28.6C: Mendel’s Law of Independent Assortment
Independent assortment allows the calculation of genotypic and phenotypic ratios based on the probability of individual gene combinations.
- Use the probability or forked line method to calculate the chance of any particular genotype arising from a genetic cross
Key Points
- Mendel’s law of independent assortment states that genes do not influence each other with regard to the sorting of alleles into gametes; every possible combination of alleles for every gene is equally likely to occur.
- The calculation of any particular genotypic combination of more than one gene is, therefore, the probability of the desired genotype at the first locus multiplied by the probability of the desired genotype at the other loci.
- The forked line method can be used to calculate the chances of all possible genotypic combinations from a cross, while the probability method can be used to calculate the chance of any one particular genotype that might result from that cross.
Key Terms
- independent assortment : separate genes for separate traits are passed independently of one another from parents to offspring
Independent Assortment
Mendel’s law of independent assortment states that genes do not influence each other with regard to the sorting of alleles into gametes: every possible combination of alleles for every gene is equally likely to occur. The independent assortment of genes can be illustrated by the dihybrid cross: a cross between two true-breeding parents that express different traits for two characteristics. Consider the characteristics of seed color and seed texture for two pea plants: one that has green, wrinkled seeds (yyrr) and another that has yellow, round seeds (YYRR). Because each parent is homozygous, the law of segregation indicates that the gametes for the green/wrinkled plant all are yr, while the gametes for the yellow/round plant are all YR. Therefore, the F 1 generation of offspring all are YyRr.
For the F2 generation, the law of segregation requires that each gamete receive either an R allele or an r allele along with either a Y allele or a y allele. The law of independent assortment states that a gamete into which an r allele sorted would be equally likely to contain either a Y allele or a y allele. Thus, there are four equally likely gametes that can be formed when the YyRr heterozygote is self-crossed as follows: YR, Yr, yR, and yr. Arranging these gametes along the top and left of a 4 × 4 Punnett square gives us 16 equally likely genotypic combinations. From these genotypes, we infer a phenotypic ratio of 9 round/yellow:3 round/green:3 wrinkled/yellow:1 wrinkled/green. These are the offspring ratios we would expect, assuming we performed the crosses with a large enough sample size.
Independent assortment of 2 genes : This dihybrid cross of pea plants involves the genes for seed color and texture.
Because of independent assortment and dominance, the 9:3:3:1 dihybrid phenotypic ratio can be collapsed into two 3:1 ratios, characteristic of any monohybrid cross that follows a dominant and recessive pattern. Ignoring seed color and considering only seed texture in the above dihybrid cross, we would expect that three-quarters of the F 2 generation offspring would be round and one-quarter would be wrinkled. Similarly, isolating only seed color, we would assume that three-quarters of the F 2 offspring would be yellow and one-quarter would be green. The sorting of alleles for texture and color are independent events, so we can apply the product rule. Therefore, the proportion of round and yellow F 2 offspring is expected to be (3/4) × (3/4) = 9/16, and the proportion of wrinkled and green offspring is expected to be (1/4) × (1/4) = 1/16. These proportions are identical to those obtained using a Punnett square. Round/green and wrinkled/yellow offspring can also be calculated using the product rule as each of these genotypes includes one dominant and one recessive phenotype. Therefore, the proportion of each is calculated as (3/4) × (1/4) = 3/16.
Forked-Line Method
When more than two genes are being considered, the Punnett-square method becomes unwieldy. For instance, examining a cross involving four genes would require a 16 × 16 grid containing 256 boxes. It would be extremely cumbersome to manually enter each genotype. For more complex crosses, the forked-line and probability methods are preferred.
To prepare a forked-line diagram for a cross between F 1 heterozygotes resulting from a cross between AABBCC and aabbcc parents, we first create rows equal to the number of genes being considered and then segregate the alleles in each row on forked lines according to the probabilities for individual monohybrid crosses. We then multiply the values along each forked path to obtain the F 2 offspring probabilities. Note that this process is a diagrammatic version of the product rule. The values along each forked pathway can be multiplied because each gene assorts independently. For a trihybrid cross, the F 2 phenotypic ratio is 27:9:9:9:3:3:3:1.
Independent assortment of 3 genes : The forked-line method can be used to analyze a trihybrid cross. Here, the probability for color in the F2 generation occupies the top row (3 yellow:1 green). The probability for shape occupies the second row (3 round:1 wrinked), and the probability for height occupies the third row (3 tall:1 dwarf). The probability for each possible combination of traits is calculated by multiplying the probability for each individual trait. Thus, the probability of F2 offspring having yellow, round, and tall traits is 3 × 3 × 3, or 27.
Probability Method
While the forked-line method is a diagrammatic approach to keeping track of probabilities in a cross, the probability method gives the proportions of offspring expected to exhibit each phenotype (or genotype) without the added visual assistance.
To fully demonstrate the power of the probability method, however, we can consider specific genetic calculations. For instance, for a tetrahybrid cross between individuals that are heterozygotes for all four genes, and in which all four genes are sorting independently in a dominant and recessive pattern, what proportion of the offspring will be expected to be homozygous recessive for all four alleles? Rather than writing out every possible genotype, we can use the probability method. We know that for each gene the fraction of homozygous recessive offspring will be 1/4. Therefore, multiplying this fraction for each of the four genes, (1/4) × (1/4) × (1/4) × (1/4), we determine that 1/256 of the offspring will be quadruply homozygous recessive.
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1.14.1: Health Literacy
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Chapter one of this book provided a brief introduction to health literacy, this chapter expands on the importance of critically consuming health information throughout your life and as you age. The CDC defines personal health literacy as the degree to which individuals have the ability to find, understand, and use information and services to inform health-related decisions and actions for themselves and others.
Finding reliable health information
Millions of consumers get health information from magazines, TV or the Internet. Some of the information is reliable and up to date; some is not. How can you tell the good from the bad to make sure you are making your own health decisions based on the best information?
First, recognize that content on the Internet is unregulated; anyone can publish anything on the Internet. There is sound medical information on the Internet along with dangerous information. You need to be able to tell the difference.
Ask yourself the following:
- Why did the person create the page?
- What’s in it for them?
- Are they trying to sell me something?
The following tips can help you to find current, unbiased information based on research [1] .
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Consider the source–Use recognized authorities
- Know who is responsible for the content.
- Look for an “about us” page. Check to see who runs the site: is it a branch of the Federal Government, a non-profit institution, a professional organization, a health system, a commercial organization or an individual.
- There is a big difference between a site that says, “I developed this site after my heart attack” and one that says, “This page on heart attack was developed by health professionals at the American Heart Association.”
- Web sites should have a way to contact the organization or webmaster. If the site provides no contact information, or if you can’t easily find out who runs the site, use caution.
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Focus on quality–All Web sites are not created equal
- Does the site have an editorial board? Is the information reviewed before it is posted?
- This information is often on the “about us” page, or it may be under the organization’s mission statement, or part of the annual report.
- See if the board members are experts in the subject of the site. For example, a site on osteoporosis whose medical advisory board is composed of attorneys and accountants is not medically authoritative.
- Look for a description of the process of selecting or approving information on the site. It is usually in the “about us” section and may be called “editorial policy” or “selection policy” or “review policy.”
- Sometimes the site will have information “about our writers” or “about our authors” instead of an editorial policy. Review this section to find out who has written the information.
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Be a cyber skeptic–Quackery abounds on the Web
- Does the site make health claims that seem too good to be true? Does the information use deliberately obscure, “scientific” sounding language? Does it promise quick, dramatic, miraculous results? Is this the only site making these claims?
- Beware of claims that one remedy will cure a variety of illnesses, that it is a “breakthrough,” or that it relies on a “secret ingredient.”
- Use caution if the site uses a sensational writing style (lots of exclamation points, for example.)
- A health Website for consumers should use simple language, not technical jargon.
- Get a second opinion! Check more than one site.
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Look for the evidence–Rely on medical research, not opinion
- Does the site identify the author? Does it rely on testimonials?
- Look for the author of the information, either an individual or an organization. Good examples are “Written by Jane Smith, R.N.,” or “Copyright 2013, American Cancer Society.”
- If there are case histories or testimonials on the Web site, look for contact information such as an email address or telephone number. If the testimonials are anonymous or hard to track down (“Jane from California”), use caution.
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Check for currency–Look for the latest information
- Is the information current?
- Look for dates on documents. A document on coping with the loss of a loved one doesn’t need to be current, but a document on the latest treatment of AIDS needs to be current.
- Click on a few links on the site. If there are a lot of broken links, the site may not be kept up-to-date.
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Beware of bias–What is the purpose? Who is providing the funding?
- Who pays for the site?
- Check to see if the site is supported by public funds, donations or by commercial advertising.
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Advertisements should be labeled. They should say “Advertisement” or “From our Sponsor.”
- Look at a page on the site, and see if it is clear when content is coming from a non-commercial source and when an advertiser provides it. For example, if a page about treatment of depression recommends one drug by name, see if you can tell if the company that manufactures the drug provides that information. If it does, you should consult other sources to see what they say about the same drug.
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Protect your privacy–Health information should be confidential
- Does the site have a privacy policy and tell you what information they collect?
- There should be a link saying “Privacy” or “Privacy Policy.” Read the privacy policy to see if your privacy is really being protected. For example, if the site says “We share information with companies that can provide you with useful products,” then your information isn’t private.
- If there is a registration form, notice what types of questions you must answer before you can view content. If you must provide personal information (such as name, address, date of birth, gender, mother’s maiden name, credit card number) you should refer to their privacy policy to see what they can do with your information.
- Consult with your health professional–Patient/provider partnerships lead to the best medical decisions.
Health Fraud
When you see statements like “miracle cure,” “revolutionary scientific breakthrough,” or “alternative to drugs or surgery,” what you should be thinking is, “bogus product,” “Danger,” or “Health fraud alert!”
Health fraud scams have been around for hundreds of years [2] . The snake oil salesmen of old have morphed into the deceptive, high-tech marketers of today. They prey on people’s desires for easy solutions to difficult health problems, from losing weight to curing serious diseases like cancer. According to the Food and Drug Administration (FDA), a health product is fraudulent if it is deceptively promoted as being effective against a disease or health condition but has not been scientifically proven safe and effective for that purpose. Health fraud scams can do more than waste your money, they can cause serious injury or even death.
Scammers promote their products through newspapers, magazines, TV infomercials and cyberspace. You can find health fraud scams in retail stores and on countless websites, in popup ads and spam, and on social media sites like Facebook and Twitter. It is important to understand the signs of fraudulent health product.
The FDA provides the following list of 6 Tip-offs to Rip-offs:
- One product does it all. Be suspicious of products that claim to cure a wide range of diseases. The agency continues to send warning letters and take enforcement action as appropriate against companies marketing fake cure-all products. These miracle cures don’t exist – they’re bogus – and the only thing these companies are selling is false hope.
- Personal “success” testimonials. Success stories, such as, “It cured my diabetes” or “It immediately stopped my COVID-19 infection,” are easy to make up and are not a substitute for scientific evidence. Reviews found on popular online marketplaces and social media can be fake.
- Quick fixes. Few diseases or conditions can be treated quickly, even with legitimate products. Beware of language such as, “Lose 30 pounds in 30 days,” “protects from viral infections,” or “eliminates skin cancer in days.”
- “All natural” cure or treatment. Don’t be fooled by descriptions like “all-natural cure.” Such phrases are often used in health fraud as an attention-grabber to suggest that a product is safer than conventional treatments. These terms don’t necessarily equate to safety. Some plants found in nature (such as poisonous mushrooms) can be harmful or even kill when consumed. Moreover, the FDA has found numerous products promoted as “all-natural” cures or treatments that contain hidden and dangerously high doses of prescription drug ingredients or other active pharmaceutical ingredients.
- “Miracle cure.” Alarms should go off when you see this claim or others like it such as, “new discovery,” “guaranteed results,” or “secret ingredient.” If a real cure for a serious disease were FDA-approved, it would be widely reported through the media and prescribed by licensed health professionals—not plastered on advertisements in social media and messaging apps, or buried in websites, print ads, and TV infomercials.
- Conspiracy theories. Claims like “This is the cure our government or Big Pharma doesn’t want you to know about” are used to distract consumers from the obvious, common-sense questions about the so-called miracle cure.


14.2: Formulate General Research Questions
Most general questions start with an inspiration or an idea about a topic or phenomenon of interest. Some examples of general questions:
- (Health Care) Would a public single payer health care system be more effective than the current private insurance system?
- (Labor) What is the effect of undocumented immigration and outsourcing of jobs on the current unemployment rate?
- (Economy) Is the federal economic stimulus package effective in lessening the impact of the recession?
- (Education) Are colleges too expensive for students today?
It is important to not be so specific in choosing these general questions. On the basis of available or potentially available data, we can decide later what specific research hypotheses will be formulated and tested to address the general question. During the data collection and testing process other ideas may come up and we may choose to redefine the general question. However, we always want to have an overriding purpose for our research.


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allostatic load
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cumulative health toll of stressors, including the body’s biological response and secondary health impacts of stress-related changes to behaviors
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burnout
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state in which chronic stress has led to lack of energy and disillusionment
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climacteric
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decline in and eventual loss of the ability to reproduce
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erectile dysfunction
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persistent inability to achieve or maintain penile erection
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Flynn effect
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phenomenon that younger generations perform better on intelligence tests than previous generations
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gut-brain axis
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bidirectional connections between the nerves of the digestive system and the central nervous system
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menopause
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cessation of the menstrual cycle
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metabolism
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chemical process by which bodies convert food to energy
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osteoporosis
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age-related loss of bone mass
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postformal thought
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intuitive thinking that combines the logic of formal operations with relativism
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presbycusis
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age-related hearing loss
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presbyopia
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age-related loss of visual acuity
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primary aging
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aging that results from inevitable and genetically influenced biological changes
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proprioception
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body’s sense of location and movement in space
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relativism
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recognizing that some problems or dilemmas may have more than one viable solution, depending on a person’s perspective and priorities
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resilience
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person’s ability to successfully adapt to difficult life experiences by making mental, emotional, and behavioral adjustments to cope
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sarcopenia
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age-related loss of muscle mass
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secondary aging
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aging that results from behavioral or environmental factors
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wisdom
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expert knowledge concerning the fundamental pragmatics of life


3.6: Solve Applications with Linear Inequalities
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Learning Objectives
By the end of this section, you will be able to:
- Solve applications with linear inequalities
Note
Before you get started, take this readiness quiz.
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Write as an inequality:
x
is at least 30.
If you missed this problem, review Exercise 2.7.34 . -
Solve \(8−3y<41\).
If you missed this problem, review Exercise 2.7.22 .
Solve Applications with Linear Inequalities
Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for $20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?
The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Example \(\PageIndex{1}\)
Emma got a new job and will have to move. Her monthly income will be $5,265. To qualify to rent an apartment, Emma’s monthly income must be at least three times as much as the rent. What is the highest rent Emma will qualify for?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the highest rent Emma will qualify for}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let r = rent}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate} \text{ into an inequality.}} &{} \\{} &{\text{Emma’s monthly income must be at least}} \\ {\text{First write a sentence that gives the information}} &{\text{three times the rent.}} \\ {\text{to find it.}} &{} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{5265 \geq 3r} \\ {\text{Remember, } a > x\text{ has the same meaning}} &{1755 \geq r} \\ {\text{as }x < a} &{r \leq 1755} \\ {\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{A maximum rent of \$1,755 seems}} &{} \\ {\text{reasonable for an income of \$5,265.}} &{} \\ {\textbf{Step 7. Answer} \text{ the answer in the problem}} &{\text{the question with a}} \\ {\text{complete sentence.}} &{\text{The maximum rent is \$1,755.}} \end{array}\)
Try It \(\PageIndex{2}\)
Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?
- Answer
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There can be no more than 20 boxes.
Try It \(\PageIndex{3}\)
The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2,100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?
- Answer
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A maximum of 14 people can safely ride in the elevator.
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down. To check applications like this, we will round our answer to a number that is easy to compute with and make sure that number makes the inequality true.
Example \(\PageIndex{4}\)
Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the maximum number of tablets Dawn can buy}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let n = the number of tablets.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{$254.12\text{ times the number of tablets is no}} \\ {} &{\text{more than \$4000.}} \\ {\text{Translate into an inequality.}} &{254.12n \leq 4000} \\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{n \leq 15.74} \\ {\text{But n must be a whole number of tablets,}} &{} \\ {\text{so round to 15.}} &{n \leq 15}\\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Rounding down the price to \$250,}} &{} \\ {\text{15 tablets would cost \$3750, while}} &{} \\ {\text{16 tablets would be \$4000. So a}} &{} \\{\text{maximum of 15 tablets at \$254.12}} &{} \\ {\text{seems reasonable.}} &{} \\{\textbf{Step 7. Answer} \text{ the answer in the problem}} &{\text{the question with a}} \\ {\text{complete sentence.}} &{\text{Dawn can buy a maximum of 15 tablets.}} \end{array}\)
Try It \(\PageIndex{5}\)
Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. What is the maximum number of packs she can buy?
- Answer
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seven packs
Try It \(\PageIndex{6}\)
Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?
- Answer
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11 people
Example \(\PageIndex{7}\)
Pete works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the total sales needed for his variable pay}} \\ {} &{\text{option to exceed the fixed amount of \$925}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let s = the total sales.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{$500\text{ plus 12% of total sales is more than \$925.}} \\ {\text{Translate into an inequality. Remember to}} &{500 + 0.12s > 925} \\{\text{convert the percent to a decimal.}} &{} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{0.12s > 425} \\ {} &{s > 3541.\overline{66}} \\ \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Rounding down the price to \$250,}} &{} \\ {\text{15 tablets would cost \$3750, while}} &{} \\ {\text{If we round the total sales up to}} &{} \\{\text{\$4000, we see that}} &{} \\ {\text{500+0.12(4000) = 980, which is more}} &{} \\ {\text{than \$925.}} &{} \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{The total sales must be more than \$3541.67}} \end{array}\)
Try It \(\PageIndex{8}\)
Tiffany just graduated from college and her new job will pay her $20000 per year plus 2% of all sales. She wants to earn at least $100000 per year. For what total sales will she be able to achieve her goal?
- Answer
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at least $4000000
Try It \(\PageIndex{9}\)
Christian has been offered a new job that pays $24000 a year plus 3% of sales. For what total sales would this new job pay more than his current job which pays $60000?
- Answer
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at least $1200000
Example \(\PageIndex{10}\)
Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can they travel and still keep within their $200 budget?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of miles Sergio and Lizeth can travel}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let m = the number of miles.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{\text{\$75 plus 0.25 times the number of miles is}} \\{\text{gives the information to find it.}} &{\text{ less than or equal to \$200.}} \\ {\text{Translate into an inequality. }} &{75 + 25m \leq 200} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{0.25m \leq 125} \\ {} &{m \leq 500 \text{ miles}} \\ \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Yes, 75 + 0.25(500) = 200.}} & {}\\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Sergio and Lizeth can travel 500 miles}} \\ {} &{\text{and still stay on budget.}} \end{array}\)
Try It \(\PageIndex{11}\)
Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. How many text messages can she use and keep her monthly phone bill no more than $50?
- Answer
-
no more than 106 text messages
Try It \(\PageIndex{12}\)
Rameen’s heating bill is $5.42 per month plus $1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of $87.50?
- Answer
-
no more than 76 therms
A common goal of most businesses is to make a profit. Profit is the money that remains when the expenses have been subtracted from the money earned. In the next example, we will find the number of jobs a small businessman needs to do every month in order to make a certain amount of profit .
Example \(\PageIndex{13}\)
Elliot has a landscape maintenance business. His monthly expenses are $1,100. If he charges $60 per job, how many jobs must he do to earn a profit of at least $4,000 a month?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of jobs Elliot needs}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {\text{Choose a variable to represent it}} &{\text{Let j = the number of jobs.}} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{\text{\$60 times the number of jobs minus \$1,100 is at least \$4,000.}} \\{\text{gives the information to find it.}} &{\text{ less than or equal to \$200.}} \\ {\text{Translate into an inequality. }} &{60j - 1100 \geq 4000} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{60j \geq 5100} \\ {} &{j \geq 85\text{ jobs}} \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{If Elliot did 90 jobs, his profit would be}} & {}\\ {\text{60(90)−1,100,or \$4,300. This is}} &{} \\ {\text{more than \$4,000.}} &{} \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Elliot must work at least 85 jobs.}} \end{array}\)
Try It \(\PageIndex{14}\)
Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are $2272. How many hours must he work in order to earn a profit of at least $800 per month?
- Answer
-
at least 96 hours
Try It \(\PageIndex{15}\)
Felicity has a calligraphy business. She charges $2.50 per wedding invitation. Her monthly expenses are $650. How many invitations must she write to earn a profit of at least $2800 per month?
- Answer
-
at least 1380 invitations
Sometimes life gets complicated! There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.
Example \(\PageIndex{16}\)
Brenda’s best friend is having a destination wedding and the event will last 3 days. Brenda has $500 in savings and can earn $15 an hour babysitting. She expects to pay $350 airfare, $375 for food and entertainment and $60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?
Solution
\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of hours Brenda must babysit}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {\text{Choose a variable to represent that quantity.}} &{\text{Let h = the number of hours.}} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{} \\ {} &{\text{The expenses must be less than or equal to}} \\ {} &{\text{the income. The cost of airfare plus the}} \\ {} &{\text{cost of food and entertainment and the}} \\ {} &{\text{hotel bill must be less than or equal to the savings}} \\ {} &{\text{plus the amount earned babysitting.}} \\ {\text{Translate into an inequality. }} &{\$350 + \$375 + \$60(3) \leq \$500 + \$15h} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{905 \leq 500 + 15h} \\{} &{405 \leq 15h} \\ {} &{27 \leq h} \\ {} &{h \geq 27} \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{We substitute 27 into the inequality.}} & {}\\{905 \leq 500 + 15h} &{} \\ {905 \leq 500 + 15(27)} &{} \\ {905 \leq 905} &{} \\ \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Brenda must babysit at least 27 hours.}} \end{array}\)
Try It \(\PageIndex{17}\)
Malik is planning a 6-day summer vacation trip. He has $840 in savings, and he earns $45 per hour for tutoring. The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?
- Answer
-
at least 23 hours
Try It \(\PageIndex{18}\)
Josue wants to go on a 10-day road trip next spring. It will cost him $180 for gas, $450 for food, and $49 per night for a motel. He has $520 in savings and can earn $30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?
- Answer
-
at least 20 driveways
Key Concepts
-
Solving inequalities
- Read the problem.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate. Write a sentence that gives the information to find it. Translate into an inequality.
- Solve the inequality.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.


6.19: Reading- The Importance of Marketing Information and Research
Fresh Customer Insights
Effective marketing starts with a strong knowledge of your customers: the kind of knowledge that gives you unique insights into what they want and how to satisfy them better than the competition. The most reliable source of fresh customer insights is good marketing information . Useful marketing information may come from a variety of sources both inside and outside your organization. Marketing information is generated by a variety of different activities, including marketing research.
Marketing research is a systematic process for identifying marketing opportunities and solving marketing problems, using customer insights that come out of collecting and analyzing marketing information. The mechanics of marketing research must be controlled so that marketers uncover the relevant facts to answer the problem at hand. Control over this fact-finding process is the responsibility of the marketing research director, who must correctly design the research and carefully supervise its execution, to ensure it yields the customer insights the organization needs.
A marketing information system is a combination of people, technologies, and processes for managing marketing information, overseeing market research activities, and using customer insights to guide marketing decisions and broader management and strategy decisions.
Knowledge Is Power Against the Competition
The business environment is increasingly competitive. With something as simple as a Google search, customers have unprecedented opportunities to explore alternatives to what any single company offers. Likewise, companies have ample opportunity to identify, track, and lure customers away from their less-vigilant competitors. A regular infusion of fresh customer insights can make all the difference between keeping customers and losing them. Marketing information and research are essential tools for marketers and the management team as they align strategy with customer wants and needs.
Consider the following examples:
- Before introducing OnStar, the first-ever embedded wireless service in cars, GM used marketing research to understand what types of applications would make consumers most interested in subscribing to the service and how much they would pay for it. Of all the benefits OnStar could offer, the research helped GM prioritize how the initial service would provide value, focusing on driver assistance and security. Research also helped determine OnStar pricing to help the company build a large subscriber base quickly. [1]
- Enterprise systems provider PeopleSoft recruited a diverse set of universities as early-adopter “Beta” partners to provide input as it designed a new student information system for higher education. This marketing research helped PeopleSoft create a versatile system that could support the needs of a variety of colleges and universities, ultimately leading to strong receptivity and market share when the new system became widely available. [2]
What Should Marketers Investigate Using Marketing Information?
An easy—and truthful—answer to this question is “everything.” There is no aspect of marketing to which information and research do not apply. Every marketing concept and every element involved in the marketing management process can be subjected to a great deal of careful marketing research and inquiry. Some important questions include:
- Who is the customer?
- What problems is the customer trying to solve with a given purchase?
- What does s/he desire in the way of satisfaction?
- How does the customer get information about available choices?
- Where does s/he choose to purchase?
- Why does s/he buy, or not buy?
- When does s/he purchase?
- How does s/he go about seeking satisfaction in the market?
Seeking answers to these questions yields insights into the customer’s needs, perceptions, and behaviors. Another area in which research is critical is profitability. Organizations need to forecast sales and related costs in order to understand how their operations will be profitable. They also need to plan competitive marketing programs that will produce the desired level of sales at an appropriate cost. The analysis of past sales and interpretation of cost information are important in evaluating performance and providing useful facts for future planning. All these activities rely on marketing information and a rigorous marketing research process to produce insights managers can trust and act on.
When to Use Marketing Information and Research
Many marketing decisions are made without consulting marketing information or the use of formal marketing research. For example, a decision maker may feel she already knows enough to make a good decision. The time required to investigate a question or conduct formal marketing research may not be available. In other cases, the cost of obtaining the data is prohibitive, or the desired data cannot be obtained in reliable form. In a few instances, there may be no choice among alternatives and therefore no decision to make because there is little value in spending time and money to study a problem if there is only one possible solution. But in most business situations, marketers and managers must choose among two or more courses of action. This is where fact-finding, marketing information, and research enter to help make the choice.
Marketing information and research address the need for quicker, yet more accurate, decision making by the marketer. These tools put marketers close to their customers to help them understand who they customers are, what they want, and what competitors are doing. When different stakeholders have very different views about a particular marketing-related decision, objective information and research can inform everyone about the issues in question and help the organization come to agreement about the path forward. Good research should help align marketing with the other areas of the business.
Marketers should always be tapping into regular sources of marketing information about their organization and industry in order to monitor what’s happening generally. For example, at any given time marketers should understand how they are doing relative to sales goals and monitor developments in their industry or competitive set.
Beyond this general level of “tuning in,” additional market research projects may also be justified. As a rule, if the research results can save the company more time, money, and/or risk than it costs to conduct the research, it is wise to proceed. If the cost of conducting the research is more than it will contribute to improving a decision, the research should not be carried out. In practice, applying this cost-test principle can be somewhat complex, but it provides useful guidance about when marketing research is worthwhile. Ultimately, successful marketing executives make decisions on the basis of a blend of facts and intuition.
Fact: Top Performers Research Customer Preferences
In 2010, the management consultancy McKinsey published research about the difference between organizations that produced top-performing products and those that produced under-performing products. The use of marketing research was a striking differentiator:
More than 80 percent of the top performers said they periodically tested and validated customer preferences during the development process, compared with just 43 percent of bottom performers. They were also twice as likely as the laggards to research what, exactly, customers wanted. [3]
The study also identified other differences between top and bottom performers, but an underlying theme was the emphasis successful projects and companies placed on understanding their customers and adjusting course when necessary to better address customers’ needs. This research provides more than anecdotal evidence that marketing research and well-applied marketing information can make a substantial contribution to an organization’s success.
- Vincent P. Barabba, Surviving Transformation: Lessons from GM's Surprising Turnaround , pp 46–50, https://books.google.com/books?id=VvbDYad7cLoC&pg ↵
- Proquest, "First We Built, Now We Buy: A Sociological Case Study for Enterprise Systems in Higher Education," pp 292–203, books.google.com/books?id=rgIAaigKQBIC&pg↵
- http://www.mckinsey.com/insights/operations/the_path_to_successful_new_products ↵
Contributors and Attributions
- Revision and Adaptation. Authored by : Lumen Learning. License : CC BY: Attribution
- Chapter 3: Marketing Research: An Aid to Decision Making, from Introducing Marketing. Authored by : John Burnett. Provided by : Global Text. Located at : http://solr.bccampus.ca:8001/bcc/file/ddbe3343-9796-4801-a0cb-7af7b02e3191/1/Core%20Concepts%20of%20Marketing.pdf . License : CC BY: Attribution


1.5: Unit conversions
Conversion of values in one unit to the same value in another unit, as illustrated in Figure \(\PageIndex{1}\) is often needed in scientific calculations. The unit conversion includes the following.
- Conversion of the same type of measurement in the same system of measurement, e.g., conversion of a measured value of the length in meters to kilometers in SI;
- conversion of the same type of measurement in different systems of measurements, e.g., conversion of a measured value of the length in kilometers from SI to miles in the English system; and
- conversion of one type of measurement to another type of measurement, e.g., conversion of a measured value of the mass in g to volume in mL of a substance using the density of the substance.
Conversion factors
The conversion factors are derived from equality between the given unit and the desired unit. For example, 1 cm = 10 -2 m is equality between centimeter and meter. The conversion factors are derived from the equality by the following steps.
Both sides of the equality are divided by one side to get one conversion factor. For example, \(1 = \frac{10^{-2} ~m}{1 ~cm}\), which is a conversion factor for cm to m.
Then both sides are divided by the other side of the equality to get the second conversion factor. For example, \(1 = \frac{1 ~cm}{10^{-2} ~m}\), which is a conversion factor for m to cm.
Since both the conversion factors are equal to one, multiplying a value with a conversion factor changes the number and the unit, such that the new number and the new unit together represent the same value. The unit of the given number should be opposite to the same unit in the conversion factor, i.e., numerator versus denominator or denominator versus numerator, to cancel them out and leave the desired unit in the answer. For example, 1.83m is converted to cm unit using the conversion factor \(\frac{1 ~cm}{10^{-2} ~m}\) as:
\begin{equation}
1.83 \cancel{\mathrm{~m}} \times \frac{1 \mathrm{~cm}}{10^{-2} \cancel{\mathrm{~m}}}=183 \mathrm{~cm}\nonumber
\end{equation}
Keep track of the units that cancel out and the unit left in the answer. If all the units cancel out, leaving only the desired unit means the chosen conversion factor is correct. An incorrect conversion factor leads to an unwanted unit in the answer, e.g., in the above calculation if incorrect conversion factor is chose, it will lead to:\[1.83 ~m \times \frac{10^{-2} ~m}{1 \mathrm{~cm}}=0.0183 \frac{\mathrm{~m}^{2}}{\mathrm{~cm}}\nonumber\], where no unit is canceled and the answer has units that are not the desired. It means an incorrect conversion factor was employed.
Converting the same type of measurement in the same system
The prefixes in the SI, listed in section 1.3 Table 2, establish equality between the base and the prefixed units. For example, centi (c) means 10 -2 . Therefore \[1 \mathrm{~cm}=10^{-2} \mathrm{~m}\nonumber\]1cm = 10 -2 m is an equality that gives two conversion factors:
\[\frac{1~cm}{10^{-2}~m}\quad\text{ and } \quad\frac{10^{-2}~m}{1~cm}\nonumber\]
Some of the common qualities in SI are listed in Table 1 and the English system in Table 2.
The prefixes are exact numbers. The equalities within the same system of measurement are exact numbers. Therefore the equalities and the conversion factors derived from them are exact numbers. Significant figures in the answers involving exact and inexact numbers are dictated by inexact numbers only.
Convert 325 cm to m units?
Solution
\[325 \cancel{~cm}\times\frac{10^{-2}~m}{1 \cancel{~cm}}=3.25 ~m\nonumber\]
Note that in this example, the answer has the significant figures the same as in the given number because the conversion factor within the same system of measurement is numbers.
| Length | Mass | Volume |
|---|---|---|
| 1 km = 1,000 m | 1 kg = 1,000 g | 1 L = 10 dL |
| 1 m = 100 cm | 1 g = 1,000 mg | 1 L = 1,000 mL |
| 1 m = 1,000 mm | 1 mg = 1,000 μg | 1 L = 1,000,000 μL |
| 1 cm = 10 mm | 10-1 | 1 dL = 100 mL |
| 1 mL = 1 cm 3 = 1 cc |
| Length | Mass | Volume |
|---|---|---|
| ft = 12 in. | 1 lb = 16 oz | 1 qt = 4 cups |
| 1 yd = 3 ft | 1 ton = 2,000 lb | 1 qt = 2 pints |
| 1 mi = 5,280 ft | 1 qt = 32 fl oz | |
| 1 gal = 4 qt |
Conversion of the same type of measurement in different systems
Table 3 lists some common qualities between SI and the English systems.
These equalities between different systems usually have one side in the equality, which is the number 1, as exact, while the other side is considered an inexact number.
For example, 1 kg = 2.205 pounds (lb) has an exact number (1 kg) on the left side but an inexact number (2.205 lb) with 4 SFs on the right. Remember that only the inexact numbers dictate the significant figures in the answer.
There are some exceptions to the above general rule. Some equalities between units in different systems are defined and considered exact. They are stated to be exact in the reference tables.
For example, 1 inch = 2.54 cm is defined, which means both sides are the exact number.
| Length | Mass | Volume |
|---|---|---|
| 2.54 cm = 1 in. defined and exact | 1 kg = 2.20 lb | 946 mL = 1 qt |
| 1 m = 39.4 in. | 454 g = 1 lb | 1 L = 1.06 qt |
| 1 km = 0.621 mi | 28.4 g = 1 oz | 29.6 mL = 1 fl oz |
Conversion of one type of measurement to another
Sometimes, equality between two different units is known under specific conditions. For example, density (d) which is mass (m) per unit volume (V), is a relationship between mass and volume of a given substance, i.e., \(d= \frac{m}{v}\). The density is a conversion factor used to convert the volume to the mass of a substance. Reciprocal density, i.e., \(\frac{1}{d}=\frac{v}{m}\), is the second conversion factor that converts the mass to the volume of the substance.
The density of ethanol at 20 o C is 0.7893 g/mL; what is the mass of 10.0 mL of ethanol?
Solution
Multiply the given volume with the conversion factor that has volume in the denumerator, i.e., \(frac{m}{v} to get the mass desired.
\begin{equation}
10.0 \cancel{\mathrm{~m L}} \times \frac{0.7893 \mathrm{~g}}{1 \cancel{\mathrm{~mL}}}=7.89 \mathrm{~g}\nonumber
\end{equation}
Note that 1mL is exact, and 10.0 and 0.7893 are inexact numbers with 3SF and 4 SF, respectively. The answer has 3 SFs.
The density of gold is 19.30 g/mL; what is the volume of 10.123 g of gold?
Solution
Multiply the given mass with the conversion factor that has a mass in the denumerator, i.e., \(frac{v}{m}\) to get the mass desired.
\begin{equation}
10.123 \cancel{\mathrm{~g}} \times \frac{1 \mathrm{~mL}}{19.30 \cancel{\mathrm{~g}}}=0.5245 \mathrm{~mL}
\end{equation}
Conversion factors derived from chemical equations
Chemical equations show relationships or equalities between reactants and products measured in moles . Mole is a SI unit for the amount of substance. Mole equals 6.02 x 10 23 particles (atoms or molecules) of the substance. Like dozen means 12 of something, mole means 6.02 x 10 23 atoms of element or molecules/formula units for compounds in a chemical equation.
For calculations in chemistry, the number of moles of a substance is considered equal to its coefficient in a balanced chemical equation.

The use of these conversion factors in the calculation is explained in the following example.
If 5 moles of oxygen (O 2 ) is consumed, how many moles of water are produced by the chemical equation mentioned above? Solution
Given: 5 mol O 2 , Desired: ? mol H 2 O
Multiply the given quantity with the conversion factor that has the given unit in denumerator and the desired unit in the numerator:
\begin{equation}
5.0 \cancel{\text { mole } \mathrm{O}_{2}} \times \frac{2 \text { mole } \mathrm{H}_{2} \mathrm{O}}{1 \cancel{\text { mole } \mathrm{O}_{2}}}=10 \text { mole } \mathrm{H}_{2} \mathrm{O}\nonumber
\end{equation}
Unit conversion involving more than one conversion factor
Often, there is no direct conversion factor between the given unit and the desired unit. In this situation, convert the given unit to another unit that can, later on, be linked with the desired unit, as explained in the following examples.
How many µg are in 10.0 mg?
Solution
Problems like this can be solved in two steps: i) by converting the given unit to the base unit and then, ii) converting the base unit to the desired unit:
\begin{equation}
10.0 \cancel{~m g} \times \frac{10^{-3} ~g}{1 \cancel{~m g}}=1.00 \times 10^{-2}~ g\nonumber
\end{equation}
\begin{equation}
1.00 \times 10^{-2} \cancel{~g}\times \frac{1 ~µ g}{10^{-6} \cancel{~g}}=1.00 \times 10^{4} ~µ g\nonumber
\end{equation}
Note that the first conversion factor converts mg to g, and then the second conversion factor converts g to the desired unit µg. The same calculation can be done using two conversion factors in a rwo:
\begin{equation}
10.0 \cancel{\mathrm{~mg}} \times \frac{10^{-3} \cancel{\mathrm{~g}}}{10 \cancel{\mathrm{mg}}} \times \frac{1 ~µ g}{10^{-6} \cancel{\mathrm{~g}}}=1.00 \times 10^{4} ~µ g\nonumber
\end{equation}
What is 100. km/h speed in m/s, where h represents hours?
Solution
This problem asks to convert two units, i.e., km to m and h to s. First convert one unit and then follow on to convert the second unit as:
\begin{equation}
\frac{100 . \cancel{\mathrm{~km}}}{\cancel{\mathrm{h}}} \times \frac{10^{3} \mathrm{~m}}{1 \cancel{\mathrm{~km}}} \times \frac{1 \cancel{\mathrm{~h}}}{60 \cancel{\mathrm{~min}}} \times \frac{1 \cancel{\mathrm{~min}}}{60 \mathrm{~s}}=27.8 \mathrm{~m} / \mathrm{s}\nonumber
\end{equation}
Note that the first conversion factor converts km to m, and then two conversion factors are needed to convert h to s via min.
A prescription says a dosage of 0.225 mg of Synthroid to be taken once a day. If tablets in stock contain 75 µg of Synthroid, how many tablets are needed per day?
Solution
The given mass is in mg, while the equality "1 tablet = 75 µg" takes mass in µg. First convert mg to base unit, i.e., g, then from g to needed unit, i.e., µg, and finally take appropriate conversion factor from the two given by the equality to convert µg to tablet, i.e., three conversion factors in a row:
\begin{equation}
0.225 \cancel{\mathrm{~mg}} \times \frac{10^{-3} \cancel{\mathrm{~g}}}{1 \cancel{\mathrm{mg}}} \times \frac{1 \cancel{\mathrm{~µg}}}{10^{-6} \cancel{\mathrm{~g}}} \times \frac{1 \text { tablet }}{75 \cancel{\mathrm{µg}}}=3.0 \text { tablet }\nonumber
\end{equation}
A healthy person has 16% body fat by mass. Calculate the mass of fat in kg of a person who weighs 180. lb?
 Take one conversion factor from each equality such that the units cancel out leaving the desired unit in the answer:
\begin{equation}
180. \cancel{\text{lb body mass }} \times \frac{16 \cancel{\mathrm{~lb} \text { body fat }}}{100 \cancel{\mathrm{~lb} \text { body mass }}} \times \frac{1 \mathrm{~kg} \text { body fat }}{2.20 \cancel{\mathrm{~lb}\text { body fat }}}=16 \mathrm{~kg} \text { body fat }\nonumber
\end{equation}
Note that there are three inexact numbers in the calculation, i.e., 180, 16, and 2.20, and the answer as two significant figures in agreement with the smallest significant figure among the inexact numbers.


14.E: Fluid Mechanics (Exercises)
Conceptual Questions
14.1 Fluids, Density, and Pressure
- Which of the following substances are fluids at room temperature and atmospheric pressure: air, mercury, water, glass?
- Why are gases easier to compress than liquids and solids?
- Explain how the density of air varies with altitude.
- The image shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.
- How is pressure related to the sharpness of a knife and its ability to cut?
- Why is a force exerted by a static fluid on a surface always perpendicular to the surface?
- Imagine that in a remote location near the North Pole, a chunk of ice floats in a lake. Next to the lake, a glacier with the same volume as the floating ice sits on land. If both chunks of ice should melt due to rising global temperatures, and the melted ice all goes into the lake, which one would cause the level of the lake to rise the most? Explain.
- In ballet, dancing en pointe (on the tips of the toes) is much harder on the toes than normal dancing or walking. Explain why, in terms of pressure.
- Atmospheric pressure exerts a large force (equal to the weight of the atmosphere above your body—about 10 tons) on the top of your body when you are lying on the beach sunbathing. Why are you able to get up?
- Why does atmospheric pressure decrease more rapidly than linearly with altitude?
- The image shows how sandbags placed around a leak outside a river levee can effectively stop the flow of water under the levee. Explain how the small amount of water inside the column of sandbags is able to balance the much larger body of water behind the levee.
- Is there a net force on a dam due to atmospheric pressure? Explain your answer.
- Does atmospheric pressure add to the gas pressure in a rigid tank? In a toy balloon? When, in general, does atmospheric pressure not affect the total pressure in a fluid?
- You can break a strong wine bottle by pounding a cork into it with your fist, but the cork must press directly against the liquid filling the bottle—there can be no air between the cork and liquid. Explain why the bottle breaks only if there is no air between the cork and liquid.
14.2 Measuring Pressure
- Explain why the fluid reaches equal levels on either side of a manometer if both sides are open to the atmosphere, even if the tubes are of different diameters.
14.3 Pascal's Principle and Hydraulics
- Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
14.4 Archimedes’ Principle and Buoyancy
- More force is required to pull the plug in a full bathtub than when it is empty. Does this contradict Archimedes’ principle? Explain your answer.
- Do fluids exert buoyant forces in a “weightless” environment, such as in the space shuttle? Explain your answer.
- Will the same ship float higher in salt water than in freshwater? Explain your answer.
- Marbles dropped into a partially filled bathtub sink to the bottom. Part of their weight is supported by buoyant force, yet the downward force on the bottom of the tub increases by exactly the weight of the marbles. Explain why.
14.5 Fluid Dynamics
- Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. ( Hint : Consider the relationship between fluid velocity and the cross-sectional area through which the fluid flows.)
14.6 Bernoulli’s Equation
- You can squirt water from a garden hose a considerably greater distance by partially covering the opening with your thumb. Explain how this works.
- Water is shot nearly vertically upward in a decorative fountain and the stream is observed to broaden as it rises. Conversely, a stream of water falling straight down from a faucet narrows. Explain why.
- Look back to Figure 14.29. Answer the following two questions. Why is p o less than atmospheric? Why is p o greater than p i ?
- A tube with a narrow segment designed to enhance entrainment is called a Venturi, such as shown below. Venturis are very commonly used in carburetors and aspirators. How does this structure bolster entrainment?
- Some chimney pipes have a T-shape, with a crosspiece on top that helps draw up gases whenever there is even a slight breeze. Explain how this works in terms of Bernoulli’s principle.
- Is there a limit to the height to which an entrainment device can raise a fluid? Explain your answer.
- Why is it preferable for airplanes to take off into the wind rather than with the wind?
- Roofs are sometimes pushed off vertically during a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.
- It is dangerous to stand close to railroad tracks when a rapidly moving commuter train passes. Explain why atmospheric pressure would push you toward the moving train.
- Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.
- David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.
- Based on Bernoulli’s equation, what are three forms of energy in a fluid? (Note that these forms are conservative, unlike heat transfer and other dissipative forms not included in Bernoulli’s equation.)
- The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy
- Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.
14.7 Viscosity and Turbulence
- Explain why the viscosity of a liquid decreases with temperature, that is, how might an increase in temperature reduce the effects of cohesive forces in a liquid? Also explain why the viscosity of a gas increases with temperature, that is, how does increased gas temperature create more collisions between atoms and molecules?
- When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it is generally better to stay near the middle. Explain why.
- Plumbing usually includes air-filled tubes near water faucets (see the following figure). Explain why they are needed and how they work.
- Doppler ultrasound can be used to measure the speed of blood in the body. If there is a partial constriction of an artery, where would you expect blood speed to be greatest: at or after the constriction? What are the two distinct causes of higher resistance in the constriction?
- Sink drains often have a device such as that shown below to help speed the flow of water. How does this work?
Problems
14.1 Fluids, Density, and Pressure
- Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?
- Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb.). What is the volume in liters of this much mercury?
- What is the mass of a deep breath of air having a volume of 2.00 L? Discuss the effect taking such a breath has on your body’s volume and density.
- A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces 89.0 cm 3 of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)
- Suppose you have a coffee mug with a circular cross-section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.
- A rectangular gasoline tank can hold 50.0 kg of gasoline when full. (a) What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume for a passenger car.
- A trash compactor can compress its contents to 0.350 times their original volume. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?
- A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?
- What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.
- The tip of a nail exerts tremendous pressure when hit by a hammer because it exerts a large force over a small area. What force must be exerted on a nail with a circular tip of 1.00-mm diameter to create a pressure of 3.00 x 10 9 N/m 2 ? (This high pressure is possible because the hammer striking the nail is brought to rest in such a short distance.)
- A glass tube contains mercury. What would be the height of the column of mercury which would create pressure equal to 1.00 atm?
- The greatest ocean depths on Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11.0 km and assuming the density of seawater is constant all the way down.
- Verify that the SI unit of h\(\rho\)g is N/m 2 .
- What pressure is exerted on the bottom of a gas tank that is 0.500-m wide and 0.900-m long and can hold 50.0 kg of gasoline when full?
- A dam is used to hold back a river. The dam has a height H = 12 m and a width W = 10 m. Assume that the density of the water is \(\rho\) = 1000 kg/m 3 . (a) Determine the net force on the dam. (b) Why does the thickness of the dam increase with depth?
14.2 Measuring Pressure
- Find the gauge and absolute pressures in the balloon and peanut jar shown in Figure 14.12, assuming the manometer connected to the balloon uses water and the manometer connected to the jar contains mercury. Express in units of centimeters of water for the balloon and millimeters of mercury for the jar, taking h = 0.0500m for each.
- How tall must a water-filled manometer be to measure blood pressure as high as 300 mm Hg?
- Assuming bicycle tires are perfectly flexible and support the weight of bicycle and rider by pressure alone, calculate the total area of the tires in contact with the ground if a bicycle and rider have a total mass of 80.0 kg, and the gauge pressure in the tires is 3.50 x 10 5 Pa .
14.3 Pascal's Principle and Hydraulics
- How much pressure is transmitted in the hydraulic system considered in Example 14.3? Express your answer in atmospheres.
- What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on a second cylinder? The master cylinder has a 2.00-cm diameter and the second cylinder has a 24.0-cm diameter.
- A host pours the remnants of several bottles of wine into a jug after a party. The host then inserts a cork with a 2.00-cm diameter into the bottle, placing it in direct contact with the wine. The host is amazed when the host pounds the cork into place and the bottom of the jug (with a 14.0-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N force.
- A certain hydraulic system is designed to exert a force 100 times as large as the one put into it. (a) What must be the ratio of the area of the cylinder that is being controlled to the area of the master cylinder? (b) What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no losses due to friction.
- Verify that work input equals work output for a hydraulic system assuming no losses due to friction. Do this by showing that the distance the output force moves is reduced by the same factor that the output force is increased. Assume the volume of the fluid is constant. What effect would friction within the fluid and between components in the system have on the output force? How would this depend on whether or not the fluid is moving?
14.4 Archimedes’ Principle and Buoyancy
- What fraction of ice is submerged when it floats in freshwater, given the density of water at 0 °C is very close to 1000 kg/m 3 ?
- If a person’s body has a density of 995 kg/m 3 , what fraction of the body will be submerged when floating gently in (a) freshwater? (b) In salt water with a density of 10 27 kg/m 3 ?
- A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water. (a) What mass of water is displaced? (b) What is the volume of the rock? (c) What is its average density? Is this consistent with the value for granite?
- Archimedes’ principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, using its density as given in Table 14.1? (c) Calculate the fluid’s density and identify it.
- (a) Calculate the buoyant force on a 2.00-L helium balloon. (b) Given the mass of the rubber in the balloon is 1.50 g, what is the net vertical force on the balloon if it is let go? Neglect the volume of the rubber.
- (a) What is the density of a woman who floats in fresh water with 4.00% of her volume above the surface? (This could be measured by placing her in a tank with marks on the side to measure how much water she displaces when floating and when held under water.) (b) What percent of her volume is above the surface when she floats in seawater?
- A man has a mass of 80 kg and a density of 955 kg/m 3 (excluding the air in his lungs). (a) Calculate his volume. (b) Find the buoyant force air exerts on him. (c) What is the ratio of the buoyant force to his weight?
- A simple compass can be made by placing a small bar magnet on a cork floating in water. (a) What fraction of a plain cork will be submerged when floating in water? (b) If the cork has a mass of 10.0 g and a 20.0-g magnet is placed on it, what fraction of the cork will be submerged? (c) Will the bar magnet and cork float in ethyl alcohol?
- What percentage of an iron anchor’s weight will be supported by buoyant force when submerged in salt water?
- Referring to Figure 14.20, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is F 2 − F 1 and that the ends of the cylinder have equal areas A. Note that the volume of the cylinder (and that of the fluid it displaces) equals (h 2 − h 1 )A.
- A 75.0-kg man floats in freshwater with 3.00% of his volume above water when his lungs are empty, and 5.00% of his volume above water when his lungs are full. (a) Calculate the volume of air he inhales—called his lung capacity—in liters. (b) Does this lung volume seem reasonable?
14.5 Fluid Dynamics
- What is the average flow rate in cm 3 /s of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?
- The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to cm 3 /s. (b) What is this rate in m 3 /s?
- The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions. On average, the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 20-m wide and averages 20-m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?
- (a) Estimate the time it would take to fill a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take if you could divert a moderate size river, flowing at 5000 m 3 /s into the pool?
- (a) What is the fluid speed in a fire hose with a 9.00-cm diameter carrying 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose?
- Water is moving at a velocity of 2.00 m/s through a hose with an internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose’s nozzle is 15.0 m/s. What is the nozzle’s inside diameter?
- Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)
- Water emerges straight down from a faucet with a 1.80-cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in cm 3 /s? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.
14.6 Bernoulli’s Equation
- Verify that pressure has units of energy per unit volume.
- Suppose you have a wind speed gauge like the pitot tube shown in Figure 14.32. By what factor must wind speed increase to double the value of h in the manometer? Is this independent of the moving fluid and the fluid in the manometer?
- If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?
- Every few years, winds in Boulder, Colorado, attain sustained speeds of 45.0 m/s (about 100 mph) when the jet stream descends during early spring. Approximately what is the force due to the Bernoulli equation on a roof having an area of 220m 2 ? Typical air density in Boulder is 1.14 kg/m 3 , and the corresponding atmospheric pressure is 8.89 x 10 4 N/m 2 . (Bernoulli’s principle as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)
- (a) What is the pressure drop due to the Bernoulli Effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum height above the nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)
- (a) Using Bernoulli’s equation, show that the measured fluid speed v for a pitot tube, like the one in Figure 14.32(b), is given by v = \(\left(\dfrac{2 \rho' gh}{\rho} \right)^{1/2}\), where h is the height of the manometer fluid, \(\rho′\) is the density of the manometer fluid, \(\rho\) is the density of the moving fluid, and g is the acceleration due to gravity. (Note that v is indeed proportional to the square root of h, as stated in the text.) (b) Calculate v for moving air if a mercury manometer’s h is 0.200 m.
- A container of water has a cross-sectional area of A = 0.1 m 2 . A piston sits on top of the water (see the following figure). There is a spout located 0.15 m from the bottom of the tank, open to the atmosphere, and a stream of water exits the spout. The cross sectional area of the spout is A s = 7.0 x 10 −4 m 2 . (a) What is the velocity of the water as it leaves the spout? (b) If the opening of the spout is located 1.5 m above the ground, how far from the spout does the water hit the floor? Ignore all friction and dissipative forces.
- A fluid of a constant density flows through a reduction in a pipe. Find an equation for the change in pressure, in terms of v 1 , A 1 , A 2 , and the density.
14.7 Viscosity and Turbulence
- (a) Calculate the retarding force due to the viscosity of the air layer between a cart and a level air track given the following information: air temperature is 20 °C, the cart is moving at 0.400 m/s, its surface area is 2.50 x 10 −2 m 2 , and the thickness of the air layer is 6.00 x 10 −5 m. (b) What is the ratio of this force to the weight of the 0.300-kg cart?
- The arterioles (small arteries) leading to an organ constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to 1.00% of its original value. By what factor do the radii of the arterioles constrict?
- A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, F s = 6\(\pi \eta\)v. Show that the terminal speed is given by v = \(\frac{2R^{2} g}{9 \eta} (\rho_{s} − \rho_{1})\), where R is the radius of the sphere, \(\rho_{s}\) is its density, and \(\rho_{1}\) is the density of the fluid, and \(\eta\) the coefficient of viscosity.
- Using the equation of the previous problem, find the viscosity of motor oil in which a steel ball of radius 0.8 mm falls with a terminal speed of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL, respectively.
- A skydiver will reach a terminal velocity when the air drag equals his or her weight. For a skydiver with a large body, turbulence is a factor at high speeds. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be F D = \(\frac{1}{2} \rho Av^{2}\), and setting this equal to the skydiver’s weight, find the terminal speed for a person falling “spread eagle.”
- (a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant. (b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?
- When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?
- An oil gusher shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. Neglecting air resistance but not the resistance of the pipe, and assuming laminar flow, calculate the pressure at the entrance of the 50.0-m-long vertical pipe. Take the density of the oil to be 900 kg/m 3 and its viscosity to be 1.00(N/m 2 ) • s (or 1.00 Pa • s). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.
- Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is 8.00 x 10 6 N/m 2 . (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? (c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.
- Verify that the flow of oil is laminar for an oil gusher that shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be 900 kg/m 3 and its viscosity to be 1.00(N/m 2 ) • s (or 1.00 Pa • s).
- Calculate the Reynolds numbers for the flow of water through (a) a nozzle with a radius of 0.250 cm and (b) a garden hose with a radius of 0.900 cm, when the nozzle is attached to the hose. The flow rate through hose and nozzle is 0.500 L/s. Can the flow in either possibly be laminar?
- A fire hose has an inside diameter of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of 1.62 x 10 6 N/m 2 . The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Calculate the Reynolds numbers for flow in the fire hose and nozzle to show that the flow in each must be turbulent.
- At what flow rate might turbulence begin to develop in a water main with a 0.200-m diameter? Assume a 20 °C temperature.
Additional Problems
- Before digital storage devices, such as the memory in your cell phone, music was stored on vinyl disks with grooves with varying depths cut into the disk. A phonograph used a needle, which moved over the grooves, measuring the depth of the grooves. The pressure exerted by a phonograph needle on a record is surprisingly large. If the equivalent of 1.00 g is supported by a needle, the tip of which is a circle with a 0.200-mm radius, what pressure is exerted on the record in Pa?
- Water towers store water above the level of consumers for times of heavy use, eliminating the need for high-speed pumps. How high above a user must the water level be to create a gauge pressure of 3.00 x 10 5 N/m 2 ?
- The aqueous humor in a person’s eye is exerting a force of 0.300 N on the 1.10-cm 2 area of the cornea. What pressure is this in mm Hg?
- (a) Convert normal blood pressure readings of 120 over 80 mm Hg to newtons per meter squared using the relationship for pressure due to the weight of a fluid (p = h\(\rho\)g) rather than a conversion factor. (b) Explain why the blood pressure of an infant would likely be smaller than that of an adult. Specifically, consider the smaller height to which blood must be pumped.
- Pressure cookers have been around for more than 300 years, although their use has greatly declined in recent years (early models had a nasty habit of exploding). How much force must the latches holding the lid onto a pressure cooker be able to withstand if the circular lid is 25.0 cm in diameter and the gauge pressure inside is 300 atm? Neglect the weight of the lid.
- Bird bones have air pockets in them to reduce their weight—this also gives them an average density significantly less than that of the bones of other animals. Suppose an ornithologist weighs a bird bone in air and in water and finds its mass is 45.0 g and its apparent mass when submerged is 3.60 g (assume the bone is watertight). (a) What mass of water is displaced? (b) What is the volume of the bone? (c) What is its average density?
- In an immersion measurement of a woman’s density, she is found to have a mass of 62.0 kg in air and an apparent mass of 0.0850 kg when completely submerged with lungs empty. (a) What mass of water does she displace? (b) What is her volume? (c) Calculate her density. (d) If her lung capacity is 1.75 L, is she able to float without treading water with her lungs filled with air?
- Some fish have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85.0-kg grouper exert to stay submerged in salt water if its body density is 1015 kg/m 3 ?
- The human circulation system has approximately 1 x 10 9 capillary vessels. Each vessel has a diameter of about 8 \(\mu\)m. Assuming cardiac output is 5 L/min, determine the average velocity of blood flow through each capillary vessel.
- The flow rate of blood through a 2.00 x 10 −6 m-radius capillary is 3.80 x 10 9 cm 3 /s. (a) What is the speed of the blood flow? (b) Assuming all the blood in the body passes through capillaries, how many of them must there be to carry a total flow of 90.0 cm 3 /s?
- The left ventricle of a resting adult’s heart pumps blood at a flow rate of 83.0 cm 3 /s, increasing its pressure by 110 mm Hg, its speed from zero to 30.0 cm/s, and its height by 5.00 cm. (All numbers are averaged over the entire heartbeat.) Calculate the total power output of the left ventricle. Note that most of the power is used to increase blood pressure.
- A sump pump (used to drain water from the basement of houses built below the water table) is draining a flooded basement at the rate of 0.750 L/s, with an output pressure of 3.00 x 10 5 N/m 2 . (a) The water enters a hose with a 3.00-cm inside diameter and rises 2.50 m above the pump. What is its pressure at this point? (b) The hose goes over the foundation wall, losing 0.500 m in height, and widens to 4.00 cm in diameter. What is the pressure now? You may neglect frictional losses in both parts of the problem.
- A glucose solution being administered with an IV has a flow rate of 4.00 cm 3 /min. What will the new flow rate be if the glucose is replaced by whole blood having the same density but a viscosity 2.50 times that of the glucose? All other factors remain constant.
- A small artery has a length of 1.1 x 10 −3 m and a radius of 2.5 x 10 −5 m. If the pressure drop across the artery is 1.3 kPa, what is the flow rate through the artery? (Assume that the temperature is 37 °C.)
- Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased in order to increase blood flow by a factor of 10?
- Suppose a blood vessel’s radius is decreased to 90.0% of its original value by plaque deposits and the body compensates by increasing the pressure difference along the vessel to keep the flow rate constant. (a) By what factor must the pressure difference increase? (b) If turbulence is created by the obstruction, what additional effect would it have on the flow rate?
Challenge Problems
- The pressure on the dam shown early in the problems section increases with depth. Therefore, there is a net torque on the dam. Find the net torque.
- The temperature of the atmosphere is not always constant and can increase or decrease with height. In a neutral atmosphere, where there is not a significant amount of vertical mixing, the temperature decreases at a rate of approximately 6.5 K per km. The magnitude of the decrease in temperature as height increases is known as the lapse rate (Γ). (The symbol is the upper case Greek letter gamma.) Assume that the surface pressure is p 0 = 1.013 x 10 5 Pa where T = 293 K and the lapse rate is (\(\Gamma\) = 6.5 K/km). Estimate the pressure 3.0 km above the surface of Earth.
- A submarine is stranded on the bottom of the ocean with its hatch 25.0 m below the surface. Calculate the force needed to open the hatch from the inside, given it is circular and 0.450 m in diameter. Air pressure inside the submarine is 1.00 atm.
- Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with 20.0% of its length above water?
- Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them at prices much below gold value but high above the cost of tungsten. With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?
- The inside volume of a house is equivalent to that of a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high. The house is heated by a forced air gas heater. The main uptake air duct of the heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 minutes?
- A garden hose with a diameter of 2.0 cm is used to fill a bucket, which has a volume of 0.10 cubic meters. It takes 1.2 minutes to fill. An adjustable nozzle is attached to the hose to decrease the diameter of the opening, which increases the speed of the water. The hose is held level to the ground at a height of 1.0 meters and the diameter is decreased until a flower bed 3.0 meters away is reached. (a) What is the volume flow rate of the water through the nozzle when the diameter is 2.0 cm? (b) What is the speed of the water coming out of the hose? (c) What does the speed of the water coming out of the hose need to be to reach the flower bed 3.0 meters away? (d) What is the diameter of the nozzle needed to reach the flower bed?
- A frequently quoted rule of thumb in aircraft design is that wings should produce about 1000 N of lift per square meter of wing. (The fact that a wing has a top and bottom surface does not double its area.) (a) At takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the bottom of the wing is 60.0 m/s. Given the sea level density of air as 1.29 kg/m 3 , how fast must it move over the upper surface to create the ideal lift? (b) How fast must air move over the upper surface at a cruising speed of 245 m/s and at an altitude where air density is one-fourth that at sea level? (Note that this is not all of the aircraft’s lift—some comes from the body of the plane, some from engine thrust, and so on. Furthermore, Bernoulli’s principle gives an approximate answer because flow over the wing creates turbulence.)
- Two pipes of equal and constant diameter leave a water pumping station and dump water out of an open end that is open to the atmosphere (see the following figure). The water enters at a pressure of two atmospheres and a speed of (v 1 = 1.0 m/s). One pipe drops a height of 10 m. What is the velocity of the water as the water leaves each pipe?
- Fluid originally flows through a tube at a rate of 100 cm 3 /s. To illustrate the sensitivity of flow rate to various factors, calculate the new flow rate for the following changes with all other factors remaining the same as in the original conditions. (a) Pressure difference increases by a factor of 1.50. (b) A new fluid with 3.00 times greater viscosity is substituted. (c) The tube is replaced by one having 4.00 times the length. (d) Another tube is used with a radius 0.100 times the original. (e) Yet another tube is substituted with a radius 0.100 times the original and half the length, and the pressure difference is increased by a factor of 1.50.
- During a marathon race, a runner’s blood flow increases to 10.0 times her resting rate. Her blood’s viscosity has dropped to 95.0% of its normal value, and the blood pressure difference across the circulatory system has increased by 50.0%. By what factor has the average radii of her blood vessels increased?
- Water supplied to a house by a water main has a pressure of 3.00 x 10 5 N/m 2 early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose. (a) What pressure is now being supplied to the house, assuming resistance is constant? (b) By what factor did the flow rate in the water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is 5.00 x 10 5 N/m 2 , and the original flow rate was 200 L/ min. (c) How many more users are there, assuming each would consume 20.0 L/min in the morning?
- Gasoline is piped underground from refineries to major users. The flow rate is 3.00 x 10 −2 m 3 /s (about 500 gal/min), the viscosity of gasoline is 1.00 x 10 −3 (N/m 2 ) • s, and its density is 680 kg/m 3 . (a) What minimum diameter must the pipe have if the Reynolds number is to be less than 2000? (b) What pressure difference must be maintained along each kilometer of the pipe to maintain this flow rate?
Contributors and Attributions
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a
Creative Commons Attribution License (by 4.0)
.