7.3H: Vitamin B₁₂ (Cobalamin)
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Vitamin B 12 , also called cobalamin, is a water-soluble vitamin that has a key role in the normal functioning of the brain and nervous system, and the formation of red blood cells. It is one of eight B vitamins. It is involved in the metabolism of every cell of the human body, especially affecting DNA synthesis, fatty acid and amino acid metabolism. No fungi, plants, nor animals (including humans) are capable of producing vitamin B 12 . Only bacteria and archaea have the enzymes needed for its synthesis. Proven sources of B 12 are animal products (meat, fish, dairy products) and supplements. Some research states that certain non-animal products possibly can be a natural source of B 12 because of bacterial symbiosis. B 12 is the largest and most structurally complicated vitamin and can be produced industrially only through a bacterial fermentation-synthesis. This synthetic B 12 is used to fortify foods and sold as a dietary supplement.
Chemical structure of cyanocobalamin. (CC BY-SA 4.0; Alsosaid1987 ).
Vitamin B 12 consists of a class of chemically related compounds (vitamers), all of which show pharmacological activity. It contains the biochemically rare element cobalt (chemical symbol Co) positioned in the center of a planar tetra-pyrrole ring called a corrin ring. The vitamer is produced by bacteria as hydroxocobalamin, but conversion between different forms of the vitamin occurs in the body after consumption.
Dietary Reference Intake
The Food and Nutrition Board of the U.S. Institute of Medicine (IOM) updated Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs) for vitamin B 12 in 1998. The current EAR for vitamin B 12 for women and men ages 14 and up is 2.0 μg/day; the RDA is 2.4 μg/day. RDAs are higher than EARs so as to identify amounts that will cover people with higher than average requirements. RDA for pregnancy equals 2.6 μg/day. RDA for lactation equals 2.8 μg/day. For infants up to 12 months the Adequate Intake (AI) is 0.4-0.5 μg/day. and for children ages 1–13 years the RDA increases with age from 0.9 to 1.8 μg/day. Because 10 to 30 percent of older people may be unable to effectively absorb vitamin B 12 naturally occurring in foods, it is advisable for those older than 50 years to meet their RDA mainly by consuming foods fortified with vitamin B 12 or a supplement containing vitamin B 12 .
For U.S. food and dietary supplement labeling purposes the amount in a serving is expressed as a percent of Daily Value (%DV). For vitamin B 12 labeling purposes 100% of the Daily Value was 6.0 μg, but as of May 2016 has been revised downward to 2.4 μg.
Deficiency
Vitamin B 12 deficiency can potentially cause severe and irreversible damage, especially to the brain and nervous system. At levels only slightly lower than normal, a range of symptoms such as fatigue, lethargy, depression, poor memory, breathlessness, headaches, and pale skin, among others, may be experienced, especially in elderly people (over age 60) who produce less stomach acid as they age, thereby increasing their probability of B 12 deficiencies.
Vitamin B 12 deficiency is most commonly caused by low intakes, but can also result from malabsorption, certain intestinal disorders, low presence of binding proteins, and use of certain medications. Vitamin B 12 is rare from plant sources, so vegetarians are most likely to suffer from vitamin B 12 deficiency. Infants are at a higher risk of vitamin B 12 deficiency if they were born to vegetarian mothers. The elderly who have diets with limited meat or animal products are vulnerable populations as well. Vitamin B 12 deficiency may occur in between 40% to 80% of the vegetarian population.
Vitamin B 12 is a co-substrate of various cell reactions involved in methylation synthesis of nucleic acid and neurotransmitters. Synthesis of the trimonoamine neurotransmitters can enhance the effects of a traditional antidepressant. The intracellular concentrations of vitamin B 12 can be inferred through the total plasma concentration of homocysteine, which can be converted to methionine through an enzymatic reaction that uses 5-methyltetrahydrofolate as the methyl donor group. Consequently, the plasma concentration of homocysteine falls as the intracellular concentration of vitamin B 12 rises. The active metabolite of vitamin B 12 is required for the methylation of homocysteine in the production of methionine, which is involved in a number of biochemical processes including the monoamine neurotransmitters metabolism. Thus, a deficiency in vitamin B 12 may impact the production and function of those neurotransmitters.
Dietary Sources
Foods having rich content of vitamin B 12 include clams, organ meats (especially liver) from lamb, veal, beef, and turkey, fish eggs, mackerel, and crab meat. B 12 is synthesized by some gut bacteria in humans and other animals, but humans cannot absorb the B 12 made in their guts, as it is made in the colon which is too far from the small intestine, where absorption of B 12 occurs.
Animals store vitamin B 12 in liver and muscle and some pass the vitamin into their eggs and milk; meat, liver, eggs and milk are therefore sources of the vitamin for other animals, including people. For humans, the bioavailability from eggs is less than 9%, compared to 40% to 60% from fish, fowl and meat. Insects are also a source of B 12 . Foods fortified with B 12 are also dietary sources of the vitamin. Foods for which B 12 -fortified versions are widely available include breakfast cereals, soy products, energy bars, and nutritional yeast.
Vitamin B 12 is an ingredient in multi-vitamin pills and in some countries used to enrich grain-based foods such as bread and pasta. In the U.S. non-prescription products can be purchased providing up to 5000 µg/serving, and it is a common ingredient in energy drinks and energy shots, usually at many times the recommended dietary allowance of B 12 . The vitamin can also be a prescription product via injection or other means.
Absorption and Distribution
Methyl-B 12 is absorbed by two processes. The first is an intestinal mechanism using intrinsic factor through which 1-2 micrograms can be absorbed every few hours. The second is a diffusion process by which approximately 1% of the remainder is absorbed. The human physiology of vitamin B 12 is complex, and therefore is prone to mishaps leading to vitamin B 12 deficiency. Protein-bound vitamin B 12 must be released from the proteins by the action of digestive proteases in both the stomach and small intestine.
Absorption of food vitamin B 12 thus requires an intact and functioning stomach, exocrine pancreas, intrinsic factor, and small bowel. Problems with any one of these organs makes a vitamin B 12 deficiency possible. Individuals who lack intrinsic factor have a decreased ability to absorb B 12 . The total amount of vitamin B 12 stored in body is about 2–5 mg in adults. Around 50% of this is stored in the liver. Approximately 0.1% of this is lost per day by secretions into the gut, as not all these secretions are reabsorbed. Bile is the main form of B 12 excretion; most of the B 12 secreted in the bile is recycled via enterohepatic circulation. Excess B 12 beyond the blood's binding capacity is typically excreted in urine. Owing to the extremely efficient enterohepatic circulation of B 12 , the liver can store 3 to 5 years’ worth of vitamin B 12 ; therefore, nutritional deficiency of this vitamin is rare. How fast B 12 levels change depends on the balance between how much B 12 is obtained from the diet, how much is secreted and how much is absorbed. B 12 deficiency may arise in a year if initial stores are low and genetic factors unfavorable, or may not appear for decades. In infants, B 12 deficiency can appear much more quickly.
Contributors and Attributions
- Wikipedia. Content is copyrighted under a CC-BY-SA 4.0 license.


1.1: Assignment- Narrative Essay—Prewriting and Drafting
For this assignment, you will begin working on a narrative essay. At this stage, you will work through the prewriting and drafting steps of the writing process.
Narrative Essay Prompt
Choose one of the following topics to write your own narrative essay. The topic you decide on should be something you care about, and the narration should be a means of communicating an idea that ties to the essay’s theme. Remember in this essay, the narration is not an end in itself.
Friends
- Gaining independence
- A friend’s sacrifice
- A significant trip with your family
- A wedding or a funeral
- An incident from family legend
The World Around You
- A storm, a flood, an earthquake, or another natural event
- A school event
- The most important minutes of a sporting event
Lessons of Daily Life
- A time you confronted authority
- A time you had to deliver bad news
- Your biggest social blunder
Firsts
- Your first day of school
- The first performance you gave
- A first date
Getting Started on Your Narrative Essay
Prewriting
STEP 1 : To get started writing, first pick at least one prewriting strategy (brainstorming, rewriting, journaling, mapping, questioning, sketching) to develop ideas for your essay. Write down what you do, as you’ll need to submit evidence of your prewrite.
Remember that “story starters” are everywhere. Think about it—status updates on social media websites can be a good place to start—you may have already started a “note” to post on social media, and now is your chance to develop that idea into a full narrative. If you keep a journal or diary, a simple event may unfold into a narrative. Simply said, your stories may be closer than you think!
Drafting
STEP 2: Next, write an outline for your essay. Organize the essay in a way that:
- Establishes the situation [introduction] ;
- Introduces the complication(s) [body] ; and
- States the lesson you learned [conclusion]
STEP 3: Lastly, write a first draft of your essay. Remember, When drafting your essay:
- Develop an enticing title—but don’t let yourself get stuck on the title! A great title might suggest itself after you’ve begun the prewriting and drafting processes.
- Use the introduction to establish the situation the essay will address.
- Avoid addressing the assignment directly. (For example, don’t write “I am going to write about my most significant experience,” because this takes the fun out of reading the work!)
- Think of things said at the moment this experience started for you—perhaps use a quote, or an interesting part of the experience that will grab the reader.
 (Is your voice serious? Humorous? Matter-of-fact?)
- To avoid just telling what happens, make sure your essay takes time to reflect on why this experience is significant.
Assignment Instructions
- Choose a writing prompt as listed above on this page.
- Review the grading rubric as listed below this page.
- Create a prewrite in the style of your choice for the prompt.
- Create an outline for your essay.
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Develop a draft essay according to the following formatting guidelines: (Papers submitted that do not meet these formatting requirements will be returned to you ungraded.)
- Minimum of 3 typed, double-spaced pages (about 600–750 words), Times New Roman, 12 pt font size
- MLA formatting
- Submit your prewriting and draft as a single file upload.
Requirements
Be sure to:
- Decide on something you care about so that the narration is a means of communicating an idea.
- Include characters, conflict, sensory details.
- Create a sequence of events in a plot.
- Develop an enticing title.
- Use the introduction to pull the reader into your singular experience.
- Avoid addressing the assignment directly. (don’t write “I am going to write about…”—this takes the fun out of reading the work!)
 Humorous? Matter-of-fact?)
- Avoid telling just what happens by making sure your essay reflects on why this experience is significant.
If you developed your prewriting by hand on paper, scan or take a picture of your prewriting, load the image onto your computer, and then insert the image on a separate page after your draft.
Rubric
Grading Rubric: Narrative Essay Prewriting and Draft |
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| Criteria | Category | Point Total: 50 |
| Presentation | Paper is double-spaced throughout using Times New Roman 12 pt. font. Full name, class, Instructor’s name, your location, and time of class, date, and description of assignment are in upper-left-hand corner. The essay should be paginated (show page numbers) as well. Title should not be bolded or underlined. | __/2 pts |
| Title | The title is three or more words and hints at the essay’s main point. | __/1 pts |
| Introduction | Introduction sets up the problem the author struggles with. This could be internal, external or both. | __/10 pts |
| Essay body | The body presents the “complication” that sets the plot in motion. | __/10 pts |
| Transformation | Conclusion shows the transformation from the introduction and thus the “moral” of the story. | __/10 pts |
| Audience | The “moral” of the story is objective and reflects a universal lesson that all readers can benefit from. | __/2 pts |
| Show, don’t tell | Author “shows” the events with vivid and compelling language rather than simply tells the story. | __/5 pts |
| Prewriting and brainstorming | Demonstrates good exploration and effort by including a prewrite and outline. | __/10 pts |
Contributors and Attributions
- Authored by : Daryl Smith O' Hare and Susan C. Hines. Provided by : Chadron State College. Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution


4: Function Limits and Continuity Last updated Save as PDF Page ID 19048 Elias Zakon University of Windsor via The Trilla Group (support by Saylor Foundation) 4.1: Basic Definitions 4.1.E: Problems on Limits and Continuity 4.2: Some General Theorems on Limits and Continuity 4.2.E: More Problems on Limits and Continuity 4.3: Operations on Limits. Rational Functions 4.3.E: Problems on Continuity of Vector-Valued Functions 4.4: Infinite Limits. Operations in E* 4.4.E: Problems on Limits and Operations in \(E^{*}\) 4.5: Monotone Function 4.5.E: Problems on Monotone Functions 4.6: Compact Sets 4.6.E: Problems on Compact Sets 4.7: More on Compactness 4.8: Continuity on Compact Sets. Uniform Continuity 4.8.E: Problems on Uniform Continuity; Continuity on Compact Sets 4.9: The Intermediate Value Property 4.9.E: Problems on the Darboux Property and Related Topics 4.10: Arcs and Curves. Connected Sets 4.10.E: Problems on Arcs, Curves, and Connected Sets 4.11: Product Spaces. Double and Iterated Limits 4.11.E: Problems on Double Limits and Product Spaces 4.12: Sequences and Series of Functions 4.12.E: Problems on Sequences and Series of Functions 4.13: Absolutely Convergent Series. Power Series 4.13.E: More Problems on Series of Functions


36.1: Sensory Processes - Reception
- Explain the process of sensory reception
Introduction to Sensation
In more advanced animals, the senses are constantly at work, making the animal aware of stimuli, such as light or sound or the presence of a chemical substance in the external environment, while monitoring information about the organism’s internal environment. All bilaterally symmetric animals have a sensory system. The development of any species ‘ sensory system has been driven by natural selection; thus, sensory systems differ among species according to the demands of their environments. For example, the shark, unlike most fish predators, is electrosensitive (i.e., sensitive to electrical fields produced by other animals in its environment). While it is helpful to this underwater predator, electrosensitivity is a sense not found in most land animals.
Senses provide information about the body and its environment. Humans have five special senses: olfaction (smell), gustation (taste), equilibrium (balance and body position), vision, and hearing. Additionally, we possess general senses, also called somatosensation, which respond to stimuli like temperature, pain, pressure, and vibration. Vestibular sensation, which is an organism’s sense of spatial orientation and balance, proprioception (position of bones, joints, and muscles), and the sense of limb position that is used to track kinesthesia (limb movement) are part of somatosensation. Although the sensory systems associated with these senses are very different, all share a common function: to convert a stimulus (light, sound, or the position of the body) into an electrical signal in the nervous system. This process is called sensory transduction.
There are two broad types of cellular systems that perform sensory transduction. In one, a neuron works with a sensory receptor, a cell, or cell process that is specialized to engage with and detect a specific stimulus. Stimulation of the sensory receptor activates the associated afferent neuron, which carries information about the stimulus to the central nervous system. In the second type of sensory transduction, a sensory nerve ending responds to a stimulus in the internal or external environment; this neuron constitutes the sensory receptor. Free nerve endings can be stimulated by several different stimuli, thus showing little receptor specificity. For example, pain receptors in your gums and teeth may be stimulated by temperature changes, chemical stimulation, or pressure.
Reception
The first step in sensation is reception: the activation of sensory receptors by stimuli such as mechanical stimuli (being bent or squished, for example), chemicals, or temperature. The receptor can then respond to the stimuli. The region in space in which a given sensory receptor can respond to a stimulus, be it far away or in contact with the body, is that receptor’s receptive field. Think for a moment about the differences in receptive fields for the different senses. For the sense of touch, a stimulus must come into contact with body. For the sense of hearing, a stimulus can be a moderate distance away. For vision, a stimulus can be very far away; for example, the visual system perceives light from stars at enormous distances.
Key Points
- Reception is the process of activating a sensory receptor by a stimuli.
- Sensory transduction is the process of converting that sensory signal to an electrical signal in the sensory neuron.
- The process of reception is dependent on the stimuli itself, the type of receptor, receptor specificity, and the receptive field, which can vary depending on the receptor type.
Key Terms
- somatosensation : general senses which respond to stimuli like temperature, pain, pressure, and vibration
- reception : the act or ability to receive signals from stimuli


<IP_ADDRESS>: Trademark
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Trademark
A trademark is a word, phrase, logo, shape or sound that identifies a source of goods or services. For example, the Nike “Swoosh,” the Facebook “f”, and Apple’s apple (with a bite taken out of it) are all trademarked. The concept behind trademarks is to protect the consumer. Imagine going to the local shopping center to purchase a specific item from a specific store and finding that there are several stores all with the same name!
Two types of trademarks exist – a common-law trademark and a registered trademark. As with copyright, an organization will automatically receive a trademark if a word, phrase, or logo is being used in the normal course of business (subject to some restrictions, discussed below). A common-law trademark is designated by placing “TM” next to the trademark. A registered trademark is one that has been examined, approved, and registered with the trademark office, such as the Patent and Trademark Office in the US. A registered trademark has the circle-R (®) placed next to the trademark.
While most any word, phrase, logo, shape, or sound can be trademarked, there are a few limitations. A trademark will not hold up legally if it meets one or more of the following conditions:
- The trademark is likely to cause confusion with a mark in a registration or prior application.
- The trademark is merely descriptive for the goods/services. For example, trying to register the trademark “blue” for a blue product you are selling will not pass muster.
- The trademark is a geographic term.
- The trademark is a surname. You will not be allowed to trademark “Smith’s Bookstore.”
- The trademark is ornamental as applied to the goods. For example, a repeating flower pattern that is a design on a plate cannot be trademarked.
As long as an organization uses its trademark and defends it against infringement, the protection afforded by it does not expire. Because of this, many organizations defend their trademark against other companies whose branding even only slightly copies their trademark. For example, Chick-fil-A has trademarked the phrase “Eat Mor Chikin” and has vigorously defended it against a small business using the slogan “Eat More Kale.” Coca-Cola has trademarked the contour shape of its bottle and will bring legal action against any company using a bottle design similar to theirs. As an example of trademarks that have been diluted and have now lost their protection in the US are “aspirin” (originally trademarked by Bayer), “escalator” (originally trademarked by Otis), and “yo-yo” (originally trademarked by Duncan).
Information Systems and Intellectual Property
The rise of information systems has forced us to rethink how we deal with intellectual property. From the increase in patent applications swamping the government’s patent office to the new laws that must be put in place to enforce copyright protection, digital technologies have impacted our behavior.


29.3: A Model of the Universe
By the end of this section, you will be able to:
- Explain how the rate of expansion of the universe affects its evolution
- Describe four possibilities for the evolution of the universe
- Explain what is expanding when we say that the universe is expanding
- Define critical density and the evidence that matter alone in the universe is much smaller than the critical density
- Describe what the observations say about the likely long-term future of the universe
Let’s now use the results about the expansion of the universe to look at how these ideas might be applied to develop a model for the evolution of the universe as a whole. With this model, astronomers can make predictions about how the universe has evolved so far and what will happen to it in the future.
The Expanding Universe
Every model of the universe must include the expansion we observe. Another key element of the models is that the cosmological principle (which we discussed in The Evolution and Distribution of Galaxies) is valid: on the large scale, the universe at any given time is the same everywhere (homogeneous and isotropic). As a result, the expansion rate must be the same everywhere during any epoch of cosmic time. If so, we don’t need to think about the entire universe when we think about the expansion, we can just look at any sufficiently large portion of it. (Some models for dark energy would allow the expansion rate to be different in different directions, and scientists are designing experiments to test this idea. However, until such evidence is found, we will assume that the cosmological principle applies throughout the universe.)
In Galaxies, we hinted that when we think of the expansion of the universe, it is more correct to think of space itself stretching rather than of galaxies moving through static space. Nevertheless, we have since been discussing the redshifts of galaxies as if they resulted from the motion of the galaxies themselves.
Now, however, it is time to finally put such simplistic notions behind us and take a more sophisticated look at the cosmic expansion. Recall from our discussion of Einstein’s theory of general relativity (in the chapter on Black Holes and Curved Spacetime) that space—or, more precisely, spacetime—is not a mere backdrop to the action of the universe, as Newton thought. Rather, it is an active participant—affected by and in turn affecting the matter and energy in the universe.
Since the expansion of the universe is the stretching of all spacetime, all points in the universe are stretching together. Thus, the expansion began everywhere at once . Unfortunately for tourist agencies of the future, there is no location you can visit where the stretching of space began or where we can say that the Big Bang happened.
To describe just how space stretches, we say the cosmic expansion causes the universe to undergo a uniform change in scale over time. By scale we mean, for example, the distance between two clusters of galaxies. It is customary to represent the scale by the factor R ; if R doubles, then the distance between the clusters has doubled. Since the universe is expanding at the same rate everywhere, the change in R tells us how much it has expanded (or contracted) at any given time. For a static universe, R would be constant as time passes. In an expanding universe, R increases with time.
If it is space that is stretching rather than galaxies moving through space, then why do the galaxies show redshift s in their spectra? When you were young and naïve—a few chapters ago—it was fine to discuss the redshifts of distant galaxies as if they resulted from their motion away from us. But now that you are an older and wiser student of cosmology, this view will simply not do.
A more accurate view of the redshifts of galaxies is that the light waves are stretched by the stretching of the space they travel through. Think about the light from a remote galaxy. As it moves away from its source, the light has to travel through space. If space is stretching during all the time the light is traveling, the light waves will be stretched as well. A redshift is a stretching of waves—the wavelength of each wave increases (Figure 29.7). Light from more distant galaxies travels for more time than light from closer ones. This means that the light has stretched more than light from closer ones and thus shows a greater redshift.
Thus, what the measured redshift of light from an object is telling us is how much the universe has expanded since the light left the object. If the universe has expanded by a factor of 2, then the wavelength of the light (and all electromagnetic waves from the same source) will have doubled.
Models of the Expansion
Before astronomers knew about dark energy or had a good measurement of how much matter exists in the universe, they made speculative models about how the universe might evolve over time. The four possible scenarios are shown in Figure 29.8. In this diagram, time moves forward from the bottom upward, and the scale of space increases by the horizontal circles becoming wider.
The simplest scenario of an expanding universe would be one in which R increases with time at a constant rate. But you already know that life is not so simple. The universe contains a great deal of mass and its gravity decelerates the expansion—by a large amount if the universe contains a lot of matter, or by a negligible amount if the universe is nearly empty. Then there is the observed acceleration, which astronomers blame on a kind of dark energy.
Let’s first explore the range of possibilities with models for different amounts of mass in the universe and for different contributions by dark energy. In some models—as we shall see—the universe expands forever. In others, it stops expanding and starts to contract. After looking at the extreme possibilities, we will look at recent observations that allow us to choose the most likely scenario.
We should perhaps pause for a minute to note how remarkable it is that we can do this at all. Our understanding of the principles that underlie how the universe works on the large scale and our observations of how the objects in the universe change with time allow us to model the evolution of the entire cosmos these days. It is one of the loftiest achievements of the human mind.
What astronomers look at in practice, to determine the kind of universe we live in, is the average density of the universe. This is the mass of matter (including the equivalent mass of energy) 1 that would be contained in each unit of volume (say, 1 cubic centimeter) if all the stars, galaxies, and other objects were taken apart, atom by atom, and if all those particles, along with the light and other energy, were distributed throughout all of space with absolute uniformity. If the average density is low, there is less mass and less gravity, and the universe will not decelerate very much. It can therefore expand forever. Higher average density, on the other hand, means there is more mass and more gravity and that the stretching of space might slow down enough that the expansion will eventually stop. An extremely high density might even cause the universe to collapse again.
For a given rate of expansion, there is a critical density —the mass per unit volume that will be just enough to slow the expansion to zero at some time infinitely far in the future. If the actual density is higher than this critical density, then the expansion will ultimately reverse and the universe will begin to contract. If the actual density is lower, then the universe will expand forever.
These various possibilities are illustrated in Figure 29.9. In this graph, one of the most comprehensive in all of science, we chart the development of the scale of space in the cosmos against the passage of time. Time increases to the right, and the scale of the universe, R , increases upward in the figure. Today, at the point marked “present” along the time axis, R is increasing in each model. We know that the galaxies are currently expanding away from each other, no matter which model is right. (The same situation holds for a baseball thrown high into the air. While it may eventually fall back down, near the beginning of the throw it moves upward most rapidly.)
The various lines moving across the graph correspond to different models of the universe. The straight dashed line corresponds to the empty universe with no deceleration; it intercepts the time axis at a time, T 0 (the Hubble time), in the past. This is not a realistic model but gives us a measure to compare other models to. The curves below the dashed line represent models with no dark energy and with varying amounts of deceleration, starting from the Big Bang at shorter times in the past. The curve above the dashed line shows what happens if the expansion is accelerating. Let’s take a closer look at the future according to the different models.
Let’s start with curve 1 in Figure 29.9. In this case, the actual density of the universe is higher than the critical density and there is no dark energy. This universe will stop expanding at some time in the future and begin contracting. This model is called a closed universe and corresponds to the universe on the left in Figure 29.8. Eventually, the scale drops to zero, which means that space will have shrunk to an infinitely small size. The noted physicist John Wheeler called this the “ big crunch ,” because matter, energy, space, and time would all be crushed out of existence. Note that the “big crunch” is the opposite of the Big Bang—it is an implosion . The universe is not expanding but rather collapsing in upon itself.
Some scientists speculated that another Big Bang might follow the crunch, giving rise to a new expansion phase, and then another contraction—perhaps oscillating between successive Big Bangs and big crunches indefinitely in the past and future. Such speculation was sometimes referred to as the oscillating theory of the universe . The challenge for theorists was how to describe the transition from collapse (when space and time themselves disappear into the big crunch) to expansion. With the discovery of dark energy, however, it does not appear that the universe will experience a big crunch, so we can put worrying about it on the back burner.
If the density of the universe is less than the critical density (curve 2 in Figure 29.9 and the universe second from the left in Figure 29.8), gravity is never important enough to stop the expansion, and so the universe expands forever. Such a universe is infinite and this model is called an open universe . Time and space begin with the Big Bang, but they have no end; the universe simply continues expanding, always a bit more slowly as time goes on. Groups of galaxies eventually get so far apart that it would be difficult for observers in any of them to see the others. (See the feature box on What Might the Universe Be Like in the Distant Future? for more about the distant future in the closed and open universe models.)
At the critical density (curve 3), the universe can just barely expand forever. The critical-density universe has an age of exactly two-thirds T 0 , where T 0 is the age of the empty universe. Universes that will someday begin to contract have ages less than two-thirds T 0 .
In an empty universe (the dashed line Figure 29.9 and the coasting universe in Figure 29.8), neither gravity nor dark energy is important enough to affect the expansion rate, which is therefore constant throughout all time.
In a universe with dark energy, the rate of the expansion will increase with time, and the expansion will continue at an ever-faster rate. Curve 4 in Figure 29.9, which represents this universe, has a complicated shape. In the beginning, when the matter is all very close together, the rate of expansion is most influenced by gravity. Dark energy appears to act only over large scales and thus becomes more important as the universe grows larger and the matter begins to thin out. In this model, at first the universe slows down, but as space stretches, the acceleration plays a greater role and the expansion speeds up.
The Cosmic Tug of War
We might summarize our discussion so far by saying that a “tug of war” is going on in the universe between the forces that push everything apart and the gravitational attraction of matter, which pulls everything together. If we can determine who will win this tug of war, we will learn the ultimate fate of the universe.
The first thing we need to know is the density of the universe. Is it greater than, less than, or equal to the critical density? The critical density today depends on the value of the expansion rate today, H 0 . If the Hubble constant is around 20 kilometers/second per million light-years, the critical density is about 10 –26 kg/m 3 . Let’s see how this value compares with the actual density of the universe.
As we discussed, the critical density is that combination of matter and energy that brings the universe coasting to a stop at time infinity. Einstein’s equations lead to the following expression for the critical density ( ρ crit ):
where H is the Hubble constant and G is the universal constant of gravity (6.67 × 10 –11 Nm 2 /kg 2 ).
Solution
Let’s substitute our values and see what we get. Take an H = 22 km/s per million light-years. We need to convert both km and light-years into meters for consistency. A million light-years = 10 6 × 9.5 × 10 15 m = 9.5 × 10 21 m. And 22 km/s = 2.2 × 10 4 m/s. That makes H = 2.3 ×10 –18 /s and H 2 = 5.36 × 10 –36 /s 2 . So,
which we can round off to the 10 –26 kg/m 3 . (To make the units work out, you have to know that N , the unit of force, is the same as kg × m/s 2 .)
Now we can compare densities we measure in the universe to this critical value. Note that density is mass per unit volume, but energy has an equivalent mass of m = E / c 2 (from Einstein’s equation E = mc 2 ).
- A single grain of dust has a mass of about 1.1 × 10 –13 kg. If the average mass-energy density of space is equal to the critical density on average, how much space would be required to produce a total mass-energy equal to a dust grain?
- If the Hubble constant were twice what it actually is, how much would the critical density be?
- Answer
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a. In this case, the average mass-energy in a volume V of space is E = ρ crit V . Thus, for space with critical density, we require that
Thus, the sides of a cube of space with mass-energy density averaging that of the critical density would need to be slightly greater than 10 km to contain the total energy equal to a single grain of dust!
b. Since the critical density goes as the square of the Hubble constant, by doubling the Hubble parameter, the critical density would increase by a factor a four. So if the Hubble constant was 44 km/s per million light-years instead of 22 km/s per million light-years, the critical density would be
We can start our survey of how dense the cosmos is by ignoring the dark energy and just estimating the density of all matter in the universe, including ordinary matter and dark matter. Here is where the cosmological principle really comes in handy. Since the universe is the same all over (at least on large scales), we only need to measure how much matter exists in a (large) representative sample of it. This is similar to the way a representative survey of a few thousand people can tell us whom the millions of residents of the US prefer for president.
There are several methods by which we can try to determine the average density of matter in space. One way is to count all the galaxies out to a given distance and use estimates of their masses, including dark matter, to calculate the average density. Such estimates indicate a density of about 1 to kg/m 3 (10 to 20% of critical), which by itself is too small to stop the expansion.
A lot of the dark matter lies outside the boundaries of galaxies, so this inventory is not yet complete. But even if we add an estimate of the dark matter outside galaxies, our total won’t rise beyond about 30% of the critical density. We’ll pin these numbers down more precisely later in this chapter, where we will also include the effects of dark energy.
In any case, even if we ignore dark energy, the evidence is that the universe will continue to expand forever. The discovery of dark energy that is causing the rate of expansion to speed up only strengthens this conclusion. Things definitely do not look good for fans of the closed universe (big crunch) model.
Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
—From the poem “Fire and Ice” by Robert Frost (1923)
Given the destructive power of impacting asteroids, expanding red giants, and nearby supernovae, our species may not be around in the remote future. Nevertheless, you might enjoy speculating about what it would be like to live in a much, much older universe.
The observed acceleration makes it likely that we will have continued expansion into the indefinite future. If the universe expands forever ( R increases without limit), the clusters of galaxies will spread ever farther apart with time. As eons pass, the universe will get thinner, colder, and darker.
Within each galaxy, stars will continue to go through their lives, eventually becoming white dwarfs, neutron stars, and black holes. Low-mass stars might take a long time to finish their evolution, but in this model, we would literally have all the time in the world.  The remains of stars will all be dark and difficult to observe.
This means that the light that now reveals galaxies to us will eventually go out. Even if a small pocket of raw material were left in one unsung corner of a galaxy, ready to be turned into a fresh cluster of stars, we will only have to wait until the time that their evolution is also complete. And time is one thing this model of the universe has plenty of. There will surely come a time when all the stars are out, galaxies are as dark as space, and no source of heat remains to help living things survive. Then the lifeless galaxies will just continue to move apart in their lightless realm.
If this view of the future seems discouraging (from a human perspective), keep in mind that we fundamentally do not understand why the expansion rate is currently accelerating. Thus, our speculations about the future are just that: speculations. You might take heart in the knowledge that science is always a progress report. The most advanced ideas about the universe from a hundred years ago now strike us as rather primitive. It may well be that our best models of today will in a hundred or a thousand years also seem rather simplistic and that there are other factors determining the ultimate fate of the universe of which we are still completely unaware.
Ages of Distant Galaxies
In the chapter on Galaxies, we discussed how we can use Hubble’s law to measure the distance to a galaxy. But that simple method only works with galaxies that are not too far away. Once we get to large distances, we are looking so far into the past that we must take into account changes in the rate of the expansion of the universe. Since we cannot measure these changes directly, we must assume one of the models of the universe to be able to convert large redshifts into distances.
This is why astronomers squirm when reporters and students ask them exactly how far away some newly discovered distant quasar or galaxy is. We really can’t give an answer without first explaining the model of the universe we are assuming in calculating it (by which time a reporter or student is long gone or asleep). Specifically, we must use a model that includes the change in the expansion rate with time. The key ingredients of the model are the amounts of matter, including dark matter, and the equivalent mass (according to E = mc 2 ) of the dark energy along with the Hubble constant.
Elsewhere in this book, we have estimated the mass density of ordinary matter plus dark matter as roughly 0.3 times the critical density, and the mass equivalent of dark energy as roughly 0.7 times the critical density. We will refer to these values as the “standard model of the universe.” The latest (slightly improved) estimates for these values and the evidence for them will be given later in this chapter. Calculations also require the current value of the Hubble constant. For Table 29.1, we have adopted a Hubble constant of 67.3 kilometers/second/million parsecs (rather than rounding it to 70 kilometers/second/million parsecs), which is consistent with the 13.8 billion-year age of the universe estimated by the latest observations.
Once we assume a model, we can use it to calculate the age of the universe at the time an object emitted the light we see. As an example, Table 29.1 lists the times that light was emitted by objects at different redshifts as fractions of the current age of the universe. The times are given for two very different models so you can get a feeling for the fact that the calculated ages are fairly similar. The first model assumes that the universe has a critical density of matter and no dark energy. The second model is the standard model described in the preceding paragraph. The first column in the table is the redshift, which is given by the equation z = Δλ/λ 0 and is a measure of how much the wavelength of light has been stretched by the expansion of the universe on its long journey to us.
| Redshift | Percent of Current Age of Universe When the Light Was Emitted (mass = critical density) | Percent of Current Age of Universe When the Light Was Emitted (mass = 0.3 critical density; dark energy = 0.7 critical density) |
|---|---|---|
| 0 | 100 (now) | 100 (now) |
| 0.5 | 54 | 63 |
| 1.0 | 35 | 43 |
| 2.0 | 19 | 24 |
| 3.0 | 13 | 16 |
| 4.0 | 9 | 11 |
| 5.0 | 7 | 9 |
| 8.0 | 4 | 5 |
| 11.9 | 2.1 | 2.7 |
| Infinite | 0 | 0 |
Notice that as we find objects with higher and higher redshifts, we are looking back to smaller and smaller fractions of the age of the universe. The highest observed redshifts as this book is being written are close to 12 (Figure 29.10). As Table 29.1 shows, we are seeing these galaxies as they were when the universe was only about 3% as old as it is now. They were already formed only about 700 million years after the Big Bang.
Footnotes
- 1 By equivalent mass we mean that which would result if the energy were turned into mass using Einstein’s formula, E = mc 2 .


9.3: Surgical Ward
Day: 12
Time: 05h00
Place: Surgical Ward
Jamie rolls over, attempting to get more comfortable, and feels a throbbing, burning sensation in her right leg. Unable to get comfortable and having a feeling of dread, she pushes the nurse call button.
 What can I help you with?”
Jamie grabs her small white board and marker and writes: My right leg is killing me. It’s throbbing and burning.
“Ok, I’ll need to have a look.” Wanda goes back to the door and turns on the lights. Removing the blanket, she examines Jamie’s right leg. Just looking at the leg, Wanda can tell it’s swollen and significantly larger than the left. A light touch reveals that it is warmer than expected. A light squeeze elicits a grimace on Jamie’s face.
“Ok, Jamie, I’m going to do the Homan’s test. I am going to push your foot up to stretch your calf. You let me know if this is painful.”
Performing Homan’s test causes Jamie extreme pain and she tries to wriggle up the bed away from Wanda.
“Ok, that was definitely painful for you. I won’t do that again. Take a couple of deep breaths and relax. Are you feeling short of breath? Any pain in your chest?”
Jamie shakes her head ‘no’ to both questions.
“Ok, I will be right back. I want to check your vitals and then I’m going to call the resident to come and see you.”
Wanda comes back to find Jamie holding her white board towards her. What is wrong with me? it says.
“Good question. I think you may have a blood clot in your calf from being on bed rest and not moving around, as you would normally do at home. This is not an uncommon thing, but we need to treat it, and for that the resident needs to come and see you.”
Jamie relaxes a little bit but still appears tense.
Wanda completes her assessment and finds Jamie’s chest is clear, RR is 20 per minute, HR 80, BP 110/64, SpO 2 95% on .35 FiO2 via tracheal mask (TM).
| Day: 12 | Pulse Rate | Blood Pressure | Respiratory Rate | Temperature | O 2 Saturation |
| Time: 05h30 | 80 | 110/64 | 20 | – | 95% on .35 FiO 2 TM |
“I’m going to call the resident and see what he or she has to say. I’ll be right back as soon as I know something.”
Wanda sits and organizes her thoughts before calling the resident. She writes the following on a scrap paper:
S: Woke complaining of pain in right leg, states it’s throbbing and burning
O: Vital signs not changed, no temp, chest is clear, SpO 2 95% on .35 FiO 2 TM, right leg is warm to touch, grimacing, positive Homan’s test
A: DVT in right leg from decreased mobility
P: Resident to see and confirm, heparin infusion?, ultrasound, VQ scan
Comfortable she has everything ready, Wanda calls the resident-on-call and communicates her concerns and findings using the SOAP notes she made.
The resident agrees and says she will be up to see both Wanda and Jamie right away.
 Looking up, the unit clerk says, “Yes?”
“Good morning, I’m Dr. Betty Johnson, resident-on call for Medicine. Wanda called me about Jamie Douglas.”
“Oh yeah, right. Ok, both Wanda and Jamie are in Room 22 down the hall to your right.”
“Thank you.”
Betty walks down the hall to Room 22 to find both Wanda and Jamie.
“Hi, I’m Dr. Johnson, resident-on-call. You called?”
“Thank you, Dr. Johnson, for coming so quickly. This is Jamie Douglas who we discussed on the phone.”
“Yes. Can I see your leg, Ms. Douglas?”
Jamie pulls the sheet off her legs and Dr. Johnson assesses and confirms Wanda’s findings.
“Ok, Wanda, I agree. Looks like a DVT. I would like to start a heparininfusion and I will check with the senior resident and the staff man to confirm starting. Pharmacy can help with the dosage. I’ll call Ultrasound and Medrad for VQ. I would recommend that you get Ms. Douglas here ready for a trip to the X-ray Department. I would like these done stat to confirm and ensure no untoward effects.”
“Thanks, Dr. Johnson. Do you have any questions, Jamie?”

Time: 06h00
“Ok, Jamie, I’m going to take you to X-ray now for the two tests that Dr. Johnson talked about. One is an ultrasound of your leg, which has no special requirements, and the second one will require an injection of dye into your IV while they take pictures of your lungs.”
Jamie nods and waves her hand, indicting she is ready to go.
Place: X-Ray Diagnostics Department
Wanda moves Jamie’s stretcher closer to the check-in desk. “Hi, I’m Wanda and this is Jamie. I called earlier to confirm an ultrasound and V/Q scan stat, as ordered by Dr. Johnson.”
“Yes, yes, right you are. Down the hallway to your right. The door should be open and Adele is waiting for you in Ultrasound, Room 2. Second door on the left.”
Wanda steers the stretcher down the hallway to find Room 2 open and Adele typing data into the ultrasound machine.
“Good morning. This is Jamie Douglas.”
Turning around, Adele smiles. “Be right with you. Just finishing up from the previous case. Please push the stretcher in here to the right of me. Thank you.”
Wanda gets the stretcher in position, applies the brakes, and slides the side rails down.
“Thank you for waiting. The last case went longer than I expected. Ok, so this is Jamie Douglas, and you are having some leg pain?”
Jamie nods and points to her right leg.
 The gel will be cool and the probe may cause you a bit of discomfort as sometimes I need to push firmly to get better pictures. Ok? Do you need any pain medication before I start?”
Jamie shakes her head.
Adele grabs the gel from the warmer on the left side of the machine and squirts a small amount on Jamie’s right femoral area. She places the probe on the gel and begins the scan.
Wanda looks at the screen and then looks at Jamie and sees a bit of worry and curiosity on her face. “Adele, what are you seeing there?” Wanda asks.
“I’m looking at the femoral vein and femoral artery to see if there are any blockages. You can tell the vein from the artery. When I push down, the vein collapses, but the artery has a muscle around it and it is resistant to collapsing easily. You see the blue and red flow that indicates venous and arterial flow. Everything looks good here. Next, I need to check her popliteal area or behind her knee. Can you help position Jamie so I can get a good view?”
Wanda helps Jamie move onto her left side so that her right popliteal area is accessible to Adele.
“Ok, Jamie, same thing. A bit of gel and some movement up and down with the probe. Might be uncomfortable since your calf is sore.”
Adele begins the scan, explaining everything she is doing. Suddenly Adele stops her explanations and stares intently at the screen.
Jamie then motions with her right hand. Wanda responds, “Yes, Jamie. It looks like something is different on the screen. Adele, what’s going on? What do you see?”
Adele looks up from her screen. “Ok, not official as this needs to be reviewed by the doctor, but it looks like there is some clotting in the popliteal artery. I need to do some more scanning of Jamie’s calf, but it looks to be that you were correct in thinking it was a DVT.”
Adele finishes off the scan and wipes all the gel off Jamie’s leg. “Are you ok, Jamie?”
Jamie grabs her white board and writes, I am never getting out of here .
“Yes, this appears to be a setback, but it’s not a bad one. Let’s see what all the tests show and what the doctors say and then go from there. Hopefully, it won’t hold you back too much.”
Adele then turns to Wanda, “Next is VQ, right?”
“Yes. Can you help me with the stretcher? It’s pretty tight in here and I feel like I would be banging back and forth against the walls.”
Adele laughs. “I’m not sure I’m a better driver. You can see a number of marks on the wall from me pushing stretchers. Maybe the two of us will be better.”
Adele helps push the stretcher to the nuclear medicine exam room for the VQ scan.
“Hey, Jenny, your patient is here. Ms. Jamie Douglas.”
A slim woman in a white lab coat looks up from her computer screen. “Great. Can you help me get her on the exam table?”
“Sure.”
The three health professionals position the stretcher beside the nuclear medicine exam table, and assist Jamie in moving across onto the table.
“Thank you, everyone,” Jenny says.
Jenny goes to a cupboard and gathers the equipment for the ventilation part of the V/Q scan.
“Ok, Jamie, I’m going to have you breathe in a special gas that will go into your lungs. As you are breathing, I will take some pictures with this large camera. The camera is going to sit very close to your chest. During this time, please do not move around or we may have to do the test again. Your nurse will be in the room, so just wave if you need something.”
Jamie nods.
“Ok, positioning the camera now. Still ok?”
Jamie makes the ok sign with her left hand.
“Now, breathe normally, and I’m going to start taking pictures now.”
Over the next five minutes, black and white pictures of Jamie’s lungs appear on the screen.
Wanda looks closely at the pictures. “To me, her lungs look like the lungs in the pictures.”
Jenny smiles. “Yes, it appears ventilation is good. Nothing abnormal to my eyes, but the doctors will have to look as well and give the final say.”
Jenny then moves to Jamie’s bedside and pushes the camera away from her chest. “Ok, done the first part. In the next part, I am going to inject a medication in your IV that will show how your lungs are perfused. This will take a bit longer and, again, the camera will be close to your chest.”
Jenny then gathers the radioactive injection from a secure area in the room. Double checking the IV and confirming its placement, she then injects the substance into Jamie’s IV port closest to the insertion site. Then she lowers the camera again to be almost sitting on Jamie’s chest.
“Ok, Jamie, starting the second test.”
Jamie makes the ok symbol again with her left hand.
Over the next 15 minutes, new pictures appear on the screen outlining the perfusion of the lungs.
“Jenny, what are you seeing?” Wanda asks.
“From my perspective, everything looks normal but, again, the doctors will have to take a look to confirm.”
Time: 07h30
Wanda, tired from night shift and from pushing a stretcher throughout the hospital, heaves a sigh of relief as she moves the stretcher into Jamie’s room.
“Jamie, do you think you can wriggle over to your bed?”
Jamie looks over at the bed and then nods.
“I need to chart all of this activity, so give me a couple of minutes to catch up, and then I will get Dr. Johnson to come and talk to you.”
Jamie grabs her white board and writes, I am ok right now, very tired, pain med when you back?
“Most definitely. I will bring something for you.”
Wanda finds an empty workstation and adds her charting to the progress notes. After completing that, she finds Dr. Johnson reviewing the V/Q scan and the ultrasound test.
“Hi, Dr. Johnson. Can you talk with Jamie Douglas about her tests?”
“Please call me Betty, and yes, I would be happy to. Did you see anything on the tests?”
“Adele in ultrasound said there were some blockages in the popliteal area and the V/Q scan looked normal.”
“I agree. I need to review the tests with the senior resident and the staff doc, but that looks like just a DVT with no complications. So what do you recommend for treatment?”
“That’s an easy one: heparin IV bolus based on weight, then a continuous infusion based on the normogram. Although I’m not sure, possible SC injection of dalteparin, or warfarin PO may be used. We should check the B.C. and Canadian guidelines.”
“The medications are very much correct. I just spoke to the pharmacist-on-call and he said that since it is a localized DVT and there is no lung involvement, dalteparin SC and warfarin PO would be his recommendation. He recommended 200 U/kg SC daily of the dalteparin and five days of warfarin until her INR [1] reaches at least 2.0.”
“Ok. Should we go discuss this with Jamie? Should we have Jamie’s husband present as well? I can set up the speaker phone so he can be present.”
“That is a great idea. Give me five minutes and I will meet you in the room.”
“Sure thing.”
Epilogue
After a further three days of hospitalization, Jamie was discharged home on warfarin PO, daily blood work, and weekly follow-up at the Respiratory Clinic for her tracheostomy.
- international normalized ratio ↵


7.3: Reactions in Supercritical Fluids (SCFs)
SCF is a substance above its critical temperature ( Tc ) and pressure ( Pc ), but below the pressure condensation leads to a solid. At the critical point, high temperature and pressure, the substance can exit both as a vapor and a liquid in equilibrium. Thus, in a closed system, as both the temperature and pressure increase, the liquid becomes less dense due to thermal expansion, and the gas becomes dense as the pressure rises. Thus, the densities of both the phases converge until they become identical at the critical point. At this point, both the phases become indistinguishable and SCF is formed. In such SCF, a high reactivity and selectivity are sometime observed.
Supercritical carbon dioxide (scCO 2 ) offers the advantages that simple depressurization leads to removal of the residual scCO 2 , and thus, no hazardous solvent is produced, providing effective route for the separation of the products. Thus the synthesis of organic compounds can be accomplished under solvent-free conditions that find wide applications in pharmaceutical, food and cosmetic industries.
7.3.1 Hydrogenation of Alkenes
Asymmetric hydrogenation of alkenes can be carried out in scCO 2 using chiral Rh complex with good enantioselectivity (Scheme \(\PageIndex{1}\)). The catalyst is dissolved in scCO 2 during the reaction making the process homogeneous.
7.3.2 Cyclopropanation
The continuous flow of scCO 2 has been used for the asymmetric cyclopropanation using an immobilized chiral Ru complex (Scheme \(\PageIndex{2}\)). The reaction has been reported 7.7 fold more efficient compared to that within dichloromethane solvent. In addition, easy product separation and environmental friendliness makes the process more attractive.


Math 142: Text (Openstax)
This text follows scope and sequence requirements of a one-semester introduction to statistics course and is geared toward students majoring in fields other than math or engineering. The text assumes some knowledge of intermediate algebra and focuses on statistics application over theory. Introductory Statistics includes innovative practical applications that make the text relevant and accessible, as well as collaborative exercises, technology integration problems, and statistics labs.
- Front Matter
- 1: Sampling and Data
- 2: Descriptive Statistics
- 3: Probability Topics
- 4: Discrete Random Variables
- 5: Continuous Random Variables
- 6: The Normal Distribution
- 7: The Central Limit Theorem
- 8: Confidence Intervals
- 9: Hypothesis Testing with One Sample
- 10: Hypothesis Testing with Two Samples
- 11: The Chi-Square Distribution
- 12: Linear Regression and Correlation
- Back Matter
Contributors and Attributions
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


2.2: What is a Vaccine-Preventable Disease (VPD)?
Vaccines are our best defense against many diseases. Vaccine-preventable diseases (VPDs) are diseases caused by bacteria and viruses that can be prevented by vaccines. VPDs can spread through different routes such as the air, respiratory droplets, and bodily contact. Some diseases, such as measles, are extremely contagious. For example, it is possible to contract measles two hours after a person with measles has left the room.
Many of the diseases prevented by vaccines have dramatically declined since the introduction of vaccination programs. However, awareness of VPDs continues to be a priority so that members of the public and health professionals understand why it is important to keep vaccinating against those diseases. When people stop vaccinating, VPDs, such as measles, can recur and spread rapidly around the world.
When a disease is eradicated around the world, vaccines are no longer needed. For example, smallpox was eradicated in 1979. As a result, the smallpox vaccine is no longer provided in routine vaccination programs. Diseases are eliminated when the disease is reduced to zero, or close to zero, in a defined geographical area. Polio, for example, is eliminated in North America and South America.


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absent FHR variability
-
lack of FHR baseline fluctuation
-
accelerations
-
abrupt increases in the FHR above the baseline 15 bpm or more lasting 15 seconds or more
-
bradycardia
-
FHR baseline less than 110 bpm lasting greater than 10 minutes
-
contraction frequency
-
how often the contractions are occurring, determined by measuring from the beginning of one contraction to the beginning of the next contraction
-
contraction length
-
how long the contraction lasts, determined by measuring the time between the beginning of the contraction to the end of the contraction
-
contraction strength
-
intensity of the contraction, determined externally by palpation or internally by the intrauterine pressure catheter
-
early decelerations
-
usually, symmetrical, gradual FHR declines that then return to baseline, mirroring the uterine contraction
-
fetal heart rate (FHR)
-
heart rate and rhythm of the fetus
-
fetal reserve
-
fetus’s capacity to tolerate the normal intermittent interruptions in oxygenation during labor and recover back to full oxygenation
-
fetal scalp electrode (FSE)
-
affixed to the scalp of the fetus and provides close monitoring of the fetal heart rate
-
FHR baseline
-
average beats per minute in a 10-minute segment, excluding periodic changes or marked variability
-
FHR variability
-
beat-to-beat fluctuations in the FHR baseline
-
intermittent auscultation (IA)
-
technique of assessing fetal well-being by listening to and counting the fetal heart rate for a specified amount of time and at specified intervals, depending upon the stage of labor
-
intrauterine pressure catheter (IUPC)
-
device placed between the uterus and fetus that measures the exact pressure inside the uterus at rest and during contractions
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intrauterine resuscitation
-
specific interventions aimed at increasing oxygen to the placenta and to the umbilical cord to reverse fetal hypoxia
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late decelerations
-
symmetrical, gradual decreases in FHR after the start of a uterine contraction, with return to baseline after the ending of the contraction
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marked FHR variability
-
FHR baseline fluctuation greater than 25 bpm
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minimal FHR variability
-
FHR fluctuations of 5 bpm or fewer
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mm Hg
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millimeters of mercury, abbreviated mm Hg; measurement unit of intrauterine pressure catheter
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moderate FHR variability
-
FHR fluctuations between 6 and 25 bpm
-
Montevideo unit
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unit of measurement of uterine contraction strength via IUPC
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periodic changes in the FHR
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accelerations and decelerations of the FHR, abrupt or gradual, in relation to the FHR baseline
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prolonged decelerations
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isolated, sporadic decelerations of at least 15 bpm from the FHR baseline that last 2 to 10 minutes from onset to return to baseline
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tachycardia
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FHR baseline greater than 160 bpm
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tocodynamometer
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device that detects the frequency of uterine contractions
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uterine contraction (UC)
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tightening and shortening of the uterine muscles
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uteroplacental insufficiency
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disruption of the delivery of oxygen and nutrients to the fetus from the placenta
-
variable decelerations
-
abrupt decelerations of the FHR of at least 15 bpm below the baseline that last at least 15 seconds and less than 2 minutes, regardless of any uterine contraction


Problem Set 1: Use Elasticities for Exogneous Variables to Estimate Shocks and Changes to an Equilibrium. In this problem set you can assume that it is appropriate to treat W and M as exogenous variables.
Exercise \(\PageIndex{1}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\phi_{1W} = -0.15\)
\(\varepsilon_{11} = -1.5\)
\(\varepsilon_{1M} = -0.5\)
\(\% \Delta M = 6\)
\(\% \Delta W = 0\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= 0\)
\(\textrm{Demand Shock}: D= -3\)
\(\% \Delta P_{1} = -1.5\)
\( \% \Delta Q_{1} = -0.75\)
Exercise \(\PageIndex{2}\)
Given the following:
\(\phi_{11} = 1.5\)
\(\phi_{1W} = -0.25\)
\(\varepsilon_{11} = -1.75\)
\(\varepsilon_{1M} = 0.5\)
\(\% \Delta M = 5\)
\(\% \Delta W = 3\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -0.75\)
\(\textrm{Demand Shock}: D= 2.5\)
\(\% \Delta P_{1} = 1\)
\( \% \Delta Q_{1} = 0.75\)
Exercise \(\PageIndex{3}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\phi_{1W} = -0.3\)
\(\varepsilon_{11} = -2\)
\(\varepsilon_{1M} = -0.4\)
\(\% \Delta M = 8\)
\(\% \Delta W = 0\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= 0\)
\(\textrm{Demand Shock}: D= -3.2\)
\(\% \Delta P_{1} = -1.28\)
\( \% \Delta Q_{1} = -0.64\)
Exercise \(\PageIndex{4}\)
Given the following:
\(\phi_{11} = 2\)
\(\phi_{1W} = -0.1\)
\(\varepsilon_{11} = -3\)
\(\varepsilon_{1M} = 0.3\)
\(\% \Delta M = 0\)
\(\% \Delta W = 8\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -0.8\)
\(\textrm{Demand Shock}: D= 0\)
\(\% \Delta P_{1} = 0.16\)
\( \% \Delta Q_{1} = -0.48\)
Exercise \(\PageIndex{5}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\phi_{1W} = -0.2\)
\(\varepsilon_{11} = -1.5\)
\(\varepsilon_{1M} = 0.4\)
\(\% \Delta M = 5\)
\(\% \Delta W = 4\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -0.8\)
\(\textrm{Demand Shock}: D= 2\)
\(\% \Delta P_{1} = 1.4\)
\( \% \Delta Q_{1} = -0.1\)
Exercise \(\PageIndex{6}\)
Given the following:
\(\phi_{11} = 0.25\)
\(\phi_{1W} = -0.3\)
\(\varepsilon_{11} = -1.75\)
\(\varepsilon_{1M} = 0.5\)
\(\% \Delta M = 0\)
\(\% \Delta W = 8\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -2.4\)
\(\textrm{Demand Shock}: D= 0\)
\(\% \Delta P_{1} = 1.2\)
\( \% \Delta Q_{1} = -2.1\)
Exercise \(\PageIndex{7}\)
Given the following:
\(\phi_{11} = 1\)
\(\phi_{1W} = -0.5\)
\(\varepsilon_{11} = -2\)
\(\varepsilon_{1M} = -0.5\)
\(\% \Delta M = 5\)
\(\% \Delta W = 8\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -4\)
\(\textrm{Demand Shock}: D= -2.5\)
\(\% \Delta P_{1} = 0.5\)
\( \% \Delta Q_{1} = -3.5\)
Exercise \(\PageIndex{8}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\phi_{1W} = -0.7\)
\(\varepsilon_{11} = -3\)
\(\varepsilon_{1M} = 0.4\)
\(\% \Delta M = 0\)
\(\% \Delta W = 2\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -1.4\)
\(\textrm{Demand Shock}: D= 0\)
\(\% \Delta P_{1} = 0.4\)
\( \% \Delta Q_{1} = -1.2\)
Exercise \(\PageIndex{9}\)
Given the following:
\(\phi_{11} = 1.5\)
\(\phi_{1W} = -0.5\)
\(\varepsilon_{11} = -1.75\)
\(\varepsilon_{1M} = -0.6\)
\(\% \Delta M = 5\)
\(\% \Delta W = 6\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= -3\)
\(\textrm{Demand Shock}: D= -3\)
\(\% \Delta P_{1} = 0\)
\( \% \Delta Q_{1} = -3\)
Exercise \(\PageIndex{10}\)
Given the following:
\(\phi_{11} = 0.75\)
\(\phi_{1W} = -0.2\)
\(\varepsilon_{11} = -1.75\)
\(\varepsilon_{1M} = -0.4\)
\(\% \Delta M = 7\)
\(\% \Delta W = 0\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S= 0\)
\(\textrm{Demand Shock}: D= -2.8\)
\(\% \Delta P_{1} = -1.12\)
\( \% \Delta Q_{1} = -0.84\)
Problem Set 2: Use Per Unit Cost or Valuation Estimates to Compute Shocks and Changes to an Equilibrium
Exercise \(\PageIndex{1}\)
Given the following:
\(\phi_{11} = 1.5\)
\(\varepsilon_{11} = -1\)
\(\textrm{Per unit change in production cost is } 8 \%\)
\(\textrm{Per unit change in willingness to pay is } = 3 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = -12\)
\(\textrm{Demand Shock}: D = 3\)
\(\% \Delta P_{1} = 6\)
\(\% \Delta Q_{1} = -3\)
Exercise \(\PageIndex{2}\)
Given the following:
\(\phi_{11} = 1\)
\(\varepsilon_{11} = -2\)
\(\textrm{Per unit change in production cost is } 0 \%\)
\(\textrm{Per unit change in willingness to pay is } = -6 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 0\)
\(\textrm{Demand Shock}: D = -12\)
\(\% \Delta P_{1} = -4\)
\(\% \Delta Q_{1} = -4\)
Exercise \(\PageIndex{3}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\varepsilon_{11} = -2\)
\(\textrm{Per unit change in production cost is } 6 \%\)
\(\textrm{Per unit change in willingness to pay is } = 7 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = -3\)
\(\textrm{Demand Shock}: D = 14\)
\(\% \Delta P_{1} = 6.8\)
\(\% \Delta Q_{1} = 0.4\)
Exercise \(\PageIndex{4}\)
Given the following:
\(\phi_{11} = 2\)
\(\varepsilon_{11} = -6\)
\(\textrm{Per unit change in production cost is } -2 \%\)
\(\textrm{Per unit change in willingness to pay is } = -3 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 4\)
\(\textrm{Demand Shock}: D = -18\)
\(\% \Delta P_{1} = -2.75\)
\(\% \Delta Q_{1} = -1.5\)
Exercise \(\PageIndex{5}\)
Given the following:
\(\phi_{11} = 2\)
\(\varepsilon_{11} = -3\)
\(\textrm{Per unit change in production cost is } 0 \%\)
\(\textrm{Per unit change in willingness to pay is } = 5 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 0\)
\(\textrm{Demand Shock}: D = 15\)
\(\% \Delta P_{1} = 3\)
\(\% \Delta Q_{1} = 6\)
Exercise \(\PageIndex{6}\)
Given the following:
\(\phi_{11} = 1.5\)
\(\varepsilon_{11} = -1.75\)
\(\textrm{Per unit change in production cost is } -5 \%\)
\(\textrm{Per unit change in willingness to pay is } = -5 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 9\)
\(\textrm{Demand Shock}: D = -10.5\)
\(\% \Delta P_{1} = -6\)
\(\% \Delta Q_{1} = 0\)
Exercise \(\PageIndex{7}\)
Given the following:
\(\phi_{11} = 1\)
\(\varepsilon_{11} = -4\)
\(\textrm{Per unit change in production cost is } -4 \%\)
\(\textrm{Per unit change in willingness to pay is } = 0 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 4\)
\(\textrm{Demand Shock}: D = 0\)
\(\% \Delta P_{1} = -0.8\)
\(\% \Delta Q_{1} = 3.2\)
Exercise \(\PageIndex{8}\)
Given the following:
\(\phi_{11} = 2\)
\(\varepsilon_{11} = -3\)
\(\textrm{Per unit change in production cost is } -6 \%\)
\(\textrm{Per unit change in willingness to pay is } = 0 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = 12\)
\(\textrm{Demand Shock}: D = 0\)
\(\% \Delta P_{1} = -2.4\)
\(\% \Delta Q_{1} = 7.2\)
Exercise \(\PageIndex{9}\)
Given the following:
\(\phi_{11} = 1.5\)
\(\varepsilon_{11} = -1.75\)
\(\textrm{Per unit change in production cost is } 5 \%\)
\(\textrm{Per unit change in willingness to pay is } = 5 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = -7.5\)
\(\textrm{Demand Shock}: D = 8.75\)
\(\% \Delta P_{1} = 5\)
\(\% \Delta Q_{1} = 0\)
Exercise \(\PageIndex{10}\)
Given the following:
\(\phi_{11} = 0.5\)
\(\varepsilon_{11} = -0.75\)
\(\textrm{Per unit change in production cost is } 6 \%\)
\(\textrm{Per unit change in willingness to pay is } = -7 \%\)
What is the supply shock?
What is the demand shock?
What is the change in equilibrium price?
What is the change in equilibrium quantity?
-
Answer
-
\(\textrm{Supply Shock}: S = -3\)
\(\textrm{Demand Shock}: D = -5.25\)
\(\% \Delta P_{1} = -1.8\)
\(\% \Delta Q_{1} = -3.9\)
Problem Set 3: Multiple Choice
Exercise \(\PageIndex{1}\)
-
Which best describes a market equilibrium?
a) A price such that quantity supplied equals quantity demanded.
b) The confluence of exogenous factors that alter conditions within the market in question.
c) Forces that cause changes in price and or quantity in a market.
d) A situation where prices reach what many consider to be reasonable levels. For instance, the market for gasoline has finally returned to an equilibrium after several years of high prices.
-
Answer
-
a
Exercise \(\PageIndex{2}\)
-
In equilibrium models, endogenous variables
a) Are variables that cause shifts in the demand and supply curves.
b) Are prices and quantities in the markets being analyzed.
c) Are non-existent because all variables are exogenous in these models.
d) Are difficult to measure but still affect the equilibrium (e.g., changes in preferences).
-
Answer
-
b
Exercise \(\PageIndex{3}\)
-
If you are modeling a market equilibrium and you need to account for feedback from other markets or sectors of the economy then you should be using
a) A partial equilibrium model.
b) A feedback accountability model.
c) A general equilibrium model.
d) A model that forces cross-price elasticities to be negative.
-
Answer
-
c
Exercise \(\PageIndex{4}\)
-
Consider a single market. Which would be most likely to cause an increase in both the equilibrium price and the equilibrium quantity?
a) A positive supply shock
b) A decrease in consumer incomes, assuming demand is for an inferior good
c) A decrease in consumer incomes, assuming demand is for a normal good
d) A negative supply shock
-
Answer
-
b
Use the scenario below to answer questions 5 .
Two products are substitutes in consumption and are unrelated in production. The Ukraine is a major supply source for Product 1. Given the current conflict in Eastern Ukraine, the supply for Product 1 (Market 1) shifts inwards (to the left).
Exercise \(\PageIndex{5}\)
-
Which of the following do you know will happen in Market 1?
a) Equilibrium quantity in Market 1 will increase
b) Equilibrium price in Market 1 will decrease
c) Equilibrium price in Market 1 will increase
d) Both a and c
-
Answer
-
c
Exercise \(\PageIndex{6}\)
-
Which of the following do you know will happen in Market 2?
a) Nothing because the supply shock was in Market 1.
b) Equilibrium price in Market 2 will increase
c) Equilibrium price in Market 2 will decrease
d) Equilibrium quantity in Market 2 will increase
e) Both b and d.
-
Answer
-
e
Exercise \(\PageIndex{7}\)
-
After considering feedback between these two markets. Which, if any, of the following schedules have not shifted once a new equilibrium is reached?
a) Supply in Market 1
b) Supply in Market 2
c) Demand in Market 1
d) Demand in Market 2
e) All of the above schedules have shifted
-
Answer
-
b
Exercise \(\PageIndex{8}\)
-
Which best describes the exogenous shock in this situation?
a) The conflict that led to the supply shift to Market 1
b) The change in equilibrium price in Market 1
c) The feedback from Market 2 into Market 1
d) All of the above.
-
Answer
-
a
Use the diagram below to answer questions 9 - 12.
Exercise \(\PageIndex{9}\)
-
Point D only
-
Points C, D, and E
-
Points B and A
-
Points C and E
-
None of the points
-
Answer
-
c
Exercise \(\PageIndex{10}\)
-
In which cases do you know for certain that demand has increased relative to the original equilibrium at point I?
a) Point B
b) Points C and E
c) Points B and A
d) Points A
e) None of the points
-
Answer
-
a
Exercise \(\PageIndex{11}\)
-
In which cases do you know for certain that supply has decreased relative to the original equilibrium at point I?
a) Point B
b) Points C and E
c) Points B and A
d) Points A
e) None of the points
-
Answer
-
b
Exercise \(\PageIndex{12}\)
-
In which cases do you know for certain that both demand and supply have shifted from the original equilibrium at point I?
a) Point B
b) Points C and E
c) Points B and A
d) Points A
e) None of the points
-
Answer
-
e


13.8.9: Chapter 9 Review Exercises
-
- Last updated
- Save as PDF
Chapter 9 Review Exercises
Simplify and Use Square Roots
Simplify Expressions with Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{1}\)
\(\sqrt{64}\)
Exercise \(\PageIndex{2}\)
\(\sqrt{144}\)
- Answer
-
12
Exercise \(\PageIndex{3}\)
\(−\sqrt{25}\)
Exercise \(\PageIndex{4}\)
\(−\sqrt{81}\)
- Answer
-
−9
Exercise \(\PageIndex{5}\)
\(\sqrt{−9}\)
Exercise \(\PageIndex{6}\)
\(\sqrt{−36}\)
- Answer
-
not a real number
Exercise \(\PageIndex{7}\)
\(\sqrt{64}+\sqrt{225}\)
Exercise \(\PageIndex{8}\)
\(\sqrt{64+225}\)
- Answer
-
17
In the following exercises, estimate each square root between two consecutive whole numbers.
Exercise \(\PageIndex{9}\)
\(\sqrt{28}\)
Exercise \(\PageIndex{10}\)
\(\sqrt{155}\)
- Answer
-
\(12<\sqrt{155}<13\)
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Exercise \(\PageIndex{11}\)
\(\sqrt{15}\)
Example \(\PageIndex{12}\)
\(\sqrt{57}\)
- Answer
-
7.55
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{13}\)
\(\sqrt{q^2}\)
Exercise \(\PageIndex{14}\)
\(\sqrt{64b^2}\)
- Answer
-
8b
Exercis\e \(\PageIndex{15}\)
\(−\sqrt{121a^2}\)
Exercise \(\PageIndex{16}\)
\(\sqrt{225m^{2}n^{2}}\)
- Answer
-
15mn
Exercise \(\PageIndex{17}\)
\(−\sqrt{100q^2}\)
Exercise \(\PageIndex{18}\)
\(\sqrt{49y^2}\)
- Answer
-
7y
Exercise \(\PageIndex{19}\)
\(\sqrt{4a^{2}b^{2}}\)
Exercise \(\PageIndex{20}\)
\(\sqrt{121c^{2}d^{2}}\)
- Answer
-
11cd
Simplify Square Roots
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{21}\)
\(\sqrt{300}\)
Exercise \(\PageIndex{22}\)
\(\sqrt{98}\)
- Answer
-
\(7\sqrt{2}\)
Exercise \(\PageIndex{23}\)
\(\sqrt{x^{13}}\)
Exercise \(\PageIndex{24}\)
\(\sqrt{y^{19}}\)
- Answer
-
\(y^{9}\sqrt{y}\)
Exercise \(\PageIndex{25}\)
\(\sqrt{16m^4}\)
Exercise \(\PageIndex{26}\)
\(\sqrt{36n^{13}}\)
- Answer
-
\(6n^{6}\sqrt{n}\)
Exercise \(\PageIndex{27}\)
\(\sqrt{288m^{21}}\)
Exercise \(\PageIndex{28}\)
\(\sqrt{150n^7}\)
- Answer
-
\(5n^3\sqrt{6n}\)
Exercise \(\PageIndex{29}\)
\(\sqrt{48r^{5}s^{4}}\)
Exercise \(\PageIndex{30}\)
\(\sqrt{108r^{5}s^{3}}\)
- Answer
-
\(6r^{2}s\sqrt{3rs}\)
Exercise \(\PageIndex{31}\)
\(\frac{10−\sqrt{50}}{5}\)
Exercise \(\PageIndex{32}\)
\(\frac{6+\sqrt{72}}{6}\)
- Answer
-
\(1+\sqrt{2}\)
In the following exercises, simplify.
Exercise \(\PageIndex{33}\)
\(\sqrt{\frac{16}{25}}\)
Exercise \(\PageIndex{34}\)
\(\sqrt{\frac{81}{36}}\)
- Answer
-
\(\frac{3}{2}\)
Exercise \(\PageIndex{35}\)
\(\sqrt{\frac{x^8}{x^4}}\)
Exercise \(\PageIndex{36}\)
\(\sqrt{\frac{y^6}{y^2}}\)
- Answer
-
\(y^2\)
Exercise \(\PageIndex{37}\)
\(\sqrt{\frac{98p^6}{2p^2}}\)
Exercise \(\PageIndex{38}\)
\(\sqrt{\frac{72q^8}{2q^4}}\)
- Answer
-
\(6q^2\)
Exercise \(\PageIndex{39}\)
\(\sqrt{\frac{65}{121}}\)
Exercise \(\PageIndex{40}\)
\(\sqrt{\frac{26}{169}}\)
- Answer
-
\(\frac{\sqrt{26}}{13}\)
Exercise \(\PageIndex{41}\)
\(\sqrt{\frac{64x^4}{25x^2}}\)
Exercise \(\PageIndex{42}\)
\(\sqrt{\frac{36r^{10}}{16r^5}}\)
- Answer
-
\(\frac{3r^2\sqrt{r}}{2}\)
Exercise \(\PageIndex{43}\)
\(\sqrt{\frac{48p^{3}q^{5}}{27pq}}\)
Exercise \(\PageIndex{44}\)
\(\sqrt{\frac{12r^{5}s^{7}}{75r^{2}s}}\)
- Answer
-
\(\frac{2rs^3\sqrt{r}}{5}\)
Add and Subtract Square Roots
Add and Subtract Like Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{45}\)
\(3\sqrt{2}+\sqrt{2}\)
Exercise \(\PageIndex{46}\)
\(5\sqrt{5}+7\sqrt{5}\)
- Answer
-
\(12\sqrt{5}\)
Exercise \(\PageIndex{47}\)
\(4\sqrt{y}+4\sqrt{y}\)
Exercise \(\PageIndex{48}\)
\(6\sqrt{m}−2\sqrt{m}\)
- Answer
-
\(4\sqrt{m}\)
Exercise \(\PageIndex{49}\)
\(−3\sqrt{7}+2\sqrt{7}−\sqrt{7}\)
Exercise \(\PageIndex{50}\)
\(8\sqrt{13}+2\sqrt{3}+3\sqrt{13}\)
- Answer
-
\(11\sqrt{13}+2\sqrt{3}\)
Exercise \(\PageIndex{51}\)
\(3\sqrt{5xy}−\sqrt{5xy}+3\sqrt{5xy}\)
Exercise \(\PageIndex{52}\)
\(2\sqrt{3rs}+\sqrt{3rs}−5\sqrt{rs}\)
- Answer
-
\(3\sqrt{3rs}−5\sqrt{rs}\)
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
Exercise \(\PageIndex{53}\)
\(\sqrt{32}+3\sqrt{2}\)
Exercise \(\PageIndex{54}\)
\(\sqrt{8}+\sqrt{32}\)
- Answer
-
\(5\sqrt{2}\)
Exercise \(\PageIndex{55}\)
\(\sqrt{72}+\sqrt{50}\)
Exercise \(\PageIndex{56}\)
\(\sqrt{48}+\sqrt{75}\)
- Answer
-
\(9\sqrt{3}\)
Exercise \(\PageIndex{57}\)
\(3\sqrt{32}+\sqrt{98}\)
Exercise \(\PageIndex{58}\)
\(\frac{1}{3}\sqrt{27}−\frac{1}{8}\sqrt{192}\)
- Answer
-
0
Exercise \(\PageIndex{59}\)
\(\sqrt{50y^5}−\sqrt{72y^5}\)
Exercise \(\PageIndex{60}\)
Add exercises text here.
- Answer
-
\(17n^2\sqrt{2}\)
Multiply Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{61}\)
\(\sqrt{2}·\sqrt{20}\)
Exercise \(\PageIndex{62}\)
\(2\sqrt{2}·6\sqrt{14}\)
- Answer
-
\(24\sqrt{7}\)
Exercise \(\PageIndex{63}\)
\(\sqrt{2m^2}·\sqrt{20m^4}\)
Exercise \(\PageIndex{64}\)
\((\sqrt{62y})(\sqrt{350y^3})\)
- Answer
-
\(180y^2\)
Exercise \(\PageIndex{65}\)
\((6\sqrt{3v^4})(5\sqrt{30v})\)
Exercise \(\PageIndex{66}\)
\((\sqrt{8})^2\)
- Answer
-
8
Exercise \(\PageIndex{67}\)
\((−\sqrt{10})^2\)
Exercise \(\PageIndex{68}\)
\((2\sqrt{5})(5\sqrt{5})\)
- Answer
-
50
Exercise \(\PageIndex{69}\)
\((−3\sqrt{3})(5\sqrt{18})\)
In the following exercises, simplify.
Exercise \(\PageIndex{70}\)
\(10(2−\sqrt{7})\)
- Answer
-
\(20−10\sqrt{7}\)
Exercise \(\PageIndex{71}\)
\(\sqrt{3}(4+\sqrt{12})\)
Exercise \(\PageIndex{72}\)
\((5+\sqrt{2})(3−\sqrt{2})\)
- Answer
-
\(13−2\sqrt{2}\)
Exercise \(\PageIndex{73}\)
\((5−3\sqrt{7})(1−2\sqrt{7})\)
Exercise \(\PageIndex{74}\)
\((1−3\sqrt{x})(5+2\sqrt{x})\)
- Answer
-
\(5−13\sqrt{x}−6x\)
Exercise \(\PageIndex{75}\)
\((3+4\sqrt{y})(10−\sqrt{y})\)
Exercise \(\PageIndex{76}\)
\((1+6\sqrt{p})^2\)
- Answer
-
\(1+12\sqrt{p}+36p\)
Exercise \(\PageIndex{77}\)
\((2−6\sqrt{5})^2\)
Exercise \(\PageIndex{78}\)
\((3+2\sqrt{7})(3−2\sqrt{7})\)
- Answer
-
−19
Exercise \(\PageIndex{79}\)
\((6−\sqrt{11})(6+\sqrt{11})\)
Divide Square Roots
Divide Square Roots
In the following exercises, simplify.
Exercise \(\PageIndex{80}\)
\(\frac{\sqrt{75}}{10}\)
- Answer
-
\(\frac{\sqrt{3}}{2}\)
Exercise \(\PageIndex{81}\)
\(\frac{2−\sqrt{12}}{6}\)
Exercise \(\PageIndex{82}\)
\(\frac{\sqrt{48}}{\sqrt{27}}\)
- Answer
-
\(\frac{4}{3}\)
Exercise \(\PageIndex{83}\)
\(\frac{\sqrt{75x^7}}{\sqrt{3x^3}}\)
Exercise \(\PageIndex{84}\)
\(\frac{\sqrt{20y^5}}{\sqrt{2y}}\)
- Answer
-
\(y^2\sqrt{10}\)
Exercise \(\PageIndex{85}\)
\(\frac{\sqrt{98p^{6}q^{4}}}{\sqrt{2p^{4}q^{8}}}\)
In the following exercises, rationalize the denominator.
Exercise \(\PageIndex{86}\)
\(\frac{10}{\sqrt{15}}\)
- Answer
-
\(\frac{2\sqrt{15}}{3}\)
Exercise \(\PageIndex{87}\)
\(\frac{6}{\sqrt{6}}\)
Exercise \(\PageIndex{88}\)
\(\frac{5}{3\sqrt{5}}\)
- Answer
-
\(\frac{\sqrt{5}}{3}\)
Exercise \(\PageIndex{89}\)
\(\frac{10}{2\sqrt{6}}\)
Exercise \(\PageIndex{90}\)
\(\sqrt{\frac{3}{28}}\)
- Answer
-
\(\frac{\sqrt{21}}{14}\)
Exercise \(\PageIndex{91}\)
\(\sqrt{\frac{9}{75}}\)
In the following exercises, rationalize the denominator.
Exercise \(\PageIndex{92}\)
\(\frac{4}{4+\sqrt{27}}\)
- Answer
-
\(\frac{16−12\sqrt{3}}{−11}\)
Exercise \(\PageIndex{93}\)
\(\frac{5}{2−\sqrt{10}}\)
Exercise \(\PageIndex{94}\)
\(\frac{4}{2−\sqrt{5}}\)
- Answer
-
\(−8−4\sqrt{5}\)
Exercise \(\PageIndex{95}\)
\(\frac{5}{4−\sqrt{8}}\)
Exercise \(\PageIndex{96}\)
\(\frac{\sqrt{2}}{\sqrt{p}+\sqrt{3}}\)
- Answer
-
\(\frac{\sqrt{2p}−\sqrt{6}}{p−3}\)
Exercise \(\PageIndex{97}\)
\(\frac{\sqrt{x}−\sqrt{2}}{\sqrt{x}+\sqrt{2}}\)
Solve Equations with Square Roots
Solve Radical Equations
In the following exercises, solve the equation.
Exercise \(\PageIndex{98}\)
\(\sqrt{7z+1}=6\)
- Answer
-
5
Exercise \(\PageIndex{99}\)
\(\sqrt{4u−2}−4=0\)
Exercise \(\PageIndex{100}\)
\(\sqrt{6m+4}−5=0\)
- Answer
-
\(\frac{7}{2}\)
Exercise \(\PageIndex{101}\)
\(\sqrt{2u−3}+2=0\)
Exercise \(\PageIndex{102}\)
\(\sqrt{u−4}+4=u\)
- Answer
-
no solution
Exercise \(\PageIndex{103}\)
\(\sqrt{v−9}+9=0\)
Exercise \(\PageIndex{104}\)
\(\sqrt{r−4}−r=−10\)
- Answer
-
13
Exercise \(\PageIndex{105}\)
\(\sqrt{s−9}−s=−9\)
Exercise \(\PageIndex{106}\)
\(2\sqrt{2x−7}−4=8\)
- Answer
-
\(\frac{43}{2}\)
Exercise \(\PageIndex{107}\)
\(\sqrt{2−x}=\sqrt{2x−7}\)
Exercise \(\PageIndex{108}\)
\(\sqrt{a}+3=\sqrt{a+9}\)
- Answer
-
0
Exercise \(\PageIndex{109}\)
\(\sqrt{r}+3=\sqrt{r+4}\)
Exercise \(\PageIndex{110}\)
\(\sqrt{u}+2=\sqrt{u+5}\)
- Answer
-
\(\frac{11}{6}\)
Exercise \(\PageIndex{111}\)
\(\sqrt{n+11}−1=\sqrt{n+4}\)
Exercise \(\PageIndex{112}\)
\(\sqrt{y+5}+1=\sqrt{2y+3}\)
- Answer
-
11
In the following exercises, solve. Round approximations to one decimal place.
Exercise \(\PageIndex{113}\)
A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula \(s=\sqrt{A}\) to find the length of each side of his lawn.
Exercise \(\PageIndex{114}\)
A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the package to reach the hiker.
- Answer
-
7.5 seconds
Exercise \(\PageIndex{115}\)
Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula \(s=\sqrt{24d}\) to find the speed of the car before the brakes were applied.
Higher Roots
Simplify Expressions with Higher Roots
In the following exercises, simplify.
Exercise \(\PageIndex{116}\)
- \(\sqrt[6]{64}\)
- \(\sqrt[3]{64}\)
- Answer
-
- 2
- 4
Exercise \(\PageIndex{117}\)
- \(\sqrt[3]{−27}\)
- \(\sqrt[4]{−64}\)
Exercise \(\PageIndex{118}\)
- \(\sqrt[9]{d^9}\)
- \(\sqrt[8]{v^8}\)
- Answer
-
- d
- |v|
Exercise \(\PageIndex{119}\)
- \(\sqrt[5]{a^{10}}\)
- \(\sqrt[3]{b^{27}}\)
Exercise \(\PageIndex{120}\)
- \(\sqrt[4]{16x^8}\)
- \(\sqrt[6]{64y^{12}}\)
- Answer
-
- \(2x^2\)
- \(2y^2\)
Exercise \(\PageIndex{121}\)
- \(\sqrt[7]{128r^{14}}\)
- \(\sqrt[4]{81s^{24}}\)
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Exercise \(\PageIndex{122}\)
- \(\sqrt[9]{d^9}\)
- Answer
-
- d
Exercise \(\PageIndex{123}\)
- \(\sqrt[3]{54}\)
- \(\sqrt[4]{128}\)
Exercise \(\PageIndex{124}\)
- \(\sqrt[5]{64c^8}\)
- \(\sqrt[4]{48d^7}\)
- Answer
-
- \(2c\sqrt[5]{2c^3}\)
- \(2d\sqrt[4]{3d^3}\)
Exercise \(\PageIndex{125}\)
- \(\sqrt[3]{343q^7}\)
- \(\sqrt[6]{192r^9}\)
Exercise \(\PageIndex{126}\)
- \(\sqrt[3]{−500}\)
- \(\sqrt[4]{−16}\)
- Answer
-
- \(−5\sqrt[3]{4}\)
- not a real number
In the following exercises, simplify.
Exercise \(\PageIndex{127}\)
\(\sqrt[5]{\frac{r^{10}}{r^5}}\)
Exercise \(\PageIndex{128}\)
\(\sqrt[3]{\frac{w^{12}}{w^2}}\)
- Answer
-
\(w^3\sqrt[3]{w}\)
Exercise \(\PageIndex{129}\)
\(\sqrt[4]{\frac{64y^8}{4y^5}}\)
Exercise \(\PageIndex{130}\)
\(\sqrt[3]{\frac{54z^9}{2z^3}}\)
- Answer
-
\(3z^2\)
Exercise \(\PageIndex{131}\)
\(\sqrt[6]{\frac{64a^7}{b^2}}\)
In the following exercises, simplify.
Exercise \(\PageIndex{132}\)
\(4\sqrt[5]{20}−2\sqrt[5]{20}\)
- Answer
-
\(2\sqrt[5]{20}\)
Exercise \(\PageIndex{133}\)
\(4\sqrt[3]{18}+3\sqrt[3]{18}\)
Exercise \(\PageIndex{134}\)
\(\sqrt[4]{1250}−\sqrt[4]{162}\)
- Answer
-
\(2\sqrt[4]{2}\)
Exercise \(\PageIndex{135}\)
\(\sqrt[3]{640c^5}−\sqrt[3]{−80c^3}\)
Exercise \(\PageIndex{136}\)
\(\sqrt[5]{96t^8}+\sqrt[5]{486t^4}\)
- Answer
-
\(2t^\sqrt[5]{3t^3}+3\sqrt[5]{2t^4}\)
Rational Exponents
Simplify Expressions with \(a^{\frac{1}{n}}\)
In the following exercises, write as a radical expression.
Exercise \(\PageIndex{137}\)
\(r^{\frac{1}{8}}\)
Exercise \(\PageIndex{138}\)
\(s^{\frac{1}{10}}\)
- Answer
-
In the following exercises, write with a rational exponent.
Exercise \(\PageIndex{139}\)
\(\sqrt[5]{u}\)
Exercise \(\PageIndex{140}\)
\(\sqrt[6]{v}\)
- Answer
-
\(v^{\frac{1}{6}}\)
Exercise \(\PageIndex{141}\)
\(\sqrt[3]{9m}\)
Exercise \(\PageIndex{142}\)
\(\sqrt[6]{10z}\)
- Answer
-
\((10z)^{\frac{1}{6}}\)
In the following exercises, simplify.
Exercise \(\PageIndex{143}\)
\(16^{\frac{1}{4}}\)
Exercise \(\PageIndex{144}\)
\(32^{\frac{1}{5}}\)
- Answer
-
2
Exercise \(\PageIndex{145}\)
\((−125)^{\frac{1}{3}}\)
Exercise \(\PageIndex{146}\)
\((125)^{−\frac{1}{3}}\)
- Answer
-
\(\frac{1}{5}\)
Exercise \(\PageIndex{147}\)
\((−9)^{\frac{1}{2}}\)
Exercise \(\PageIndex{148}\)
\((36)^{−\frac{1}{2}}\)
- Answer
-
\(\frac{1}{6}\)
In the following exercises, write with a rational exponent.
Exercise \(\PageIndex{149}\)
\(\sqrt[3]{q^5}\)
Exercise \(\PageIndex{150}\)
\(\sqrt[5]{n^8}\)
- Answer
-
\(n^{\frac{8}{5}}\)
In the following exercises, simplify.
Exercise \(\PageIndex{151}\)
\(27^{−\frac{2}{3}}\)
Exercise \(\PageIndex{152}\)
\(64^{\frac{5}{2}}\)
- Answer
-
32,768
Exercise \(\PageIndex{153}\)
\(36^{\frac{3}{2}}\)
Exercise \(\PageIndex{154}\)
\(81^{−\frac{5}{2}}\)
- Answer
-
\(\frac{1}{59,049}\)
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.
Exercise \(\PageIndex{155}\)
\(3^{\frac{4}{5}}·3^{\frac{6}{5}}\)
Exercise \(\PageIndex{156}\)
\((x^6)^{\frac{4}{3}}\)
- Answer
-
\(x^8\)
Exercise \(\PageIndex{157}\)
\(\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}\)
Exercise \(\PageIndex{158}\)
\((16s^{\frac{9}{4}})^{\frac{1}{4}}\)
- Answer
-
\(2s^{\frac{9}{16}}\)
Exercise \(\PageIndex{159}\)
\((m^{8}n^{12})^{\frac{1}{4}}\)
Exercise \(\PageIndex{160}\)
\(\frac{z^{\frac{2}{3}}·z^{−\frac{1}{3}}}{z^{−\frac{5}{3}}}\)
- Answer
-
\(z^2\)
Practice Test
In the following exercises, simplify.
Exercise \(\PageIndex{161}\)
\(\sqrt{81+144}\)
Exercise \(\PageIndex{162}\)
\(\sqrt{169m^{4}n^{2}}\)
- Answer
-
\(13m^{2}|n|\)
Exercise \(\PageIndex{163}\)
\(\sqrt{36n^{13}}\)
Exercise \(\PageIndex{164}\)
\(3\sqrt{13}+5\sqrt{2}+\sqrt{13}\)
- Answer
-
\(4\sqrt{13}+5\sqrt{2}\)
Exercise \(\PageIndex{165}\)
\(5\sqrt{20}+2\sqrt{125}\)
Exercise \(\PageIndex{166}\)
\((3\sqrt{6y})(\sqrt{250y^3})\)
- Answer
-
\(180y^2\sqrt{3}\)
Exercise \(\PageIndex{167}\)
\((2−5\sqrt{x})(3+\sqrt{x})\)
Exercise \(\PageIndex{168}\)
\((1−2\sqrt{q})^2\)
- Answer
-
\(1−4\sqrt{q}+4q\)
Exercise \(\PageIndex{169}\)
- \(\sqrt{a^{12}}\)
- \(\sqrt[3]{b^{21}}\)
Exercise \(\PageIndex{170}\)
- \(\sqrt[4]{81x^{12}}\)
- \(\sqrt[6]{64y^{18}}\)
- Answer
-
- \(3x^3\)
- \(2y^3\)
Exercise \(\PageIndex{171}\)
\(\sqrt[6]{\frac{64r^{12}}{25r^6}}\)
Exercise \(\PageIndex{172}\)
\(\sqrt{\frac{14y^3}{7y}}\)
- Answer
-
\(y\sqrt{2}\)
Exercise \(\PageIndex{173}\)
\(\frac{\sqrt{256x^7}}{\sqrt{54x^2}}\)
Exercise \(\PageIndex{174}\)
\(\sqrt[4]{512}−2\sqrt[4]{32}\)
- Answer
-
0
Exercise \(\PageIndex{175}\)
- \(256^{\frac{1}{4}}\)
- \(243^{\frac{1}{5}}\)
Exercise \(\PageIndex{176}\)
\(49^{\frac{3}{2}}\)
- Answer
-
343
Exercise \(\PageIndex{177}\)
\(25^{−\frac{5}{2}}\)
Exercise \(\PageIndex{178}\)
\(\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}\)
- Answer
-
\(\frac{1}{w}\)
Exercise \(\PageIndex{179}\)
\((27s^{\frac{3}{5}})^{\frac{1}{3}}\)
In the following exercises, rationalize the denominator.
Exercise \(\PageIndex{180}\)
\(\frac{3}{2\sqrt{6}}\)
- Answer
-
\(\frac{\sqrt{6}}{4}\)
Exercise \(\PageIndex{181}\)
\(\frac{\sqrt{3}}{\sqrt{x}+\sqrt{5}}\)
In the following exercises, solve.
Exercise \(\PageIndex{182}\)
\(3\sqrt{2x−3}−20=7\)
- Answer
-
42
Exercise \(\PageIndex{183}\)
\(\sqrt{3u−2}=\sqrt{5u+1}\)
In the following exercise, solve.
Exercise \(\PageIndex{184}\)
A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.
- Answer
-
6.1 seconds


4.3: Exercises
- True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let \(G\) and \(G'\) be groups.
- Every group contains at least two distinct subgroups.
- If \(H\) is a proper subgroup of group \(G\) and \(G\) is finite, then we must have \(|H|\lt |G|\text{.}\)
- \(7\mathbb{Z}\) is a subgroup of \(14\mathbb{Z}\text{.}\)
- A group \(G\) may have two distinct proper subgroups which are isomorphic (to one another).
2. Give specific, precise examples of the following groups \(G\) with subgroups \(H\text{:}\)
- A group \(G\) with a proper subgroup \(H\) of \(G\) such that \(|H|=|G|\text{.}\)
- A group \(G\) of order \(12\) containing a subgroup \(H\) with \(|H|=3\text{.}\)
- A nonabelian group \(G\) containing a nontrivial abelian subgroup \(H\text{.}\)
- A finite subgroup \(H\) of an infinite group \(G\text{.}\)
3. Let \(n\in \mathbb{Z}^+\text{.}\)
- Prove that \(n\mathbb{Z} \leq \mathbb{Z}\text{.}\)
- Prove that the set \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A=\pm 1\}\) is a subgroup of \(GL(n,\mathbb{R})\text{.}\)
( Note: Your proofs do not need to be long to be correct!)
4. Let \(n\in \mathbb{Z}^+\text{.}\) For each group \(G\) and subset \(H\text{,}\) decide whether or not \(H\) is a subgroup of \(G\text{.}\) In the cases in which \(H\) is not a subgroup of \(G\text{,}\) provide a proof. ( Note. Your proofs do not need to be long to be correct!)
- \(G=\mathbb{R}\text{,}\) \(H=\mathbb{Z}\)
- \(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,5,10\}\)
- \(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,4,8,12\}\)
- \(G=\mathbb{C}\text{,}\) \(H=\mathbb{R}^*\)
- \(G=\mathbb{C}^*\text{,}\) \(H=\{1,i,-1,-i\}\)
- \(G=\mathbb{M}_n(\mathbb{R})\text{,}\) \(H=GL(n,\mathbb{R})\)
- \(G=GL(n,\mathbb{R})\text{,}\) \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A = -1\}\)
5. Let \(G\) and \(G'\) be groups, let \(\phi\) be a homomorphism from \(G\) to \(G'\text{,}\) and let \(H\) be a subgroup of \(G\text{.}\) Prove that \(\phi(H)\) is a subgroup of \(G'\text{.}\)
6. Let \(G\) be an abelian group, and let \(U=\{g\in G\,:\, g^{-1}=g\}.\) Prove that \(U\) is a subgroup of \(G\text{.}\)