5.4: The Secrecy Requirement
After completing this section, you will be able to
- Explain the steps that businesses can take in order to protect trade secrets.
- Describe how trade secrets can be accessed through passive and active disclosure.
The second element necessary to gain trade secret protection is evidence that the trade secret’s owner made a reasonable effort to maintain secrecy. This flexible standard set out by the UTSA is often a point of contention, which results in litigation. To determine whether a piece of information rises to the level of a trade secret, courts often engage in a highly factual inquiry into the security measures taken by the owner to protect a trade secret. For as the Supreme Court has noted, if an owner has not taken steps to protect the trade secret, their proprietary interests may be diminished. xi
Under Company Lock and Key
Because the UTSA may be varied by any state, the level of secrecy required varies from state to state. For example, a version of the UTSA adopted by some states contains a provision that reduces the burden of secrecy placed on the owner of a trade secret. Specifically, this version of the UTSA provides that the existence of a trade secret is not negated merely because an employee or other person has acquired the trade secret without express or specific notice that it is a trade secret if, under all the circumstances, the employee or other person knows or has reason to know that the owner intends or expects the secrecy of the type of information comprising the trade secret to be maintained. Although this provision tips the scales in favor of trade secret protection, it should not be viewed as removing the affirmative obligation the UTSA places on one seeking protection of trade secrets. In both states that have and have not modified the UTSA in this manner, the measures taken to secure the information are key to finding whether the information will be protected as a trade secret.
Because trade secret protection is only given to information that is subject to reasonable efforts to maintain secrecy, every person or company that desires to protect its trade secrets under the UTSA should adopt and implement a trade secret protection plan. The specific elements of such a plan will vary depending on the nature of the intellectual property to be protected, as well as the circumstances surrounding its development and use. However, in taking reasonable efforts to protect the secrecy of the information, many different factors are considered. Because no rigid formula exists to determine if reasonable efforts to maintain secrecy are in place, courts look to many of the best practices, discussed below, to determine whether an owner of information made reasonable efforts to protect its secrecy.
Protection Plans
Memorializing a trade secret protection plan in writing helps to ensure that the business or person is indeed taking reasonable measures to protect the trade secret. The person or company seeking to protect its intellectual property through the use of trade secret law should maintain a written statement of its trade secret security policy. This statement can be written in the form of a policy manual and helps prevent situations in which “unwritten rules” are followed or rules are enforced in a lax manner and then eventually ignored. Moreover, should litigation become necessary, it is easier to persuade a court that the security procedures were indeed a primary concern if they are enumerated in writing. However, if security measures are merely written, but not followed, it may be difficult to prove reasonable efforts to maintain secrecy. xii
A trade secret protection plan should include a program for ensuring that employees are made aware of the trade secret status of information and that the employees are periodically reminded of their confidentiality obligations. Discussions between the employer and the employee may be appropriate. Writing an annual conflict of interest letter to remind management-level employees of their responsibilities in keeping information confidential may also suffice. An essential part of this program is making certain that employees are aware of and familiar with the written trade secret protection plan discussed above.
Employees should also be required to sign confidentiality agreements acknowledging that they have access to trade secret information, that they will not disclose or misappropriate that information, and that they will report all disclosures. Finally, “exit interviews” should be conducted for employees leaving the company. This interview provides an opportunity to remind employees of their written obligation not to disclose or misuse any trade secret information, and to extend an open door to discuss any issues or concerns that may arise in connection with their future employment. xiii
In addition to educating employees, reasonable efforts to protect information often include physically restricting access to trade secrets. Trade secret information should be treated differently from nonproprietary information and should, if possible, be physically segregated so that it is not readily accessible. For example, it is not uncommon to restrict visitor access to sales and administrative headquarters. In addition, access to the intellectual property should be restricted to those with a need to know or access the information. Depending on the nature of the intellectual property to be protected, reasonably restricting access to trade secrets may be as simple as segregating the information in separate file drawers, or may require measures as extreme as designating an entirely separate facility to house the intellectual property. xiv
If possible under the circumstances, physical security measures should be instituted, including the use of locks on doors, gates, and cabinets containing trade secrets. These physical security steps should be obvious to both employees and third parties who may have occasion to visit the area in which trade secrets are stored. xv These types of security measures communicate that an effort is being made to protect the information.
Strategies for Protecting Trade Secrets
Given the ease with which documents can be reproduced, trade secrets that are reduced to writing should be treated with special care. Documents containing trade secrets should be clearly labeled with shorthand instructions that are consistent with the written trade secret protection plan. For example, documents may be marked “secret” or “confidential.” xvi Access to these documents should be monitored and all persons allowed access to the documents should be informed of the meaning of the trade secret designation on the document and the relevant provisions of the written trade secret protection plan. This attention to detail should also be followed in arranging for proper destruction of the documents. xvii
Like trade secrets that are reduced to writing, trade secrets stored on a computer system are particularly susceptible to theft. Access to a computer system containing trade secrets should be restricted in accordance with the written trade secret protection plan and all physical computer- related items such as discs and printouts should contain the same trade secret legend used to identify documents containing intellectual property.
When allowing third parties, be they members of the public or business associates, access to a facility in which trade secrets are stored or used, care must be taken to ensure that trade secrets are not inadvertently disclosed. Disclosure can occur either passively or actively. Passive disclosure might occur by carelessly leaving documents containing trade secrets in open view when business associates from another company visit your facility. Active disclosure might occur during a guided tour of the facility when a tour guide reveals the nature of a project or process the company is working on or has recently perfected. xviii
The type of active trade secret disclosure that can occur outside of the facility where trade secrets are housed is through the dissemination of confidential information at trade shows, in magazine articles, in publications, in press releases, or during public speeches. Such disclosures often occur unwittingly because they are made by marketing executives or mid-level managers who are not themselves instrumental in the development of the trade secret and who are often charged with a goal somewhat at odds with maintaining the secrecy of intellectual property—that is, their duty often is to publicly disclose the company’s achievements. Similar difficulties might arise when engineers or those responsible for the creation of trade secrets exchange ideas with their colleagues from other companies at professional seminars. xix One method of curtailing such disclosures is to insist that all publications, articles, and speeches be reviewed and approved by a “filter” committee prior to dissemination. xx
Although the very nature of many businesses requires that a company disclose its trade secrets to prospective buyers, licensees, joint venturers, or other outsiders, this does not mean that a company must surrender all rights to its trade secrets. Rather, when the company enters into a transaction with a third party, it should cautiously monitor the flow of information to the third party and document the nature of the trade secrets exposed and the specific limited use to which they may be put. xxi You might consider putting outsiders on notice regarding the need to keep information confidential, and even require the signing of confidentiality agreements. The standard of what constitutes reasonable efforts to maintain secrecy varies substantially depending upon the circumstances and the information that is being protected. However, the different methods of maintaining secrecy discussed above all often help to show that the information in question qualifies as a trade secret, and deserves trade secret protection.
Footnotes
- xi See Ruckelshaus v. Monsanto Co. , 467 U.S. 986, 1002 (1984).
- xii See Lexis-Nexis v. Beer , 41 F. Supp.2d 950 (D. Minn. 1999).
- xiii Compare Electro-Craft Corp. v. Controlled Motion, Inc. , 332 N.W.2d 890 (Minn. 1983) (confidentiality agreement signed by employees failed to identify with sufficient particularity what were and were not trade secrets; plaintiff began conducting “exit interviews” ten days after litigation commenced) and Gordon Employment, Inc. v. Jewel , 356 N.W.2d 738 (Minn. Ct. App. 1984) (confidentiality never discussed with employees).
- xiv Id.
- xv Id.
- xvi See Gemisys Corp. v. Phoenix American, Inc. , 186 F.R.D. 551, 558-62 (N.D. Cal. 1999).
- xvii Electro -Craft Corp. v. Controlled Motion, Inc. , 332 N.W.2d 890 (Minn. 1983) (no technical documents were marked “confidential”).
- xviii Id. (confidentiality procedures inadequate where, among other things, plaintiff allowed public tour of its plants).
- xix See e.g. Mobile Med. Int’l Corp. v. United States , 95 Fed. Cl. 706, 739 (Fed. Cl. 2010) (finding that although plaintiff may have taken some measures to maintain the secrecy of its trade information, plaintiff’s assertions of confidentiality and the steps taken to protect its information were wholly undermined by its publicly displaying a model unit at a trade show).
- xx Gallowhur Chem. Corp. v. Schwerdle , 37 N.J. Super. 385, 117 A.2d 416, 423, 108 U.S.P.Q. 260 (1955) (information communicated in a public lecture revealed trade secret); Fishing Concepts, Inc. v. Ross , 1985 WL 1549 (D. Minn. 1985) (information claimed secret was revealed in article published in advertising magazine).
- xxi Id.


5.5: The Upper Half-Plane Model
The Poincaré disk model is one way to represent hyperbolic geometry, and for most purposes it serves us very well. However, another model, called the upper half-plane model, makes some computations easier, including the calculation of the area of a triangle.
Definition: Upper Half-Plane Model of Hyperbolic Geometry
The upper half-plane model of hyperbolic geometry has space \(\mathbb{U}\) consisting of all complex numbers \(z\) such that Im(\(z) \gt 0\text{,}\) and transformation group \(\cal U\) consisting of all Möbius transformations that send \(\mathbb{U}\) to itself. The space \(\mathbb{U}\) is called the upper half-plane of \(\mathbb{C}\text{.}\)
The Poincaré disk model of hyperbolic geometry may be transferred to the upper half-plane model via a Möbius transformation built from two inversions as follows:
- Invert about the circle \(C\) centered at \(i\) passing through \(-1\) and \(1\) as in Figure \(5.5.1\).
- Reflect about the real axis.
Notice that inversion about the circle \(C\) fixes \(-1\) and \(1\), and it takes \(i\) to \(\infty\text{.}\) Since reflection across the real axis leaves these image points fixed, the composition of the two inversions is a Möbius transformation that takes the unit circle to the real axis. The map also sends the interior of the disk into the upper half plane. Notice further that the Möbius transformation takes \(\infty\) to \(-i\text{;}\) therefore, by Theorem \(3.5.1\), the map can be written as
\[ V(z) = \dfrac{-iz + 1}{z - i}\text{.} \]
This Möbius transformation is the key to transferring the disk model of the hyperbolic plane to the upper half-plane model. In fact, when treading back and forth between these models it is convenient to adopt the following convention for this section: Let \(z\) denote a point in \(\mathbb{D}\text{,}\) and \(w\) denote a point in the upper half-plane \(\mathbb{U}\text{,}\) as in Figure \(5.5.2\). We record the transformations linking the spaces below.
The Möbius transformation \(V\) mapping \(\mathbb{D}\) to \(\mathbb{U}\text{,}\) and its inverse \(V^{-1}\text{,}\) are given by:
\[ w = V(z) = \dfrac{-iz + 1}{z - i} ~~~~\text{and}~~~~z = V^{-1}(w) = \dfrac{iw+1}{w+i}\text{.} \]
Some features of the upper half-plane model immediately come to light. Since \(V\) is a Möbius transformation, it preserves clines and angles. This means that the ideal points in the disk model, namely the points on the circle at infinity, \(\mathbb{S}^1_\infty\text{,}\) have moved to the real axis and that hyperbolic lines in the disk model have become clines that intersect the real axis at right angles.
Define the hyperbolic distance between two points \(w_1, w_2\) in the upper half-plane model, denoted \(d_U(w_1, w_2)\text{,}\) to be the hyperbolic distance between their pre-images in the disk model.
Suppose \(w_1\) and \(w_2\) are two points in \(V\) whose pre-images in the unit disk are \(z_1\) and \(z_2\text{,}\) respectively. Then,
\[ d_U(w_1, w_2) = d_H(z_1,z_2) = \ln((z_1, z_2 ; u, v))\text{,} \]
where \(u\) and \(v\) are the ideal points of the hyperbolic line through \(z_1\) and \(z_2\text{.}\) But, since the cross ratio is preserved under Möbius transformations,
\[ d_U(w_1,w_2) = \ln((w_1,w_2; p, q))\text{,} \]
where \(p,q\) are the ideal points of the hyperbolic line in the upper half-plane through \(w_1\) and \(w_2\text{.}\) In particular, going from \(w_1\) to \(w_2\) we're heading toward ideal point \(p\text{.}\)
For \(r > s > 0\) we compute the distance between \(ri\) and \(si\) in the upper half-plane model.
The hyperbolic line through \(ri\) and \(si\) is the positive imaginary axis, having ideal points \(0\) and \(\infty\text{.}\) Thus,
\begin{align*} d_U(ri, si) & = \ln((ri, si; 0, \infty))\\ & = \dfrac{ri - 0}{ri - \infty}\cdot\dfrac{si-\infty}{si-0}\\ & = \ln\left(\dfrac{r}{s}\right)\text{.} \end{align*}
Example \(5.5.2\): The Distance Between Any Two Points.
To find the distance between any two points \(w_1\) and \(w_2\) in \(\mathbb{U}\text{,}\) we first build a map in the upper half-plane model that moves these two points to the positive imaginary axis. To build this map, we work through the Poincaré disk model.
By the transformation \(V^{-1}\) we send \(w_1\) and \(w_2\) back to \(\mathbb{D}\text{.}\) We let \(z_1 = V^{-1}(w_1)\) and \(z_2 = V^{-1}(w_2)\text{.}\) Then, let \(S(z) = e^{i\theta}\dfrac{z-z_1}{1-\overline{z_1}z}\) be the transformation in \((\mathbb{D},{\cal H})\) that sends \(z_1\) to \(0\) with \(\theta\) chosen carefully so that \(z_2\) gets sent to the positive imaginary axis. In fact, \(z_2\) gets sent to the point \(ki\) where \(k = |S(z_2)| = |S(V^{-1}(w_2))|\) (and \(0 \lt k \lt 1\)). Then, applying \(V\) to the situation, \(0\) gets sent to \(i\) and \(ki\) gets sent to \(\dfrac{1+k}{1-k}i\text{.}\) Thus, \(V \circ S \circ V^{-1}\) sends \(w_1\) to \(i\) and \(w_2\) to \(\dfrac{1+k}{1-k}i\text{,}\) where by the previous example the distance between the points is known:
\[ d_U(w_1,w_2) = \ln(1+k) - \ln(1-k)\text{.} \]
Describing \(k\) in terms of \(w_1\) and \(w_2\) is left for the adventurous reader. We do not need to pursue that here.
We now derive the hyperbolic arc-length differential for the upper half-plane model working once again through the disk model. Recall the arc-length differential in the disk model is
\[ ds = \dfrac{2|dz|}{1-|z|^2}\text{.} \]
Since \(z = V^{-1}(w) = \dfrac{iw+1}{w+i}\) we may work out the arc-length differential in terms of \(dw\text{.}\) We will need to take the derivative of a complex expression, which can be done just as if it were a real valued expression. Here we go:
\begin{align*} ds & = \dfrac{2|dz|}{1-|z|^2}\\ & =\dfrac{2|d\bigg(\dfrac{iw+1}{w+i}\bigg)|}{1-\bigg|\dfrac{iw+1}{w+i} \bigg|^2} \tag{$z = \dfrac{iw+1}{w+i}$}\\ & =\dfrac{2|i(w+i)dw-(iw+1)dw|}{|w+i|^2}\bigg/\bigg[1-\dfrac{|iw+1|^2}{|w+i|^2}\bigg]\tag{chain rule}\\ & = \dfrac{4|dw|}{|w+i|^2-|iw+1|^2}\\ & = \dfrac{4|dw|}{(w+i)(\overline{w}-i)-(iw+1)(-i\overline{w}+1)}\\ & = \dfrac{4|dw|}{2i(\overline{w}-w)}\\ & = \dfrac{|dw|}{{ Im}(w)}\text{.} \end{align*}
This leads us to the following definition:
Definition: Length of a Smooth Curve
The length of a smooth curve \(\boldsymbol{r}(t)\) for \(a \leq t \leq b\) in the upper half-plane model \((\mathbb{U},{\cal U})\text{,}\) denoted \({\cal L}(\boldsymbol{r})\text{,}\) is given by
\[ {\cal L}(\boldsymbol{r}) = \int_a^b \dfrac{|\boldsymbol{r}^\prime(t)|}{\text{Im}(\boldsymbol{r}(t))}~dt\text{.} \]
Example \(5.5.3\): The Length of a Curve.
To find the length of the horizontal curve \(\boldsymbol{r}(t) = t + ki\) for \(a \leq t \leq b\text{,}\) note that \(\boldsymbol{r}^\prime(t) = 1\) and \(\text{ Im}(\boldsymbol{r}(t)) = k\text{.}\) Thus,
\[ {\cal L}(\boldsymbol{r}) = \int_a^b \dfrac{1}{k}~dt = \dfrac{b-a}{k}\text{.} \]
From the arc-length differential \(ds = \dfrac{dw}{\text{Im}(w)}\) comes the area differential:
Definition: Area of a Region
In the upper half-plane model \((\mathbb{U},{\cal U})\) of hyperbolic geometry, the area of a region \(R\) described in cartesian coordinates, denoted \(A(R)\text{,}\) is given by
\[ A(R) = \iint_R \dfrac{1}{y^2}~dxdy\text{.} \]
Example \(5.5.4\): The Area of a \(\dfrac{2}{3}\)-Ideal Triangle.
Suppose \(w \in \mathbb{U}\) is on the unit circle, and consider the \(\dfrac{2}{3}\)-ideal triangle \(1w\infty\) as pictured.
In particular, suppose the interior angle at \(w\) is \(\alpha\text{,}\) so that \(w = e^{i(\pi-\alpha)}\) where \(0 \lt \alpha \lt \pi\text{.}\)
The area of this \(\dfrac{2}{3}\)-ideal triangle is thus
\begin{align*} A & = \int_{\cos(\pi - \alpha)}^1 \int_{\sqrt{1-x^2}}^\infty \dfrac{1}{y^2}~dydx\\ & = \int_{\cos(\pi - \alpha)}^1 \dfrac{1}{\sqrt{1-x^2}}~dx\text{.} \end{align*}
With the trig substituion \(\cos(\theta) = x\text{,}\) so that \(\sqrt{1-x^2} = \sin(\theta)\) and \(-\sin(\theta)d\theta = dx\text{,}\) the integral becomes
\begin{align*} & =\int_{\pi - \alpha}^0\dfrac{-\sin(\theta)}{\sin(\theta)}~d\theta\\ & = \pi - \alpha\text{.} \end{align*}
It turns out that any \(\dfrac{2}{3}\)-ideal triangle is congruent to one of the form \(1w\infty\) where \(w\) is on the upper half of the unit circle (Exercise \(5.5.3\)), and since our transformations preserve angles and area, we have proved the area formula for a \(\dfrac{2}{3}\)-ideal triangle.
Theorem \(5.5.1\)
The area of a \(\dfrac{2}{3}\)-ideal triangle having interior angle \(\alpha\) is equal to \(\pi - \alpha\text{.}\)
Exercises
What becomes of horocycles when we transfer the disk model of hyperbolic geometry to the upper half-plane model?
What do hyperbolic rotations in the disk model look like over in the upper half-plane model? What about hyperbolic translations?
Give an explicit description of a transformation that takes an arbitrary \(\dfrac{2}{3}\)-ideal triangle in the upper half-plane to one with ideal points \(1\) and \(\infty\) and an interior vertex on the upper half of the unit circle.
Determine the area of the “triangular” region pictured below. What is the image of this region under \(V^{-1}\) in the disk model of hyperbolic geometry? Why doesn't this result contradict Theorem \(5.4.4\)?
Another type of block. Consider the four-sided figure \(pqst\) in \((\mathbb{D},{\cal H})\) shown in the following diagram. This figure is determined by two horocycles \(C_1\) and \(C_2\text{,}\) and two hyperbolic lines \(L_1\) and \(L_2\) all sharing the same ideal point. Note that the lines are orthogonal to the horocycles, so that each angle in the four-sided figure is \(90^{\circ}\text{.}\)
- By rotation about the origin if necessary, assume the common ideal point is \(i\) and use the map \(V\) to transfer the figure to the upper half-plane. What does the transferred figure look like in \(\mathbb{U}\text{?}\) Answer parts (b)-(d) by using this transferred version of the figure.
- Prove that the hyperbolic lengths of sides \(pq\) and \(st\) are equal.
- Let \(c\) equal the hyperbolic length of the leg \(pt\) along the larger radius horocycle \(C_1\text{,}\) and let \(d\) equal the hyperbolic length of the leg \(sq\) on \(C_2\text{.}\) Show that \(c = e^x d\) where \(x\) is the common length found in part (b).
- Prove that the area of the four-sided figure is \(c - d\text{.}\)


1.4: Fractions
By the end of this section, you will be able to:
- Simplify fractions
- Multiply and divide fractions
- Add and subtract fractions
- Use the order of operations to simplify fractions
- Evaluate variable expressions with fractions
A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.
Simplify Fractions
A fraction is a way to represent parts of a whole. The fraction \(\frac{2}{3}\) represents two of three equal parts (Figure \(\PageIndex{1}\)). In the fraction \(\frac{2}{3}\), the 2 is called the numerator and the 3 is called the denominator . The line is called the fraction bar.
Figure \(\PageIndex{1}\):
In the circle, \(\frac{2}{3}\) of the circle is shaded—2 of the 3 equal parts.
A fraction is written \(\dfrac{a}{b}\), where \(b\neq 0\) and
\(a\) is the numerator and \(b\) is the denominator .
A fraction represents parts of a whole. The denominator \(b\) is the number of equal parts the whole has been divided into, and the numerator \(a\) indicates how many parts are included.
Fractions that have the same value are equivalent fractions . The Equivalent Fractions
Property allows us to find equivalent fractions and also simplify fractions.
If \(a\), \(b\), and \(c\) are numbers where \(b\neq 0,c\neq 0\),
then \(\dfrac{a}{b}=\dfrac{a·c}{b·c}\) and \(\dfrac{a·c}{b·c}=\dfrac{a}{b}.\)
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
\(\dfrac{2}{3}\) is simplified because there are no common factors of \(2\) and \(3\).
\(\dfrac{10}{15}\) is not simplified because \(5\) is a common factor of \(10\) and \(15\).
We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.
Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.
Simplify \(\dfrac{−315}{770}\).
- Answer
-
Simplify \(−\dfrac{69}{120}\).
- Answer
-
\(−\dfrac{23}{40}\)
Simplify \(−\dfrac{120}{192}\).
- Answer
-
\(−\dfrac{5}{8}\)
We now summarize the steps you should follow to simplify fractions.
-
Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first. - Simplify using the Equivalent Fractions Property by dividing out common factors.
- Multiply any remaining factors.
Multiply and Divide Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions.
To multiply fractions, we multiply the numerators and multiply the denominators.
If \(a\), \(b\), \(c\), and \(d\) are numbers where \(b≠0\), and \(d≠0\), then
\[\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}\]
To multiply fractions, multiply the numerators and multiply the denominators.
When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example , we will multiply negative and a positive, so the product will be negative.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a , can be written as \(\dfrac{a}{1}\). So, for example, \(3=\dfrac{3}{1}\).
Multiply: \(−\dfrac{12}{5}(−20x).\)
- Answer
-
The first step is to find the sign of the product. Since the signs are the same, the product is positive.
Determine the sign of the product. The signs are the same, so the product is positive.
Write 20 x as a fraction. Multiply. Rewrite 20 to show the common factor 5 and divide it out.
Simplify.
Multiply: \(\dfrac{1}{13}(−9a)\).
- Answer
-
\(−33a\)
Multiply: \(\dfrac{13}{7}(−14b)\).
- Answer
-
\(−26b\)
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). Since 4 is written in fraction form as \(\frac{4}{1}\), the reciprocal of 4 is \(\frac{1}{4}\).
To divide fractions, we multiply the first fraction by the reciprocal of the second.
If \(a\), \(b\), \(c\), and \(d\) are numbers where \(b≠0\),\(c≠0\), and \(d≠0\), then
\[\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}⋅\frac{d}{c}\]
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say \(b≠0\), \(c≠0\), and \(d≠0\), to be sure we don’t divide by zero!
Find the quotient: \(−\dfrac{7}{18}÷(−\dfrac{14}{27}).\)
- Answer
-
\(−\dfrac{7}{18}÷(−\dfrac{14}{27})\) To divide, multiply the first fraction by the reciprocal of the second.
Determine the sign of the product, and then multiply.
Rewrite showing common factors. Remove common factors. Simplify.
Divide: \(−\dfrac{7}{27}÷(−\dfrac{35}{36})\).
- Answer
-
\(\dfrac{4}{15}\)
Divide: \(−\dfrac{5}{14}÷(−\dfrac{15}{28}).\)
- Answer
-
\(\dfrac{2}{3}\)
The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction .
A complex fraction is a fraction in which the numerator or the denominator contains a fraction.
\[\dfrac{\frac{6}{7}}{3} \quad \dfrac{\frac{3}{4}}{\frac{5}{8}} \quad \dfrac{\frac{x}{2}}{ \frac{5}{6}}\]
To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction \(\dfrac{\frac{3}{4}}{\frac{5}{8}}\) means \(\dfrac{3}{4}÷\frac{5}{8}.\)
Simplify: \(\dfrac{\dfrac{x}{2}}{ \dfrac{xy}{6}}\).
- Answer
-
\(\begin{array}{lc} \text{} & \dfrac{\dfrac{x}{2}}{ \dfrac{xy}{6}} \\[6pt] \text{Rewrite as division.} & \dfrac{x}{2}÷\dfrac{xy}{6} \\[6pt] \text{Multiply the first fraction by the reciprocal of the second.} & \dfrac{x}{2}·\dfrac{6}{xy} \\[6pt] \text{Multiply.} & \dfrac{x·6}{2·xy} \\[6pt] \text{Look for common factors.} & \dfrac{ \cancel{x}·3·\cancel{2}}{\cancel{2}·\cancel{x}·y} \\[6pt] \text{Divide common factors and simplify.} & \dfrac{3}{y} \end{array}\)
Simplify: \(\dfrac{\dfrac{a}{8}}{ \dfrac{ab}{6}}\).
- Answer
-
\(\dfrac{3}{4b}\)
Simplify: \(\dfrac{\dfrac{p}{2}}{ \dfrac{pq}{8}}\).
- Answer
-
\(\dfrac{4}{q}\)
Add and Subtract Fractions
When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.
If \(a\), \(b\), and \(c\) are numbers where \(c≠0\), then
\[\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} \text{ and } \dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}\]
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!
Add: \(\dfrac{7}{12}+\dfrac{5}{18}\).
- Answer
-
Add: \(\dfrac{7}{12}+\dfrac{11}{15}\).
- Answer
-
\(\dfrac{79}{60}\)
Add: \(\dfrac{13}{15}+\dfrac{17}{20}\).
- Answer
-
\(\dfrac{103}{60}\)
-
Do they have a common denominator?
- Yes—go to step 2.
-
No—rewrite each fraction with the LCD (least common denominator).
- Find the LCD.
- Change each fraction into an equivalent fraction with the LCD as its denominator.
- Add or subtract the fractions.
- Simplify, if possible.
We now have all four operations for fractions. Table summarizes fraction operations.
| Fraction Multiplication | Fraction Division |
|---|---|
| \(\dfrac{a}{b}⋅\dfrac{c}{d}=\dfrac{ac}{bd}\) | \(\dfrac{a}{b}÷\dfrac{c}{d}=\dfrac{a}{b}⋅\dfrac{d}{c}\) |
| Multiply the numerators and multiply the denominators | Multiply the first fraction by the reciprocal of the second. |
| Fraction Addition | Fraction Subtraction |
| \(\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}\) | \(\dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}\) |
| Add the numerators and place the sum over the common denominator. | Subtract the numerators and place the difference over the common denominator. |
|
To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed. |
When starting an exercise, always identify the operation and then recall the methods needed for that operation.
Simplify: ⓐ \(\dfrac{5x}{6}−\dfrac{3}{10}\) ⓑ \(\dfrac{5x}{6}·\dfrac{3}{10}\).
- Answer
-
First ask, “What is the operation?” Identifying the operation will determine whether or not we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.
ⓐ
\(\begin{array}{lc} \text{What is the operation? The operation is subtraction.} \\[6pt] \text{Do the fractions have a common denominator? No.} & \dfrac{5x}{6} − \dfrac{3}{10} \\[6pt] \text{Find the LCD of }6 \text{ and }10 & \text{The LCD is 30.}\\[6pt] {\begin{align*} 6 & =2·3 \\[6pt]
\; \; \underline{\; \; \; \; 10 \; \; \; \; } & \underline{=2·5 \; \; \; \;} \; \\[6pt]
\text{LCD} & =2·3·5 \\[6pt]
\text{LCD} & =30 \end{align*}} \\[6pt] \\ \\
\text{Rewrite each fraction as an equivalent fraction with the LCD.} & \dfrac{5x·5}{6·5}−\dfrac{3·3}{10·3}\\[6pt]
\text{} & \dfrac{25x}{30}−\dfrac{9}{30} \\[6pt]
\text{Subtract the numerators and place the} \\[6pt]
\text{difference over the common denominators.} & \dfrac{25x−9}{30} \\[6pt] \\ \\
\text{Simplify,if possible.There are no common factors.} \\[6pt]
\text{The fraction is simplified.} \end{array}\)ⓑ
\(\begin{array}{lc} \text{What is the operation? Multiplication.} & \dfrac{25x}{6}·\dfrac{3}{10} \\ \text{To multiply fractions,multiply the numerators} \\ \text{and multiply the denominators.} & \dfrac{25x·3}{6·10} \\ \text{Rewrite, showing common factors.} \\ \text{Remove common factors.} & \dfrac{\cancel{5} x · \cancel{3}}{2·\cancel{3}·2·\cancel{5}} \\ \text{Simplify.} & \dfrac{x}{4} \end{array}\)
Notice, we needed an LCD to add \(\dfrac{25x}{6}−\dfrac{3}{10}\), but not to multiply \(\dfrac{25x}{6}⋅\dfrac{3}{10}\).
Simplify: ⓐ \(\dfrac{3a}{4}−\dfrac{8}{9}\) ⓑ \(\dfrac{3a}{4}·\dfrac{8}{9}\).
- Answer
-
ⓐ \(\dfrac{27a−32}{36}\) ⓑ \(\dfrac{2a}{3}\)
Simplify: ⓐ\(\dfrac{4k}{5}−\dfrac{1}{6}\) ⓑ \(\dfrac{4k}{5}⋅\dfrac{1}{6}\).
- Answer
-
ⓐ \(\dfrac{24k−5}{30}\) ⓑ \(\dfrac{2k}{15}\)
Use the Order of Operations to Simplify Fractions
The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.
- Simplify the expression in the numerator. Simplify the expression in the denominator.
- Simplify the fraction.
Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.
\[\dfrac{−1}{3}=−\dfrac{1}{3} \; \; \; \; \; \; \dfrac{\text{negative}}{\text{positive}}=\text{negative}\]
\[\dfrac{1}{−3}=−\dfrac{1}{3} \; \; \; \; \; \; \dfrac{\text{positive}}{\text{negative}}=\text{negative}\]
For any positive numbers \(a\) and \(b\),
\[\dfrac{−a}{b}=\dfrac{a}{−b}=−\dfrac{a}{b}\]
Simplify: \(\dfrac{4(−3)+6(−2)}{−3(2)−2}\).
- Answer
-
The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.
\(\begin{array}{lc} \text{} & \dfrac{4(−3)+6(−2)}{−3(2)−2} \\[5pt] \text{Multiply.} & \dfrac{−12+(−12)}{−6−2} \\[5pt] \text{Simplify.} & \dfrac{−24}{−8} \\[5pt] \text{Divide.} & 3 \end{array}\)
Simplify:\(\dfrac{8(−2)+4(−3)}{−5(2)+3}\).
- Answer
-
4
Simplify: \(\dfrac{7(−1)+9(−3)}{−5(3)−2}\).
- Answer
-
2
Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.
Simplify: \(\dfrac{\left(\frac{1}{2}\right)^2}{4+3^2}\).
- Answer
-
Simplify: \(\dfrac{\left(\frac{1}{3}\right)^2}{2^3+2}\).
- Answer
-
\(\frac{1}{90}\)
Simplify: \(\dfrac{1+4^2}{\left(\frac{1}{4}\right)^2}\).
- Answer
-
272
- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator. Simplify if possible.
Simplify: \(\dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{3}{4}−\dfrac{1}{6}}\).
- Answer
-
It may help to put parentheses around the numerator and the denominator.
\(\begin{array}{lc}\text{} & \dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{3}{4}−\dfrac{1}{6}} \\[6pt] \text{Simplify the numerator }(LCD=6)\text{ and } \\[6pt] \text{simplify the denominator }(LCD=12). & \dfrac{\left(\dfrac{3}{6}+\dfrac{4}{6}\right)}{\left(\dfrac{9}{12}−\dfrac{2}{12}\right)} \\[6pt] \text{Simplify.} & \left(\dfrac{7}{6}\right)\left(\dfrac{7}{12}\right) \\[6pt] \text{Divide the numerator by the denominator.} & \dfrac{7}{6}÷\dfrac{7}{12} \\[6pt] \text{Simplify.} & \dfrac{7}{6}⋅\dfrac{12}{7} \\[6pt] \text{Divide out common factors.} & \dfrac{\cancel{7}⋅\cancel{6}⋅2}{ \cancel{6}⋅\cancel{7}⋅1} \\[6pt] \text{Simplify.} & 2 \end{array}\)
Simplify: \( \dfrac{\dfrac{1}{3}+\dfrac{1}{2}}{ \dfrac{3}{4}−\dfrac{1}{3}}\).
- Answer
-
2
Simplify: \(\dfrac{\dfrac{2}{3}−\dfrac{1}{2}}{ \dfrac{1}{4}+\dfrac{1}{3}}\).
- Answer
-
\(\frac{2}{7}\)
Evaluate Variable Expressions with Fractions
We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.
Evaluate \(2x^2y\) when \(x=\frac{1}{4}\) and \(y=−\frac{2}{3}\).
- Answer
-
Substitute the values into the expression.
Simplify exponents first. Multiply; divide out the common factors. Notice we write 16 as 2⋅2⋅42·2·4 to make it easy to remove common factors. Simplify.
Evaluate \(3ab^2\) when \(a=−\frac{2}{3}\) and \(b=−\frac{1}{2}\).
- Answer
-
\(−\dfrac{1}{2}\)
Evaluate \(4c^3d\) when \(c=−\frac{1}{2}\) and \(d=−\frac{4}{3}\).
- Answer
-
\(\dfrac{2}{3}\)
Access this online resource for additional instruction and practice with fractions.
- Adding Fractions with Unlike Denominators
Key Concepts
- If \(a\), \(b\), and \(c\) are numbers where \(b≠0,c≠0\), then
\(\dfrac{a}{b}=\dfrac{a·c}{b·c}\) and \(\dfrac{a·c}{b·c}=\dfrac{a}{b}.\)
-
How to simplify a fraction.
-
Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first. - Simplify using the Equivalent Fractions Property by dividing out common factors.
- Multiply any remaining factors.
-
Rewrite the numerator and denominator to show the common factors.
-
If \(a\), \(b\), \(c\), and \(d\) are numbers where \(b≠0\), and \(d≠0\), then
\(\dfrac{a}{b}·\dfrac{c}{d}=\dfrac{ac}{bd}\)
To multiply fractions, multiply the numerators and multiply the denominators.
-
If \(a\), \(b\), \(c\), and \(d\) are numbers where \(b≠0\),\(c≠0\), and \(d≠0\), then
\(\dfrac{a}{b}÷\dfrac{c}{d}=\dfrac{a}{b}⋅\dfrac{d}{c}\)
To divide fractions, we multiply the first fraction by the reciprocal of the second.
-
If \(a\), \(b\), and \(c\) are numbers where \(c≠0\), then
\(\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} \text{ and } \dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}\)
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
-
How to add or subtract fractions.
-
Do they have a common denominator?
- Yes—go to step 2.
-
No—rewrite each fraction with the LCD (least common denominator).
- Find the LCD.
- Change each fraction into an equivalent fraction with the LCD as its denominator.
- Add or subtract the fractions.
- Simplify, if possible.
-
Do they have a common denominator?
-
How to simplify an expression with a fraction bar.
- Simplify the expression in the numerator. Simplify the expression in the denominator.
- Simplify the fraction.
-
For any positive numbers \(a\) and \(b\),
\(\dfrac{−a}{b}=\dfrac{a}{−b}=−\dfrac{a}{b}\)
-
How to simplify complex fractions.
- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator. Simplify if possible.
Glossary
- complex fraction
- A fraction in which the numerator or the denominator is a fraction is called a complex fraction.
- denominator
- In a fraction, written \(\dfrac{a}{b}\), where \(b≠0\), the denominator \(b\) is the number of equal parts the whole has been divided into.
- equivalent fractions
- Equivalent fractions are fractions that have the same value.
- fraction
- A fraction is written \(\dfrac{a}{b}\), where \(b≠0\), and a is the numerator and \(b\) is the denominator. A fraction represents parts of a whole.
- least common denominator
- The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
- numerator
- In a fraction, written \(\dfrac{a}{b}\), where \(b≠0\), the numerator a indicates how many parts are included.
- reciprocal
- The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.


7: Atmospheric Moisture
By the end of the chapter you should be able to:
- Explain the phase changes of water.
- Determine the stability of air and its likelihood for cloud development
- Identify common cloud types.
- Describe how precipitation forms.
- Explain the global pattern of precipitation.
"It is better to read the weather forecast before we pray for rain." - Notebook; More Maxims of Mark , Johnson, 1927
Water in its various forms sustains life, transports energy and erodes the surface beneath our feet. Water is needed for cell growth, photosynthesis, the formation of soil, and to absorb and transport nutrients in plants and animals. Without water, living things could not survive. Energy is transported between the various spheres of the Earth system via phase changes of water. Nearly every portion of the Earth has been sculpted by the movement of water across the surface at some point in geologic history. Here we'll look at water in its various forms, as a gas, liquid, and solid. We'll investigate how it moves through and the vital role it plays in the Earth system. You will become familiar with the geographic distribution of precipitation and its impact on the environment.
The animation below shows the global pattern of cloud cover measured as the fraction of sky covered by clouds from January 2005 to April 2011. Cloud fraction ranges from 0 for cloudless skies (dark blue) to 1 for total cloud cover (white).
Seasonal changes in cloud fraction (Courtesy NASA Earth Observatory ( Source ))
Tropical regions appear to uniformly cloudy through out the year. The noticeable band of clouds circling the tropics shifts slightly to the north and south through the year. Much of subtropical Africa and Antarctica have relatively low cloud cover throughout the year. The midlatitudes have more variable cloud conditions. Why does this spatial and temporal pattern in cloud cover occur and what are the implications? Will this pattern change in the future?
Assess if you are ready for this chapter by "Getting Ready for Chapter 7".
Thumbnail: Cumulus clouds over the Atlantic Ocean. (Courtesy NOAA)


Page 3.1: Difference Quotient
A big part of calculus is being able to understand the notation when it comes to dealing with functions. For this reason, we will spend some time on discussing how to think about our notation.
Suppose we have the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) specified by \(f(x) = x^2 \). To recap a few things, this notation is telling us that the domain is \( \mathbb{R} \) meaning that the \(x\) values can be any real number. When we plug in these \(x\) values into \(f\), the outputs are also real numbers and hence our codomain is \( \mathbb{R} \).
Allow us to consider the following exercise.
Suppose \( f: \mathbb{R} \rightarrow \mathbb{R} \) specified by \(f(x) = x^2 \). Find the following:
1) \( f(1) \)
2) \( f(2) \)
3) \( f(3) \)
4) \( f(a) \)
5) \( f(b) \)
6) \( f(a+h) \)
7) \( \frac{f(a+h) - f(a)}{h} \)
- Answer
-
1) The notation \(f(1)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by a 1. Doing so yields \( f(1) = 1^2 = 1 \)
2) The notation \(f(2)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by a 2. Doing so yields \( f(2) = 2^2 = 4 \)
3) The notation \(f(3)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by a 3. Doing so yields \( f(3) = 3^2 = 9\)
4) The notation \(f(a)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by an \(a\). Doing so yields \(f(a) = a^2 \).
5) The notation \(f(b)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by a \(b\). Doing so yields \(f(b) = b^2 \).
6) The notation \( f(a+h)\) is telling us that wherever we see \(x\) in the function \(f(x) = x^2 \), we should replace the \(x\) by an \( a+h\). Doing so yields \(f(a+h) = (a+h)^2 = a^2 + 2ah + h^2 \).
7) Putting Parts 4 and 6 together yields: \begin{align*} \frac{f(a+h) - f(a)}{h} &= \frac{(a+h)^2 - a^2}{h } \\ &= \frac{a^2 + 2ah + h^2 - a^2}{h} \\&= \frac{2ah + h^2}{h} \\ &= 2a + h \end{align*}
The quantity we considered in Part 7 has a special name - the difference quotient.
Definition: Given a function \(f\), we refer to \( \dfrac{f(a+h) - f(a)}{h} \) as the difference quotient.
More will be said about the difference quotient in the coming days but for now, allow us to compute a few more difference quotients.
Compute the difference quotient of \( f(x) = x \).
- Answer
-
\( 1 \).
Compute the difference quotient of \( f(x) = x^3 - x \).
- Answer
-
\( 3a^2 + 3ah + h^2 -1 \).
Compute the difference quotient of \( f(x) = \sqrt{x} \). Hint: to simplify, rationalize the numerator!
- Answer
-
\( \frac{1}{ \sqrt{a+h} - \sqrt{a}} \).
Compute the difference quotient of \( f(x) = \dfrac{1}{x} \).
- Answer
-
\( - \dfrac{1}{(a+h)(a)} \).
Compute the difference quotient of \( f(x) = \dfrac{1-x}{2+x} \).
- Answer
-
\( \dfrac{-3}{(2+a+h)(2+a)} \).


2.5: Formation of Minerals
In order for a mineral crystal to grow, the elements needed to make it must be present in the appropriate proportions, the physical and chemical conditions must be favourable, and there must be sufficient time for the atoms to become arranged.
Physical and chemical conditions include factors such as temperature, pressure, presence of water, pH, and amount of oxygen available. Time is one of the most important factors because it takes time for atoms to become ordered. If time is limited, the mineral grains will remain very small. The presence of water enhances the mobility of ions and can lead to the formation of larger crystals over shorter time periods.
Most of the minerals that make up tehe rocks around us formed through the cooling of molten rock, known as magma . At the high temperatures that exist deep within Earth, some geological materials are liquid. As magma rises up through the crust, either by volcanic eruption or by more gradual processes, it cools and minerals crystallize. If the cooling process is rapid (minutes, hours, days, or years), the components of the minerals will not have time to become ordered and only small crystals can form before the rock becomes solid. The resulting rock will be fine-grained (i.e., with crystals less than 1 mm). If the cooling is slow (from decades to millions of years), the degree of ordering will be higher and relatively large crystals will form. In some cases, the cooling will be so fast (seconds) that the texture will be glassy, which means that no crystals at all form. Volcanic glass is not composed of minerals because the magma has cooled too rapidly for crystals to grow, although over time (millions of years) the volcanic glass may crystallize into various silicate minerals.
Minerals can also form in several other ways:
- Precipitation from aqueous solution (i.e., from hot water flowing underground, from evaporation of a lake or inland sea, or in some cases, directly from seawater) (see Exercise 2.7 below),
- Precipitation from gaseous emanations,
- Metamorphism: formation of new minerals directly from the elements within existing minerals under conditions of elevated temperature and pressure,
- Weathering: during which minerals unstable at Earth’s surface may be altered to other minerals,
- Organic formation: formation of minerals within shells (primarily calcite) and teeth and bones (primarily apatite) by organisms (these organically formed minerals are still called minerals because they can also form inorganically).
Opal is a mineraloid (i.e., not an actual mineral) because although it has all of the other properties of a mineral, it does not have a specific structure. Pearl is not a mineral because it can only be produced by organic processes.
Place about ½ teaspoon (~2.5 cm 3 ) of any kind of table salt into a small bowl. Add about 2 teaspoons (~10 mL) of very hot water and swirl it around for a few minutes until all or almost all of the salt has dissolved. (Be careful not to splash yourself with the hot water.)
Place the bowl in a safe place (windowsill, bookshelf), and check back every 24 hours to see what has happened. Depending on the level of humidity in the room, you should see crystals forming within 24 hours, and all of the water should be gone, with reasonably large crystals formed, within about 3 days. They should look a little like those shown here. In other words, they should be cubes.
Now try the same experiment again, but this time put the salt and water into a small sauce pan on the stove top at the lowest heat possible. Within 10 to 20 minutes all of the water should be gone and you should be left with some very small salt crystals—too small to even see their shapes. It takes time for mineral crystals to form.
Where does lithium come from?
The global demand for lithium has increased dramatically in the past decade, and will increase even more in the future as long as there is increasing demand for lithium-ion batteries in electronic devices, electric vehicles and for storage of solar- and wind-generated energy. Most of the world’s lithium supply comes from salt lakes ( salars in Spanish) like the one shown below in southwestern Bolivia.
Media Attributions
- Figure \(\PageIndex{1}\): © Steven Earle. CC BY.
- Figure \(\PageIndex{2}\): © Isaac Earle. CC BY.


9.10: The E2 Reaction and Cyclohexane Conformation
-
- Last updated
- Save as PDF
Objectives
After completing this section, you should be able to
- identify anti periplanar arrangements of atoms in substituted cyclohexanes.
- determine which cyclohexane conformation will generate a specific anti periplanar arrangement.
Stereochemical Requirements of the E2 Reaction
E2 elimination reactions of certain isomeric cycloalkyl halides show unusual rates and regioselectivity which can provide important supporting evidence that anti-periplanar is the preferred orientation of reactant species in the E2 transition state. Unlike open chain structures, cyclic compounds generally restrict the spatial orientation of ring substituents to relatively few arrangements. The compounds used here all have six-membered rings, so to achieve the anti-periplanar orientation required for the E2 reaction both the halogen and the adjacent hydrogen much assume an axial conformation.
For the compound, 2-methyl-1-chlorocyclohexane, the cis to the trans isomers have distinctly different reaction rates and form different preferred products. For example, trans -2-methyl-1-chlorocyclohexane reacts with alcoholic KOH at a much slower rate than does its cis -isomer. For trans - methyl-1-chlorocyclohexane, to obtains a chair conformation which places the chlorine substituent in the axial orientation required for E2 elimination, the methyl substituents is also forced into an axial position. Having both substituents in the axial position makes this chair conformer of the trans -isomer much less stable and presents an energy barrier that must be overcome for an E2 reaction to occur. When cis- 2-methyl-1-chlorocyclohexane obtains a chair conformation which places its chlorine substituent in an axial orientation, the methyl substituent is forced into an equatorial orientation. Having one substituent axial and one equatorial makes this chair conformation of the cis-isomer lower in energy and thus easier to form. Consequently, the E2 reaction rates with the trans- isomer are slower than with the cis -isomer.
Furthermore, the product from E2 elimination of the trans -isomer is 3-methylcyclohexene (not predicted by Zaitsev's rule ), whereas the cis -isomer gives the 1-methylcyclohexene as the preferred product (as predicted by Zaitsev's rule). The trans-isomer only has one adjacent hydrogen which can obtain an axial orientation along with the chlorine. The adjacent hydrogen which would lead to a 1-methylcyclohexene product cannot obtain the diaxial, anti-periplanar orientation with chlorine so an E2 reaction cannot occur. This makes 3-methylcyclohexene the preferred E2 elimination product of the trans -isomer showing that the anti-periplanar orientation requirement of E2 reactions is more important in determining products for this reaction than Zaitsev's rule.
In the cis- isomer the smaller chlorine atom assumes an axial position in the more stable chair conformation, and here there are two adjacent axial hydrogens. Removing one hydrogen forms the product, 3-methylcyclohexene, while removing the other hydrogen forms 1-methylcyclohexene. Here Zaitsev's rule determines that the more substituted alkene, 1-methylcyclohexene, is the preferred product.
Which isomer would be expected to undergo E2 elimination quicker cis or trans-1-bromo-4-tert-butylcyclohexane? Explain your answer.
In the case of the 1-bromo-4-tert-butylcyclohexane isomers, the tert-butyl group is so large that it will always assume an equatorial orientation, leaving the bromine to be axial in the
cis
-isomer and equatorial in the
trans
-isomer. Because of symmetry, the two axial adjacent hydrogens in the
cis
-isomer react equally with base, resulting in rapid elimination to the same alkene (actually a racemic mixture). This reflects the fixed anti orientation of these hydrogens to the bromine atom. To assume a conformation having an axial bromine, the
trans
-isomer must tolerate serious crowding distortions. Such conformers are therefore present in extremely low concentration, and the rate of elimination is very slow. Indeed, substitution by hydroxide anion predominates in this situation.
Which of the following compounds will react faster in an E2 reaction;
trans
-1-bromo-2-isopropylcyclohexane or
cis
-1-bromo-2-isopropylcyclohexane?
The
cis
isomer will react faster than the
trans.
The
cis
isomer has two possible perpendicular hydrogen in which it can eliminate from.
Worked Exercise\(\PageIndex{1}\)
Exercise \(\PageIndex{1}\)


4.8: Methodology- Sources and Problem
How do we know what we know about the Greek world? Studying Greek history, especially in its earliest periods, is like putting together a puzzle, most of whose pieces are missing, and some pieces from another puzzle have also been added in for good measure. Greek history requires careful consideration of a wide range of sources, which fall into two broad categories: literary sources (including both fiction and non-fiction), and material culture. In a world where social media, television, print newspapers, and radio did not exist, historians need to use other sources, such as epics, lyric poetry, and drama.
The earliest literary sources for Greek history are the Homeric epics, Iliad and Odyssey. Composed orally before the existence of the Greek alphabet, the epics were not written down until sometime in the sixth century BCE. The stories most likely do not reflect the society of any particular Greek city-state in any one period, but rather consist of an amalgam of features from the Bronze Age to the early Archaic Period. More than any other literary source, the Homeric Epics influenced the mentality of the Greeks in thinking about war and what it means to be a hero.
The overwhelming majority of surviving literature is from Athens, which brings it own challenges. The Greeks of the Classical period considered Sparta to be as great a city as Athens. However, Spartans valued military valor over all else, so they did not cultivate arts and letters the way Athenians did. As a result, the only literary sources from Sparta are the works of two poets, Alcman and Tyrtaeus. Tyrtaeus’ military elegies glorify heroic death in battle over life without honor and were likely sung by Spartan warriors as they marched into battle.
Several genres of non-fiction survive too. The works of three major historians survive from Classical Athens. Herodotus , dubbed the Father of History, wrote the Histories about the Persian Wars in mid-fifth century BCE. Thucydides , an Athenian general in the Peloponnesian War, wrote a history of the Peloponnesian War over the course of the war (431 – 404 BCE). Finally, Xenophon wrote a history of the end of the Peloponnesian War, starting with 411 BCE, where Thucydides’ work ended, and into the fourth century.
In addition to the works of the historians, philosophical treatises – most notably, those of Plato and Aristotle – provide crucial insight into the political thought, moral values, and perceptions of the world in the late fifth and fourth centuries BCE. Approximately 100 surviving courtroom speeches provide a window into the Athenian legal system. Finally, the Hippocratic corpus, a series of medical treatises and physicians’ journals from the Classical period, help us to understand the Greeks’ views of the human body and diseases.
In addition to the geographical restrictions of these sources, it is also important to note two other key limitations of the available evidence. First, virtually all of the literary sources were written by men and provide very little evidence of the lives and perspectives of women in the Greek world, except as seen through the eyes of men. Second, most of the authors were wealthy and socially prominent individuals; thus, their perspective does not reflect that of less affluent citizens and slaves.
Archaeological evidence allows historians to fill some of the gaps in the literary evidence, as well as provide key information about different aspects of everyday life in different city-states. For example, the excavations of the sixth-century BCE colony Megara Hyblaea in Sicily show that Greek colonists were interested in city planning and in equality of citizens, as demonstrated by the equal size of the lots.
Material finds, such as pottery remains, in different sites across the Mediterranean allow historians to map trading routes. For instance, Figures \(\PageIndex{1}\) and \(\PageIndex{2}\) show vases that were made in Euboea and Corinth respectively, but were found in Greek colonies in Italy. In addition, images on pottery provide information about stories and myths that have entered popular culture and that sometimes reflect further aspects of everyday life. The prevalence of images of women gathering at public fountains on Athenian hydriae (water pots) from the late sixth century BCE shows the importance of the public fountains for the social life of women in Athens in the period.
Finally, written archaeological sources, such as inscriptions on stone or pottery shards from all over the Greek world, and papyri from Hellenistic Egypt, are the equivalent of documentary archives from the ancient world. The evidence from epigraphy (inscriptions) includes laws that were written on large stones and set up in public, such as the monumental law-code from Gortyn, Crete, and lists of war-dead, as well as private tomb inscriptions.
Papyri include such private documents as prenuptial agreements (among the strangest are prenuptial documents for brother-sister marriages – legal in Egypt but nowhere else in the Greek world), divorce documents, loans, and village police reports (cattle theft appears to have been a serious problem in the Faiyum in the Hellenistic Period!).
Taken together, the literary and archaeological sources allow the historian to complete much more of the puzzle than would have been possible with just one of these types of sources. Still, significant gaps in knowledge remain nevertheless, and are, perhaps, one of the joys of studying ancient history: the historian gets to play the part of a sleuth, attempting to reconstruct the history of events based on just a few available clues.


1.7: Summary
The topics covered in this chapter can be summarized as follows:
| Section | Summary |
|---|---|
| 1.1 What is Geology? | Geology is the study of Earth. It is an integrated science that involves the application of many of the other sciences, but geologists also have to consider geological time because most of the geological features that we see today formed thousands, millions, or even billions of years ago. |
| 1.2 Why Study Earth? | Geologists study Earth out of curiosity and for other more practical reasons, including understanding the evolution of life on Earth, searching for resources, understanding risks from geological events such as earthquakes, volcanoes, and slope failures, and documenting past environmental and climate changes so that we can understand how human activities are affecting Earth. |
| 1.3 What Do Geologists Do? | Geologists work in the resource industries and in efforts to protect our natural resources and the environment in general. They are involved in ensuring that risks from geological events (e.g., earthquakes) are minimized and that the public understands what the risks are. Geologists are also engaged in fundamental research about Earth and in teaching. |
| 1.4 Minerals and Rocks | Minerals are naturally occurring, specific combinations of elements that have particular three-dimensional structures. Rocks are made up of mixtures of minerals and can form though igneous, sedimentary, or metamorphic processes. |
| 1.5 Fundamentals of Plate Tectonics | The Earth’s mantle is convecting because it is being heated from below by the hot core. Those convection currents contribute to the movement of tectonic plates (which are composed of the crust and the uppermost rigid mantle). Plates are formed at divergent boundaries and consumed (subducted) at convergent boundaries. Many important geological processes take place at plate boundaries. |
| 1.6 Geological Time | Earth is approximately 4,570,000,000 years old; that is, 4.57 billion years or 4.57 Ga or 4,570 Ma. It’s such a huge amount of time that even extremely slow geological processes can have an enormous impact. |
Answers to Review Questions at the end of each chapter are provided in Appendix 2 .
- In what way is geology different from the other sciences, such as chemistry and physics?
- How would some familiarity with biology be helpful to a geologist?
- List three ways in which geologists can contribute to society.
- Describe the lattice structure and elemental composition of the mineral halite.
- In what way is a mineral different from a rock?
- What is the main component of Earth’s core?
- What process leads to convection in the mantle?
- How does mantle convection contribute to plate tectonics?
- What are some of the processes that take place at a divergent plate boundary?
- Dinosaurs first appear in the geological record in rocks at 225 Ma and then disappear at 65 Ma. For what proportion (%) of geological time did dinosaurs exist?
- If a typical rate for the accumulation of sediments is 1 mm/year, what thickness (meters) of sedimentary rock could accumulate over a period of 30 million years?


5.4: Becker’s (1965) Household Production Model
-
- Last updated
- Save as PDF
Another model of consumer behavior is Becker’s (1965) household production model. Here, the consumer has a utility function that depends on two things:
- household produced goods and services, and
- time spent in leisure activities.
To illustrate the idea of Becker’s model, suppose that the household produced good in question is a meal. To be more specific, let us suppose this meal consists of pasta with a marinara sauce. Ultimate utility comes from consuming the meal. However, in order to obtain this utility, the consumer will need to:
- purchase ingredients from the marketplace,
- spend time in the kitchen and dining room, and
- have some knowledge about cooking. This can be called human capital.
A key point of the Becker model is that items the consumer buys at the supermarket provide him or her with no utility per se . In other words, Inputs such as a bag of pasta, tomatoes, garlic cloves, and oregano leaves do not satisfy the consumer. Satisfaction is not obtained until these ingredients have been combined with time and human capital to produce a meal.
Almost any human activity could be characterized in terms of the household production model. Even a good night’s rest could be viewed as household production. A good night’s requires purchased inputs (shelter, a bed, linens, and possibly sleeping pills). It requires an investment in time (normally 6 to 8 hours). Finally, it might require some human capital (the ability to clear one’s mind of the pressures of the day and relax into sleep). Similarly a sporting event, like a football game, could be characterized as another household produced good. This requires purchased inputs (tickets, television, cable or satellite subscription, snacks), time (several hours), and human capital (some knowledge of the rules and strategies of the game). The point being made here is that the household production idea is broadly applicable to human activities. However, in actual implementations of the Becker model, activities such as sleep or enjoying sporting events might be lumped into leisure time.
It would be wrong to suppose that only activities within the home would classify as household production and fall within the Becker framework. Suppose that instead of making the pasta dinner in the home kitchen, the consumer decides to purchase it from a restaurant. Utility is ultimately derived from consuming the prepared meal. In this case, the consumer requires purchased goods (transportation to the restaurant and a pasta dinner ordered from the menu). The consumer must also invest time to enjoy the meal. Human capital in the case of a restaurant meal involves knowledge about quality and service of competing restaurants and related products that go along with the meal (e.g., a suitable beverage or appetizer that complements the main course). Even though a restaurant meal is not “homemade”, it could easily conceptualize it within the household production model.
The formal elements of the Becker model include a utility function:
\(U = f[(hh \: produced \: goods), \: (leisure \: time)]\)
The consumer maximizes utility subject to:
-
A hh production function:
\(hh \: goods = g[(time \: in \: hh \: production), \: (market \: goods), \: (human \: capital)]\)
- Income-use constraint:
\(money \: income \geq spending \: on \: market \: goods\)
- Income-source constraint:
\(money \: income = (wage) \times (time \: in \: labor \: force) + (other \: income)\)
-
Time-use constraint:
\(total \: time = (time \: in \: hh \: production) + (time \: in \: labor \: force) + (leisure \: time)\)
A main advantage of Becker’s model is that it incorporates the value of the consumer’s time. The income-use constraint in the Becker model is essentially the same as in the neoclassical model. The consumer cannot spend more on purchased inputs than his or her available income. However, the model differs in that it incorporates where the consumer gets income (the income-source constraint) along with the potential uses for the consumer’s time. The income-use, income-source, and time-use constraints can be combined into one overall constraint known as the full time-income constraint below.
\((wage) \times (total \: time) + (other \: income) \geq (wage) \times (time \: in \: hh \: production) + (wage) \times (leisure \: time) + (spending \: on \: market \: goods)\)
The left side of the full time-income constraint represents income potential. The right side of the full time-income constraint represents how income potential is used.
A key insight from Becker’s model for food marketing is that time spent in the kitchen has a very real cost. Many consumers are spending more time in the workforce to generate income. They are using this income to save time in household production by buying goods that require minimal preparation. Senauer, Asp, and Kinsey (1991) make a distinction between time-intensive goods and expenditure-intensive goods. A cake from the bakery costs more money than a couple of cups of flour, vegetable oil, eggs, and a bit of baking powder and cocoa. However, when you consider the opportunity cost of making a cake from scratch, the bakery cake might be the cheapest way to obtain this good. In this example, the bakery cake is an expenditure-intensive good while the raw ingredients to make a cake from scratch are time-intensive goods.
One of the best examples of Becker’s model in action is the growth in food-away-from-home expenditures (the overwhelming majority of which is accounted for by food consumed at restaurants) relative to growth in food purchased for at-home consumption. Figure \(\PageIndex{1}\) shows these data from 1970 through 2014. In 2014 food-away from home expenditures actually surpassed food-at-home expenditures. Insights from Becker’s model are also reflected in the product assortments contained in supermarkets. New or recently renovated supermarkets tend to have larger deli and prepared food sections.


11.1: Reading- Competitor Impact on Pricing
Introduction
It’s important to remem ber that pricing is just one component of the marketing mix, and even very specific pricing decisions need to take into account the other components. This is particularly true in a competitive marketplace. Actions by different competitors integrate all elements of the marketing mix and do not focus on price alone. A competitor might make a change to a product or initiate a promotion that impacts customers’ perceptions of value and, therefore, their perceptions of price.
Competitive Pricing
Once a business decides to use price as a primary competitive strategy, there are many well-established tools and techniques that can be employed. The pricing process normally begins with a decision about the company’s pricing approach to the market. Price is a very important decision criterion that customers use to compare alternatives. It also contributes to the company’s position. In general, a business can price its offering to match its competition, or it can price higher or price lower. Each has its pros and cons.
Pricing to Meet Competition
Many organizations attempt to establish prices that, on average, are the same as those set by their more important competitors. Automobiles of the same size with comparable equipment and features tend to have similar prices, for instance. This strategy means that the organization uses price as an indicator or baseline. Quality in production, better service, creativity in advertising, or some other element of the marketing mix is used to attract customers who are interested in products in a particular price category.
The key to implementing a strategy of meeting competitive prices is to have an accurate definition of competition and a knowledge of competitors’ prices. A maker of handcrafted leather shoes is not in competition with mass producers. If he/she attempts to compete with mass producers on price, higher production costs will make the business unprofitable. A more realistic definition of competition in this case would be other makers of handcrafted leather shoes. Such a definition along with an understanding of competitors’ prices would enable management to put the strategy into effect.
The banking industry often uses this strategy by using technology to actively monitor competitors’ rates, fees, and packages in order to adjust their own prices.
Pricing Above Competitors
Pricing above competitors can be rewarding to organizations, provided that the objectives of the policy are clearly understood and the marketing mix is developed in such a way that the policy can be successfully implemented by management.
Pricing above competition generally requires a clear advantage on some nonprice element of the marketing mix. In some cases, that advantage may be due to a high price-quality association on the part of potential buyers.
Betting on that advantage is increasingly dangerous in today’s information-rich environment, however. Online shoppers can get quick price comparisons and read customer or expert reviews to evaluate other elements of the value proposition. This is true for both business-to-consumer and business-to-business offerings. Many consumers also take advantage of their smartphones when they shop: it’s easy enough to stand in one store and compare price and distribution options for the same product and for competitive products. Customers’ access to information puts more pressure on marketers to understand customer value and provide an offering whose price, relative to competitors’ prices, contributes to the value.
You’ll recall our earlier example of Nike using a strategy of raising prices—while its competitors were holding pricing flat or reducing prices—because its analysis showed that it was providing sufficient value to sustain a higher price.
Pricing Below Competitors
While some firms are positioned to price above competition, others wish to carve out a market niche by pricing below competitors. The goal of such a policy is to realize a large sales volume through a lower price and lower profit margins. By controlling costs and reducing services, these firms are able to earn an acceptable profit, even though profit per unit is usually less.
Such a strategy can be effective if a significant segment of the market is price sensitive and/or the organization’s cost structure is lower than competitors’. Costs can be reduced by increased efficiency, economics of scale, or by reducing or eliminating such things as credit, delivery, and advertising. For example, if a firm could replace its field sales force with telemarketing or online access, this function might be performed at lower cost. Such reductions often involve some loss in effectiveness, so the trade-off must be considered carefully.
One of the worst outcomes that can result from pricing lower than a competitor is a “price war.” Price wars usually occur when a company believes that price-cutting will increase market share, but it doesn’t have a true cost advantage. Price wars are often caused by companies misreading or misunderstanding competitors. Typically, they are overreactions to threats that either are nonexistent or are not as big as they seem. You will remember our example of the airline price war, in which the stock price of airlines plummeted because stockholders reacted negatively to price reductions, fearing that a price war would eliminate profits and put the health of the industry at risk.
In the module on product marketing we described the ride-sharing service Uber. Uber has successfully undercut the taxi industry with a product that improves services while lowering prices, which has led to extremely rapid growth and success for the company. When lower prices are part of a complete, compelling value proposition, pricing can provide a powerful solution and create a challenging competitive environment for existing players.
Contributors and Attributions
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Chapter 9, Pricing, from Introducing Marketing. Authored by : John Burnett. Provided by : Global Text. Located at : solr.bccampus.ca:8001/bcc/file/ddbe3343-9796-4801-a0cb-7af7b02e3191/1/Core%20Concepts%20of%20Marketing.pdf.. License : CC BY: Attribution
- Shoe Store in Trinidad. Authored by : Bud Ellison. Located at : https://www.flickr.com/photos/budellison/15920712043/ . License : CC BY: Attribution


32.2: Biological Terrorism
By the end of this section, you should be able to:
- Define biological terrorism.
- Categorize bioterrorism agents and diseases.
- Describe how Category A agents have been or could be used as bioterrorist weapons.
- Describe strategies for identifying and managing casualties of biological terrorism.
- Explain the nurse’s role in identifying and managing casualties of biological terrorism as a component of emergency preparedness and disaster response.
- Explain the nurse’s role as a first line of defense against biological terrorism.
Terrorist attacks use force or violence against civilians, cause mass property damage, and create fear and panic among the public. Chemical, biological, radiological, and nuclear (CBRN) agents have been used as weapons against humans, causing devastating health consequences (Bland, 2014). Chemical agents include nerve agents, vesicant agents, cyanides, pulmonary agents, incapacitants, toxic industrial chemicals, pharmaceuticals, and riot-control agents. Biological agents include live pathogens, such as bacteria, fungi, and viruses, and toxins derived from bacteria, fungi, plants, and animals. Radiological agents have ionizing radiation present, such as alpha, beta, gamma, and neutron participles. Nuclear agents are materials used to generate nuclear power.
Biological terrorism , or bioterrorism , is the intentional release of biological agents into the atmosphere or environment to threaten a civilian population. Humans and animals absorb the biological agent through inhalation, ingestion, skin, mucous membranes, or eyes (Bland, 2014). Bioterrorism can cause disease or death among a large number of people or livestock and can contaminate food or water supplies. Inhalation of biological agents poses the greatest risk for multiple casualties because the agent is transferred through the air and can reach larger numbers of the population (Bland, 2014).
Categorization of Bioterrorism Agents
As discussed in Pandemics and Infectious Disease Outbreaks, biological agents that would likely be used as agents of bioterrorism include bacteria, viruses, and toxins. The CDC categorizes these agents by risk threat (see Table 32.2 for examples). Most biological agents are difficult to grow and maintain, as many are destroyed quickly when exposed to sunlight and other environmental factors (CDC, 2018; Siegel et al., 2023). Other agents, such as anthrax spores, are heartier and live much longer, especially on surfaces. Biological agents can be dispersed by spraying them into the air, infecting animals that carry the disease to humans, and contaminating food and water (Bland, 2014). Delivery methods include:
- Aerosols: Biological agents are circulated into the air via a fine mist that may carry for miles. Inhaling the agent may cause disease in people or animals.
- Animals: Diseases are spread by insects and animals, such as fleas, mice, flies, mosquitoes, and livestock.
- Food and water contamination: Pathogenic organisms and toxins are released into food and water supplies. If contamination is known, most microorganisms can be killed and toxins deactivated by cooking food and boiling water for 1 minute or longer.
- Person-to-person: Communicable diseases such as smallpox and plague can be spread via close human contact.
| Category | Definition/Impact on Public Health | Agent/Disease |
|---|---|---|
| Category A—Highest priority |
|
|
| Category B—Second-highest priority |
|
|
| Category C—Third-highest priority |
|
Emerging infectious diseases (Nipah virus and Hantavirus) |
Anthrax, a Category A agent, has been used for almost a century as an effective biological weapon , as it can be released quietly and discreetly. Anthrax spores are easily available in nature, can be produced in a laboratory, and remain persistent in the environment for a long period. The microscopic spores can be deployed in a powder or spray and can be used to contaminate the air, food, and water supplies. They may be invisible to the naked eye, odorless, and tasteless (CDC, 2020). The German Army used biological agents against enemy countries during World War I, attempting to directly infect animals and contaminate their feed with anthrax (Frischknecht, 2003).
Anthrax was used in the worst biological attack in U.S. history, known as “Amerithrax” (FBI, n.d.). Following the terrorist attacks of September 11, 2001, letters laced with anthrax began circulating through the U.S. Postal Service (USPS), addressed to the offices of U.S. senators and national media outlets in Washington D.C., South Florida, and New York. Opening the letters released anthrax powder into the offices, where it was inhaled, ultimately causing the deaths of five victims and illness in 17 individuals. Panic and fear spread around the country, as it was unknown how many people had come into contact with these letters or how many other letters might be contaminated. This biological attack resulted in one of the largest and most complex law enforcement investigations to date (FBI, n.d.). The costs of human life, emotional distress, and resources were significant. Over 32,000 people exposed to anthrax were required to take prophylactic antibiotics, the USPS purchased 4.8 million masks and 88 million gloves for its employees, and 300 postal facilities were tested for anthrax. The investigation required extensive law enforcement resources. The Amerithrax Task Force comprised 25 to 30 full-time FBI investigators, the U.S. Postal Inspection Service, other law enforcement agencies, and federal prosecutors from the District of Columbia, and the Justice Department’s Counterterrorism Section spent hundreds of thousands of investigative hours on the case. These efforts involved more than 10,000 witness interviews on six different continents, the execution of 80 searches, and the recovery of more than 6,000 items of potential evidence during the investigation. More than 5,750 grand jury subpoenas were issued, and 5,730 environmental samples were collected from 60 sites. These efforts ultimately did not result in a conviction, as the suspect took his own life (FBI, n.d.).
This video gives an overview of how and why anthrax has been used in biological warfare globally.
Watch the video, and then respond to the following questions.
- What makes anthrax an effective biological agent for terrorists?
- What do you think are the risks that stores of anthrax, intended for use as a biological agent in warfare, pose to the community?
- Who do you think might be at risk of contamination when biological agents are dispersed via a federal communication system such as the USPS?
Identification and Management of Biological Casualties
Nurses within the community and acute care facilities must be prepared to identify and manage biological casualties. The United States remains vulnerable to bioterrorism. Biological agents like anthrax, botulinum toxin, and bubonic plague can cause a large number of illnesses or deaths in a short amount of time. An attack using a biological agent can mimic a natural outbreak or event, complicating public health assessment and response. Biological agents capable of secondary transmission can lead to epidemics, so early recognition of unusual circumstances is essential in prevention and response (WHO, 2023a).
The first step in identification is recognizing potential exposure to a biological casualty. Indications of biological agent exposure include (Bland, 2014):
- Reports of unexplained or unusual symptoms, such as headache, burns, vision problems, chest tightness, difficulty breathing, excessive secretions, nausea, vomiting, convulsions
- Multiple casualties with an unidentified cause
- Unusual taste, smell, or mist
- Unexplained dead animals or plants
Whenever the nurse suspects individual exposure to a biological agent, a thorough assessment should be completed only after ensuring personal safety. For example, the biological agent may have contaminated the individual’s clothing, putting the nurse in danger of exposure. The nurse in this situation should don appropriate PPE, direct the individual to remove contaminated clothing, and follow procedures for decontamination, which include washing with soap and water.
In the event of biological exposure and contamination, decontamination may be necessary. Biological contamination occurs when infectious agents come into contact with a body surface, inanimate objects, or food or water supplies. Biological decontamination includes disinfection or sterilization of the area to reduce the number of microorganisms to a safe level (Table 32.3) (Dembeck et al., 2011).
| Method | Examples |
|---|---|
Mechanical
|
|
Chemical
|
|
Physical
|
|
The nurse compares individual assessment data to signs and symptoms of common biological agents to assist in diagnosis and potential treatment options. Table 32.4 describes common agents of bioterrorism, method of transmission, signs and symptoms, and treatments. The nurse reports biological agent exposure to the local health department and the CDC.
| Agent of Bioterrorism | Transmission | Signs & Symptoms | Treatment |
|---|---|---|---|
|
Anthrax
Bacillus anthracis |
|
Cutaneous
Ingestion
Inhalation
|
60 days of doxycycline,
ciprofloxacin, or levofloxacin |
|
Botulism
Clostridium botulinum |
|
|
|
|
Ebola
Ebola virus |
|
|
|
|
Plague
Yersinia pestis |
|
|
|
|
Smallpox
Variola virus |
|
|
|
|
Tuleremia
Francisella tularensis |
|
|
Streptomycin, gentamicin, doxycycline, ciprofloxacin |
Mass exposure to biological agents must be identified and managed as quickly as possible to prevent additional exposure. Principles of mass exposure management include (Bland, 2014):
- Recognition of exposure
- Ensuring safety of the area through evacuation of nonessential personnel, use of PPE, and limiting additional exposure
- Establishing zones for decontamination, treatment, and safe or clean zones; see Zoning of Care Areas
- Setting up a command center and establishing a chain of communication
- Triage, assess, treat, and transport exposed individuals
The U.S. government remains in a constant state of readiness to protect Americans against the potential release of biological agents in terrorist threats and attacks. The Strategic National Stockpile (SNS) is part of the federal medical response infrastructure. The SNS supplements MCMs for states, tribal nations, territories, and metropolitan areas during public health emergencies. The stockpile has supplies, medicines, and devices for lifesaving care that can be used in the short term when the immediate supply of critical medical assets is unavailable or insufficient to meet the community’s needs (Administration for Strategic Preparedness and Response [ASPRN], n.d.).
Federal law enforcement and health and safety agencies focus on risk assessments and ensuring that preventative medical countermeasures (MCM) are safe, effective, and secure. The U.S. Food and Drug Administration (FDA) Medical Countermeasures Initiative (MCMi) works at local, state, national, and international levels of government to support MCM-related public health preparedness and response efforts (FDA, n.d.). MCMs are used to prevent, diagnose, or treat conditions associated with biological agents and include:
- Biologic products, such as vaccines, blood products, and antibodies
- Drugs, such as antimicrobial medications, antiviral medications, antidotes, or radiation treatments
- Devices, including diagnostic tests to identify threat agents, and personal protective equipment (PPE), such as gloves, respirators, face masks, and gowns
The Nurse’s Role in Bioterrorism Preparedness and Response
Nurses are essential members of the public health system. Public health and community health nurses must be prepared for the possibility of terrorist activity at any time. The American Nurses Association (2016) has developed policies, resources, and educational opportunities for nurses on disaster preparedness and acknowledges the importance of preparation so that nurses are equipped to respond to critical events. Nurses have a role in primary, secondary, and tertiary prevention for bioterrorism preparedness and response (Table 32.5). Refer to Pandemics and Infectious Disease Outbreaks for more information on conducting surveillance.
| Primary Prevention |
|
| Secondary Prevention |
|
| Tertiary Prevention |
|


2.16: Evaluating Rhetorical Context
- Identify the rhetorical context of a passage
For example, suppose in a music education course you are asked to read the following speech and then compose an opposing argument:
It is high time for music education to enter the digital age. In its current form, high school music education focuses almost exclusively on large instrumental and vocal ensembles grounded in classical music and conducted by one individual, typically the school’s music teacher. However, today’s average teenager listens to music for four hours a day, most of which is created digitally and produced through computer software, drum kits, and keyboards. Additionally, teens are taking to the internet themselves, recording their own work and sending it out to the world, with approximately 12,000 covers of songs being uploaded every 24 hours. As a former high school band conductor and current music professor at a state university, I train professional musicians and study music education curriculum, and I believe that current music classes are not providing what most students desire and what most future professionals need. As a consequence, high school students are abandoning school music classes. Initiative 952, with its emphasis on digital recording and production, would entice students back to music class and set them on a lifelong love of musicianship. I respectfully urge the board to vote yes on Initiative 952 and fund the education of tomorrow’s musicians.
To understand the rhetorical context of the speech, you must ask yourself the following questions:
- Who is the author, speaker, or composer?
- What is the author trying to accomplish? What is his or her purpose?
- What is the author writing about? What is his or her topic?
- Who is the audience?
- What is the occasion, or external motivation, for writing?
In the speech above, the author is a music professor who was formerly a high school band director. His purpose in writing this speech is to persuade school board members to fund Initiative 952. His audience includes, narrowly, school board members and, more broadly, anyone interested in music education. His topic is changing the focus of high school music education to digital composition through the funding of an initiative. The occasion for the speech is a meeting at which the school board will vote on funding the music education initiative.
Now, how does knowing the speech’s rhetorical context help you in writing an opposing argument? Let’s consider your rhetorical context.
- Who are you as a reader of a text and an author of a response?
- What is your purpose in reading and then writing?
- What are you reading and writing about?
- Who is your audience?
- What is the occasion, or external motivation, for your reading and writing?
Your assignment requires you to read and respond as an author opposing the original speech. Your purpose is to persuade readers that the speech’s argument is flawed. Your topic will be the speech and the proposed initiative. Your audience is your professor. The occasion , for you, is a course assignment and probably the desire to do well on the paper.
How can you use the rhetorical context of the music professor’s argument to help you meet the rhetorical context of your assignment?
Knowing that the author is a music professor, you decide to Google him to learn more about him. On his university’s website, you learn that his specialty is contemporary, digital music. You wonder if his scholarly interests might have affected his position on this argument and begin to consider ways that you could address his bias in your own paper. You also note that several other professors in his department are specialists in classical music and decide to investigate what they have written on the topic, finding several have written in support of the classical approach to music education. Your own professor has emphasized using academic sources, so you decide to use some of the classical music specialists as sources for your paper.
You also find the author’s LinkedIn page where he mentions an online product he has developed to bring digital tools to music classrooms. Since the occasion of his writing is a school board meeting where members could potentially vote to purchase such a product, you wonder if his motivation for funding the initiative might be linked to his desire to sell his product. Since your assignment requires you to oppose his argument, you decide to raise the possibility that the speechwriter may be motivated by selling digital tools rather than improving music education.
The following video demonstrates how external factors influence a piece of writing, even before the first word is written down.
The video focuses on the use of rhetoric from the viewpoint as a writer. As you watch, consider how the same elements hold true from the viewpoint as a reader.
You can view the transcript for “The Rhetorical Situation” here (opens in new window) .”
As a reader, pay attention to these factors:
- What do I know about the author , just by reading this text? How does the author’s experience or education shape the reading?
- What do I know about the goals for this work, just by reading this text? How do these goals influence what actually appears in the text?
- What do I know about the intended audience for this work? Am I a member of that intended audience?
Examining the rhetorical context in which a writer is operating helps you understand an author’s biases and agendas as well as the influences surrounding the writer that may have affected his or her composition. Examining the rhetorical context in which you, as a reader/responder, are operating helps you situate the text rhetorically, become aware of your own position, and respond to the text appropriately.
Below is an excerpt from the advance text of a speech then-Senator John F. Kennedy gave just a few days before he won the election to become the 35th U.S. president. Read the excerpt, and then answer the questions.
We live in a fast-moving nation. But one thing constant from the birth of our Republic has been our faith in education and our determination to make it available to all our citizens.
It was Aristotle, more than 2,000 years ago, who said: “The neglect of education ruins the constitution of the country.” And Thomas Jefferson echoed these principles when he wrote to a friend in 1786 that “the most important bill is that for the diffusion of knowledge among the people. No other sure foundation can be devised for the preservation of freedom and happiness.”
Thus the value and importance of education was at the foundation of Western thought—and was again present at the foundation of the American Republic. . . .
Today I want to discuss education with you—the current crisis in our educational system is a crisis caused by our failure to meet our responsibilities over the past 8 years; and tell you what I think we must do in the future to build an educational system to meet our expanding needs. . . .
Where, then, have we failed in the past eight years? And what must we do for the future?
First, we have failed to provide adequate classrooms for our expanding school population. Today we have 131,000 classrooms fewer than we need—and, at our current rate of construction shortage is actually increasing. The result is double shifts, obsolete, overcrowded, and even dangerous classrooms.
In one community a dog kennel was converted into a school where four classes were being held. In an adjoining town the school superintendent said, “I only wish I had a dog kennel to use.” In another area the school board is renting 2 windowless, cinder block factories to house 883 children—while in other cities kindergarten children are being taught in firetraps. . . .
Second, we have failed to provide enough well-trained and well-paid teachers. Today we need 135,000 more teachers. Almost three million schoolchildren are being taught by teachers working on substandard certificates. And as our school population expands in the next decade, one-and-a-half million more teachers—one-third of all our college graduates—will be needed to keep our educational system going. We are not attracting bright young men and women into teaching because the salaries which we pay our teachers are shamefully low. . . .
Therefore, I propose the enactment of a Student Loan Insurance Act—modeled on the highly successful program which has been adopted by my native State of Massachusetts. Under this program the Federal Government—in return for a small premium—would guarantee student loans made by colleges and universities. Thus colleges would be able to secure funds adequate to meet the pressing financial needs of all of its students—so that no able student would have to leave school because he could not pay his expenses. Although a small special revolving fund would be required, Federal payments would be made only in the unlikely event of default. Basic responsibility for repayment would be in the hands of the student, and the loan program itself would be administered by the individual college or university. In this way we can make sure that no bright young American is denied a college education. . . .
Abraham Lincoln once said that “He has the right to criticize who has the heart to help.” We of the Democratic Party criticize our educational system—and the leadership which has permitted it to falter—because we have the heart to help, and, even more, the programs and the leadership which can build an educational system of which all Americans can be proud. [1]
https://assessments.lumenlearning.co...sessments/5599
- John F. Kennedy: "Speech by Senator John F. Kennedy, Beverly Hilton Hotel, Los Angeles, CA - (Advance Release Text)," November 2, 1960. Online by Gerhard Peters and John T. Woolley, The American Presidency Project. www.presidency.ucsb.edu/ws/?pid=25930. ↵
Contributors and Attributions
- Rhetorical Context Practice Activity. Provided by : Ole Miss. License : CC BY: Attribution
- Rhetorical Context. Authored by : Karen Forgette. Provided by : University of Mississippi. Project : PLATO Project. License : CC BY-SA: Attribution-ShareAlike
- The Rhetorical Situation. Provided by : University of Maryland, Baltimore Writing Center. Located at : https://youtu.be/A-eRycqjzbg . License : All Rights Reserved . License Terms : Standard YouTube License