20.3: Multiplication Rule
What is \(\vert A \times B \vert\) for \(A = \{ 0, 1, 2, 3 \}\) and \(B = \{ -1, 0, 1 \}\text{?}\)
Solution
We can solve this by just writing out the elements of \(A \times B\) and counting them.
\begin{align*} A \times B & = \{ (0,-1), (0,0), (0,1), (1,-1), (1,0), (1,1),\\ & \quad (2,-1), (2,0), (2,1), (3,-1), (3,0), (3,1) \} \end{align*}
So \(\vert A \times B \vert = 12\text{.}\)
What is \(\vert C \times D \vert\) for \(C = \{ a,b,c,\ldots,z \}\) and \(D = \{0,1,2,\cdots,99 \}\text{?}\)
Solution
Writing out all the elements of \(C \times D\) and then counting them all seems like a lot of work. Instead, using our experience from Worked Example \(\PageIndex{1}\), notice that we usually perform the task of writing the elements of a Cartesian product in a pattern to make sure we get them all. One-by-one we pick a single element of the first set \(C\text{,}\) and pair it up with every element of the second set \(D\text{.}\) From this pattern we see that for each \(c \in C\text{,}\) there are \(\vert D \vert\) elements of \(C \times D\) with \(c\) as the first coordinate, and there are \(\vert C \vert\) such groupings of elements from \(C \times D\text{.}\) So we arrive at
\begin{equation*} \vert C \times D \vert = \vert C \vert \cdot \vert D \vert = 26 \cdot 100 = 2600 \text{.} \end{equation*}
For sets \(X\) and \(Y\text{,}\) define an equivalence relation on \(X \times Y\) whose equivalence classes partition \(X \times Y\) in the manner described in the provided solution to Worked Example \(\PageIndex{2}\). Then describe how the number of classes and the number of objects in each class correspond to \(\vert X \vert\) and \(\vert Y \vert\text{.}\)

Warning \(\PageIndex{1}\)
The Multiplication Rule only applies to consecutive tasks \(S,T\) such that the number of ways of performing task \(T\) is independent of the choice made in performing task \(S\text{.}\)
To create a specific example of an element from \(A \times B\text{,}\) we must first choose an element of \(A\) to be the first coordinate (task \(S\)), then choose an element of \(B\) to be the second coordinate (task \(T\)).  Therefore, the Multiplication Rule says there are \(m n\) ways to construct an element of \(A \times B\text{,}\) which means \(\vert A\times B \vert = m n\text{.}\)
Suppose you are a casting director and need to select both a primary actor and an understudy for the lead role in a play. If \(n\) actors audition for the role, then there are \(n\) different ways to select the primary actor. Once this choice is made, there remain \(n-1\) different ways to the select the understudy. Hence there are \(n (n - 1)\) ways to cast the role.
Now, the actual pool of candidates for understudy will differ based on which actor is offered the lead role. However, no matter who is chosen for the lead, the number of remaining candidates for understudy is the same.
We may extend the Multiplication Rule to any (finite) number of consecutive tasks.
If \(A_1,A_2,\ldots,A_m\) are finite sets with \(\vert A_j \vert= m_j\text{,}\) then
\begin{equation*} \vert A_1 \times A_2 \times \cdots \times A_\ell \vert = m_1 m_2 \cdots m_\ell \text{.} \end{equation*}
Recall that, given alphabet \(\Sigma\) and number \(n \in \mathbb{N} \text{,}\) \(\Sigma^\ast_n\) is the set of words of length \(n\text{.}\) If \(\vert \Sigma \vert= m\text{,}\) what is \(\vert \Sigma^\ast_n \vert\text{?}\)
Solution
To construct a specific example word \(w \in \Sigma^\ast_n\text{,}\) there are:
- \(m\) ways to choose the first letter,
- \(m\) ways to choose the second letter,
- …,
- \(m\) ways to choose the \(n^{th}\) letter.
So there are
\begin{equation*} \underbrace{m \cdot m \cdot m \cdot \cdots \cdot m}_{n \text{ factors}} = m^n \end{equation*}
ways to construct \(w\text{.}\) We conclude \(\vert \Sigma^\ast_n \vert = m^n\text{.}\)
Suppose \(\vert \Sigma \vert = 5\text{.}\) How many words in \(\vert \Sigma^\ast_5 \vert\) have no repeated letters? (That is, in which no two letters are the same?)
Solution
To construct a specific example word \(w \in \Sigma^\ast_5\) in which no two letters are the same, there are
- \(5\) ways to choose the first letter,
- \(4\) remaining ways to choose the second letter,
- \(3\) remaining ways to choose the third letter,
- \(2\) remaining ways to choose the fourth letter, and
- only \(1\) remaining way to choose the last letter.
So there are
\begin{equation*} 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \end{equation*}
ways to construct \(w\text{.}\)
Similar to Example \(\PageIndex{4}\), while the actual pool of candidates for the next letter at each step will differ based on which letters have been chosen already, the number of remaining letters is always independent of which letters have actually been chosen so far. So the Multiplication Rule can be applied to this problem exactly as we have applied it.
Let \(\Sigma = \{a, b, c, \ldots, y, z \} \text{.}\) How many palindromes \(w\) with \(3 \le \vert w \vert \le 6\) are there in \(\Sigma^\ast\text{?}\)
Solution
Break into cases based on the length of \(w\text{.}\)
Case \(\vert w \vert = 3\).
Once we choose the first letter, the last is chosen for us, but we are still free to choose the middle letter. So there are \(26^2\) palindromes of length \(3\text{.}\)
Case \(\vert w \vert = 4\).
Once we choose the first two letters, the last two are chosen for us. So there are also \(26^2\) palindromes of length \(4\text{.}\)
Case \(\vert w \vert = 5\).
Once we choose the first two letters, the last two are chosen for us, but we are still free to choose the middle letter. So there are \(26^3\) palindromes of length \(5\text{.}\)
Case \(\vert w \vert = 6\).
Once we choose the first three letters, the last three are chosen for us. So there are also \(26^3\) palindromes of length \(6\text{.}\)
Total.
Applying the
Addition Rule
to these non-overlapping cases, we obtain
\begin{align*} 26^2 + 26^2 + 26^3 + 26^3 & = 26^2(1+1+26+26) \\ & = 54 \cdot 26^2 \\ & = 36,504 \end{align*}
as the number of palindromes length \(3\) to \(6\text{.}\)
Set \(A = \{a,b,c\}\) and \(B = \{0,1,2,3,4\}\text{.}\) How many functions \(A \to B\) exist? How many of these are injections? How many are surjections?
Solution
Number of functions.
A function \(f: A \rightarrow B\) can be constructed in three steps: choose \(f(a)\text{,}\) then choose \(f(b)\text{,}\) then choose \(f(c)\text{.}\) Each of the steps can be carried out in \(\vert B \vert = 5\) ways. So the number of functions is \(5^3 = 125\text{.}\)
Number of injections.
An injection \(f: A \hookrightarrow B\) can be constructed in three steps: choose \(f(a)\text{,}\) then choose \(f(b)\) to be
different from
\(f(a)\text{,}\) then choose \(f(c)\) to be different from both \(f(a)\) and \(f(b)\text{.}\) First step has \(\vert B \vert = 5\) choices. Second step has \(\vert B \setminus \{f(a)\} \vert = 4\) choices. Third step has \(\vert B \setminus \{f(a),f(b)\} \vert = 3\) choices. So the number of injections is \(5 \cdot 4 \cdot 3 = 60\text{.}\)
- A look ahead.
-
Notice that the number of injections has turned out to be
\begin{equation*} \dfrac{\vert B \vert!}{(\vert B \vert - \vert A \vert)!} \text{.} \end{equation*}
We will understand better how this formula arises in Section 21.4.
Number of surjections.
Suppose \(f: A \rightarrow B\text{.}\) Since \(\vert A \vert = 3\text{,}\) the
largest
that \(\vert f(A)\vert\) can be is \(3\text{,}\) which occurs when \(f\) is injective. However, even in such a largest case it is still smaller then \(\vert B \vert\text{,}\) so no surjections exist. That is, the number of surjections is \(0\text{.}\)


3.7: Data to Support the Multichannel Shopping Experience
- Summarize the data that could be collected to support the multichannel shopping experience of the future
Measurement of retail performance is important, but especially for multi-channel retailers as it provides a basis for comparing performance across channels. Among a number of other details, brick & mortar stores track:
- Traffic: the number of shoppers in-store within a given period
- Average Inventory: (beginning inventory for a period + ending inventory for the same period) / 2
- Return on Inventory: Revenue / Average Inventory
- Inventory Turn-over: cost of goods sold in a period / average cost of goods in inventory
- Gross Margin Return on Inventory Investment: (sales – product costs) / average inventory
- Cost Per Sale: marketing costs / sales revenue
Many of these same metrics are the same for e-tailing. However, fragmented channels, especially those that are not necessarily transaction-oriented, require different measures. Think about social networks or websites that provide engagement or information, but are not necessarily intended for transaction. How can we assess their impact? What about apps? How can their impact be measured, when they do not generally have comparable functionality to the firm’s full website?
First, a firm will need to establish whether it is interested in having ongoing monitoring of performance or monitoring a specific campaign with a defined beginning and end.
Marketers consider a number of metrics that assess interaction and engagement online, using these as indicators of impact. Traffic is measured through website visits, separated by unique and returning, as well as hits, which are requests for a file from a web server, and impressions, which is the number of times an ad loads on a viewer’s screen. They measure engagement through page views, duration of time spent on the site, and events, which includes clicks, page views, downloads, video plays, etc. Engagement is also measured by the click-through rate (CTR), which is the number of visitors who click on a link divided by the number of visitors who were served the link, and conversion rate, which is the number of visitors who complete any desired action divided by the total number of visitors.
For social networks, marketers track followers or friends to measure reach. They measure engagement in the context of likes, shares, mentions or retweets. For apps, meaningful measures might be downloads and the number of users in a given period. They also often measure daily active users (DAU) and monthly active users (MAU), which are the number of users that opened the app in a given day and month, respectively. Another helpful way marketers assess active users is to calculate the level of “stickiness” and app has for potential users. That is DAU/ MAU. This reflects how indispensible the app is implying that the higher the percentage the more users are returning to the app and engaging with it at a high frequency.
A final set of metrics for marketers to use to assess apps relates to profitability. The measures are cost per user (acquisition costs divided by total users), average revenue per user (revenue in a period divided by users during that same period) and gross margin per user, which is the average revenue per user (ARPU) minus the cost per user (CPU).
Overall, metrics for assessing retail performance are similar online and offline, focused upon measuring traffic and engagement. In tracking these metrics, retailers can be most effective in prioritizing channels and investing appropriately to support customer interaction.
Contributors and Attributions
- Data to Support the Multichannel Shopping Experience. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


19.2: Naming Aldehydes and Ketones
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Objectives
After completing this section, you should be able to
- provide the IUPAC name of an aldehyde or ketone, given its Kekulé, condensed, or skeletal structure.
- draw the structure of an aldehyde or ketone, given its IUPAC name.
- draw the structure of the following aldehydes and ketones, given their common names: formaldehyde, acetaldehyde, benzaldehyde, acetone, acetophenone, benzophenone.
Study Notes
We will only use those common names listed under Objective 3, above. We will use systematic names in all other cases. For example, the systematic name of the compound shown below is benzenecarbaldehyde, but it has the common name of benzaldehyde.
When naming unsaturated aldehydes and ketones, you must give the carbonyl group “priority” over the double bond when you are deciding which end of the carbon chain to begin numbering The carbonyl‑carbon of an aldehyde will always be at the end of the carbon chain in an acyclic compound; and therefore numbering always starts at this carbon. It is for this reason, too, that the number “1” is not required when naming a compound such as 2‑ethyl‑4‑methylpentanal.
The most potent and varied odors are aldehydes. Ketones are widely used as industrial solvents. Aldehydes and ketones contain the carbonyl group. Aldehydes are considered the most important functional group. They are often called the formyl or methanoyl group. Aldehydes derive their name from the dehyd ration of al cohols. Aldehydes contain the carbonyl group bonded to at least one hydrogen atom. Ketones contain the carbonyl group bonded to two carbon atoms.
Introduction to Naming Carbonyls
Aldehydes and ketones are organic compounds which incorporate a carbonyl functional group , C=O. The carbon atom of this group has two remaining bonds that may be occupied by hydrogen, alkyl or aryl substituents. If at least one of these substituents is hydrogen, the compound is an aldehyde . If neither is hydrogen, the compound is a ketone .
Naming Aldehydes
The IUPAC system of nomenclature assigns a characteristic suffix - al to aldehydes. For example, H 2 C=O is methan al , more commonly called formaldehyde. Since an aldehyde carbonyl group must always lie at the end of a carbon chain, it is always is given the #1 location position in numbering and it is not necessary to include it in the name. There are several simple carbonyl containing compounds which have common names which are retained by IUPAC.
Also, there is a common method for naming aldehydes and ketones. For aldehydes common parent chain names, similar to those used for carboxylic acids, are used and the suffix –aldehyde is added to the end. In common names of aldehydes, carbon atoms near the carbonyl group are often designated by Greek letters. The atom adjacent to the carbonyl function is alpha, the next removed is beta and so on.
If the aldehyde group (-CHO) is attached to a ring the suffix –carbaldehyde is added to the name of the ring. The carbon attached to this group will get the #1 location number in naming the ring.
Summary of Aldehyde Nomenclature rules
- Aldehydes take their name from their parent alkane chains. The - e is removed from the end and is replaced with -al .
- The aldehyde funtional group is given the #1 numbering location and this number is not included in the name.
- For the common name of aldehydes start with the common parent chain name and add the suffix -aldehyde . Substituent positions are shown with Greek letters.
- When the -CHO functional group is attached to a ring the suffix - carbaldehyde is added , and the carbon attached to that group is C1.
Example \(\PageIndex{1}\)
The IUPAC system names are given on top while the common name is given on the bottom in parentheses.
|
methanal (formaldehyde) |
ethanal (acetaldehyde) |
|
pentanal (valeraldehyde) |
2-chloropentanal (⍺-chlorovaleraldehyde) |
|
butanal (butyraldehyde) |
3-methylbutanal (isovaleraldehyde) |
|
cyclohexanecarbaldehyde |
3-methylcyclohexanecarbaldehyde |
|
3,3-dibromocyclopentanecarbaldehyde |
trans -2-methylcyclopentanecarbaldehyde |
Naming Ketones
The IUPAC system of nomenclature assigns a characteristic suffix of - one to ketones. A ketone carbonyl function may be located anywhere within a chain or ring, and its position is usually given by a location number. Chain numbering normally starts from the end nearest the carbonyl group. Very simple ketones, such as propanone and phenylethanone do not require a locator number, since there is only one possible site for a ketone carbonyl function
The common names for ketones are formed by naming both alkyl groups attached to the carbonyl then adding the suffix - ketone . The attached alkyl groups are arranged in the name alphabetically.
Summary of Ketone Nomenclature rules
- Ketones take their name from their parent alkane chains. The ending - e is removed and replaced with -one .
- The common name for ketones are simply the substituent groups listed alphabetically + ketone .
- Some simple ketones are known by their common names. Such as propanone which is commonly referred to as acetone .
Example \(\PageIndex{2}\)
The IUPAC system names are given on top while the common name is given on the bottom in parentheses.
|
propanone (acetone) |
acetophenone (methyl phenyl ketone) |
benzophenone (diphenyl ketone) |
|
2-pentanone (methyl propyl ketone) |
3-methyl-2-butanone (methyl isopropyl ketone) |
3-hexanone (ethyl propyl ketone) |
Naming Aldehydes and Ketones in the Same Molecule
As with many molecules with two or more functional groups, one is given priority while the other is named as a substituent. Because aldehydes have a higher priority than ketones, molecules which contain both functional groups are named as aldehydes and the ketone is named as an "oxo" substituent. It is not necessary to give the aldehyde functional group a location number, however, it is usually necessary to give a location number to the ketone.
Example \(\PageIndex{3}\)
|
4-oxopentanal |
3,4-dioxopentanal |
2-methyl-3-oxobutanal |
Naming Dialdehydes and Diketones
For dialdehydes the location numbers for both carbonyls are omitted because the aldehyde functional groups are expected to occupy the ends of the parent chain. The ending –dial is added to the end of the parent chain name.
Example \(\PageIndex{4}\)
|
pentanedial |
butanedial |
For diketones both carbonyls require a location number. The ending -dione is added to the end of the parent chain.
Example \(\PageIndex{5}\)
|
2,4-pentanedione |
2,3-hexanedione |
Naming Cyclic Ketones and Diketones
In cyclic ketones the carbonyl group is assigned location position #1, and this number is not included in the name, unless more than one carbonyl group is present. The rest of the ring is numbered to give substituents the lowest possible location numbers. Remember the prefix cyclo is included before the parent chain name to indicate that it is in a ring. As with other ketones the –e ending is replaced with the –one to indicate the presence of a ketone.
With cycloalkanes which contain two ketones both carbonyls need to be given a location numbers. Also, an –e is not removed from the end but the suffix –dione is added.
Example \(\PageIndex{6}\)
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cyclopentanone |
2-bromocyclopentanone |
4-hydroxycyclohexanone |
|
2-bromo-5-methylcyclohexanone |
1,4-cyclohexanedione |
1,2-cyclopentanedione |
Naming Carbonyls and Hydroxyls in the Same Molecule
When an aldehyde or ketone is present in a molecule which also contains an alcohol functional group the carbonyl is given nomenclature priority by the IUPAC system. This means that the carbonyl is given the lowest possible location number and the appropriate nomenclature suffix is included. In the case of alcohols the OH is named as a hydroxyl substituent. However, the l in hydroxyl is generally removed.
Example \(\PageIndex{7}\)
|
4-hydroxybutanal |
4-hydroxy-3-methylbutanal |
|
4-hydroxy-2-butanone |
1,5-dihydroxy-3-pentanone |
Naming Carbonyls and Alkenes in the Same Molecule
When an aldehyde or ketone is present in a molecule which also contains an alkene functional group the carbonyl is given nomenclature priority by the IUPAC system. This means that the carbonyl is given the lowest possible location number and the appropriate nomenclature suffix is included.
When carbonyls are included with an alkene the following order is followed:
(Location number of the alkene)-(Prefix name for the longest carbon chain minus the -ane ending)-(an -en ending to indicate the presence of an alkene)-(the location number of the carbonyl if a ketone is present)-(either an –one or and -anal ending).
Remember that the carbonyl has priority so it should get the lowest possible location number. Also, remember that cis/tran or E/Z nomenclature for the alkene needs to be included if necessary.
Example \(\PageIndex{8}\)
|
4-penten-2-one |
trans -3-pentenal |
|
( E )-3-methyl-3-pentenal |
( Z )-5-bromo-4-hexen-3-one |
Aldehydes and Ketones as Fragments
- Alkanoyl is the common name of the R-C=O fragment, though the older naming, acyl , is still widely used.
- Formyl is the common name of the H-C=O fragment.
- Acetyl is the common name of the CH 3 -C=O- fragment.
Example \(\PageIndex{9}\)
|
acetyl chloride |
acetylaldehyde |
formyl fluoride |
Exercises
Exercise \(\PageIndex{1}\)
Give the IUPAC name for each compound:
- Answer
-
- 3,4-dimethylhexanal
- 5-bromo-2-pentanone
- 2,4-hexanedione
- cis -3-pentenal
- 6-methyl-5-hepten-3-one
- 3-hydroxy-2,4-pentanedione
- 1,2-cyclobutanedione
- 2-methyl-propanedial
- 3-methyl-5-oxo-hexanal
- cis -2,3-dihydroxycyclohexanone
- 3-bromo-2-methylcyclopentanecarboaldehyde
- 3-bromo-2-methylpropanal
Exercise \(\PageIndex{2}\)
Give the structure corresponding to each IUPAC name:
- butanal
- 2-hydroxycyclopentanone
- 2,3-pentanedione
- 1,3-cyclohexanedione
- 3,4-dihydoxy-2-butanone
- 3-methyl-2-hepten-4-one
- 3-oxobutanal
- cis -3-bromocyclohexanecarboaldehyde
- butanedial
- trans -2-methyl-3-hexenal
- Answer
-
References
- Vollhardt, K. Peter C., and Neil E. Schore. Organic Chemistry . 5th ed. New York: W.H. Freeman, 2007.
- Zumdahl, Steven S., and Susan A. Zumdahl. Chemistry . 6th ed. Boston: Houghton Mifflin College Division, 2002.


5.5: Perpendicular is shortest
Assume \(Q\) is the foot point of \(P\) on the line \(\ell\). Then the inequality
\(PX > PQ\)
holds for any point \(X\) on \(\ell\) distinct from \(Q\).
If \(P, Q\), and \(\ell\) are as above, then \(PQ\) is called the distance from \(P\) to \(\ell\).
- Proof
-
If \(P \in \ell\), then the result follows since \(PQ = 0\). Further we assume that \(P \not\in \ell\).
Let \(P'\) be the reflection of \(P\) across the line \(\ell\). Note that \(Q\) is the midpoint of \([PP']\) and \(\ell\) is the perpendicular bisector of \([PP']\). Therefore
\(PX = P'X\) and \(PQ = P'Q = \dfrac{1}{2} \cdot PP'\)
Note that \(\ell\) meets \([PP']\) only at the point \(Q\). Therefore, \(X \not\in [PP']\); by triangle inequality and Corollary 4.4.1 ,
\(PX + P'X > PP'\)
and hence the result: \(PX > PQ\).
Assume \(\angle ABC\) is right or obtuse. Show that
\(AC > AB.\)
- Hint
-
If \(\angle ABC\) is right, the statement follows from Lemma \(\PageIndex{1}\). Therefore, we can assume that \(\angle ABC\) is obtuse.
Draw a line \((BD)\) perpendicular to \((BA)\). Since \(\angle ABC\) is obtuse, the angles \(DBA\) and \(DBC\) have opposite signs.
By Corollary 3.4.1 , \(A\) and \(C\) lies on opposite sides of \((BD)\). In particular, \([AC]\) intersects \((BD)\) at a point; denote it by \(X\).
Note that \(AX < AC\) and by Lemma \(\PageIndex{1}\), \(AB \le AX\).
Suppose that \(\triangle ABC\) has right angle at \(C\). Show that for any \(X \in [AC]\) the distance from \(X\) to \((AB)\) is smaller than \(AB\).
- Hint
-
Let \(Y\) be the foot point of \(X\) on \((AB)\). Apply Lemma \(\PageIndex{1}\) to show that \(XY < AX \le AC < AB\).


3.2: Global Trade in the United States
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1. Why is global trade important to the United States, and how is it measured?
No longer just an option, having a global vision has become a business imperative. Having a global vision means recognizing and reacting to international business opportunities, being aware of threats from foreign competitors in all markets, and effectively using international distribution networks to obtain raw materials and move finished products to the customer.
U.S. managers must develop a global vision if they are to recognize and react to international business opportunities, as well as remain competitive at home. Often a U.S. firm’s toughest domestic competition comes from foreign companies. Moreover, a global vision enables a manager to understand that customer and distribution networks operate worldwide, blurring geographic and political barriers and making them increasingly irrelevant to business decisions. Over the past three decades, world trade has climbed from $200 billion a year to more than $1.4 trillion. 1 U.S. companies play a major role in this growth in world trade, with 113 of the Fortune 500 companies making over 50 percent of their profits outside the United States. Among these companies are recognizable names such as Apple, Microsoft, Pfizer, Exxon Mobil, and General Electric. 2
Starbucks Corp. is among the fastest growing global consumer brands and one of the most visible emblems of U.S. commercial culture overseas. Of Starbucks’s 24,000 total stores, almost 66 percent are international stores that contribute a substantial amount to the company’s revenues, which have grown from $4.1 billion in 2003 to $21.3 billion in 2016. 3
Go into a Paris McDonald’s and you may not recognize where you are. There are no Golden Arches or utilitarian chairs and tables and other plastic features. The restaurants have exposed brick walls, hardwood floors, and armchairs. Some French McDonald’s even have faux marble walls. Most restaurants have TVs with continuous music videos. You can even order an espresso, beer, and a chicken on focaccia bread sandwich. It’s not America.
Global business is not a one-way street, where only U.S. companies sell their wares and services throughout the world. Foreign competition in the domestic market used to be relatively rare but now occurs in almost every industry. In fact, U.S. makers of electronic goods, cameras, automobiles, fine china, tractors, leather goods, and a host of other consumer and industrial products have struggled to maintain their domestic market shares against foreign competitors. Toyota now has 14 percent of the U.S. auto market, followed by Honda at 9 percent and Nissan with 8 percent. 4 Nevertheless, the global market has created vast new business opportunities for many U.S. firms.
The Importance of Global Business to the United States
Many countries depend more on international commerce than the United States does. For example, France, Great Britain, and Germany all derive more than 55 percent of their gross domestic product (GDP) from world trade, compared to about 28 percent for the United States. 5 Nevertheless, the impact of international business on the U.S. economy is still impressive:
- Trade-dependent jobs have grown at a rate three times the growth of U.S.-dependent jobs.
- Every U.S. state has realized a growth of jobs attributable to trade.
- Trade has an effect on both service and manufacturing jobs. 6
These statistics might seem to imply that practically every business in the United States is selling its wares throughout the world, but most is accounted for by big business. About 85 percent of all U.S. exports of manufactured goods are shipped by 250 companies. Yet, 98 percent of all exporters are small and medium-size firms. 7
The Impact of Terrorism on Global Trade
The terrorist attacks on America on September 11, 2001, and the Charlie Hebdo terrorist attacks in Paris in 2015 have changed the way the world conducts business. The immediate impacts of these events have included a short-term shrinkage of global trade. Globalization, however, will continue because the world’s major markets are too vitally integrated for globalization to stop. Nevertheless, terrorism has caused the growth to be slower and costlier. 8
Companies are paying more for insurance and to provide security for overseas staff and property. Heightened border inspections slow movements of cargo, forcing companies to stock more inventory. Tighter immigration policies curtail the liberal inflows of skilled and blue-collar workers that allowed companies to expand while keeping wages in check. The impact of terrorism may lessen over time, but multinational firms will always be on guard. 9
Measuring Trade between Nations
International trade improves relationships with friends and allies; helps ease tensions among nations; and—economically speaking—bolsters economies, raises people’s standard of living, provides jobs, and improves the quality of life. The value of international trade is over $16 trillion a year and growing. This section takes a look at some key measures of international trade: exports and imports, the balance of trade, the balance of payments, and exchange rates.
Exports and Imports
The developed nations (those with mature communication, financial, educational, and distribution systems) are the major players in international trade. They account for about 70 percent of the world’s exports and imports. Exports are goods and services made in one country and sold to others. Imports are goods and services that are bought from other countries. The United States is both the largest exporter and the largest importer in the world.
Each year the United States exports more food, animal feed, and beverages than the year before. A third of U.S. farm acreage is devoted to crops for export. The United States is also a major exporter of engineering products and other high-tech goods, such as computers and telecommunications equipment. For more than 60,000 U.S. companies (the majority of them small), international trade offers exciting and profitable opportunities. Among the largest U.S. exporters are Apple, General Motors Corp., Ford Motor Co., Procter & Gamble, and Cisco Systems. 10
Despite our impressive list of resources and great variety of products, imports to the United States are also growing. Some of these imports are raw materials that we lack, such as manganese, cobalt, and bauxite, which are used to make airplane parts, exotic metals, and military hardware. More modern factories and lower labor costs in other countries make it cheaper to import industrial supplies (such as steel) and production equipment than to produce them at home. Most of Americans’ favorite hot beverages—coffee, tea, and cocoa—are imported. Lower manufacturing costs have resulted in huge increases in imports from China.
Balance of Trade
The difference between the value of a country’s exports and the value of its imports during a specific time is the country’s balance of trade . A country that exports more than it imports is said to have a favorable balance of trade, called a trade surplus . A country that imports more than it exports is said to have an unfavorable balance of trade, or a trade deficit . When imports exceed exports, more money from trade flows out of the country than flows into it.
Although U.S. exports have been booming, we still import more than we export. We have had an unfavorable balance of trade throughout the 1990s, 2000s and 2010s. In 2016, our exports totaled $2.2 trillion, yet our imports were $2.7 trillion. Thus, in 2016 the United States had a trade deficit of $500 billion. 11 America’s exports continue to grow, but not as fast as our imports: The export of goods, such as computers, trucks, and airplanes, is very strong. The sector that is lagging in significant growth is the export of services. Although America exports many services—ranging from airline trips to education of foreign students to legal advice—part of the problem is due to piracy, which leads companies to restrict the distribution of their services to certain regions. The FBI estimates that the theft of intellectual property from products, books and movies, and pharmaceuticals totals in the billions every year. 12
Balance of Payments
Another measure of international trade is called the balance of payments , which is a summary of a country’s international financial transactions showing the difference between the country’s total payments to and its total receipts from other countries. The balance of payments includes imports and exports (balance of trade), long-term investments in overseas plants and equipment, government loans to and from other countries, gifts and foreign aid, military expenditures made in other countries, and money transfers in and out of foreign banks.
From 1900 until 1970, the United States had a trade surplus, but in the other areas that make up the balance of payments, U.S. payments exceeded receipts, largely due to the large U.S. military presence abroad. Hence, almost every year since 1950, the United States has had an unfavorable balance of payments. And since 1970, both the balance of payments and the balance of trade have been unfavorable. What can a nation do to reduce an unfavorable balance of payments? It can foster exports, reduce its dependence on imports, decrease its military presence abroad, or reduce foreign investment. The U.S. balance of payments deficit was over $504 billion in 2016. 13
The Changing Value of Currencies
The exchange rate is the price of one country’s currency in terms of another country’s currency. If a country’s currency appreciates , less of that country’s currency is needed to buy another country’s currency. If a country’s currency depreciates , more of that currency will be needed to buy another country’s currency.
How do appreciation and depreciation affect the prices of a country’s goods? If, say, the U.S. dollar depreciates relative to the Japanese yen, U.S. residents have to pay more dollars to buy Japanese goods. To illustrate, suppose the dollar price of a yen is $0.012 and that a Toyota is priced at 2 million yen. At this exchange rate, a U.S. resident pays $24,000 for a Toyota ($0.012 × 2 million yen = $24,000). If the dollar depreciates to $0.018 to one yen, then the U.S. resident will have to pay $36,000 for a Toyota.
As the dollar depreciates, the prices of Japanese goods rise for U.S. residents, so they buy fewer Japanese goods—thus, U.S. imports decline. At the same time, as the dollar depreciates relative to the yen, the yen appreciates relative to the dollar. This means prices of U.S. goods fall for the Japanese, so they buy more U.S. goods—and U.S. exports rise.
Currency markets operate under a system called floating exchange rates . Prices of currencies “float” up and down based upon the demand for and supply of each currency. Global currency traders create the supply of and demand for a particular currency based on that currency’s investment, trade potential, and economic strength. If a country decides that its currency is not properly valued in international currency markets, the government may step in and adjust the currency’s value. In a devaluation , a nation lowers the value of its currency relative to other currencies. This makes that country’s exports cheaper and should, in turn, help the balance of payments.
In other cases, a country’s currency may be undervalued, giving its exports an unfair competitive advantage. Many people believe that China’s huge trade surplus with the United States is partially because China’s currency was undervalued. In 2017, the U.S. Department of Commerce issued a fact sheet detailing how it accused China of dumping steel on the U.S. market as well as providing financial assistance to Chinese companies to produce, manufacture, and export stainless steel to the United States from the People’s Republic of China. 14
CONCEPT CHECK
- What is global vision, and why is it important?
- What impact does international trade have on the U.S. economy?
- Explain the impact of a currency devaluation.


7.4: The Fundamental Goal of EEG Preprocessing
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This chapter and the following two chapters describe EEG preprocessing steps that are used to deal with problem in the data, such as bad channels and artifacts. First, we need to think about what our goal is for these steps. I find that most people haven’t thought clearly about what they are trying to achieve in their preprocessing pipeline, and this leads to pipelines that are less than ideal.
So, what is the goal of EEG preprocessing? It’s simple: our fundamental goal is to accurately answer the scientific question that the experiment was designed to address . If we include bad channels or large artifacts, this will likely move us away from this goal. However, if throw out a lot of trials or participants, this will also tend to move us away from our goal by reducing our statistical power. Most preprocessing steps—such as interpolating bad channels or rejecting trials with artifacts—have both costs and benefits, and we need to ask whether the benefits of a given processing step outweigh the costs in terms of answering the scientific question that the experiment was designed to address. We will discuss how to assess these costs and benefits as we go through this chapter and the next two chapters.


10.1: Recipes
Lesson
Let's explore how ratios affect the way a recipe tastes.
Exercise \(\PageIndex{1}\): Flower Pattern
This flower is made up of yellow hexagons, red trapezoids, and green triangles.
- Write sentences to describe the ratios of the shapes that make up this pattern.
- How many of each shape would be in two copies of this flower pattern?
Exercise \(\PageIndex{2}\): Powdered Drink Mix
Here are diagrams representing three mixtures of powdered drink mix and water:
- How would the taste of Mixture A compare to the taste of Mixture B?
-
Use the diagrams to complete each statement:
- Mixture B uses ______ cups of water and ______ teaspoons of drink mix. The ratio of cups of water to teaspoons of drink mix in Mixture B is ________.
- Mixture C uses ______ cups of water and ______ teaspoons of drink mix. The ratio of cups of water to teaspoons of drink mix in Mixture C is ________.
- How would the taste of Mixture B compare to the taste of Mixture C?
Are you ready for more?
Sports drinks use sodium (better known as salt) to help people replenish electrolytes. Here are the nutrition labels of two sports drinks.
- Which of these drinks is saltier? Explain how you know.
- If you wanted to make sure a sports drink was less salty than both of the ones given, what ratio of sodium to water would you use?
Exercise \(\PageIndex{3}\): Batches of Cookies
A recipe for one batch of cookies calls for 5 cups of flour and 2 teaspoons of vanilla.
- Draw a diagram that shows the amount of flour and vanilla needed for two batches of cookies.
- How many batches can you make with 15 cups of flour and 6 teaspoons of vanilla? Show the additional batches by adding more ingredients to your diagram.
- How much flour and vanilla would you need for 5 batches of cookies?
-
Whether the ratio of cups of flour to teaspoons of vanilla is \(5:2\), \(10:4\), or \(15:6\), the recipes would make cookies that taste the same. We call these
equivalent ratios
.
- Find another ratio of cups of flour to teaspoons of vanilla that is equivalent to these ratios.
- How many batches can you make using this new ratio of ingredients?
Summary
A recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.”
To double this recipe, we would use 10 cups of cranberry juice with 4 cups of soda water. To triple this recipe, we would use 15 cups of cranberry juice with 6 cups of soda water.
This diagram shows a single batch of the recipe, a double batch, and a triple batch:
We say that the ratios \(5:2\), \(10:4\), and \(15:6\) are equivalent . Even though the amounts of each ingredient within a single, double, or triple batch are not the same, they would make fizzy juice that tastes the same.
Practice
Exercise \(\PageIndex{4}\)
A recipe for 1 batch of spice mix says, “Combine 3 teaspoons of mustard seeds, 5 teaspoons of chili powder, and 1 teaspoon of salt.” How many batches are represented by the diagram? Explain or show your reasoning.
Exercise \(\PageIndex{5}\)
Priya makes chocolate milk by mixing 2 cups of milk and 5 tablespoons of cocoa powder. Draw a diagram that clearly represents two batches of her chocolate milk.
Exercise \(\PageIndex{6}\)
In a recipe for fizzy grape juice, the ratio of cups of sparkling water to cups of grape juice concentrate is 3 to 1.
- Find two more ratios of cups of sparkling water to cups of juice concentrate that would make a mixture that tastes the same as this recipe.
- Describe another mixture of sparkling water and grape juice that would taste different than this recipe.
Exercise \(\PageIndex{7}\)
Write the missing number under each tick mark on the number line.
(From Unit 2.1.1)
Exercise \(\PageIndex{8}\)
At the kennel, there are 6 dogs for every 5 cats.
- The ratio of dogs to cats is ______ to ______.
- The ratio of cats to dogs is ______ to ______.
- For every ______ dogs there are ______ cats.
- The ratio of cats to dogs is ______ : ______.
(From Unit 2.1.1)
Exercise \(\PageIndex{9}\)
Elena has 80 unit cubes. What is the volume of the largest cube she can build with them?
(From Unit 1.6.1)
Exercise \(\PageIndex{10}\)
Fill in the blanks to make each equation true.
- \(3\cdot\frac{1}{3}=\underline{\qquad}\)
- \(10\cdot\frac{1}{10}=\underline{\qquad}\)
- \(19\cdot\frac{1}{19}=\underline{\qquad}\)
- \(a\cdot\frac{1}{a}=\underline{\qquad}\)
(As long as \(a\) does not equal \(0\).)
- \(5\cdot\underline{\qquad}=1\)
- \(17\cdot\underline{\qquad}=1\)
- \(b\cdot\underline{\qquad}=1\)
(From Unit 2.1.1)


14.5: Evaluation of the Standard Cell Potential
As we have seen, the value of the standard cell potential \(\Eeq\st\) of a cell reaction has useful thermodynamic applications. The value of \(\Eeq\st\) for a given cell reaction depends only on temperature. To evaluate it, we can extrapolate an appropriate function to infinite dilution where ionic activity coefficients are unity.
To see how this procedure works, consider again the cell reaction \(\ce{H2}\tx{(g)} + \ce{2AgCl}\tx{(s)} \arrow \ce{2H+}\tx{(aq)}+\ce{2Cl-}\tx{(aq)}+\ce{2Ag}\tx{(s)}\). The cell potential depends on the molality \(m\B\) of the HCl solute according to Eq. 14.4.5. We can rearrange the equation to \begin{equation} \Eeq\st = \Eeq + \frac{2RT}{F}\ln\g_{\pm} + \frac{2RT}{F}\ln\frac{m\B}{m\st} - \frac{RT}{2F}\ln\frac{\fug\subs{H\(_2\)}}{p\st} \tag{14.5.1} \end{equation} For given conditions of the cell, we can measure all quantities on the right side of Eq. 14.5.1 except the mean ionic activity coefficient \(\g_\pm\) of the electrolyte. We cannot know the exact value of \(\ln\g_{\pm}\) for any given molality until we have evaluated \(\Eeq\st\). We do know that as \(m\B\) approaches zero, \(\g_{\pm}\) approaches unity and \(\ln\g_{\pm}\) must approach zero. The Debye–Hückel formula of Eq. 10.4.7 is a theoretical expression for \(\ln\g_{\pm}\) that more closely approximates the actual value the lower is the ionic strength. Accordingly, we define the quantity \begin{equation} E\cell' = \Eeq + \frac{2RT}{F}\left(-\frac{A\sqrt{m\B}}{1+Ba\sqrt{m\B}}\right) +\frac{2RT}{F}\ln\frac{m\B}{m\st} - \frac{RT}{2F}\ln\frac{\fug\subs{H\(_2\)}}{p\st} \tag{14.5.2} \end{equation} The expression in parentheses is the Debye–Hückel formula for \(\ln\g_{\pm}\) with \(I_m\) replaced by \(m\B\). The constants \(A\) and \(B\) have known values at any temperature (Sec. 10.4), and \(a\) is an ion-size parameter for which we can choose a reasonable value. At a given temperature, we can evaluate \(E\cell'\) experimentally as a function of \(m\B\).
The expression on the right side of Eq. 14.5.1 differs from that of Eq. 14.5.2 by contributions to \((2RT/F)\ln\g_{\pm}\) not accounted for by the Debye–Hückel formula. Since these contributions approach zero in the limit of infinite dilution, the extrapolation of measured values of \(E\cell'\) to \(m\B{=}0\) yields the value of \(\Eeq\st\).
Figure 14.5 \(E\cell'\) (defined by Eq. 14.5.2) as a function of HCl molality for the cell of Fig. 14.1 at \(298.15\K\). Data from Herbert S. Harned and Russell W. Ehlers, J. Am. Chem. Soc. , 54 , 1350–1357, 1932, with \(\fug\subs{H\(_2\)}\) set equal to \(p\subs{H\(_2\)}\) and the parameter \(a\) set equal to \(4.3\timesten{-10}\units{m}\). The dashed line is a least-squares fit to a linear relation.
Figure 14.5 shows this extrapolation using data from the literature. The extrapolated value indicated by the filled circle is \(\Eeq\st=0.2222\V\), and the uncertainty is on the order of only \(0.1\units{mV}\).


39.7: Gas Exchange across Respiratory Surfaces - Lung Volumes and Capacities
- Distinguish between lung volume and lung capacity
Lung Volumes and Capacities
Different animals exhibit different lung capacities based on their activities. For example, cheetahs have evolved a much higher lung capacity than humans in order to provide oxygen to all the muscles in the body, allowing them to run very fast. Elephants also have a high lung capacity due to their large body and their need to take up oxygen in accordance with their body size.
Human lung size is determined by genetics, gender, and height. At maximal capacity, an average lung can hold almost six liters of air; however, lungs do not usually operate at maximal capacity. Air in the lungs is measured in terms of lung volumes and lung capacities. Volume measures the amount of air for one function (such as inhalation or exhalation) and capacity is any two or more volumes (for example, how much can be inhaled from the end of a maximal exhalation).
Lung Volumes
The volume in the lung can be divided into four units: tidal volume, expiratory reserve volume, inspiratory reserve volume, and residual volume. Tidal volume (TV) measures the amount of air that is inspired and expired during a normal breath. On average, this volume is around one-half liter, which is a little less than the capacity of a 20-ounce drink bottle. The expiratory reserve volume (ERV) is the additional amount of air that can be exhaled after a normal exhalation. It is the reserve amount that can be exhaled beyond what is normal. Conversely, the inspiratory reserve volume (IRV) is the additional amount of air that can be inhaled after a normal inhalation. The residual volume (RV) is the amount of air that is left after expiratory reserve volume is exhaled. The lungs are never completely empty; there is always some air left in the lungs after a maximal exhalation. If this residual volume did not exist and the lungs emptied completely, the lung tissues would stick together. The energy necessary to re-inflate the lung could be too great to overcome. Therefore, there is always some air remaining in the lungs. Residual volume is also important for preventing large fluctuations in respiratory gases (O 2 and CO 2 ). The residual volume is the only lung volume that cannot be measured directly because it is impossible to completely empty the lung of air. This volume can only be calculated rather than measured..
Lung volumes are measured by a technique called spirometry. An important measurement taken during spirometry is the forced expiratory volume (FEV), which measures how much air can be forced out of the lung over a specific period, usually one second (FEV1). In addition, the forced vital capacity (FVC), which is the total amount of air that can be forcibly exhaled, is measured. The ratio of these values (FEV1/FVC ratio) is used to diagnose lung diseases including asthma, emphysema, and fibrosis. If the FEV1/FVC ratio is high, the lungs are not compliant (meaning they are stiff and unable to bend properly); the patient probably has lung fibrosis. Patients exhale most of the lung volume very quickly. Conversely, when the FEV1/FVC ratio is low, there is resistance in the lung that is characteristic of asthma. In this instance, it is difficult for the patient to get the air out of his or her lungs. It takes a long time to reach the maximal exhalation volume. In either case, breathing is difficult and complications arise.
Lung Capacities
The lung capacities are measurements of two or more volumes. The vital capacity (VC) measures the maximum amount of air that can be inhaled or exhaled during a respiratory cycle. It is the sum of the expiratory reserve volume, tidal volume, and inspiratory reserve volume. The inspiratory capacity (IC) is the amount of air that can be inhaled after the end of a normal expiration. It is, therefore, the sum of the tidal volume and inspiratory reserve volume. The functional residual capacity (FRC) includes the expiratory reserve volume and the residual volume. The FRC measures the amount of additional air that can be exhaled after a normal exhalation. The total lung capacity (TLC) is a measurement of the total amount of air that the lung can hold. It is the sum of the residual volume, expiratory reserve volume, tidal volume, and inspiratory reserve volume..
Key Points
- The lung volumes that can be measured using a spirometer include tidal volume (TV), expiratory reserve volume (ERV), and inspiratory reserve volume (IRV).
- Residual volume (RV) is a lung volume representing the amount of air left in the lungs after a forced exhalation; this volume cannot be measured, only calculated.
- The lung capacities that can be calculated include vital capacity (ERV+TV+IRV), inspiratory capacity (TV+IRV), functional residual capacity (ERV+RV), and total lung capacity (RV+ERV+TV+IRV).
Key Terms
- tidal volume : the amount of air breathed in or out during normal respiration
- residual volume : the volume of unexpended air that remains in the lungs following maximum expiration
- spirometry : the measurement of the volume of air that a person can move into and out of the lungs


4.4: Bayesian approach to evolutionary rates
Finally, we can also use a Bayesian approach to fit Brownian motion models to data and to estimate the rate of evolution. This approach differs from the ML approach in that we will use explicit priors for parameter values, and then run an MCMC to estimate posterior distributions of parameter estimates. To do this, we will modify the basic algorithm for Bayesian MCMC (see Chapter 2) as follows:
-
Sample a set of starting parameter values,
σ
2
and $\bar{z}(0)$ from their prior distributions. For this example, we can set our prior distribution as uniform between 0 and 0.5 for
σ
2
and uniform from 0 to 10 for $\bar{z}(0)$.
-
Given the current parameter values, select new proposed parameter values using the proposal density
Q
(
p
′|
p
). For both parameter values, we will use a uniform proposal density with width
w
p
, so that: (eq. 4.10)
\[ Q(p'|p) \sim U(p-\frac{w_p}{2}, p+\frac{w_p}{2}) \]
-
Calculate three ratios:
- The prior odds ratio, R p r i o r . This is the ratio of the probability of drawing the parameter values p and p ’ from the prior. Since our priors are uniform, this is always 1.
 This is the ratio of probability of proposals going from p to p ’ and the reverse.
-
The likelihood ratio,
R
l
i
k
e
l
i
h
o
o
d
. This is the ratio of probabilities of the data given the two different parameter values. We can calculate these probabilities from equation 4.5 above. (eq. 4.11)
\[ R_{likelihood} = \frac{L(p'|D)}{L(p|D)} = \frac{P(D|p')}{P(D|p)} \]
-
Find the acceptance ratio,
R
a
c
c
e
p
t
, which is product of the prior odds, proposal density ratio, and the likelihood ratio. In this case, both the prior odds and proposal density ratios are 1, so
R
a
c
c
e
p
t
=
R
l
i
k
e
l
i
h
o
o
d
.
-
Draw a random number
x
from a uniform distribution between 0 and 1. If
x
<
R
a
c
c
e
p
t
, accept the proposed value of both parameters; otherwise reject, and retain the current value of the two parameters.
-
Repeat steps 2-5 a large number of times.
Using the mammal body size data, I ran an MCMC with 10,000 generations, discarding the first 1000 as burn-in. Sampling every 10 generations, I obtain parameter estimates of $\hat{\sigma}_{bayes}^2 = 0.10$ (95% credible interval: 0.066 − 0.15) and $\hat{\bar{z}}(0) = 3.5$ (95% credible interval: 2.3 − 5.3; Figure 4.5).
Note that the parameter estimates from all three approaches (REML, ML, and Bayesian) were similar. Even the confidence/credible intervals varied a little bit but were of about the same size in all three cases. All of the approaches above are mathematically related and should, in general, return similar results. One might place higher value on the Bayesian credible intervals over confidence intervals from the Hessian of the likelihood surface, for two reasons: first, the Hessian leads to an estimate of the CI under certain conditions that may or may not be true for your analysis; and second, Bayesian credible intervals reflect overall uncertainty better than ML confidence intervals (see chapter 2).


1.4: Terms- Claims
Finding an Answer
Once you’ve established your question at issue, it’s time to find an answer. To do this, you’ll conduct research (a process discussed later), synthesize others’ ideas, and reach a point where you have a clear side for your argument. When you have a one-sentence answer to your question at issue, you’ve found your claim .
The claim, which is also part of our argumentative thesis, can take several different forms, as both professors Jennifer Fletcher and John Gage point out in their books on argumentative writing. These forms are:
- Claim of fact
- Claim of definition
- Claim of value
- Claim of rule/policy/action
A claim of fact may ask about consequences or reasons, or it may try to establish whether something can be true. For example, I might claim Providing free textbooks would enable more students to stay enrolled in college. That’s a claim of fact: I can prove (or fail to prove) that free books will mean more enrolled students by looking at numbers. Likewise, I can make claims of fact about existing policies or procedures (Most students do not understand our college plagiarism policies) or about reasons or consequences (Netflix users are less likely to go to the movies than those without Netflix).
A claim of definition tries to explain something or a part of something in a new or potentially controversial way. Let’s take a break to talk about controversy: Usually, the word “controversy” is now attached to the idea of scandal or impropriety. For our use, controversy reverts to its original meaning: “disagreement.” So defining something in a new way, or in a way about which there could be disagreement, is making a claim of definition.
Why would we want to argue over words, though? Well, for some of us, it’s because that’s what we love to do (think of your instructor). For the rest of us, claims of definition are interesting because they offer the chance to redefine how people think and talk about a topic you’re interested in. Think of the dozens of words you use each day that have a meaning that’s specific to you, or just to you and your friends. When you use the word “leader” or “leadership,” for instance, what do you mean by it? When you picture “success,” what are you picturing? What about words like “accessible,” or “free,” or “entitled”? Do you think you define them the same way as everyone else, or do you think your own experiences have led you to a different understanding of what being free or being successful really means? If so, perhaps you want to make a claim of definition about those words.
Other kinds of claims of definitions may work to narrow or broaden well-known terms. For example, if you want to argue that cheerleading should be considered an Olympic sport, you may have a claim of definition about what sports are (or what the Olympics should be).
Claims of value might sound like the kind of claims that get you into arguments over Thanksgiving dinner — and that’s not untrue. However, in academic argument, remember, we’re not trying to get into any fights. We don’t want anyone to stop speaking to anyone else. We’re trying to find reasonable ways to disagree, and sometimes, our disagreements are based on shared values or ethics, not on facts. Shared values might be personally held beliefs, such as religious/spiritual beliefs, but they might also be ethical codes that professional groups adhere to. As a student, you are likely subject to a student Code of Conduct, which is a statement about the shared values or standards upheld by students at your institution. A claim of value, also called a “claim of quality” (Fletcher), makes an argument based on these shared values that something is good or bad.
That might sound pretty heavy, but you’ve likely seen more claims of value in your life than anything else — particularly if you’ve ever looked at reviews online. Restaurant and product reviews are often based on claims of value: This restaurant has the best burger in the history of ever is a claim of value. Can we factually prove that any restaurant has the best burger in history? No. Can we define what “the best burger” is? Perhaps. Can we appeal to other burger lovers and claim that this restaurant is great? Sure. That’s a claim of value.
Likewise, most reviews of art, music, movies, and dance rely on claims of value or quality. The reviewer may establish shared values with the audience by explaining what they’ll be evaluating the performance on, but the audience must share those values in order to believe the claim. Put another way: If I love explosions and spaceships, and a reviewer says “This new movie is great” and supports that claim by talking about all of the fireworks and space adventures, I’m going to realize we share some values and be more likely to agree with the reviewer’s claim.
Claims of value often deal with gray-area issues, as well. Consider the earlier question about whether drugs should be decriminalized. Questions about what behavior should be seen as criminal come down to questions of value and may require require claims of value as their answers. Comparing two different issues may also be a claim of value: Drinking while under the influence of alcohol is more irresponsible than drinking while stoned is a claim of value, because the argument hinges on what the writer (and reader) consider “irresponsible” behavior to be.
Finally, Claims of rule/policy/action (which I’ll abbreviate to Claims of Rule ) call for changes to be made to existing ways of doing things. These types of claims nearly always begin with a “should” statement: The college should make all first-year textbooks free is a Claim of Rule because it argues for changing an existing policy.
Claims of rule are perhaps the easiest to establish — just find a law you disagree with, and get going — but they are sometimes the hardest to narrow down. Arguing against a national policy, particularly one that’s been around for years, would likely take a book. Many students often decide they want to engage in debate around the Bill of Rights and the U.S. Constitution. While there’s always plenty of debate to be had over what these policies mean, there’s never enough time to thoroughly cover even a less popular amendment (say, the 17th) in a short paper. Trying to argue for or against changes to something like the First or Second Amendment would require giving up the next few years of your life to read nothing but federal court case law. Again, find narrow topics whenever possible: You may not be able to argue broadly about free speech restrictions across the United States, but there may be local issues or campus issues about which research and argument would be doable.
How do you know what type of claim to make? The type of claim you have may depend on the type of question you’ve asked. Look at how these questions and claims match up:
Question of fact:
- Are more accidents caused by drivers under the influence of alcohol or marijuana?
- Claim: More accidents are caused by…
Question of definition:
- What does it mean to be “intoxicated” or “under the influence”?
- Claim: Intoxication means…
Question of value:
- Which is worse, drinking and driving or driving while stoned?
- Claim: It is worse to….
Question of rules:
- Should we change the local policy regarding sales of recreational marijuana?
- Claim: We should change/not change local policy to state that…
Once you have your claim established, it’s time to find the fun stuff: Evidence!
References
Fletcher, J. (2015). T eaching Arguments: Rhetorical Comprehension, Critique, and Response. Portland, ME: Stenhouse.
Gage, J. (2006). The Shape of Reason. New York, NY: Pearson.


15: DNA Technologies
Learning Objectives
- Suggest molecular techniques to design experiments (e.g., how would you use cDNA or a PCR product to clone a gene).
- Determine when to make or use a cDNA library or a genomic library
- Outline an experiment to purify rRNA from eukaryotic cells.
- Outline an experiment to isolate a cDNA for a human protein and clone it so you can manufacture insulin for the treatment of disease.
- Explain why you might want to clone and express a human growth hormone gene.
- List components needed to make a cDNA library using purified poly(A) RNA.
- List the components needed to make a genomic library from isolated genomic DNA.
- Compare PCR and genomic cloning as strategies for isolating a gene.
- Outline a strategy for using fly DNA to obtain copies of a human DNA sequence.
- Ask a question that requires screening a genomic library to obtain a gene you want to study
- Ask a question that requires using a microarray to obtain a gene you want to study
-
- 15.1: Overview
- We start this chapter by looking at technologies that led to genetic engineering. The ability of make recombinant DNA is such a seminal technology that just realizing it could be done and then doing it in a test tube for the first time earned Paul berg a half-share in the 1980 Nobel Prize in Chemistry (the other half was shared by Walter Gilbert and Frederick Sanger for studies that enabled efficient DNA sequencing).
-
- 15.2: Make and Screen a cDNA Library
- The first step in making a cDNA library is to isolate cellular mRNA. This mRNA extract should represent all of the transcripts in the cells at the time of isolation, or the cell’s transcriptome. This term is used by analogy to genome. However, a genome is all of the genetic information of an organism. In contrast, a transcriptome (usually eukaryotic) reflects all of the genes expressed in a given cell type at a moment in time.
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- 15.3: DNA Sequencing
- RNA sequencing came first, when Robert Holley sequenced a tRNA in 1965. The direct sequencing of tRNAs was possible because tRNAs are small, short nucleic acids, and because many of the bases in tRNAs are chemically modified after transcription. An early method for DNA sequencing developed by Walter Gilbert and colleagues involved DNA fragmentation, sequencing of the small fragments of DNA, and then aligning the overlapping sequences of the short fragments to assemble longer sequences.
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- 15.4: Genomic Libraries
- A genomic library might be a tube full of recombinant bacteriophage. Each phage DNA molecule contains a fragmentary insert of cellular DNA from a foreign organism. The library is made to contain a representation of all of possible fragments of that genome. The need for vectors like bacteriophage that can accommodate long inserts becomes obvious from the following bit of math.
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- 15.5: The Polymerase Chain Reaction (PCR)
- The polymerase chain reaction (PCR) can amplify a region of DNA from any source, even from a single cell’s worth of DNA or from fragments of DNA obtained from a fossil. This amplification usually takes just a few hours, generating millions of copies of the desired target DNA sequence. The effect is to purify the DNA from surrounding sequences in a single reaction!
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- 15.6: Genomic Approaches- The DNA Microarray
- Traditionally, when cellular levels of a protein were known to change in response to a chemical effector, molecular studies focused on control of the transcription of its gene. These studies often revealed that the control of gene expression was at the level of transcription, turning a gene on or off through interactions of transcription factors with DNA.
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- 15.7: Ome-Sweet Ome
- Early molecular technologies, including the ones described in this chapter, were applied to understanding the structure, function and regulation of specific genes. Some of the more recent technologies (e.g., microarrays) are well adapted to holistic approaches to understanding cell function. Terms we have already seen (genome, epigenome, transcriptome) were coined in an effort to define the different objects of study whose underlying network of molecular interactions can more accurately explain
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- 15.8: From Genetic Engineering and Genetic Modification
- By enabling us to focus on how genes and their regulation have evolved, these genomic, transcriptomic and proteomic technologies have vastly increased our knowledge of how cells work at a molecular level. We continue to add to our knowledge of disease process and in at least a few cases, how we can treat disease.


23.1A: Digestive Systems
Animals use the organs of their digestive systems to extract important nutrients from food they consume, which can later be absorbed.
- Summarize animal nutrition and the digestive system
Key Points
- Animals obtain lipids, proteins, carbohydrates, essential vitamins, and minerals from the food they consume.
- The digestive system is composed of a series of organs, each with a specific, yet related function, that work to extract nutrients from food.
- Organs of the digestive system include the mouth, esophagus, stomach, small intestine, and the large intestine.
- Accessory organs, such as the liver and pancreas, secrete digestive juices into the gastrointestinal tract to assist with food breakdown.
Key Terms
- digestion : the process, in the gastrointestinal tract, by which food is converted into substances that can be utilized by the body
- macromolecule : a very large molecule, especially used in reference to large biological polymers (e.g. nucleic acids and proteins)
- alimentary canal : the organs of a human or an animal through which food passes; the digestive tract
All living organisms need nutrients to survive. While plants can obtain the molecules required for cellular function through the process of photosynthesis, most animals obtain their nutrients by the consumption of other organisms. At the cellular level, the biological molecules necessary for animal function are amino acids, lipid molecules, nucleotides, and simple sugars. The food consumed consists of protein, fat, and complex carbohydrates, but the requirements of each are different for each animal.
Balanced human diet : For humans, fruits and vegetables are important in maintaining a balanced diet. Both of these are an important source of vitamins and minerals, as well as carbohydrates, which are broken down through digestion for energy.
Animals must convert these macromolecules into the simple molecules required for maintaining cellular functions, such as assembling new molecules, cells, and tissues. The conversion of the food consumed to the nutrients required is a multi-step process involving digestion and absorption. During digestion, food particles are broken down to smaller components which will later be absorbed by the body.
Digestive System
The digestive system is one of the largest organ systems in the human body. It is responsible for processing ingested food and liquids. The cells of the human body all require a wide array of chemicals to support their metabolic activities, from organic nutrients used as fuel to the water that sustains life at the cellular level. The digestive system not only effectively chemically reduces the compounds in food into their fundamental building blocks, but also acts to retain water and excrete undigested materials. The functions of the digestive system can be summarized as follows: ingestion (eat food), digestion (breakdown of food), absorption (extraction of nutrients from the food), and defecation (removal of waste products).
The digestive system consists of a group of organs that form a closed tube-like structure called the gastrointestinal tract (GI tract) or the alimentary canal. For convenience, the GI tract is divided into upper GI tract and lower GI tract. The organs that make up the GI tract include the mouth, the esophagus, the stomach, the small intestine, and the large intestine. There are also several accessory organs that secrete various enzymes into the GI tract. These include the salivary glands, the liver, and the pancreas.
Generalized animal digestive system : This diagram shows a generalized animal digestive system, detailing the different organs and their functions.
Challenges to Human Nutrition
One of the challenges in human nutrition is maintaining a balance between food intake, storage, and energy expenditure. Imbalances can have serious health consequences. For example, eating too much food while not expending much energy leads to obesity, which in turn will increase the risk of developing illnesses such as type-2 diabetes and cardiovascular disease. The recent rise in obesity and related diseases means that understanding the role of diet and nutrition in maintaining good health is more important than ever.


3.4: Demographic Transition, Migration, and Political Policy
3.4.1 What is the relationship between population structure and migration?
Even though people generally migrate to find/make a better life for themselves and for their families, the benefits and pitfalls of migration affect different countries and regions in very different ways. Demography (age structure) has a lot to do with understanding how such a differentiation occurs. (Hint: You may need to go back to Chapter 2 to review your understanding of the demographic transition model or click here): www.bbc.co.uk/schools/gcsebit...hy/population/population_change_structure_rev4.shtml.
The demographic transition model explains how countries experience different stages of population growth and family sizes, but the model also works well to understand sources and destinations for migrants. Before the explanation continues, take a look at the model to see if you can predict the stages during which you would expect large-scale emigration versus immigration. Do you feel confident in your guesses? Why or why not?
Geographers note that countries of emigration tend to be late in stage 2 or early stage 3, while countries of immigration tend to be late stage 4 or stage 5 countries. Let’s think about this for just a moment. At what age are people most likely to take the risk to move to a completely different country to seek their fortune? Would you move to Slovenia next year, if you could earn triple your current salary? Most international migrants tend to be relatively young (18-35) during the prime of their working years. Countries that tend to have an abundance of working-age people also tend to be early in stage 3 of the model. Countries like Mexico, Guatemala, the Philippines, and India had large families a generation ago. As those children enter their working years, there are not enough jobs created due to an “oversupply” of laborers. Meanwhile, countries in late stage 4 find that because of low fertility rates a generation earlier, the economy now faces a shortage of working-age residents to do all kinds of jobs. Birth rates were high during the 1950’s in the US, but as women had fewer children in subsequent generations, fewer workers were entering the workforce every year. Unsurprisingly, young workers from Mexico, El Salvador, Jamaica, etc., began to immigrate as the demand for their labor increased. In Europe, the same pattern occurred as those from higher fertility countries like Turkey migrated to lower fertility countries like Germany.
3.4.2 Why do some countries benefit from migration while others do not?
Demographic realities can be push factors or pull factors that serve to push people away from a place or pull them towards a place as explained in the previous paragraph. Push and pull factors can be cultural, economic, or ecological. Baptists might be “pulled” towards the American South, and Mormons might be “pulled” to Utah for cultural reasons. Meanwhile, the devastation of Hurricane Maria (2017) “pushed” hundreds of thousands of Puerto Ricans off the island. The most common destination for Puerto Ricans was Florida for economic (jobs), ecological (warm weather), and cultural (existing Spanish speakers) reasons. When the push factors and pull factors fit together nicely, migration can benefit both the sending and receiving regions.
Very often in the case of forced migration , however, migration does not benefit both parties. Those migrants who flee their country based upon claims of danger based upon race, religion, nationality, or other pertinent identifiers are known as asylum seekers. They seek a country willing to take them in permanently for fear of imprisonment, retribution, or death in their country of origin. When asylum seekers have satisfactorily demonstrated a claim in court, their status changes to refugee. Under international agreement, refugees cannot be forced to return to any country where they are deemed to be in danger, so refugee status provides displaced people with a legal protection against deportation. The number of refugees worldwide at the end of 2016 was 22.5 million, the highest total of refugees since the end of WWII. There were also 40.3 million people uprooted within their own country, known as internally displaced people (IDP), bringing the total number of displaced people to an all-time high of nearly 63 million. In recent years, large numbers of asylum seekers have left Syria, Iraq, Afghanistan, Palestine, Somalia, Sudan, Cuba, Venezuela, and Myanmar (formerly Burma) as wars and political conflict have endangered millions. During previous wars, wealthier countries in Europe, North America, and Australia accepted large numbers of asylum seekers, but anti-immigrant sentiments have risen in many parts of the world, leaving the majority of would-be refugees without refuge. Opponents of resettlement argue that the cost is just too great, and they fear that accepting refugees would encourage more unwelcome immigration in the future, so this has significantly reduced the number of migrants that countries are willing to accept.
Though twenty-first century anti-immigrant sentiment remains high in countries like the US, Australia, and the U.K., the benefits for receiving countries are well documented. Apart from the demographic advantages already described, employers and consumers tend to benefit markedly from the low-cost, readily- available supply of labor provided by immigrants. Everything from the cost of fruit, construction, fast food, and lawn care tends to be cheaper owing to an immigrant labor force. Evidence also indicates that wages in low-skilled jobs may be suppressed by immigrants, but the overall economic benefit is widely reported by economists to be favorable when unemployment rates are low. Immigrants have higher rates of employment, are more likely to start businesses and are less likely to commit crimes than the native-born population in the US Sending countries also tend to benefit from emigration in two ways. First, emigration provides an opportunity for young workers who cannot find employment at home. Second, emigrants tend to send home the majority of their overseas earnings. Money sent home by overseas workers is called remittances. Countries that send large numbers of workers overseas benefit from the large infusion of foreign capital into the local economy, which tends to spur new investment opportunities. When migration is working smoothly, both sending and receiving countries can benefit.
Besides low-skilled workers and refugees, a third category of immigrant has increased dramatically under globalization. Highly-skilled immigrants represent a unique contradiction. On the one hand, countries seek to increase border security, limit asylum seekers, and build walls. On the other, those same countries actively recruit and seek to attract immigrants with specific skills, training, and educational levels. Most commonly, wealthy countries like the U.K. regularly recruit nurses, scientists, and engineers from poorer countries to meet the needs of an aging population. As British residents have fewer children and society gets older, there is more demand specifically for healthcare workers of all kinds. As such, the best and brightest minds from poorer countries become attracted to the much higher wages outside of their countries and they leave—resulting in a brain drain. Nearly sixty percent of all doctors born in Ghana and eighty-five percent of nurses born in the Philippines have left the country to work elsewhere! Wealthy countries reap the benefits of this brain gain. Although highly-skilled and educated individuals and families that emigrate for higher pay undoubtedly benefit from emigration, the countries that experience brain drain persistently lose a very valuable resource. Besides losing most of its doctors, Ghana also now faces a major shortage of those qualified to teach the next generation of medical professionals, so the negative effect crosses multiple generations.


11.2: Horizon Development Processes
Ever dug into in the soil and noticed how it seems to change color the deeper you go? Some soils are dark brown near the surface and get lighter in color as you go deeper. Others display a sandy, light-gray layer near the surface with a reddish layer beneath. These different colored layers are known as horizons . All the horizons taken together comprise the soil profile . Soil horizons form as a result of the four horizon development processes, additions, transformation, translocation, and removal.
Additions
Additions can be materials that are transported into the location where a soil is forming. For instance, dust with a high calcium carbonate content could be blown on to the developing soil adding calcium to the evolving profile. This is one idea as to how soils in the Midwestern part of the United States called mollisols were enriched with CaCO 3 . When plants die or leaves fall on to the surface they decompose adding organic matter to the soil.
Transformation
Transformation of the materials added to the developing soil occur by chemical and biological processes acting on them. For instance, leaves falling on the surface and plant roots dying beneath may decompose into a dark brown, nutrient-rich material called humus . Humus is responsible for the dark brown to black color of many soils, especially near the top of the soil profile. Iron and aluminum can be oxidized under warm, moist climates. Soil material is constantly being transformed in one way or another.
Translocation
Translocation involves the movement of soil-forming materials through the developing soil profile. Translocation occurs by water running through the soil transferring materials from upper to lower portions of the profile. Burrowing animals like earth worms, ants, etc., move soil materials within the profile. Burrowing animals create passage ways through which air and water can travel promoting soil development.
Removal
Removal of soil forming materials means that they are completely removed from the soil profile. Easily dissolved elements like calcium carbonate can be removed from the soil profile under rainy climates.
Soil horizons develop in response to the relative importance of each of the above processes. All soils are impacted by the horizon development processes to one degree or another. These processes determine the characteristics or properties of soil in each of the horizons.


5.6: Trees
Terminology
A graph is a tree if and only if it is connected and contains no cycles.
A vertex in a tree is a leaf if and only if it has degree one.
A graph \(G\) is a spanning tree of a graph \(H\) if and only if \(G\) is a subgraph of HH that contains all the vertices of \(H\) and is a tree.
Practice
Checkpoint \(\PageIndex{4}\)
Determine which graphs in Figure Figure 5.1.1 are trees.
Checkpoint \(\PageIndex{4}\)
Draw all non-isomorphic trees on 3 vertices.
Checkpoint \(\PageIndex{4}\)
Prove that every tree with at least two vertices has at least one leaf.
Checkpoint \(\PageIndex{4}\)
Draw a tree with 7 vertices. Determine for what \(n\) the tree is \(n\)-connected.
Checkpoint \(\PageIndex{4}\)
Explain why for any tree removal of a leaf produces another tree.
Checkpoint \(\PageIndex{4}\)
Find a spanning tree for every graph in Figure 5.2.43 .


6.9.E: Problems on Maxima and Minima
Verify Note 1.
Complete the missing details in the proof of Theorems 2 and 3.
Verify Examples (A) and (B). Supplement Example (A) by applying Theorem 2.
Test \(f\) for extrema in \(E^{2}\) if \(f(x, y)\) is
(i) \(\frac{x^{2}}{2 p}+\frac{y^{2}}{2 q}(p>0, q>0)\);
(ii) \(\frac{x^{2}}{2 p}-\frac{y^{2}}{2 q}(p>0, q>0)\);
(iii) \(y^{2}+x^{4}\);
(iv) \(y^{2}+x^{3}\).
(i) Find the maximum volume of an interval \(A \subset E^{3}\) (see Chapter 3, §7) whose edge lengths \(x, y, z\) have a prescribed sum: \(x+y+z=a\).
(ii) Do the same in \(E^{4}\) and in \(E^{n};\) show that \(A\) is a cube.
(iii) Hence deduce that
\[\sqrt[n]{x_{1} x_{2} \cdots x_{n}} \leq \frac{1}{n} \sum_{1}^{n} x_{k} \quad\left(x_{k} \geq 0\right),\]
i.e., the geometric mean of \(n\) nonnegative numbers is \(\leq\) their arithmetic mean.
Find the minimum value for the sum \(f(x, y, z, t)=x+y+z+t\) of four positive numbers on the condition that \(x y z t=c^{4}\) (constant).
[Answer: \(x=y=z=t=c; f_{\max }=4c\).]
Among all triangles inscribed in a circle of radius \(R,\) find the one of maximum area.
[Hint: Connect the vertices with the center. Let \(x, y, z\) be the angles at the center. Show that the area of the triangle \(=\frac{1}{2} R^{2}(\sin x+\sin y+\sin z),\) with \(z=2 \pi-(x+y)\).]
Among all intervals \(A \subset E^{3}\) inscribed in the ellipsoid
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\]
find the one of largest volume.
[Answer: the edge lengths are \(\frac{2 a}{\sqrt{3}}, \frac{2 b}{\sqrt{3}}, \frac{2 c}{\sqrt{3}}\).]
Let \(P_{i}=\left(a_{i} \cdot b_{i}\right), i=1,2,3,\) be 3 points in \(E^{2}\) forming a triangle in which one angle (say, \(\,easureangle P_{1})\) is \(\geq 2 \pi / 3\).
Find a point \(P=(x, y)\) for which the sum of the distances,
\[P P_{1}+P P_{2}+P P_{3}=\sum_{i=1}^{3} \sqrt{\left(x-a_{i}\right)^{2}+\left(y-b_{i}\right)^{2}},\]
is the least possible.
[Outline: Let \(f(x, y)=\sum_{i=1}^{3} \sqrt{\left(x-a_{i}\right)^{2}+\left(y-b_{i}\right)^{2}}\).
Show that \(f\) has no partial derivatives at \(P_{1}, P_{2},\) or \(P_{3}\) (and so \(P_{1}, P_{2},\) and \(P_{3}\) are critical points at which an extremum may occur), while at other points \(P,\) partials do exist but never vanish simultaneously, so that there are no other critical points.
Indeed, prove that \(D_{1} f(P)=0=D_{2} f(P)\) would imply that
\[\sum_{i=1}^{3} \cos \theta_{i}=0=\sum_{1}^{3} \sin \theta_{I},\]
where \(\theta_{i}\) is the angle between \(\overline{P P_{i}}\) and the \(x\)-axis; hence
\[\sin \left(\theta_{1}-\theta_{2}\right)=\sin \left(\theta_{2}-\theta_{3}\right)=\sin \left(\theta_{3}-\theta_{1}\right) \quad\text {(why?),}\]
and so \(\theta_{1}-\theta_{2}=\theta_{2}-\theta_{3}=\theta_{3}-\theta_{1}=2 \pi / 3,\) contrary to \(\angle P_{1} \geq 2 \pi / 3.\) (Why?)
From geometric considerations, conclude that \(f\) has an absolute minimum at \(P_{1}\).
(This shows that one cannot disregard points at which \(f\) has no partials.)]
Continuing Problem 8, show that if none of \(\angle P_{1}, \angle P_{2},\) and \(\angle P_{3}\) is \(\geq\) \(2 \pi / 3,\) then \(f\) attains its least value at some \(P\) (inside the triangle) such that \(\angle P_{1} P P_{2}=\angle P_{2} P P_{3}=\angle P_{3} P P_{1}=2 \pi / 3\).
[Hint: Verify that \(D_{1} f=0=D_{2} f\) at \(P\).
Use the law of cosines to show that \(P_{1} P_{2}>P P_{2}+\frac{1}{2} P P_{1}\) and \(P_{1} P_{3}>P P_{3}+\frac{1}{2} P P_{1}\).
Adding, obtain \(P_{1} P_{3}+P_{1} P_{2}>P P_{1}+P P_{2}+P P_{3},\) i.e., \(f\left(P_{1}\right)>f(P).\) Similarly, \(f\left(P_{2}\right)>f(P)\) and \(f\left(P_{3}\right)>f(P).\)
Combining with Problem 8, obtain the result.]
In a circle of radius \(R\) inscribe a polygon with \(n+1\) sides of maximum area.
[Outline: Let \(x_{1}, x_{2}, \ldots, x_{n+1}\) be the central angles subtended by the sides of the polygon. Then its area \(A\) is
\[\frac{1}{2} R^{2} \sum_{k=1}^{n+1} \sin x_{k},\]
with \(x_{n+1}=2 \pi-\sum_{k=1}^{n} x_{k}.\) (Why?) Thus all reduces to maximizing
\[f\left(x_{1}, \ldots, x_{n}\right)=\sum_{k=1}^{n} \sin x_{k}+\sin \left(2 \pi-\sum_{k=1}^{r_{k}} x_{k}\right),\]
on the condition that \(0 \leq x_{k}\) and \(\sum_{k=1}^{n} x_{k} \leq 2 \pi.\) (Why?)
These inequalities define a bounded set \(D \subset E^{n}\) (called a simplex). Equating all partials of \(f\) to \(0,\) show that the only critical point interior to \(D\) is \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right),\) with \(x_{k}=\frac{2 \pi}{n+1}, k \leq n\) (implying that \(x_{n+1}=\frac{2 \pi}{n+1},\) too). For that \(\vec{x},\) we get
\[f(\vec{x})=(n+1) \sin [2 \pi /(n+1)].\]
This value must be compared with the "boundary" values of \(f,\) on the "faces" of the simplex D (see Note 4).
Do this by induction. For \(n=2,\) Problem 6 shows that \(f(\vec{x})\) is indeed the largest when all \(x_{k}\) equal \(\frac{2 \pi}{n+1}.\) Now let \(D_{n}\) be the "face" of \(D,\) where \(x_{n}=0.\) On that face, treat \(f\) as a function of only \(n-1\) variables, \(x_{1}, \ldots, x_{n-1}\).
By the inductive hypothesis, the largest value of \(f\) on \(D_{n}\) is \(n \sin (2 \pi / n).\) Similarly for the other "faces." As \(n \sin (2 \pi / n)<(n+1) \sin 2 \pi /(n+1),\) the induction is complete.
Thus, the area \(A\) is the largest when the polygon is regular, for which
\[\left.A=\frac{1}{2} R^{2}(n+1) \sin \frac{2 \pi}{n+1}.\right]\]
Among all triangles of a prescribed perimeter \(2p,\) find the one of maximum area.
[Hint: Maximize \(p(p-x)(p-y)(p-z)\) on the condition that \(x+y+z=2 p\).]
Among all triangles of area \(A,\) find the one of smallest perimeter.
Find the shortest distance from a given point \(\vec{p} \in E^{n}\) to a given plane \(\vec{u} \cdot \vec{x}=c\) (Chapter 3, §§4-6). Answer:
\[\pm \frac{\vec{u} \cdot \vec{p}-c}{|\vec{u}|}.\]
[Hint: First do it in \(E^{3},\) writing \((x, y, z)\) for \(\vec{x}\).]