2.8: Geography of Health
Although the health of populations has been considered in most sections of this chapter, there are some aspects that can be addressed most appropriately on their own. The geography of health or medical geography is the study sickness or health across space. In the same way that it is possible to compare places by their population characteristics, it is possible to compare places by their health characteristics. Also in the same way that societies tend to pass through the stages of demographic transition, they also tend to pass through an epidemiological transition. The demographic transition and epidemiological transition are somewhat related. The improvements in food supply and better sanitation that lead to larger populations also lead to healthier populations. They are not exactly the same thing however. Epidemiology is the study of diseases themselves and the way that they function. The epidemiological transition is a way of representing the relationship between development and disease. In less developed places, infant mortality is high ( Figure 2.20 ) and infectious disease is the greatest threat. In more developed societies, the health threats tend to be chronic afflictionscancer, diabetes, or heart disease. One of the reasons for the difference between the developed and less developed places is the fact that in developed countries people tend to live longer. Countries in which people die in their forties will not have a problem with Alzheimer’s disease.
Mortality refers to death and morbidity refers to sickness. We can look at different rates of mortality and morbidity to gain some insight into the health of a population.
Diabetes was once related to development, but the relationship is weakening ( Figure 2.21 ). As obesity becomes a greater problem in developing countries, so does diabetes.
On the other hand, malaria is a disease of poor countries, or at least it is a disease of poor places ( Figure 2.22 ). The poorer parts of developing countries will likely have malaria problems, while the wealthier parts may not.
One of the changes that has occurred in medical geography is that the differences between the developed and less developed countries have been narrowing. Obesity was once considered fully within the sphere of developed world problems, but the obesity crisis has diffused into many parts of the developing world now. In the same way, infectious diseases such as HIV and Hepatitis C have become problems in developed countries.
Due to the work of numerous organizations, a number of diseases in many parts of the world have been reduced and some have been eradicated. Smallpox no longer exists in the wild and Dracunculiasis (Guinea worm) has gone from millions of cases thirty years ago to single digits in recent years. Other diseases have been dramatically reduced. Polio once infected millions of people per year across the world, but through vaccination programs, its range has been reduced to just two countries.
There is now even a category of diseases known as re-emerging diseases. These are diseases that were previously thought to be eradicable but are now returning to populations that had previously been largely free of them. Re-emerging diseases are diseases like tuberculosis and diphtheria that had been declining for decades that have begun to increase in prevalence.


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16: Social and Emotional Development in Late Adulthood (Age 60 and Beyond)
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16.0: What Does Psychology Say?
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Research indicates that while later adulthood involves social and emotional adjustments, these changes can be positive, often leading to more free time and positive emotions. LeShaun, a sixty-two-year-old nurse considering retirement, ponders how to fill her time without work colleagues after losing her spouse. Despite concerns over loneliness, she is exploring new activities, like volunteering abroad, though her health condition poses challenges.
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16.1: The Meaning of Aging in Late Adulthood
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This section explores the social and cultural aspects of aging, examining perspectives on older age, stereotypes, and discrimination. It covers how aging relates to Erikson's integrity versus despair challenge and how self and socioemotional development evolve in late adulthood. Older adults, despite stereotypes, often maintain independence and report increased happiness.
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16.2: Contexts - Family, Friendships, Romantic Relationships, and Social Communities in Late Adulthood
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This page explores social relationships and support systems in late adulthood, discussing evolving dynamics with family, friends, and romantic partners. Key theories, including disengagement, activity, and socioemotional selectivity theories, are examined to explain changes in social networks as people age. The text further highlights differences in family roles, such as grandparents and kinkeepers, and describes how marriage, divorce, and widowhood affect older adults.
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16.3: Retirement in Late Adulthood
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The section discusses key considerations in retirement planning, such as the financial and nonfinancial factors influencing the decision to retire, styles of retirement, and common barriers. It highlights the importance of early financial planning and discusses retirement styles, including phased and bridge employment. Barriers to retirement include financial constraints, health issues, and societal changes in retirement systems.
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16.4: Living Environments and Aging in Late Adulthood
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The section covers various housing environments for older adults, emphasizing the concept of aging in place and the role of caregivers. It explores options like independent living communities, assisted living, and nursing homes, considering the benefits and concerns of each. The financial implications and safety considerations are also discussed. The information highlights the challenges and resources in maintaining older adults' independence and socialization.
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16.5: Successful Social and Emotional Aging in Late Adulthood
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The page discusses successful emotional and social aging, emphasizing the importance of optimism, social support, and spiritual contentment in psychosocial well-being. It highlights the roles of resilience and gratitude in emotional aging, noting that older adults typically show increased life satisfaction. Theories like the selection, optimization, and compensation (SOC) framework outline strategies for maintaining high functioning despite physical declines.
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16.6: Key Terms
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This page provides a glossary of terms and concepts related to aging, retirement, and elder care. It includes definitions of various retirement plans such as 401(k) and pension plans, theories of aging like activity and disengagement theories, and concepts such as aging in place and bridge employment.


<IP_ADDRESS>: Dashes
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The dash is kind of like the superhero of punctuation. Not only does it have a cool, sleek name, it’s often referred to as the “super comma.” But you can’t just go around using dashes because you want super commas. There are some rules you should follow, and you don’t want to overuse any one type of punctuation.
The following two rules should help you make good decisions regarding that dashing dash:
His ideas regarding an evacuation in case of a zombie attack are certainly controversial—even radical.
I think the reason those books became so popular—so much so that they became a cultural phenomenon—is that the world was ready for a little bit of magic.
The changes that came with the arrival of the eleventh doctor—the screwdriver, the TARDIS, and all of the side characters—certainly surprised many fans.
Of course, while the rules related to using dashes are relatively simple, there seems to be great confusion over how the dash is actually created in word processing programs.
To be sure, the dash is no hyphen. The hyphen is smaller and comes between words like mother-in-law. The dash is longer and is created by placing two hyphens in a row. Then, after you space after the word after the dash, most word processing programs will turn those two little hyphens — into a dash—like so.


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Electrical and magnetic interactions
don't
quite behave like this, which is a point we'll take up later in the book.
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We can now see that the derivation would have been equally valid for \(U_i\ne U_f\). The two observers agree on the distance between the particles, so they also agree on the interaction energies, even though they disagree on the kinetic energies.
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Recall that uppercase \(P\) is power, while lowercase \(p\) is momentum.
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This is with the benefit of hindsight. At the time, the word “force” already had certain connotations, and people thought they understood what it meant and how to measure it, e.g., by using a spring scale. From their point of view, \(F=d{}p/d{}t\) was not a definition but a testable --- and controversial! --- statement.
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This pathological solution was first noted on page 83, and discussed in more detail on page 910.
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The converse isn't true, because kinetic energy doesn't depend on the direction of motion, but momentum does. We can change a particle's momentum without changing its energy, as when a pool ball bounces off a bumper, reversing the sign of \(p\).
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The part of the definition about “by a force” is meant to exclude the transfer of energy by heat conduction, as when a stove heats soup.
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“Black box” is a traditional engineering term for a device whose inner workings we don't care about.
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For conceptual simplicity, we ignore the transfer of heat energy to the outside world via the exhaust and radiator. In reality, the sum of these energies plus the useful kinetic energy transferred would equal \(W\).
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This subroutine isn't as accurate a way of calculating the period as the energy-based one we used in the undamped case, since it only checks whether the mass turned around at some point during the time interval \(\Delta{}t\).
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The relationship is \(\omega_{max\ A}/\omega_\text{o}=\sqrt{1-1/2Q^2}\), which is similar in form to the equation for the frequency of the free vibration, \(\omega_{f}/\omega_\text{o}=\sqrt{1-1/4Q^2}\). A subtle point here is that although the maximum of \(A\) and the maximum of \(A^2\) must occur at the same frequency, the maximum energy does not occur, as we might expect, at the same frequency as the maximum of \(A^2\). This is because the interaction energy is proportional to \(A^2\) regardless of frequency, but the kinetic energy is proportional to \(A^2\omega^2\). The maximum energy actually occurs are precisely \(\omega_\text{o}\).
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For example, the graphs calculated for sinusoidal driving have resonances that are somewhat below the natural frequency, getting lower with increasing damping, until for \(Q\le1\) the maximum response occurs at \(\omega=0\). In figure
m
, however, we can see that impulsive driving at \(\omega=2\omega_\text{o}\) produces a steady state with more energy than at \(\omega=\omega_\text{o}\).
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If you've learned about differential equations, you'll know that any second-order differential equation requires the specification of two boundary conditions in order to specify solution uniquely.
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Actually, if you know about complex numbers and Euler's theorem, it's not quite so nonsensical.
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Of course, you could tell in a sealed laboratory which way was down, but that's because there happens to be a big planet nearby, and the planet's gravitational field reaches into the lab, not because space itself has a special down direction. Similarly, if your experiment was sensitive to magnetic fields, it might matter which way the building was oriented, but that's because the earth makes a magnetic field, not because space itself comes equipped with a north direction.
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The zero here is really a zero
vector
, i.e., a vector whose components are all zero, so we should really represent it with a boldface \vc{0}. There's usually not much danger of confusion, however, so most books, including this one, don't use boldface for the zero vector.
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There is, however, a different operation, discussed in the next chapter, which multiplies two vectors to give a vector.
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The symbol \(\nabla\) is called a “nabla.” Cool word!


4.1: Chapter Summary
You now have an initial understanding of the types of writing that may be required of you during your post-secondary nursing program. When starting an assignment, the first thing you should do is closely read the assignment guidelines (and the marking rubric if you are provided one). You need to know what you are supposed to do before you do it. The next chapters will help you understand and advance your capacity to write, revise, and edit a paper, and ensure academic integrity.
Your Writing Journey
Now that you have a good sense of the different types of writing, you should focus on what is being asked of you in assignments. Pay attention to the language that your instructor provides – are they asking you to persuade, describe, or reflect on a particular topic? This will help you establish a writing style and is a crucial step to determine before you write an outline. Remember, the type of writing will influence what information you choose to put forward to the reader. It will be an important basis for the whole paper, so as you formulate your ideas, you should reflect on whether you are trying to influence an opinion, merely share information, or engage in self-analysis. You’re almost ready to start an outline!


1.3: Structural Arithmetic
Whenever the answer to a question turns out to be unexpectedly nice, one should ask oneself whether this is an accident, or whether there is some explanation which should perhaps have led one to expect such a result. For example:
- Exactly \(25\) of the integers up to \(100\) are prime numbers – and \(25\) is exactly one quarter of \(100\). This is certainly a beautifully memorable fact. But it is a numerical fluke, with no hidden mathematical explanation.
- \(11\) and \(101\) are prime numbers. Is this perhaps a way of generating lots of prime numbers:
\(11, 101, 1001, 10 001, 100 001, . . .?\)
It may at first be tempting to think so – until, that is, you remember what you found in Problem 6 (a)(iii).
Write out the first \(12\) or so powers of \(4\):
\(4, 16, 64, 256, 1024, 4096, 16384, 65536, ...\)
Now create two sequences:
the sequence of final digits: \(4, 6, 4, 6, 4, 6, . . .\)
the sequence of leading digits: \(4, 1, 6, 2, 1, 4, 1, 6, . . .\)
Both sequences seem to consist of a single “block”, which repeats over and over for ever.
(a) How long is the apparent repeating block for the first sequence? How long is the apparent repeating block for the second sequence?
(b) It may not be immediately clear whether either of these sequences really repeats forever. Nor may it be clear whether the two sequences are alike, or whether one is quite different from the other. Can you give a simple proof that one of these sequences recurs , that is, repeats forever?
(c) Can you explain why the other sequence seems to recur, and decide whether it really does recur forever?
The \(4\) by \(4\) “multiplication table” below is completely familiar.
\(\begin{array} &1&2&3&4 \\ 2&4&6&8 \\ 3&6&9&12 \\ 4&8&12&16\end{array}\)
What is the total of all the numbers in the \(4\) by \(4\) square? How should one write this answer in a way that makes the total obvious?


2.6: Systems Testing
Software Testing
Software testing is an investigation conducted to provide stakeholders with information about the quality of the software product or service under test. [1] Software testing can also provide an objective, independent view of the software to allow the business to appreciate and understand the risks of software implementation. Test techniques include the process of executing a program or application with the intent of finding software bugs (errors or other defects), and verifying that the software product is fit for use.
Software testing involves the execution of a software component or system component to evaluate one or more properties of interest. In general, these properties indicate the extent to which the component or system under test:
- meets the requirements that guided its design and development,
- responds correctly to all kinds of inputs,
- performs its functions within an acceptable time,
- is sufficiently usable,
- can be installed and run in its intended environments , and
- achieves the general result its stakeholders desire.
As the number of possible tests for even simple software components is practically infinite, all software testing uses some strategy to select tests that are feasible for the available time and resources. As a result, software testing typically (but not exclusively) attempts to execute a program or application with the intent of finding software bugs (errors or other defects). The job of testing is an iterative process as when one bug is fixed, it can illuminate other, deeper bugs, or can even create new ones.
Software testing can provide objective, independent information about the quality of software and risk of its failure to users or sponsors. [1]
Software testing can be conducted as soon as executable software (even if partially complete) exists. The overall approach to software development often determines when and how testing is conducted. For example, in a phased process, most testing occurs after system requirements have been defined and then implemented in testable programs. In contrast, under an agile approach , requirements, programming, and testing are often done concurrently.
Test plans
A test plan documents the strategy that will be used to verify and ensure that a product or system meets its design specifications and other requirements. A test plan is usually prepared by or with significant input from test engineers .
Depending on the product and the responsibility of the organization to which the test plan applies, a test plan may include a strategy for one or more of the following:
- Design Verification or Compliance test – to be performed during the development or approval stages of the product, typically on a small sample of units.
- Manufacturing or Production test – to be performed during preparation or assembly of the product in an ongoing manner for purposes of performance verification and quality control.
- Acceptance or Commissioning test – to be performed at the time of delivery or installation of the product.
- Service and Repair test – to be performed as required over the service life of the product.
- Regression test – to be performed on an existing operational product, to verify that existing functionality was not negatively affected when other aspects of the environment were changed (e.g., upgrading the platform on which an existing application runs).
A complex system may have a high level test plan to address the overall requirements and supporting test plans to address the design details of subsystems and components.
Test plan document formats can be as varied as the products and organizations to which they apply. There are three major elements that should be described in the test plan: Test Coverage, Test Methods, and Test Responsibilities. These are also used in a formal test strategy .
System testing is testing conducted on a complete integrated system to evaluate the system's compliance with its specified requirements .
System testing takes, as its input, all of the integrated components that have passed integration testing . The purpose of integration testing is to detect any inconsistencies between the units that are integrated together (called assemblages ). System testing seeks to detect defects both within the "inter-assemblages" and also within the system as a whole. The actual result is the behavior produced or observed when a component or system is tested. [1]
System testing is performed on the entire system in the context of either functional requirement specifications (FRS) or system requirement specification (SRS), or both. System testing tests not only the design, but also the behaviour and even the believed expectations of the customer. It is also intended to test up to and beyond the bounds defined in the software or hardware requirements specification(s).
Integration Testing
Integration testing (sometimes called integration and testing , abbreviated I&T ) is the phase in software testing in which individual software modules are combined and tested as a group. Integration testing is conducted to evaluate the compliance of a system or component with specified functional requirements . [1] It occurs after unit testing and before validation testing . Integration testing takes as its input modules that have been unit tested, groups them in larger aggregates, applies tests defined in an integration test plan to those aggregates, and delivers as its output the integrated system ready for system testing . [2]
Unit Testing
In computer programming , unit testing is a software testing method by which individual units of source code —sets of one or more computer program modules together with associated control data, usage procedures, and operating procedures—are tested to determine whether they are fit for use. [1]
User Acceptance Testing
User acceptance testing (UAT) consists of a process of verifying that a solution works for the user. [8] It is not system testing (ensuring software does not crash and meets documented requirements) but rather ensures that the solution will work for the user (i.e. tests that the user accepts the solution); software vendors often refer to this as "Beta testing".
This testing should be undertaken by a subject-matter expert (SME), preferably the owner or client of the solution under test, and provide a summary of the findings for confirmation to proceed after trial or review. In software development , UAT as one of the final stages of a project often occurs before a client or customer accepts the new system. Users of the system perform tests in line with what would occur in real-life scenarios. [9]
It is important that the materials given to the tester be similar to the materials that the end user will have. Testers should be given real-life scenarios such as the three most common or difficult tasks that the users they represent will undertake.
The UAT acts as a final verification of the required business functionality and proper functioning of the system, emulating real-world conditions on behalf of the paying client or a specific large customer. If the software works as required and without issues during normal use, one can reasonably extrapolate the same level of stability in production. [10]
User tests, usually performed by clients or by end-users, do not normally focus on identifying simple cosmetic problems such as spelling errors, nor on showstopper defects, such as software crashes ; testers and developers identify and fix these issues during earlier unit testing , integration testing , and system testing phases.
UAT should be executed against test scenarios.Test scenarios usually differ from System or Functional test cases in that they represent a "player" or "user" journey. The broad nature of the test scenario ensures that the focus is on the journey and not on technical or system-specific details, staying away from "click-by-click" test steps to allow for a variance in users' behaviour. Test scenarios can be broken down into logical "days", which are usually where the actor (player/customer/operator) or system (backoffice, front end) changes.
In industry, a common UAT is a factory acceptance test (FAT). This test takes place before installation of the equipment. Most of the time testers not only check that the equipment meets the specification, but also that it is fully functional. A FAT usually includes a check of completeness, a verification against contractual requirements, a proof of functionality (either by simulation or a conventional function test) and a final inspection.
The results of these tests give clients confidence in how the system will perform in production. There may also be legal or contractual requirements for acceptance of the system.
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1.1: Solving Linear Equations
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Solve Equations Using the General Strategy
Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.
Beginning by simplifying each side of the equation makes the remaining steps easier.
Example \(\PageIndex{1}\): How to Solve Linear Equations Using the General Strategy
Solve: \(-6(x + 3) = 24\).
Solution
Try It \(\PageIndex{2}\)
Solve: \(5(x + 3)=35\)
- Answer
-
\(x = 4\)
Try It \(\PageIndex{3}\)
Solve: \(6(y - 4) = -18\)
- Answer
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\(y = 1\)
GENERAL STRATEGY FOR SOLVING LINEAR EQUATIONS.
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Simplify each side of the equation as much as possible.
Use the Distributive Property to remove any parentheses.
Combine like terms. -
Collect all the variable terms on one side of the equation.
Use the Addition or Subtraction Property of Equality. -
Collect all the constant terms on the other side of the equation.
Use the Addition or Subtraction Property of Equality. -
Make the coefficient of the variable term to equal to 1.
Use the Multiplication or Division Property of Equality.
State the solution to the equation. - Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
Example \(\PageIndex{4}\)
Solve: \(-(y + 9) = 8\)
Solution
| Simplify each side of the equation as much as possible by distributing. | ||
| The only y term is on the left side, so all variable terms are on the left side of the equation. | ||
| Add 9 to both sides to get all constant terms on the right side of the equation. | ||
| Simplify. | ||
| Rewrite −y as −1y. | ||
| Make the coefficient of the variable term to equal to 1 by dividing both sides by −1. | ||
| Simplify. | ||
| Check: | ||
| Let y=−17. | ||
Try It \(\PageIndex{5}\)
Solve: \(-(y + 8) = -2\)
- Answer
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\(y = -6\)
Try It \(\PageIndex{6}\)
Solve: \(-(z + 4) = -12\)
- Answer
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\(z = 8\)
Example \(\PageIndex{7}\)
Solve: \(5(a - 3) + 5 = -10\)
Solution
| Simplify each side of the equation as much as possible. | ||
| Distribute. | ||
| Combine like terms. | ||
| The only a term is on the left side, so all variable terms are on one side of the equation. | ||
| Add 10 to both sides to get all constant terms on the other side of the equation. | ||
| Simplify. | ||
| Make the coefficient of the variable term to equal to 11 by dividing both sides by 55. | ||
| Simplify. | ||
| Check: | ||
| Let a=0. | ||
Try It \(\PageIndex{8}\)
Solve: \(2(m - 4) + 3 = -1\)
- Answer
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\(m = 2\)
Try It \(\PageIndex{9}\)
Solve:\(7(n - 3) - 8 = -15\)
- Answer
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\(n = 2\)
Example \(\PageIndex{10}\)
Solve: \(\frac{2}{3}(6m - 3) = 8 - m\)
Solution
| Distribute. | ||
| Add m to get the variables only to the left. | ||
| Simplify. | ||
| Add 2 to get constants only on the right. | ||
| Simplify. | ||
| Divide by 5. | ||
| Simplify. | ||
| Check: | ||
| Let m=2. | ||
Try It \(\PageIndex{11}\)
Solve: \(\frac{1}{3}(6u + 3) = 7 - u\)
- Answer
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\(u = 2\)
Try It \(\PageIndex{12}\)
Solve: \(\frac{2}{3}(9x - 12) = 8 + 2x\)
- Answer
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\(x = 4\)
Example \(\PageIndex{13}\)
Solve: \(8 - 2(3y + 5) = 0\)
Solution
| Simplify—use the Distributive Property. | |
| Combine like terms. | |
| Add 2 to both sides to collect constants on the right. | |
| Simplify. | |
| Divide both sides by −6−6. | |
| Simplify. | |
|
Check: Let y=−13.
|
Try It \(\PageIndex{14}\)
Solve: \(12 - 3(4j + 3) = -17\)
- Answer
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\(j = \frac{5}{3}\)
Try It \(\PageIndex{15}\)
Solve: \(-6 - 8(k - 2) = -10\)
- Answer
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\(k = \frac{5}{2}\)
Example \(\PageIndex{16}\)
Solve: \(4(x - 1)-2=5(2x+3)+6\)
Solution
| Distribute. | ||
| Combine like terms. | ||
| Subtract 4x to get the variables only on the right side since \(10>4\). | ||
| Simplify. | ||
| Subtract 21 to get the constants on left. | ||
| Simplify. | ||
| Divide by 6. | ||
| Simplify. | ||
| Check: | ||
| Let \(x=-\frac{9}{2}\). | ||
Try It \(\PageIndex{17}\)
Solve: \(6(p-3)-7=5(4p+3)-12\)
- Answer
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\(p = -2\)
Try It \(\PageIndex{18}\)
Solve: \(8(q +1)-5=3(2q-4)-1\)
- Answer
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\(q = -8\)
Example \(\PageIndex{19}\)
Solve: \(10[3 - 8(2s-5)] = 15(40 - 5s)\)
Solution
| Simplify from the innermost parentheses first. | ||
| Combine like terms in the brackets. | ||
| Distribute. | ||
| Add 160s to get the s’s to the right. | ||
| Simplify. | ||
| Subtract 600 to get the constants to the left. | ||
| Simplify. | ||
| Divide. | ||
| Simplify. | ||
| Check: | ||
| Substitute s=−2. | ||
Try It \(\PageIndex{20}\)
Solve: \(6[4−2(7y−1)]=8(13−8y)\).
- Answer
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\(y = -\frac{17}{5}\)
Try It \(\PageIndex{21}\)
Solve: \(12[1−5(4z−1)]=3(24+11z)\).
- Answer
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\(z = 0\)
Example \(\PageIndex{22}\)
Solve: \(0.36(100n+5)=0.6(30n+15)\).
Solution
| Distribute. | ||
| Subtract 18n to get the variables to the left. | ||
| Simplify. | ||
| Subtract 1.8 to get the constants to the right. | ||
| Simplify. | ||
| Divide. | ||
| Simplify. | ||
| Check: | ||
| Let n=0.4. | ||
Try It \(\PageIndex{23}\)
Solve: \(0.55(100n+8)=0.6(85n+14)\).
- Answer
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\(n = 1\)
Try It \(\PageIndex{24}\)
Solve: \(0.15(40m−120)=0.5(60m+12)\).
- Answer
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\(m = -1\)
Classify Equations
Consider the equation we solved at the start of the last section, 7x+8=−13. The solution we found was x=−3. This means the equation 7x+8=−13 is true when we replace the variable, x , with the value −3. We showed this when we checked the solution x=−3 and evaluated 7x+8=−13 for x=−3.
If we evaluate 7x+8 for a different value of x , the left side will not be −13.
The equation 7x+8=−13 is true when we replace the variable, x , with the value −3, but not true when we replace x with any other value. Whether or not the equation 7x+8=−13 is true depends on the value of the variable. Equations like this are called conditional equations.
All the equations we have solved so far are conditional equations.
CONDITIONAL EQUATION
An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation .
Now let’s consider the equation 2y+6=2(y+3). Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y .
| Distribute. | |
| Subtract 2y to get the y’s to one side. | |
| Simplify—the y’s are gone! |
But 6=6 is true.
This means that the equation 2y+6=2(y+3) is true for any value of y . We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity .
IDENTITY
An equation that is true for any value of the variable is called an identity .
The solution of an identity is all real numbers.
What happens when we solve the equation 5z=5z−1?
| Subtract 5z to get the constant alone on the right. | |
| Simplify—the z’s are gone! |
But \(0\neq −1\).
Solving the equation 5z=5z−1 led to the false statement 0=−1. The equation 5z=5z−1 will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction .
CONTRADICTION
An equation that is false for all values of the variable is called a contradiction .
A contradiction has no solution.
Example \(\PageIndex{25}\)
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
\(6(2n−1)+3=2n−8+5(2n+1)\)
Solution
| Distribute. | |
| Combine like terms. | |
| Subtract 12n to get the nn’s to one side. | |
| Simplify. | |
| This is a true statement. |
The equation is an identity.
The solution is all real numbers. |
Try It \(\PageIndex{26}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
\(4+9(3x−7)=−42x−13+23(3x−2)\)
-
Answer -
identity; all real numbers
Try It \(\PageIndex{27}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
\(8(1−3x)+15(2x+7)=2(x+50)+4(x+3)+1\)
-
Answer -
identity; all real numbers
Example \(\PageIndex{28}\)
Classify as a conditional equation, an identity, or a contradiction. Then state the solution.
\(10+4(p−5)=0\)
Solution
| Distribute. | |
| Combine like terms. | |
| Add 10 to both sides. | |
| Simplify. | |
| Divide. | |
| Simplify. | |
| The equation is true when \(p = frac{5}{2}\). |
This is a conditional equation.
The solution is \(p = frac{5}{2}\). |
Try It \(\PageIndex{29}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \(11(q+3)−5=19\)
- Answer
-
conditional equation; \(q = \frac{9}{11}\
Try It \(\PageIndex{30}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: \(6+14(k−8)=95\)
- Answer
-
conditional equation; \(k = \frac{193}{14}\)
Example \(\PageIndex{31}\)
Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution.
\(5m+3(9+3m)=2(7m−11)\)
Solution
| Distribute. | |
| Combine like terms. | |
| Subtract 14m from both sides. | |
| Simplify. | |
| But \(27\neq −22\). |
The equation is a contradiction.
It has no solution. |
Try It \(\PageIndex{32}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
\(12c+5(5+3c)=3(9c−4)\)
-
Answer -
contradiction; no solution
Try It \(\PageIndex{33}\)
Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution:
\(4(7d+18)=13(3d−2)−11d\)
-
Answer -
contradiction; no solution
| Type of equation | What happens when you solve it? | Solution |
|---|---|---|
| Conditional Equation | True for one or more values of the variables and false for all other values | One or more values |
| Identity | True for any value of the variable | All real numbers |
| Contradiction | False for all values of the variable | No solution |
Key Concepts
-
General Strategy for Solving Linear Equations
-
Simplify each side of the equation as much as possible.
Use the Distributive Property to remove any parentheses.
Combine like terms. -
Collect all the variable terms on one side of the equation.
Use the Addition or Subtraction Property of Equality. -
Collect all the constant terms on the other side of the equation.
Use the Addition or Subtraction Property of Equality. -
Make the coefficient of the variable term to equal to 1.
Use the Multiplication or Division Property of Equality.
State the solution to the equation. -
Check the solution.
Substitute the solution into the original equation.
-
Simplify each side of the equation as much as possible.


18.6B: Short-Term Neural Control
Neural regulation of blood pressure is achieved through the role of cardiovascular centers and baroreceptor stimulation.
- Describe the role of baroreceptors and cardiovascular centers (ANS) in blood pressure control
Key Points
- The cardioaccelerator center, the cardioinhibitor center, and the vasomotor center form the cardiovascular center, a cluster of neurons that function independently to regulate blood pressure and flow.
- The release of the neurotransmitter norepinephrine from sympathetic neurons directs the majority of neurons associated with the cardiovascular center.
- Baroreceptors respond to the degree of stretch caused by the presence of blood; this stimulates impulses to be sent to the cardiovascular center to regulate blood pressure to achieve homeostasis when needed.
Key Terms
- autonomic nervous system : The part of the nervous system that regulates the involuntary activity of the heart, intestines, and glands. These activities include digestion, respiration, perspiration, metabolism, and blood pressure modulation.
- norepinephrine : A catecholamine with multiple roles including as a hormone and neurotransmitter. Areas of the body that produce or are affected by this substance are described as noradrenergic.
- sympathetic : Of or related to the part of the autonomic nervous system that under stress raises blood pressure and heart rate, constricts blood vessels, and dilates the pupils.
- baroreceptor : A nerve ending that is sensitive to changes in blood pressure.
- parasympathetic : Of or relating to the part of the autonomic nervous system that inhibits or opposes the effects of the sympathetic nervous system.
The autonomic nervous system plays a critical role in the regulation of vascular homeostasis. The primary regulatory sites include the cardiovascular centers in the brain that control both cardiac and vascular functions.
Neurological regulation of blood pressure and flow depends on the cardiovascular centers located in the medulla oblongata. This cluster of neurons responds to changes in blood pressure as well as blood concentrations of oxygen, carbon dioxide, and other factors such as pH.
Baroreceptor Function
Baroreceptors are specialized stretch receptors located within thin areas of blood vessels and heart chambers that respond to the degree of stretch caused by the presence of blood. They send impulses to the cardiovascular center to regulate blood pressure. Vascular baroreceptors are found primarily in sinuses (small cavities) within the aorta and carotid arteries. The aortic sinuses are found in the walls of the ascending aorta just superior to the aortic valve, whereas the carotid sinuses are located in the base of the internal carotid arteries. There are also low-pressure baroreceptors located in the walls of the venae cavae and right atrium.
When blood pressure increases, the baroreceptors are stretched more tightly and initiate action potentials at a higher rate. At lower blood pressures, the degree of stretch is lower and the rate of firing is slower. When the cardiovascular center in the medulla oblongata receives this input, it triggers a reflex that maintains homeostasis.
Baroreceptor Reflexes : Increased blood pressure results in increased rates of baroreceptor firing, whereas decreased blood pressure results in slower rates of fire, both initiating the homeostatic mechanism to restore blood pressure.
Baroreceptor Reflexes
When blood pressure rises too high, baroreceptors fire at a higher rate and trigger parasympathetic stimulation of the heart. As a result, cardiac output falls. Sympathetic stimulation of the peripheral arterioles will also decrease, resulting in vasodilation. Combined, these activities cause blood pressure to fall.
When blood pressure drops too low, the rate of baroreceptor firing decreases. This triggers an increase in sympathetic stimulation of the heart, causing cardiac output to increase. It also triggers sympathetic stimulation of the peripheral vessels, resulting in vasoconstriction. Combined, these activities cause blood pressure to rise.
The baroreceptors in the venae cavae and right atrium monitor blood pressure as the blood returns to the heart from the systemic circulation. Normally, blood flow into the aorta is the same as blood flow back into the right atrium. If blood is returning to the right atrium more rapidly than it is being ejected from the left ventricle, the atrial receptors will stimulate the cardiovascular centers to increase sympathetic firing and cardiac output until homeostasis is achieved. The opposite is also true. This mechanism is referred to as the atrial reflex.
Other neural mechanisms can also have a significant impact on cardiovascular function. These include the limbic system, which links physiological responses to psychological stimuli, chemoreceptor reflexes, generalized sympathetic stimulation, and parasympathetic stimulation.


9.9: Ohm’s Law- Resistance and Simple Circuits
Learning Objectives
- Explain the origin of Ohm’s law.
- Calculate voltages, currents, or resistances with Ohm’s law.
- Explain what an ohmic material is.
- Describe a simple circuit.
What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference \(V\) that creates an electric field. The electric field in turn exerts force on charges, causing current.
Ohm’s Law
The current that flows through most substances is directly proportional to the voltage \(V\) applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied :
\[I \propto V. \nonumber \]
This important relationship is known as Ohm’s law . It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn’t always occur.
Resistance and Simple Circuits
If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance \(R\). Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or
\[I \propto \frac{1}{R}. \nonumber \]
Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives
\[I=\frac{V}{R}. \nonumber \]
This relationship is also called Ohm’s law. Ohm’s law in this form really defines resistance for certain materials. Ohm’s law (like Hooke’s law) is not universally valid. The many substances for which Ohm’s law holds are called ohmic . These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance \(R\) that is independent of voltage \(V\) and current \(I\). An object that has simple resistance is called a resistor , even if its resistance is small. The unit for resistance is an ohm and is given the symbol \(\Omega\) (upper case Greek omega). Rearranging \(I=V / R\) gives \(R=V / I\), and so the units of resistance are 1 ohm = 1 volt per ampere:
\[1 \ \Omega=1 \frac{V}{A}. \nonumber \]
Figure \(\PageIndex{1}\) shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in \(R\).
Example \(\PageIndex{1}\): Calculating Resistance: An Automobile Headlight
What is the resistance of an automobile headlight through which 2.50 A flows when 12.0 V is applied to it?
Strategy
We can rearrange Ohm’s law as stated by \(I=V / R\) and use it to find the resistance.
Solution
Rearranging \(I=V / R\) and substituting known values gives
\[R=\frac{V}{I}=\frac{12.0 \mathrm{~V}}{2.50 \mathrm{~A}}=4.80 \Omega. \nonumber\]
Discussion
This is a relatively small resistance, but it is larger than the cold resistance of the headlight. Resistance usually increases with temperature, and so the bulb has a lower resistance when it is first switched on and will draw considerably more current during its brief warm-up period.
Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of \(10^{12} \ \Omega\) or more. A dry person may have a hand-to-foot resistance of \(10^{5} \ \Omega\), whereas the resistance of the human heart is about \(10^{3} \ \Omega\).  Resistance is related to the shape of an object and the material of which it is composed.
Additional insight is gained by solving \(I=V / R\) for \(V\) yielding
\[V=I R. \nonumber \]
This expression for \(V\) can be interpreted as the voltage drop across a resistor produced by the flow of current \(I\). The phrase \(IR\) drop is often used for this voltage. For instance, the headlight in Example \(\PageIndex{1}\) has an \(IR\) drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since \(\mathrm{PE}=q \Delta V\), and the same \(q\) flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See Figure \(\PageIndex{2}\).)
MAKING CONNECTIONS: CONSERVATION OF ENERGY
In a simple electrical circuit, the sole resistor converts energy supplied by the source into another form. Conservation of energy is evidenced here by the fact that all of the energy supplied by the source is converted to another form by the resistor alone. We will find that conservation of energy has other important applications in circuits and is a powerful tool in circuit analysis.
Section Summary
- A simple circuit is one in which there is a single voltage source and a single resistance.
- One statement of Ohm’s law gives the relationship between current \(I\), voltage \(V\), and resistance \(R\) in a simple circuit to be \(I=\frac{V}{R}\)
- Resistance has units of ohms (\(\Omega\)), related to volts and amperes by \(1 \ \Omega=1 \mathrm{~V} / \mathrm{A}\).
- There is a voltage or \(IR\) drop across a resistor, caused by the current flowing through it, given by \(V=I R\).
Glossary
- Ohm’s law
- an empirical relation stating that the current I is proportional to the potential difference V . It is often written as I = V/R , where R is the resistance
- resistance
- the electric property that impedes current; for ohmic materials, it is the ratio of voltage to current, R = V/I
- ohm
- the unit of resistance, given by \(1 ~\Omega=1 \mathrm{~V} / \mathrm{A}\)
- ohmic
- a type of a material for which Ohm's law is valid
- simple circuit
- a circuit with a single voltage source and a single resistor


2.1: Random Experiments
Experiments
Probability theory is based on the paradigm of a random experiment ; that is, an experiment whose outcome cannot be predicted with certainty, before the experiment is run. In classical or frequency-based probability theory, we also assume that the experiment can be repeated indefinitely under essentially the same conditions. The repetitions can be in time (as when we toss a single coin over and over again) or in space (as when we toss a bunch of similar coins all at once). The repeatability assumption is important because the classical theory is concerned with the long-term behavior as the experiment is replicated. By contrast, subjective or belief-based probability theory is concerned with measures of belief about what will happen when we run the experiment. In this view, repeatability is a less crucial assumption. In any event, a complete description of a random experiment requires a careful definition of precisely what information about the experiment is being recorded, that is, a careful definition of what constitutes an outcome .
The term parameter refers to a non-random quantity in a model that, once chosen, remains constant. Many probability models of random experiments have one or more parameters that can be adjusted to fit the physical experiment being modeled.
The subjects of probability and statistics have an inverse relationship of sorts. In probability, we start with a completely specified mathematical model of a random experiment. Our goal is perform various computations that help us understand the random experiment, help us predict what will happen when we run the experiment. In statistics, by contrast, we start with an incompletely specified mathematical model (one or more parameters may be unknown, for example). We run the experiment to collect data, and then use the data to draw inferences about the unknown factors in the mathematical model.
Compound Experiments
Suppose that we have \(n\) experiments \((E_1, E_2, \ldots, E_n)\). We can form a new, compound experiment by performing the \(n\) experiments in sequence, \(E_1\) first, and then \(E_2\) and so on, independently of one another. The term independent means, intuitively, that the outcome of one experiment has no influence over any of the other experiments. We will make the term mathematically precise later.
In particular, suppose that we have a basic experiment. A fixed number (or even an infinite number) of independent replications of the basic experiment is a new, compound experiment. Many experiments turn out to be compound experiments and moreover, as noted above, (classical) probability theory itself is based on the idea of replicating an experiment.
In particular, suppose that we have a simple experiment with two outcomes. Independent replications of this experiment are referred to as Bernoulli trials, named for Jacob Bernoulli. This is one of the simplest, but most important models in probability. More generally, suppose that we have a simple experiment with \(k\) possible outcomes. Independent replications of this experiment are referred to as multinomial trials.
Sometimes an experiment occurs in well-defined stages, but in a dependent way, in the sense that the outcome of a given stage is influenced by the outcomes of the previous stages.
Sampling Experiments
In most statistical studies, we start with a population of objects of interest. The objects may be people, memory chips, or acres of corn, for example. Usually there are one or more numerical measurements of interest to us—the height and weight of a person, the lifetime of a memory chip, the amount of rain, amount of fertilizer, and yield of an acre of corn.
Although our interest is in the entire population of objects, this set is usually too large and too amorphous to study. Instead, we collect a random sample of objects from the population and record the measurements of interest of for each object in the sample.
There are two basic types of sampling. If we sample with replacement , each item is replaced in the population before the next draw; thus, a single object may occur several times in the sample. If we sample without replacement , objects are not replaced in the population. The chapter on Finite Sampling Models explores a number of models based on sampling from a finite population.
Sampling with replacement can be thought of as a compound experiment, based on independent replications of the simple experiment of drawing a single object from the population and recording the measurements of interest. Conversely, a compound experiment that consists of \(n\) independent replications of a simple experiment can usually be thought of as a sampling experiment. On the other hand, sampling without replacement is an experiment that consists of dependent stages, because the population changes with each draw.
Examples and Applications
Probability theory is often illustrated using simple devices from games of chance: coins, dice, card, spinners, urns with balls, and so forth. Examples based on such devices are pedagogically valuable because of their simplicity and conceptual clarity. On the other hand, it would be a terrible shame if you were to think that probability is only about gambling and games of chance. Rather, try to see problems involving coins, dice, etc. as metaphors for more complex and realistic problems.
Coins and Dice
In terms of probability, the important fact about a coin is simply that when tossed it lands on one side or the other. Coins in Western societies, dating to antiquity, usually have the head of a prominent person engraved on one side and something of lesser importance on the other. In non-Western societies, coins often did not have a head on either side, but did have distinct engravings on the two sides, one typically more important than the other. Nonetheless, heads and tails are the ubiquitous terms used in probability theory to distinguish the front or obverse side of the coin from the back or reverse side of the coin.
Consider the coin experiment of tossing a coin \(n\) times and recording the score (1 for heads or 0 for tails) for each toss.
- Identify a parameter of the experiment.
- Interpret the experiment as a compound experiment.
- Interpret the experiment as a sampling experiment.
- Interpret the experiment as \(n\) Bernoulli trials.
Answer
- The number of coins \(n\) is the parameter.
- The experiment consists of \(n\) independent replications of the simple experiment of tossing the coin one time.
- The experiment can be thought of as selecting a sample of size \( n \) with replacement from he population \(\{0, 1\}\).
- There are two outcomes on each toss and the tosses are independent.
In the simulation of the coin experiment, set \(n = 5\). Run the simulation 100 times and observe the outcomes.
Dice are randomizing devices that, like coins, date to antiquity and come in a variety of sizes and shapes. Typically, the faces of a die have numbers or other symbols engraved on them. Again, the important fact is that when a die is thrown, a unique face is chosen (usually the upward face, but sometimes the downward one). For more on dice, see the introductory section in the chapter on Games of Chance.
Consider the dice experiment of throwing a \(k\)-sided die (with faces numbered 1 to \(k\)), \(n\) times and recording the scores for each throw.
- Identify the parameters of the experiment.
- Interpret the experiment as a compound experiment.
- Interpret the experiment as a sampling experiment.
- Identify the experiment as \(n\) multinomial trials.
Answer
- The parameters are the number of dice \(n\) and the number of faces \(k\).
- The experiment consists of \(n\) independent replications of the simple experiment of throwing one die.
- The experiment can be thought of as selecting a sample of size \( n \) with replacement form the population \(\{1, 2, \ldots, k\}\).
- The same \( k \) outcomes occur for each die the throws are independent.
In reality, most dice are Platonic solids (named for Plato of course) with 4, 6, 8, 12, or 20 sides.
In the simulation of the dice experiment, set \(n = 5\). Run the simulation 100 times and observe the outcomes.
In the die-coin experiment , a standard die is thrown and then a coin is tossed the number of times shown on the die. The sequence of coin scores is recorded (1 for heads and 0 for tails). Interpret the experiment as a compound experiment.
Answer
The first stage consists rolling the die and the second stage consists of tossing the coin. The stages are dependent because the number of tosses depends on the outcome of the die throw.
Note that this experiment can be obtained by randomizing the parameter \(n\) in the basic coin experiment in (1) .
Run the simulation of the die-coin experiment 100 times and observe the outcomes.
In the coin-die experiment , a coin is tossed. If the coin lands heads, a red die is thrown and if the coin lands tails, a green die is thrown. The coin score (1 for heads and 0 for tails) and the die score are recorded. Interpret the experiment as a compound experiment.
Answer
The first stage consists of tossing the coin and the second stage consists of rolling the die. The stages are dependent because different dice (that may behave differently) are thrown, depending on the outcome of the coin toss.
Run the simulation of the coin-die experiment 100 times and observe the outcomes.
Cards
Playing cards, like coins and dice, date to antiquity. From the point of view of probability, the important fact is that a playing card encodes a number of properties or attributes on the front of the card that are hidden on the back of the card. (Later in this chapter, these properties will become random variables.) In particular, a standard card deck can be modeled by the Cartesian product set \[ D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k \} \times \{\clubsuit, \diamondsuit, \heartsuit, \spadesuit \} \] where the first coordinate encodes the denomination or kind (ace, 2–10, jack, queen, king) and where the second coordinate encodes the suit (clubs, diamonds, hearts, spades). Sometimes we represent a card as a string rather than an ordered pair (for example \(q \heartsuit\) rather than \((q, \heartsuit)\) for the queen of hearts). Some other properties, derived from the two main ones, are color (diamonds and hearts are red, clubs and spades are black), face (jacks, queens, and kings have faces, the other cards do not), and suit order (from least to highest rank: \( (\clubsuit, \diamondsuit, \heartsuit, \spadesuit) \)).
Consider the card experiment that consists of dealing \(n\) cards from a standard deck (without replacement).
- Identify a parameter of the experiment.
- Interpret the experiment as a compound experiment.
- Interpret the experiment as a sampling experiment.
Answer
- The parameter is \( n \), the number of cards dealt.
- At each stage, we draw a card from a deck, but the deck changes from one draw to the next, so the stages are dependent.
- The experiment is to select a sample of size \( n \) from the population \(D\), without replacement.
In the simulation of the card experiment, set \(n = 5\). Run the simulation 100 times and observe the outcomes.
The special case \(n = 5\) is the poker experiment and the special case \(n = 13\) is the bridge experiment .
Open each of the following to see depictions of card playing in some famous paintings.
- Cheat with the Ace of Clubs by Georges de La Tour
- The Cardsharps by Michelangelo Carravagio
- The Card Players by Paul Cézanne
- His Station and Four Aces by CM Coolidge
- Waterloo by CM Coolidge
Urn Models
Urn models are often used in probability as simple metaphors for sampling from a finite population.
An urn contains \(m\) distinct balls, labeled from 1 to \(m\). The experiment consists of selecting \(n\) balls from the urn, without replacement, and recording the sequence of ball numbers.
- Identify the parameters of the experiment.
- Interpret the experiment as a compound experiment.
- Interpret the experiment as a sampling experiment.
Answer
- The parameters are the number of balls \(m\) and the sample size \(n\).
- At each stage, we draw a ball from the urn, but the contents of the urn change from one draw to the next, so the stages are dependent
- The experiment is to select a sample of size \( n \) from the balls in the urn (the population), without replacement.
Consider the basic urn model of the previous exercise.  Identify an additional parameter of the model. This experiment is a metaphor for sampling from a general dichotomous population
Answer
The parameters are the population size \(m\), the sample size \(n\), and the number of red balls \(r\).
In the simulation of the urn experiment, set \(m = 100\), \(r = 40\), and \(n = 25\). Run the experiment 100 times and observe the results.
An urn initially contains \(m\) balls; \(r\) are red and \(m - r\) are green. A ball is selected from the urn and removed, and then replaced with \(k\) balls of the same color. The process is repeated. This is known as Pólya's urn model , named after George Pólya.
- Identify the parameters of the experiment.
- Interpret the case \(k = 0\) as a sampling experiment.
- Interpret the case \(k = 1\) as a sampling experiment.
Answer
- The parameters are the population size \(m\), the initial number of red balls \(r\), and the number new balls added \(k\).
- When \(k = 0\), each ball drawn is removed and no new balls are added, so the experiment is to select a sample of size \( n \) from the urn, without replacement.
- When \(k = 1\), each ball drawn is replaced with another ball of the same color. So at least in terms of the colors of the balls, the experiment is equivalent to selecting a sample of size \( n \) from the urn, with replacement.
Open the image of the painting Allegory of Fortune by Dosso Dossi. Presumably the young man has chosen lottery tickets from an urn.
Buffon's Coin Experiment
Buffon's coin experiment consists of tossing a coin with radius \(r \leq \frac{1}{2} \) on a floor covered with square tiles of side length 1. The coordinates of the center of the coin are recorded, relative to axes through the center of the square, parallel to the sides. The experiment is named for comte de Buffon.
- Identify a parameter of the experiment
- Interpret the experiment as a compound experiment.
- Interpret the experiment as sampling experiment.
Answer
- The parameter is the coin radius \(r\).
- The experiment can be thought of as selecting the coordinates of the coin center independently of one another.
- The experiment is equivalent to selecting a sample of size 2 from the population \(\left[-\frac{1}{2}, \frac{1}{2}\right]\), with replacement.
In the simulation of Buffon's coin experiment, set \(r = 0.1\). Run the experiment 100 times and observe the outcomes.
Reliability
In the usual model of structural reliability , a system consists of \(n\) components, each of which is either working or failed . The states of the components are uncertain, and hence define a random experiment. The system as a whole is also either working or failed, depending on the states of the components and how the components are connected. For example, a series system works if and only if each component works, while a parallel system works if and only if at least one component works. More generally, a \(k\) out of \(n\) system works if at least \(k\) components work.
Consider the \(k\) out of \(n\) reliability model.
- Identify two parameters.
- What value of \(k\) gives a series system?
- What value of \(k\) gives a parallel system?
Answer
- The parameters are \(k\) and \(n\)
- \(k = n\) gives a series system.
- \(k = 1\) gives a parallel system.
The reliability model above is a static model. It can be extended to a dynamic model by assuming that each component is initially working, but has a random time until failure. The system as a whole would also have a random time until failure that would depend on the component failure times and the structure of the system.
Genetics
In ordinary sexual reproduction, the genetic material of a child is a random combination of the genetic material of the parents. Thus, the birth of a child is a random experiment with respect to outcomes such as eye color, hair type, and many other physical traits. We are often particularly interested in the random transmission of traits and the random transmission of genetic disorders.
For example, let's consider an overly simplified model of an inherited trait that has two possible states ( phenotypes ), say a pea plant whose pods are either green or yellow. The term allele refers to alternate forms of a particular gene, so we are assuming that there is a gene that determines pod color, with two alleles: \(g\) for green and \(y\) for yellow. A pea plant has two alleles for the trait (one from each parent), so the possible genotypes are
- \(gg\), alleles for green pods from each parent.
- \(gy\), an allele for green pods from one parent and an allele for yellow pods from the other (we usually cannot observe which parent contributed which allele).
- \(yy\), alleles for yellow pods from each parent.
The genotypes \(gg\) and \(yy\) are called homozygous because the two alleles are the same, while the genotype \(gy\) is called heterozygous because the two alleles are different. Typically, one of the alleles of the inherited trait is dominant and the other recessive . Thus, for example, if \(g\) is the dominant allele for pod color, then a plant with genotype \(gg\) or \(gy\) has green pods, while a plant with genotype \(yy\) has yellow pods. Genes are passed from parent to child in a random manner, so each new plant is a random experiment with respect to pod color.
Pod color in peas was actually one of the first examples of an inherited trait studied by Gregor Mendel, who is considered the father of modern genetics. Mendel also studied the color of the flowers (yellow or purple), the length of the stems (short or long), and the texture of the seeds (round or wrinkled).
For another example, the \(ABO\) blood type in humans is controlled by three alleles: \(a\), \(b\), and \(o\). Thus, the possible genotypes are \(aa\), \(ab\), \(ao\), \(bb\), \(bo\) and \(oo\). The alleles \(a\) and \(b\) are co-dominant and \(o\) is recessive . Thus there are four possible blood types (phenotypes):
- Type \(A\): genotype \(aa\) or \(ao\)
- Type \(B\): genotype \(bb\) or \(bo\)
- Type \(AB\): genotype \(ab\)
- type \(O\): genotype \(oo\)
Of course, blood may be typed in much more extensive ways than the simple \(ABO\) typing. The RH factor (positive or negative) is the most well-known example.
For our third example, consider a sex-linked hereditary disorder in humans. This is a disorder due to a defect on the X chromosome (one of the two chromosomes that determine gender). Suppose that \(h\) denotes the healthy allele and \(d\) the defective allele for the gene linked to the disorder. Women have two X chromosomes, and typically \(d\) is recessive . Thus, a woman with genotype \(hh\) is completely normal with respect to the condition; a woman with genotype \(hd\) does not have the disorder, but is a carrier , since she can pass the defective allele to her children; and a woman with genotype \(dd\) has the disorder. A man has only one X chromosome (his other sex chromosome, the Y chromosome , typically plays no role in the disorder). A man with genotype \(h\) is normal and a man with genotype \(d\) has the disorder. Examples of sex-linked hereditary disorders are dichromatism , the most common form of color-blindness, and hemophilia , a bleeding disorder. Again, genes are passed from parent to child in a random manner, so the birth of a child is a random experiment in terms of the disorder.
Point Processes
There are a number of important processes that generate
random points in time
. Often the random points are referred to as
arrivals
. Here are some specific examples:
- times that a piece of radioactive material emits elementary particles
- times that customers arrive at a store
- times that requests arrive at a web server
- failure times of a device
To formalize an experiment, we might record the number of arrivals during a specified interval of time or we might record the times of successive arrivals.
There are other processes that produce
random points in space
. For example,
- flaws in a piece of sheet metal
- errors in a string of symbols (in a computer program, for example)
- raisins in a cake
- misprints on a page
- stars in a region of space
Again, to formalize an experiment, we might record the number of points in a given region of space.
Statistical Experiments
In 1879, Albert Michelson constructed an experiment for measuring the speed of light with an interferometer. The velocity of light data set contains the results of 100 repetitions of Michelson's experiment. Explore the data set and explain, in a general way, the variability of the data.
Answer
The variablility is due to measurement and other experimental errors beyond the control of Michelson.
In 1998, two students at the University of Alabama in Huntsville designed the following experiment: purchase a bag of M&Ms (of a specified advertised size) and record the counts for red, green, blue, orange, and yellow candies, and the net weight (in grams). Explore the M&M data. set and explain, in a general way, the variability of the data.
Answer
The variability in weight is due to measurement error on the part of the students and to manufacturing errors on the part of the company. The variability in color counts is less clear and may be due to purposeful randomness on the part of the company.
In 1999, two researchers at Belmont University designed the following experiment: capture a cicada in the Middle Tennessee area, and record the body weight (in grams), the wing length, wing width, and body length (in millimeters), the gender, and the species type. The cicada data set contains the results of 104 repetitions of this experiment. Explore the cicada data and explain, in a general way, the variability of the data.
Answer
The variability in body measurements is due to differences in the three species, to all sorts of envirnomental factors, and to measurement errors by the researchers.
On June 6, 1761, James Short made 53 measurements of the parallax of the sun, based on the transit of Venus. Explore the Short data set and explain, in a general way, the variability of the data.
Answer
The variability is due to measurement and other experimental errors beyond the control of Short.
In 1954, two massive field trials were conducted in an attempt to determine the effectiveness of the new vaccine developed by Jonas Salk for the prevention of polio. In both trials, a treatment group of children were given the vaccine while a control group of children were not. The incidence of polio in each group was measured. Explore the polio field trial data set and explain, in a general way, the underlying random experiment.
Answer
The basic random experiment is to observe whether a given child, in the treatment group or control group, comes down with polio in a specified period of time. Presumabley, a lower incidence of polio in the treatment group compared with the control group would be evidence that the vaccine was effective.
Each year from 1969 to 1972 a lottery was held in the US to determine who would be drafted for military service. Essentially, the lottery was a ball and urn model and became famous because many believed that the process was not sufficiently random. Explore the Vietnam draft lottery data set and speculate on how one might judge the degree of randomness.
Answer
This is a difficult problem, but presumably in a sufficiently random lottery, one would not expect to see dates in the same month clustered too closely together. Observing such clustering, then, would be evidence that the lottery was not random.
Deterministic Versus Probabilistic Models
One could argue that some of the examples discussed above are inherently deterministic . In tossing a coin, for example, if we know the initial conditions (involving position, velocity, rotation, etc.), the forces acting on the coin (gravity, air resistance, etc.), and the makeup of the coin (shape, mass density, center of mass, etc.), then the laws of physics should allow us to predict precisely how the coin will land. This is true in a technical, theoretical sense, but false in a very real sense. Coins, dice, and many more complicated and important systems are chaotic in the sense that the outcomes of interest depend in a very sensitive way on the initial conditions and other parameters. In such situations, it might well be impossible to ever know the initial conditions and forces accurately enough to use deterministic methods.
In the coin experiment, for example, even if we strip away most of the real world complexity, we are still left with an essentially random experiment. Joseph Keller in his article
The Probability of Heads
deterministically analyzed the toss of a coin under a number of ideal assumptions:
- The coin is a perfect circle and has negligible thickness
- The center of gravity of the coin is the geometric center.
- The coin is initially heads up and is given an initial upward velocity \(u\) and angular velocity \(\omega\).
- In flight, the coin rotates about a horizontal axis along a diameter of the coin.
- In flight, the coin is governed only by the force of gravity. All other possible forces (air resistance or wind, for example) are neglected.
- The coin does not bounce or roll after landing (as might be the case if it lands in sand or mud).
Of course, few of these ideal assumptions are valid for real coins tossed by humans. Let \(t = u / g\) where \(g\) is the acceleration of gravity (in appropriate units). Note that the \(t\) just has units of time (in seconds) and hence is independent of how distance is measured. The scaled parameter \(t\) actually represents the time required for the coin to reach its maximum height.
Keller showed that the regions of the parameter space \((t, \omega)\) where the coin lands either heads up or tails up are separated by the curves \[ \omega = \left( 2n \pm \frac{1}{2} \right) \frac{\pi}{2t}, \quad n \in \N \] The parameter \(n\) is the total number of revolutions in the toss. A plot of some of these curves is given below. The largest region, in the lower left corner, corresponds to the event that the coin does not complete even one rotation, and so of course lands heads up, just as it started. The next region corresponds to one rotation, with the coin landing tails up. In general, the regions alternate between heads and tails.
The important point, of course, is that for even moderate values of \(t\) and \(\omega\), the curves are very close together, so that a small change in the initial conditions can easily shift the outcome from heads up to tails up or conversely. As noted in Keller's article, the probabilist and statistician Persi Diaconis determined experimentally that typical values of the initial conditions for a real coin toss are \(t = \frac{1}{4}\) seconds and \(\omega = 76 \pi \approx 238.6\) radians per second. These values correspond to \(n = 19\) revolutions in the toss. Of course, this parameter point is far beyond the region shown in our graph, in a region where the curves are exquisitely close together.


12.8: Recommendations for Managing Diversity
- What can organizations do to ensure applicants, employees, and customers from all backgrounds are valued?
Organizations that are committed to equality and inclusion must take steps to combat the examples of discrimination and harassment that have been covered in this chapter. And they must take steps to make diversity a goal in the pre-employment stages as well as in the post-employment stages. Anyone with managerial or supervisory responsibilities should pay careful attention to hiring and performance-rewarding practices, and make sure to rely on relevant information for making decisions and ignore race-based stereotypes. The following are examples of what leaders and organizations can do make sure employees feel valued.
Interview Selection Process
To ensure fairness for all applicants, organizations should use highly structured interviews during the selection process to avoid bias based on race or gender. 146 Highly structured interviews consists of the following 15 characteristics: “(1) job analysis, (2) same questions, (3) limited prompting, (4) better questions, (5) longer interviews, (6) control of ancillary information, (7) limited questions from candidates, (8) multiple rating scales, (9) anchored rating scales, (10) detailed notes, (11) multiple interviewers, (12) consistent interviewers, (13) no discussion between interviews, (14) training, and (15) statistical prediction.” 147 Similarity bias can occur when interviewers prefer interviewees with whom they share similar traits. Organizations can mitigate this challenge if all 15 characteristics of a structured interview are used consistently with each job applicant.
Diversified Mentoring Relationships
Thanks to the rapid growth of international travel and globalization, managers are often called upon to manage a workforce that is increasingly diverse. Research has shown that racially and ethnically diverse firms have better financial performance than more homogeneous firms, because, as mentioned, employees from different backgrounds and with different experiences can give the firm a competitive advantage in various ways. It is necessary, however, that managers and those in positions of power are adequately equipped to manage diverse workforces in ways that are beneficial to all. Diversified mentoring relationships are relationships in which the mentor and the mentee differ in terms of their status within the company and within larger society. The differences could be in terms of race, gender, class, disability, sexual orientation, or other status. Research has found that these types of relationships are mutually beneficial and that the mentor and the mentee both have positive outcomes in terms of knowledge, empathy, and skills related to interactions with people from different power groups. 148
Diversity Training Programs
As the workforce becomes increasingly more diverse, managers will face a major challenge in understanding how to manage diversity. One of many decisions to be made is whether an organization should offer diversity training and, if so, what topics and issues should be addressed based on the organizational goals.
There has been a debate over the effectiveness of corporate diversity training since the Civil Rights Act of 1964 helped prompt corporate diversity training with the organizational goal of simply being compliant with the law. Prior research shows that it can be effective, ineffective, or even detrimental for employees, but as diversity training has evolved through the years, it has become an important factor in helping employers manage diversity.
In the 1980s through the late 1990s, diversity training evolved from focusing solely on compliance to addressing the needs of women and minorities as they entered the workforce at a faster rate. Unfortunately, this type of training was perceived by Whites and men as singling them out as the problem; sometimes such training was even formatted as “confession” sessions for White employees to express their complicity in institutional racism. Not unexpectedly, this type of training would often backfire and would further separate employees from each other, the exact opposite of its intention.
Recently, diversity training has evolved to focus on (1) building cultural competencies regarding fellow employees, (2) valuing differences, and (3) learning how diversity helps make better business decisions. This perspective toward diversity training is more effective than simply focusing on causes of a lack of diversity and the historical roots of discrimination. Understanding how to comply with the law is still important, but training has a greater effect when the other factors are also included.
A recent study investigated various diversity-training methods, including having participants engage in activities on perspective taking and goal setting. For perspective-taking activities, participants were asked to write a few sentences about the challenges they believed minority group members might experience. Goal-setting activities involved writing specific and measurable goals related to workplace diversity such as crafting future policies or engaging in future behaviors. Researchers found that when these activities were used as a diversity-training method, pro-diversity attitudes and behavioral intentions persisted months later.
Issues regarding employee sexual orientation have also been introduced into corporate diversity training in recent years. Because employees’ religious beliefs are protected by Title VII of the Civil Rights Act, employers should be sensitive to balancing the rights of lesbian, gay, and bisexual employees and employees’ religious rights. Attempting to protect the rights of one group and not be perceived to disrespect another is a difficult situation for managers. In order to mitigate any backlash from some employees, employers should seek feedback from all groups to learn the best ways to accommodate them, and should assess the organizational climate. Additionally, managers should explain how diversity based on sexual orientation aligns with the company’s strategic objectives and explain the company’s legal position with supportive reasoning. Lastly, based on their organizational climate and how it reshapes itself over time, some companies may wish to address diversity training on sexual orientation in a voluntary training separate from other diversity issues.
Sources: Young, Cheri A., Badiah Haffejee, and David L. Corsun. "Developing Cultural Intelligence and Empathy Through Diversified Mentoring Relationships."Journal of Management Education(2017):<PHONE_NUMBER>710687; Bezrukova, K., Jehn, K.A., & Spell, C.S. (2012). Reviewing diversity training: Where we have been and where we should go.Academy of Management Learning & Education, 11 (2): 207-227; Anand, R., & Winters, M. (2008). A retrospective view of corporate diversity training from 1964 to the present.Academy of Management Learning & Education, 7 (3): 356-372; Lindsey, A., King, E., Membere, A., & Cheung, H.K. (July 28, 2017). Two types of diversity training that really work.Harvard Business Review.
Discussion Questions
- Why do you believe diversity training is resisted by some employees?
- Do you believe there will always be a need for workplace diversity training?
- How would you determine what types of diversity training are needed at your company?
Visible Leadership
Another key to ensure that employees are treated fairly is utilizing appropriate leadership strategies. 149 Leadership must sincerely value variety of opinions, and organizational culture must encourage openness and make workers feel valued. Organizations must also have a well-articulated and widely understood mission and a relatively egalitarian, nonbureaucratic structure. Having such a work environment will ensure that the attitudes and values of employees are aligned with those of the organization. In this way culture serves as a control mechanism for shaping behaviors.
Strategies for Employees
Individuals can increase positive employment outcomes by obtaining high levels of education, because for all groups education is a predictor of employment and increased earnings. Individuals can also seek employment in larger firms, which are more likely to have formal hiring programs and specific diversity provisions in place. Individuals of any race or ethnic background can also take steps to eliminate discrimination by being aware of their own personal stereotypes or biases and taking steps to challenge and address them.
- How can managers ensure fairness in the interviewing and selection process regarding diversity?
- What is the role of leadership regarding diversity?
References
146. McCarthy, J.M, Van Iddekinge, C.H., & Campion, M.(2010). Are highly structured job interviews resistant to demographic similarity effects?: Personnel Psychology , 63: 325-359.
147. McCarthy, J.M, Van Iddekinge, C.H., & Campion, M.(2010). Are highly structured job interviews resistant to demographic similarity effects?:Personnel Psychology, 63: 325-359, p.333.; Campion M.A., Palmer D.K., Campion J.E. (1997). A review of structure in the selection interview. Personnel Psychology, 50, 655–702.
148. Young, Cheri A., Badiah Haffejee, and David L. Corsun. "Developing Cultural Intelligence and Empathy Through Diversified Mentoring Relationships." Journal of Management Education (2017):<PHONE_NUMBER>710687.
149. Thoms, D.A., & Ely, R.J. (Sep 1996). Making differences matter: A new paradigm for managing diversity. Harvard Business Review.


9.3: Coulomb’s Law
Learning Objectives
- State Coulomb’s law in terms of how the electrostatic force changes with the distance between two objects.
- Calculate the electrostatic force between two charged point forces, such as electrons or protons.
- Compare the electrostatic force to the gravitational attraction for a proton and an electron; for a human and the Earth.
Through the work of scientists in the late 18th century, the main features of the electrostatic force —the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb’s law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.
Definition: COULOMB’S LAW
\[F=k \frac{\left|q_{1} q_{2}\right|}{r^{2}}. \nonumber \]
Coulomb’s law calculates the magnitude of the force F between two point charges, \(q_{1}\) and \(q_{2}\), separated by a distance \(r\). In SI units, the constant \(k\) is equal to
\[k=8.988 \times 10^{9} \frac{\mathrm{N} \cdot \mathrm{m}^{2}}{\mathrm{C}^{2}} \approx 8.99 \times 10^{9} \frac{\mathrm{N} \cdot \mathrm{m}^{2}}{\mathrm{C}^{2}}. \nonumber \]
The electrostatic force is a vector quantity and is expressed in units of newtons. The force is understood to be along the line joining the two charges. (See Figure \(\PageIndex{2}\).)
Although the formula for Coulomb’s law is simple, it was no mean task to prove it. The experiments Coulomb did, with the primitive equipment then available, were difficult. Modern experiments have verified Coulomb’s law to great precision. For example, it has been shown that the force is inversely proportional to distance between two objects squared (\(F \propto 1 / r^{2}\)) to an accuracy of 1 part in 10 16 .
Example \(\PageIndex{1}\): How Strong is the Coulomb Force Relative to the Gravitational Force?
Compare the electrostatic force between an electron and proton separated by 0.530×10 − 10 m with the gravitational force between them. This distance is their average separation in a hydrogen atom.
Strategy
To compare the two forces, we first compute the electrostatic force using Coulomb’s law, \(F=k \frac{\left|q_{1} q_{2}\right|}{r^{2}}\). We then calculate the gravitational force using Newton’s universal law of gravitation. Finally, we take a ratio to see how the forces compare in magnitude.
Solution
Entering the given and known information about the charges and separation of the electron and proton into the expression of Coulomb’s law yields
\[\begin{gathered}
F=k \frac{\left|q_{1} q_{2}\right|}{r^{2}} \\
=\left(8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\right) \times \frac{\left(1.60 \times 10^{-19} \mathrm{C}\right)\left(1.60 \times 10^{-19} \mathrm{C}\right)}{\left(0.530 \times 10^{-10} \mathrm{~m}\right)^{2}}
\end{gathered} \nonumber\]
Thus the Coulomb force is
\[F=8.19 \times 10^{-8} \mathrm{~N}. \nonumber\]
The charges are opposite in sign, so this is an attractive force. This is a very large force for an electron—it would cause an acceleration of 8.99×10 22 m/s 2 (verification is left as an end-of-section problem).The gravitational force is given by Newton’s law of gravitation as:
\[F_{G}=G \frac{m M}{r^{2}}, \nonumber\]
where \(G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}\). Here \(m\) and \(M\) represent the electron and proton masses, which can be found in the appendices. Entering values for the knowns yields
\[F_{G}=\left(6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}\right) \times \frac{\left(9.11 \times 10^{-31} \mathrm{~kg}\right)\left(1.67 \times 10^{-27} \mathrm{~kg}\right)}{\left(0.530 \times 10^{-10} \mathrm{~m}\right)^{2}}=3.61 \times 10^{-47} \mathrm{~N} \nonumber\]
This is also an attractive force, although it is traditionally shown as positive since gravitational force is always attractive. The ratio of the magnitude of the electrostatic force to gravitational force in this case is, thus,
\[\frac{F}{F_{G}}=2.27 \times 10^{39}. \nonumber\]
Discussion
This is a remarkably large ratio! Note that this will be the ratio of electrostatic force to gravitational force for an electron and a proton at any distance (taking the ratio before entering numerical values shows that the distance cancels). This ratio gives some indication of just how much larger the Coulomb force is than the gravitational force between two of the most common particles in nature.
As the example implies, gravitational force is completely negligible on a small scale, where the interactions of individual charged particles are important. On a large scale, such as between the Earth and a person, the reverse is true. Most objects are nearly electrically neutral, and so attractive and repulsive Coulomb forces nearly cancel. Gravitational force on a large scale dominates interactions between large objects because it is always attractive, while Coulomb forces tend to cancel.
Section Summary
- Frenchman Charles Coulomb was the first to publish the mathematical equation that describes the electrostatic force between two objects.
-
Coulomb’s law gives the magnitude of the force between point charges. It is
\[F=k \frac{\left|q_{1} q_{2}\right|}{r^{2}}, \nonumber\]
where \(q_{1}\) and \(q_{1}\) are two point charges separated by a distance \(r\), and \(k \approx 8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\)
- This Coulomb force is extremely basic, since most charges are due to point-like particles. It is responsible for all electrostatic effects and underlies most macroscopic forces.
- The Coulomb force is extraordinarily strong compared with the gravitational force, another basic force—but unlike gravitational force it can cancel, since it can be either attractive or repulsive.
- The electrostatic force between two subatomic particles is far greater than the gravitational force between the same two particles.
Glossary
- Coulomb’s law
- the mathematical equation calculating the electrostatic force vector between two charged particles
- Coulomb force
- another term for the electrostatic force
- electrostatic force
- the amount and direction of attraction or repulsion between two charged bodies


2.10: Proficiency Exam
Proficiency Exam
For the following problems, simplify each of the expressions.
\(8(6−3)−5\cdot4+3(8)(2)\div4\cdot3\)
- Answer
-
\(40\)
\(\{2(1+7)^2\}^0\)
- Answer
-
\(1\)
\(\dfrac{1^8 + 4^0 + 3^3(1 + 4)}{2^2(2 + 15)}\)
- Answer
-
\(\dfrac{137}{68}\)
\(\dfrac{2 \cdot 3^4 - 10^2}{4 - 3} + \dfrac{5(2^2 + 3^2)}{11 - 6}\)
- Answer
-
\(75\)
Write the appropriate relation symbol (>, <) in place of the *.
\(5(2+11)∗2(8−3)−2\)
- Answer
-
>
For the following problems, use algebraic notation.
\((x-1)\) times \((3x \text{plus} 2)\)
- Answer
-
\((x-1)(3x+2)\)
A number divided by twelve is less than or equal to the same number plus four.
- Answer
-
\(\dfrac{x}{12} \le (x+4)\)
Locate the approximate position of \(−1.6\) on the number line.
- Answer
-
Is \(0\) a positive number, a negative number, neither, or both?
- Answer
-
Zero is neither positive nor negative.
Draw a portion of the number line and place points at all even integers strictly between 14 and 20.
- Answer
-
Draw a portion of the number line and place points at all real numbers strictly greater than −1 but less than or equal to 4.
- Answer
-
What whole numbers can replace x so that the following statement is true? \(-4 \le x \le 5\).
- Answer
-
\(0,1,2,3,4,5\)
Is there a largest real number between and including 6 and 10? If so, what is it?
- Answer
-
Yes, 10.
Use the commutative property of multiplication to write \(m(a+3)\) in an equivalent form.
- Answer
-
\((a + 3)m\)
Use the commutative properties to simplify \(3a4b8cd\).
- Answer
-
\(96abcd\)
Use the commutative properties to simplify \(4(x−9)2y(x−9)3y\).
- Answer
-
\(24y^2(x-9)^2\)
Simplify \(4\) squared times \(x\) cubed times \(y\) to the fifth.
- Answer
-
\(16x^3y^5\)
Simplify \((3)(3)(3)aabbbbabba(3)a\).
- Answer
-
\(81a^5b^6\)
For the following problems, use the rules of exponents to simplify each of the expressions.
\((3ab^2)^2(2a^3)^3\)
- Answer
-
\(71a^{11}b^7\)
\(\dfrac{x^{10}y^{12}}{x^2y^5}\)
- Answer
-
\(x^8y^7\)
\(\dfrac{52x^7y^{10}(y-x^4)^{12}(y+x)^5}{4y^6(y-x^4)^{10}(y+x)}\)
- Answer
-
\(13x^7y^4(y-x^4)^2(y+x)^4\)
\((x^ny^{3m}z^{2p})^4\)
- Answer
-
\(x^{4bn}y^{12m}z^{8p}\)
\(\dfrac{(5x+4)^0}{(3x^2-1)^0}\)
- Answer
-
\(1\)
\(\dfrac{x^∇x^□y^Δ}{x^Δy^∇}\)
- Answer
-
\(x^{∇+□-Δ}y^{Δ-∇}\)
What word is used to describe the letter or symbol that represents an unspecified member of a particular collection of two or more numbers that are clearly defined?
- Answer
-
A variable


3.7.8: Nutrition and Food Safety in The United States
Cigarettes are not the only products that have labeling requirements. In 1906, the Pure Food and Drugs Act was passed by Congress, setting governmental regulations and oversight over the food industry. This first iteration of what eventually became the Food and Drug Administration (FDA) oversight was quite simple and required accurate labeling of foods. Any food or medicine that contained certain psychoactive drugs (like caffeine, morphine, cocaine or cannabis), had to be accurately labeled with amounts and dosage. At the time, medicines and “cure-alls” containing these ingredients were often sold as proprietary formulations (Contributors to Wikimedia projects, Pure Food and Drug Act, 2023c). The USDA also began inspecting meat transported between states thanks to a new law in the same year. Later, in 1938 the Federal Food, Drug, and Cosmetic act was passed, and amendments were made afterward, which provided federal oversight on drug testing and approval, and further established the role of the FDA in ensuring safety and quality of foods, medicine, and cosmetic products (Boston University , n.d.). Nutritional supplements like vitamins and herbal remedies have some FDA oversight to ensure appropriate labeling but are not tested for efficacy and side effects in the same way that pharmaceutical drugs are.


1.17.1: Isentropic
The term ‘adiabatic’ means that for a closed system no heat passes between system and surroundings; \(\mathrm{q} = 0\). The term ‘isentropic’ introduces the further constraint that the system remains at equilibrium with the surroundings; i.e. the affinity for spontaneous change is zero. From the Second Law,
\[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi \quad \text { where } \mathrm{A} \, \mathrm{d} \xi \geq 0 \nonumber \]
The isentropic condition means that both \(\mathrm{A}\) and \(\mathrm{q}\) are zero. Hence \(\mathrm{dS}\) is zero, indicating that the entropy of the system remains constant. In other words, ‘isentropic’ describes an adiabatic change along an equilibrium and therefore reversible pathway.