it travels down through the media and then out to either a finished water clear well for distribution or perhaps disinfected prior to delivery.
This is the formula used for Filtration Rates:
\[\mathrm{FR}=\dfrac{\text { Flow }}{\text { SurfaceArea }} \nonumber \]
Since Filtration Rates are expressed as gpm/ft\(^2\) then it only makes sense to convert the flow rate to gpm prior to dividing by the filter surface area. However, in this equation it is not necessary to do so prior to dividing.
The “Pie-Wheel” for Filtration Rates looks like this:
Remember that the units for the flow and area will determine the unit for filtration rate.
Example \(\PageIndex{1}\)
A water treatment plant with a 100 ft2 filter has a flow of 1,650 gpm. What is the filtration rate expressed in gpm/ft\(^2\).
Solution
\[\dfrac{1,650 \mathrm{gpm}}{100 \mathrm{ft}^{2}}=16.5 \mathrm{gpm} / \mathrm{ft}^{2} \nonumber \]
This is a straight forward example with the units given in the same units requested to be expressed in the answer. Let’s try another.
Example \(\PageIndex{2}\)
A water treatment plant has an average daily flow of 10.5 MGD. The filter at the end of the treatment process is 22 ft by 15 ft. What is the filtration rate expressed as gpm/ft\(^2\)?
Solution
In this question you need to calculate the area of the filter and determine if you want to convert the daily flow to gpm before or after you divide it by the area. Both ways of solving are explained below.
\[\dfrac{10.5 \mathrm{MGD}}{22 \mathrm{ft} \times 15 \mathrm{ft}}=\dfrac{10.5 \mathrm{MGD}}{330 \mathrm{ft}^{2}}=0.032 \mathrm{MGD} / \mathrm{ft}^{2} \nonumber \]
Note that the units are million gallons per day per square feet and need to be converted to gpm/ft\(^2\).
\[\dfrac{0.032 \mathrm{MG}}{\text { Day }}=\dfrac{32,000 \mathrm{gal}}{\text { Day }} \times \dfrac{1 \text { day }}{24 \text { hours }} \times \dfrac{1 \text { hour }}{60 \mathrm{min}}=22.2 \mathrm{gpm} / \mathrm{ft}^{2} \nonumber \]
Another way to solve this is to convert the flow to gpm before you divide it by the area.
\[\dfrac{10,500,000 \text { gal }}{\text { day }} \times \frac{1 \text { day }}{24 \text { hours }} \times \frac{1 \text { hour }}{60 \text { min }}=\frac{7,291.7 \text { gallons }}{\text { minute }} \nonumber \]
Then plug this into the filtration rate formula.
\[\dfrac{7,292 \text { gal }}{22 \mathrm{ft} \times 15 \mathrm{ft}}=\dfrac{7,291 \mathrm{gal}}{330 \mathrm{ft}^{2}}=22.1 \mathrm{gpm} / \mathrm{ft}^{2} \nonumber \]
Take note that the answers are slightly different (0.1 off.) This is only due to rounding. If both solutions for this problem are rounded to the same number of digits you would get the exact same answer.
Exercise 9.3
(Express all filtration rates as gpm/ft\(^2\))
- A water treatment plant has a typical flow through a 175 ft\(^2\) filter of 755 gpm. What is the filtration rate?
- A circular filter with a diameter of 74 feet receives a flow of 2,200 gpm. What is the filtration rate?
- A three-hour test revealed that 75,000 gallons passed through a filter with dimensions of 10 ft by 20 ft. Find the filtration rate.
- A water treatment plant has a total of 4 filters. The plant is designed to treat a daily flow of 7.25 million gallons. The target filtration rate is 2.25 gpm/ft\(^2\). Assuming that all 4 filters are identical in size, what is the area of each filter?
- What is the rated capacity in MGD of a rapid sand filter that is 25 ft long and 16 feet wide when operated at a filtration rate of 1.75 gpm/ft\(^2\)?
- It was determined that the optimum filtration rate of a filter with the dimensions of 12 ft wide and 18 feet long is 2.45 gpm/ft\(^2\). What is the maximum flow rate in cfs?
- A water treatment plant treated 70 million gallons through 3 identical filters in one week. If the average filtration rate was 3.15 gpm/ft\(^2\), what is the area of each filter?
Periodically, filters will need to be cleaned. Cleaning is done through a routine process called backwash. As filters remove particles, the efficiency decreases and the flow rates slow. At predetermined intervals, treatment operators will backwash the filters. Backwashing is nothing more than reversing the flows. However, backwashing rates are typically 10 or more times greater than filtration rates.
- After 6 hours in operation a 20ft by 30 ft filter required backwashing. Assuming a flow rate of 10,000 gpm, what was the backwash rate?
- A 15 ft by 17 ft filter was backwashed at a rate of 22.5 gpm/ft\(^2\) for 20 minutes. How many gallons were used for the backwashing process?
- It was determined that a filter with a surface area of 725 ft\(^2\) needs to be backwashed for 12 minutes using 94,500 gallons. What would the corresponding backwash rate be in gpm/ft\(^2\)?


3.8: Multiply and Divide Integers (Part 2)
Evaluate Variable Expressions with Integers
Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.
Evaluate \(2x^2 − 3x + 8\) when \(x = −4\).
Solution
| Substitute \(\textcolor{red}{-4}\) for x. | \(2(\textcolor{red}{-4})^{2} - 3(\textcolor{red}{-4}) + 8\) |
| Simplify exponents. | \(2(16) - 3(-4) + 8\) |
| Multiply. | \(32 - (-12) + 8\) |
| Subtract. | \(44 + 8\) |
| Add. | \(52\) |
Keep in mind that when we substitute \(−4\) for \(x\), we use parentheses to show the multiplication. Without parentheses, it would look like \(2 • −4^2 − 3 • −4 + 8\).
Evaluate: \(3x^2 − 2x + 6\) when \(x = −3\)
- Answer
-
\(39\)
Evaluate: \(4x^2-x-5\) when \(x = −2\)
- Answer
-
\(13\)
Evaluate \(3x + 4y − 6\) when \(x = −1\) and \(y = 2\).
Solution
| Substitute x = \(\textcolor{red}{-1}\) and y = \(\textcolor{blue}{2}\). | \(3(\textcolor{red}{-1}) + 4(\textcolor{blue}{2}) - 6\) |
| Multiply. | \(-3 + 8 - 6\) |
| Simplify. | \(-1\) |
Evaluate: \(7x + 6y − 12\) when \(x = −2\) and \(y = 3\)
- Answer
-
\(-8\)
Evaluate: \(8x − 6y + 13\) when \(x = −3\) and \(y = −5\)
- Answer
-
\(19\)
Translate Word Phrases to Algebraic Expressions
Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient .
Translate to an algebraic expression and simplify if possible: the product of \(−2\) and \(14\).
Solution
The word product tells us to multiply.
| Translate. | (−2)(14) |
| Simplify. | −28 |
Translate to an algebraic expression and simplify if possible: the product of \(−5\) and \(12\)
- Answer
-
\(-5(12)=-60\)
Translate to an algebraic expression and simplify if possible: the product of \(8\) and \(−13\)
- Answer
-
\(8(-13)=-104\)
Translate to an algebraic expression and simplify if possible: the quotient of \(−56\) and \(−7\).
Solution
The word quotient tells us to divide.
| Translate. | −56 ÷ (−7) |
| Simplify. | 8 |
Translate to an algebraic expression and simplify if possible: the quotient of \(−63\) and \(−9\)
- Answer
-
\(-63 \div -9 = 7\)
Translate to an algebraic expression and simplify if possible: the quotient of \(−72\) and \(−9\)
- Answer
-
\(-72 \div -9 = 8\)
Access Additional Online Resources
Key Concepts
-
Multiplication of Signed Numbers
-
To determine the sign of the product of two signed numbers:
Same Signs Product Two positives
Two negativesPositive
PositiveDifferent Signs Product Positive • negative
Negative • positiveNegative
Negative
-
To determine the sign of the product of two signed numbers:
-
Division of Signed Numbers
-
To determine the sign of the quotient of two signed numbers:
Same Signs Quotient Two positives
Two negativesPositive
PositiveDifferent Signs Quotient Positive • negative
Negative • PositiveNegative
Negative
-
To determine the sign of the quotient of two signed numbers:
-
Multiplication by \(-1\)
- Multiplying a number by \(-1\) gives its opposite: \(-1a=-a\)
-
Division by \(-1\)
- Dividing a number by \(-1\) gives its opposite: \(a \div (-1) = -a\)
Practice Makes Perfect
Multiply Integers
In the following exercises, multiply each pair of integers.
- −4 • 8
- −3 • 9
- −5(7)
- −8(6)
- −18(−2)
- −10(−6)
- 9(−7)
- 13(−5)
- −1 • 6
- −1 • 3
- −1(−14)
- −1(−19)
Divide Integers
In the following exercises, divide.
- −24 ÷ 6
- −28 ÷ 7
- 56 ÷ (−7)
- 35 ÷ (−7)
- −52 ÷ (−4)
- −84 ÷ (−6)
- −180 ÷ 15
- −192 ÷ 12
- 49 ÷ (−1)
- 62 ÷ (−1)
Simplify Expressions with Integers
In the following exercises, simplify each expression.
- 5(−6) + 7(−2)−3
- 8(−4) + 5(−4)−6
- −8(−2)−3(−9)
- −7(−4)−5(−3)
- (−5) 3
- (−4) 3
- (−2) 6
- (−3) 5
- −4 2
- −6 2
- −3(−5)(6)
- −4(−6)(3)
- −4 • 2 • 11
- −5 • 3 • 10
- (8 − 11)(9 − 12)
- (6 − 11)(8 − 13)
- 26 − 3(2 − 7)
- 23 − 2(4 − 6)
- −10(−4) ÷ (−8)
- −8(−6) ÷ (−4)
- 65 ÷ (−5) + (−28) ÷ (−7)
- 52 ÷ (−4) + (−32) ÷ (−8)
- 9 − 2[3 − 8(−2)]
- 11 − 3[7 − 4(−2)]
- (−3) 2 −24 ÷ (8 − 2)
- (−4) 2 − 32 ÷ (12 − 4)
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
- −2x + 17 when (a) x = 8 (b) x = −8
- −5y + 14 when (a) y = 9 (b) y = −9
- 10 − 3m when (a) m = 5 (b) m = −5
- 18 − 4n when (a) n = 3 (b) n = −3
- p 2 − 5p + 5 when p = −1
- q 2 − 2q + 9 when q = −2
- 2w 2 − 3w + 7 when w = −2
- 3u 2 − 4u + 5 when u = −3
- 6x − 5y + 15 when x = 3 and y = −1
- 3p − 2q + 9 when p = 8 and q = −2
- 9a − 2b − 8 when a = −6 and b = −3
- 7m − 4n − 2 when m = −4 and n = −9
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
- The product of −3 and 15
- The product of −4 and 16
- The quotient of −60 and −20
- The quotient of −40 and −20
- The quotient of −6 and the sum of a and b
- The quotient of −7 and the sum of m and n
- The product of −10 and the difference of p and q
- The product of −13 and the difference of c and d
Everyday Math
- Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio?
- Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?
Writing Exercises
- In your own words, state the rules for multiplying two integers.
- In your own words, state the rules for dividing two integers.
- Why is −2 4 ≠ (−2) 4 ?
- Why is −4 2 ≠ (−4) 2 ?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
Contributors and Attributions
- Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.


11.4: Achieved Status vs. Ascribed Status
Achieved status refers to the status level an individual in society has earned through work, education, luck, and/or social climbing. Achieved status is changeable throughout one's life. An example would be the status one earns when they become a doctor after years of studying and preparation. Having the credibility of being a doctor is a higher achieved status than the credibility of being a medical school student.
Ascribed status refers to the status that an individual acquires by virtue or by birth. The individual has no control over this status, it is simply the social position they are born into (James 2017). In many instances, this status is a social construct already pre-determined before one is born into the specific culture; it is nearly impossible to move up. One examples of ascribed status is eye color. When a baby is born, they have a certain eye color. Because the baby has no control over its eye color and can't change this feature it is considered an ascribed characteristic. Another example of an ascribed characteristic is kinship. When a baby is born, it is related by blood to a certain group of people, its kin, and nothing can change this.
Cultural Example of Achieved Status
American society possesses a number of examples of achieved status. In America, it is culturally acceptable (if you have the necessary resources) to begin life at the low end of the social ladder and to work your way up, by means of achieving a proper education, making useful social connections, and getting promoted within your career. Achieved status is not a position that a person is born into, but rather, it is attained through effort; this includes becoming an Olympic athlete, a doctor, or even a criminal. Although this struggle from the low end of the social ladder to the upper has become ingrained in the idea of America (The American Dream), the actual occurrence of someone rising from lower class to higher class is extremely rare. The number and severity of the obstacles one faces to climb the social ladder often depends on one's race, ethnicity, and beginning economic status.
Examples of Ascribed Status
Caste System
A caste is a system of social stratification found in India (as well as other parts of the world) dividing people into categories based on moral purity and pollution (James 2017). Abiding by the Caste System ultimately allows the people in the highest caste to control the rest of society and keep social barriers from being crossed. In India, the caste system consists of five different levels. The highest caste is considered the most "pure"- ritually and morally; the castes beneath it decline in "purity" and increase in "pollution". The Castes are as follow:
- Vedas(The Enlightened)
- Brahmins (priests and teachers)[6]
- Kshatriyas (rulers and soldiers)[7]
- Vaishyas (merchants and traders)[8]
- Shudras (laborers)[9]
Below these castes are the "Untouchables" or the Achuta (Dalit).
An “untouchable” or Dalit is considered outside of the caste system. They are the lowest in the Indian social stratification and treated very poorly often segregated from the rest of society. The "Untouchables" are taught early on that they are born into their caste to pay for bad behavior in their previous lives. They are limited to jobs considered ritually polluting such as taking care of human waste, metal work, street sweeping. Some insist that the Indian caste system doesn't exist anymore due to the incorporation of democracy, change in government programs and the implementation of rights for the "untouchables"; however, this is mostly only seen in the urban areas.
Gender
An ascribed status of an individual can be based on the sex that they are born. Gender typing is known as the process in which a child starts becoming aware of their gender. They slowly are socially constructed into the norm of that gender. This comes from an infant maturing and trying to focus and figure out their human behavior.Often there are certain activities that are reserved for males or females. Crossing the gender roles set forth by society is often frowned upon in communities that gender type. The vast majority of gender typing is culturally generated and not a creation of inborn biological distinctions between the sexes.[10]
An ethnographic example of gender typing can be observed in the early development of children in the United States. From birth, some U.S. parents set their children up for certain sexual categories by giving their babies gender-distinct names, clothes, and environments. The gender roles ascribed by the parents can lead to differences in intellectual and emotional development. For example, girls are provided with toys such as Barbies that encourage them to learn social rules and imitate behaviors. In contrast, boys are given more active toys and encouraged to explore. As a result of this early childhood gender typing, elementary school girls typically say they would choose lower paid, lower status careers such as nurse, teacher, or stewardess and boys are more likely to obtain higher paid, higher status careers such as pilot, architect, doctor, or lawyer, largely influence by their toys and surroundings. [11]
Political Organization
Political organization gives thorough information on the values/ideas of separate individuals. In modern human societies, people have organized in groups, usually according to their status/role in society. Some examples include:
- Political parties
- Non-governmental organizations
- Advocacy groups
- Special interest groups
Types of Political Organization
There are four types of political organization within groups and they are split between centralized or non-centralized political systems. An uncentralized political system is a political organization that requires several different parties to make a decision/law where as the centralized system is a political organization that is made up of one group that holds all authority within a government.[8]
Between the centralized and non-centralized forms of political organization, there are four groups:
- Band Society - a foraging group and the smallest group of political organization ranging anywhere from 20 to 200 people but typically consisting of about 80 people. Most of the people within this group are relatives either by birth or marriage. Since a band is a foraging society they do not have a place of permanent residence because they are constantly moving around. A band is referred to as egalitarian because there is no distinction between an upper and a lower class but they have a leader. The leader doesn't exhibit typical leadership by lacking power and influence over the members.
- Tribe - comprised of several bands. Leadership is based on ascribed and achieved statuses, some tribes may have a chief, and their organization is based on kinship. A tribe is more reliant on horticulture and pastoralism rather than foraging like bands and are usually a larger group than bands. A sub division of a tribe is the “Big Man” system which has a very influential leader who has no formal authority.
- Bushman - traditionally a society of people that are comprised of a band and thus egalitarian, which is defined as, relating to or believing in the principle that all people are equal and deserve equal rights and opportunities. Since they moved around a lot they had traditional gear that they wear which included a hide sling, blanket, and cloak in order to carry their food, firewood, a digging stick and even a separate smaller cloak to carry a baby. The woman gather and the men typically hunt in this society and the children do not have jobs.
- Chiefdom - the people are led by one person known as a chief. The chief governs over a group of tribes which are related through blood or marriage. In many chiefdoms, the chief is looked upon as the sole decider of what goes on in the society, and holds much sway with the members of the chiefdom. This centralized style of government has a social hierarchy and economic stratification unlike bands and tribes. On the other hand, a state is much more centralized than a chiefdom and has formal laws and authority. They have power to tax, maintain law and order, and to keep track of their citizens.
Nation, Nationalities and Nation-State
In the past, nations came about when groups of people who were similar in ways such as language, appearance, religious beliefs, and history came together to form territories, nation-states, and eventually countries. Out of these nations came the sense of nationalities and nationalism. Nationalism can be defined as a sense of belonging to a particular nation that comes with birth (loyalty and devotion). An example would be patriotism in the United States.
Nation: A group of people believed to share the same history, culture, identity, and oftentimes ethnicity.
Nation-State : A political unit consisting of an autonomous state inhabited predominantly by a people sharing a common culture, history and language. [12]
Nationality ( Nation-Building ): The sense of belonging and loyalty to a particular nation that comes about through origin, birth or naturalization.[13] Often, government officials will encourage citizens to feel loyalty and devotion for their nation-states; this is called nationalism.
Nation-building: An effort to instill a sense of nationality into the citizens of a state


3.11: Electron density map
A three-dimensional description of the electron density in a crystal structure, determined from X-ray diffraction experiments. X-rays scatter from the electron clouds of atoms in the crystal lattice; the diffracted waves from scattering planes h , k , l are described by structure factors \(\mathbf{F}_{hkl}.\) The electron density as a function of position x , y , z is the Fourier transform of the structure factors:
\(\rho(xyz) = {1\over V}\sum_{hkl} F(hkl)\exp[-2\pi i(hx + ky + lz)]\).
The electron density map describes the contents of the unit cells averaged over the whole crystal and not the contents of a single unit cell (a distinction that is important where structural disorder is present).
Three-dimensional maps are often evaluated as parallel two-dimensional contoured sections at different heights in the unit cell.
Units
Electron density is measured in electrons per cubic ångström, e Å -3 .


3.3: Patient Risk Assessment
To prevent and minimize MSI injuries related to patient handling activities, a risk assessment must be done to determine a patient’s ability to move, the need for assistance, and the most appropriate means of assistance (Provincial Health Services Authority [PHSA], 2010). There are four important areas to assess:
- The patient
- The environment
- The health care provider
- The organization of the work
Checklist 24 outlines what to assess and how to assess a patient prior to positioning, ambulation, and transfers.
Checklist 24: Risk AssessmentDisclaimer: Always review and follow your hospital policy regarding this specific skill. |
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Safety considerations:
|
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Steps |
Additional Considerations |
||
| 1. Assess your patient. |
There are three areas to assess:
1. Is the patient cooperative and able to follow directions? Ask patient to squeeze your hands. Is the behaviour predictable (non-aggressive, fearful, or fatigued)? Is the patient able to follow directions with cues? If yes, proceed to next question. If no, use a mechanical lift for transfers and/or assistive devices for repositioning in bed if patient has some abilities. 2. Can the patient bear weight? Ask patient to lift buttocks off the bed (also known as “bridging”) and hold the position for 5 seconds. The health care provider may give cues on how to lift buttocks off the bed. After bridging, ask the patient to perform a straight leg raise by lifting one leg up off the bed and holding it for 5 seconds while the other leg is kept bent. Repeat with the opposite leg. If yes, proceed to next question. If no, use an appropriate moving technique, such as a mechanical lift and/or assistive device, to transfer a non-weight-bearing patient. 3. Can the patient sit up on the side of the bed without support? Can the patient sit forward on a chair or the edge of the bed without support? If yes, decide on the amount of assist required (minimum, moderate, or maximum) according to your agency policy. If no, use a mechanical lift for transfers and/or an assistive device for repositioning if patient has some movement abilities. |
||
| 2. Assess your environment. |
Is there adequate space?
Is available equipment in proper working order? Have all hazards been removed? |
||
| 3. Assess yourself and readiness to perform procedures. |
Complete all required training according to health agency regulations.
Wear non-slip footwear. Maintain a neutral spine; do not twist or side bend, and use proper body mechanics when moving or positioning patients. Designate a leader if working in a team to mobilize or position a patient. Always use proper weight-shift techniques (side to side, front to back, and up and down). |
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| 4. Assess your work organization. |
Ensure adequate number of caregivers.
Ensure there is enough time to perform the procedure. Take rest breaks and vary activities to promote optimal back health. If patient is complex or bariatric, consult additional resources, seek assistance, and use assistive devices. |
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| Data source: Interior Health, 2012; National Institute of Occupational Safety and Health, 2010; PHSA, 2010; WorkSafeBC, 2010 |
To help you assess and make decisions about moving a patient, refer to these two useful tools.
Critical Thinking Exercises
- A patient requires repositioning in bed. After your assessment, you determine the patient is cooperative and predictable, able to weight bear, but unable to sit up unassisted. What are your two options to reposition the patient?
- When assessing your abilities to perform a patient-handling procedure, what five things must you consider?
- Vision and hearing impairments, along with language barriers, are risk factors when performing patient-handling procedures. What additional risk factors should be considered?


2.4: What Does “Evidence-Based Practice” Mean?
We often use this term “evidence-based practice” in public health when it comes to policies, programs, or healthcare services. A lot of people have what seem to be great ideas on how to solve health problems, but as the saying goes “ideas are a dime a dozen”. How do we know if this idea will really work? And especially when it comes to funding with government grants - which are taxpayer dollars - we need to make sure that we’re not just throwing money at random ideas and hoping they are effective.
Evidence-based practice means “systematically finding, appraising, and using evidence as the basis for decision making” (Seabert et al., 2021, p. 115). This means that before we create programs or propose policies, we should gather as much evidence on the problem and potential solutions as possible, assess the applicability of available evidence to the current problem, and then use this knowledge to help guide decisions on whether or how to move forward. There may have been programs or policies similar to our ideas that have already been tried in the target population, or in other communities. What if there is evidence that this particular solution doesn’t work well? We certainly wouldn’t want to propose a program or policy that had already been proven to be ineffective. Or perhaps there is evidence that certain implementation methods were more effective than others. We would want to use that insight to make our interventions or policies more likely to work.
Two types of evidence can be used in this process: objective and subjective . Objective evidence comes in the form of scientific studies and data collection, whereas subjective evidence may be in the form of expert advice or personal experiences and observations from community stakeholders. Typically objective evidence includes things that can be measured or observed, and is often expressed in numbers or percentages. Subjective evidence includes personal accounts, individual experiences and stories - this type of evidence is often based on the perceptions of the person observing something or being interviewed. In practice, objective evidence is often valued more highly than subjective evidence, in large part because well-done research uses several methods to reduce bias and misinterpretation. Objective data also tends to be more reliable than individual experiences or opinions. For example, a community member might perceive that homelessness is not a problem for their city, because they used to see many tents and makeshift shelters on sidewalks in their neighborhood, and recently the sidewalks have been clear. But the objective data may show a more complex picture: perhaps the individuals who had been living in those makeshift shelters had been forced to relocate, but were still unhoused.
Care should also be taken in interpreting research and applying objective evidence to different situations. Just because a particular program worked for one community doesn’t mean it will work in another. Subjective evidence from stakeholders in the community, or professional experience from public health workers, should still be valued. For example, a community health worker may know from their work that many of the unhoused are not living on the street - they are living in their cars or “couch-surfing”, and therefore they might not be aware of or have access to social services. Scientific studies may leave out certain populations, or we may not yet have scientific research available for a specific health problem. Therefore, it’s important to consider all sources of evidence when using an evidence-based approach to public health.