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behavior
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the way a person acts, the mannerisms they display, and the conduct they employ
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federally qualified community health center (FQHC)
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clinics providing primary care services to all clients regardless of their ability to pay
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genetics
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the study of how genes, traits, and diseases are passed from one generation to the next
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health behavior
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the actions, habits, activities, policies, and procedures of individual clients, communities, organizations, and governments that can either support or undermine health
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intersectionality
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a concept that considers how disadvantages interact for clients at both an individual level and systems level to identify and describe the inequities, injustices, oppressions, and privileges perpetuated by structures including racism, classism, colonialism, sexism, heterosexism, ableism, and others
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medical care
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services rendered by a health care clinician during visits to clinics, offices, hospitals, surgical centers, schools, laboratories, and other places
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socio-ecological model (SEM)
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a model that facilitates the study and organization of how client and community health behaviors and outcomes are affected by personal, situational, community, societal, political, and other contexts
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theories
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used to explain phenomena in a systematic way and to guide thoughts and decision-making processes


1.22: Slideshow- The Elements of Music
This slideshow was designed to serve as a study guide on the musical elements of melody, rhythm, harmony, texture, and form. While we will review all the musical elements covered in this slideshow over the course of the semester, they may not necessarily be relevant in this first module. As such, this slideshow refers to musical elements that we will not cover for some time — this is particularly true of the musical element harmony.
That said, this slideshow is included here as a resource in case any of you are interested in exploring the elements of form and harmony more fully, but please understand that the documents provided above align more closely with the information covered on the Medieval & Renaissance Music Exam.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution


1.5: Sphingolipids
Sphingolipids are a type of lipids made up of fatty acid chains that were first mentioned in 1884 in J.L.W. Thudichum’s A Treatise on the chemical composition of the brain. They were named after the Greek mythological creature, the sphinx, due to the unknown riddle of their function. [1] Sphingolipids are found in essentially all plants, animals, fungi and in some prokaryotes and viruses. Specifically, they are found in membranes and as a major component of lipoproteins. [2]
Structure
Sphingolipids are amphipathic molecules; they have hydrophobic and hydrophilic properties. In the hydrophobic region there is a sphingoid long chain base (aliphatic chains with attached hydroxyl groups) with a fatty acid chain attached by amide bond at carbon 2. In the hydrophilic region, there are phosphate groups, sugar residues, and/or hydroxyl groups. [3] This amphipathic nature allows for the diffusion of sphingolipids between membranes and for the flipping of sphingolipids between membrane leaflets. However, sphingolipids are still more likely to accumulate in acidic environments because of the possible ionization of a free amino group. [4] There are several classes of sphingolipids: the sphingoid base and simple derivatives, ceramides, and complex sphingolipids (Figure \(\PageIndex{1}\)).
Sphingoid base & simple derivatives
Sphingolipids are composed of many backbone “sphingoid bases” which are synthesized from serine and a long-chain fatty acyl-CoA. [2,5] If a backbone has not yet been analyzed, it is referred to as a sphinosine , after the original name for the fundamental component of these lipids. Sphingoid bases can vary in alkyl chain length/branching, the number and position of double bonds, the number and locations of hydroxyl groups, etc. A simple sphinosine base can be seen in Figure \(\PageIndex{1}\).
These simple derivatives exist mainly as the backbone of more complex sphingolipids, but they do exist on their own. They are mainly used in intra-and extra-cellular signaling. [2] The exact structure of the sphingolipid determines the function. For example, in the dermis, there are additional hydroxyl groups at positions 4 and/or 6 that interact with neighboring molecules to strengthen the permeability of the skin barrier. [5]
Ceramides
Ceramides, which are synthesized in the endoplasmic reticulum, are the simplest sphingolipid after the backbone. [3] They are fatty acid derivatives of sphingoid bases, linked by amide bonds. They are usually fully saturated or mono-unsaturated and consist of 14-26 carbons, but are sometimes longer. There is sometimes a hydroxyl group on the ω or ω-carbon atom. [2,5] The structure can be seen in Figure \(\PageIndex{2}\). Ceramides have high phase transition temperatures of greater than 37°C. This is due to their predominately saturated nature, which allows for the tight packing of lipids. This also favors the segregation of ceramides (and more complex sphingolipids) into regions called membrane “ rafts .” [2,6]
Ceramides are key to cell signaling processes. [3] They generally serve as second messengers by regulating cell growth, senescence, and apoptosis. The exact function depends on the type of sphingolipid base and fatty acid. [2] For example, ceramides containing palmitic acid for a fatty acid chain are generated in response to apoptosis. Much of the specifics of ceramide function is still under investigation. Ceramides also serve as a precursor to more complex sphingolipids. [3]
Ceramides can only exist in membranes but they have a rapid flip rate between leaflets. This means they cannot leave the organelle they are created in, but will have access to binding proteins and enzymes on both sides of the bilayer. As such, ceramides can usually only be modified by enzymes that exist in the membrane compartment that the ceramide was produced in. [4]
Complex Sphingolipids
The synthesis of complex sphingolipids occurs in the Golgi apparatus . [3] The bulky headgroups of these sphingolipids make flipping between membrane leaflets very unlikely without the aid of flippases. This means that without flippases, sphingolipids are restricted to the lumenal Golgi leaflet/outer leaflet of the plasma membrane. [4] The two main type of complex sphingolipids are phospho- and glycol-sphingolipids.
Phosphosphingolipids
Phosphosphingolipids are linked by phosphodiester bonds. [5] In mammals, the major phosphosphingolipids are gomyelins (ceramide phosphocholines). In insects, it is ceramide phosphoethanolamines. The major phosphosphingolipids in fungi are phytoceramidesphosphoinositols and inositol phosphates. Aquatic organisms typically have sphingolipids where the phosphate has been replaced with a phosphono- or arsenate group. [2] Example phosphosphingolipids can be seen in Figure \(\PageIndex{3}\).
Glycosphingolipids
Glycosphingolipids have one or more carbohydrate groups in a glycosidic linkage (usually; ceramide phophonositols are an exception). The structure can be seen in Figure \(\PageIndex{4}\). There are four classifications of glycosphingolipids:
- Neutral glycosphingolipids: 1 or more uncharged sugars (i.e. glucose, galactose, N-acetylglucosamine, N-acetylgalacoseamine, fucose) [2]
- Acid glycosphingolipids: ionized functional groups (i.e. phosphate or sulfate) attached to neutral or charged sugars [2]
- Basic glycosphingolipids [5]
- Amphoteric glycosphingolipids [5]
Naming of Sphingolipids
Sphingolipids are named using IUPAC nomenclature for lipids ( http://www.chem.qmul.ac.uk/iupac/lipid/ ). The most common sphingoid base is sphingosine. The official names of sphingosine are (2S,3R,4E)-2-aminooctadec-4-ene-1,3-diol or D- erythro -sphingosine & E-sphing-4-enine. The abbreviation is d18:1. The first number represents the number of carbon atoms. The second number is the number of double bonds. In this case, the letter ‘d’ refers to the 2 (di-) hydroxyl groups. [2,5]
Biological Functions
There are many biological functions associated with sphingolipids, including cellular regulation, signaling and metabolism.
Biophysical Properties
Sphingolipids tend to form clustered structured called microdomains, rafts, or caveolae (when containing caveolin). Saturated alkyl sidechains allow for strong van der Waals interactions. Ceramide hydroxyls, amide bonds, and polar headgroups allow for hydrogen bonding and dipole interactions. Sphingolipids are enriched in growth factor receptors, transporteres and proteins with glycosylphosphatidylinositiol-lipid anchors. Sphingolipids help stabilize the membrane bilayer which helps structures such as the lamellar bodies that maintain the permeability of skin. Their water solubility allows for rapid movement between membranes. Sphingolipids can exist in cationic and neutral forms. [2]
There are three major phases that a lipid membrane can take (Figure \(\PageIndex{5}\)).
- In the gel phase , the lipid bilayers are in the solid state. The packing of the lipid molecules is very tight and ordered, leading to little movement of lipids across the membrane surface.
- In the liquid disordered phase, are highly fluid states in which the individual lipids can move unhindered across the surface of the membrane. The packing of the lipids is highly irregular and the lipids themselves are often kinked due to unsaturated fatty acids.
- The liquid ordered phase in a cross between the gel phase and the liquid disordered phase. There is still the relatively easy movement of lipid molecules across the membrane, but the membrane itself is more tightly packed, leading to a more solid structure. This phase comes about through the presence of sterol molecules within the membrane.
Membrane rafts are caused by a phase separation in the membrane. They occur when the liquid ordered phase and the disordered phase coexist. Tight packing of lipids is key to raft formation. In the liquid ordered phase, like the ordered gel phase, lipids are elongated and tightly packed, but they still have a higher degree of lateral mobility. [6] Figure \(\PageIndex{7}\) shows a membrane with a lipid raft.
Membrane rafts are abundant in membranes that are cholesterol and sphingolipid rich (i.e. the plasma membrane). They are important in sorting in the trans-Golgi network, sorting in the endocytic pathway, as a docking site for pathogens and toxins, and as integrin receptors. [6]
Cellular Regulation
Cellular sphingolipids are located in many places throughout the cell. They are found in the outer leaflet of the plasma membrane, in the lumen of intracellular vesicles/organelles (i.e. endosomes, Golgi membranes), and in undefined locations in the mitochondria, nuclei, and intracellular compartments. This abundance makes them ideal candidates for cell regulation. [2]
Cellular sphingolipids interact with complementary ligands (i.e. extracellular matrix proteins/receptors), with other carbohydrates (i.e. glycosynapse) and with proteins on the same cell surface. Through these interactions, they can control the location of a protein (i.e. in membrane rafts) and modify the conformation of the protein and its activity. They can also be recognized by viruses, bacteria and bacterial toxins as a means of attachment and entry. [2]
Ceramides have multiple roles in membrane biogenesis and regulation. They can influence ER to Golgi trafficking. They can help with the formation of autophagosomes. It is also likely that they influence the ER interactions with the nuclear membrane, outer membrane of mitochondria, and the plasma membrane. [2]
Cellular Signaling
Sphingolipids affect cell signaling by influencing the properties of receptors directly. The exact pathway for signal transduction varies depending on the exact function of the sphingolipids but there is a general pathway. [2]
- Receptors are activated by cytokines (i.e. IL1, tumor necrosis factor-ω), growth factors (i.e. platelet derived growth factor), or other agonists.
- Hydrolysis of membrane sphingolipid to ceramide is induced.
- Sphingolipids either serve as a lipid second messenger, or can be converted to downstream metabolites (i.e. sphingosine, sphingosine 1-phosphate, ceramide 1-phosphate).
- Downstream targets (i.e. proteins kinases, phosphoprotein phosphatases) that control cell behavior are either activated or inhibited.
Sometimes, multiple sphingolipids within the same cell have opposing signaling function. For example, ceraminde and sphingosine 1-phosphate signal for conflicting functions of cell growth. One signals for the inhibition of growth, while the other signals for the stimulation of growth. (Or in other words, the induction vs. inhibition of apoptosis.) The cellular pathway taken is determined by a ceraminde/sphingosine 1-phosophate “rheostat” in deciding between growth arrest/apoptosis and proliferation/survial. [2]
Because sphingolipids are complex molecules, they can take diverse input signals and produce multiple coordinated outputs. Signaling from sphingolipids also alters membrane structure and behavior of other membrane receptors and/or proteins. [2]
Metabolism
Sphingolipid base acylation is done through ceramide synthases which are highly specific for acyl chain lengths. [2] Regulation occurs through control of enzyme expression, post-translational modifications, and allosteric mechanisms. These can be cell specific – they can control sphingolipid synthesis during specific phases of the cell cycle, in response to specific signals. [3] Figure \(\PageIndex{8}\) shows several metabolic pathways of sphingolipids.
During sphingolipid metabolism, sphingomyelin turnover produces signaling metabolites and alters the structure of membrane domains, depending on its presence. In addition, changing the glycosphingolipid composition affects the properties of the cell surface and its interaction with the extracellular matrix and neighboring cells. Because of this, sphingolipid metabolism and signaling are basically the same. [2]
Sources
- Thudichum, J. L. W. 1884. “A Treatise on the Chemical Constitution of Brain”. Bailliere, Tindall, and Cox, London.
- Hirabayashi, Y., Igarashi, Y., Merrill, A.H. (2006). Sphingolipid Biology Springer, Japan.
- Futerman, A. H., Hannun, Y. A. (2004). "The complex life of simple sphingolipids." EMBO Rep 5 (8): 777-782.
- Gault, C., et al. (2010). "An overview of sphingolipid metabolism: from synthesis to breakdown." Adv Exp Med Biol 688 : 1-23.
- Fahy, E., et al. (2005). "A comprehensive classification system for lipids." J Lipid Res 46 (5): 839-861.
- Brown, D. A. and E. London (2000). "Structure and Function of Sphingolipid- and Cholesterol-rich Membrane Rafts." J Biol. Chem. 275 (23): 17221-17224.


38.2: Types of Skeletal Systems - Types of Skeletal Systems
- Differentiate among skeletal types: hydrostatic skeleton, exoskeleton, and endoskeleton
Types of Skeletal Designs
A skeletal system is necessary to support the body, protect internal organs, and allow for the movement of an organism. There are three different skeleton designs that provide organisms these functions: hydrostatic skeleton, exoskeleton, and endoskeleton.
Hydrostatic Skeleton
A hydrostatic skeleton is one formed by a fluid-filled compartment within the body: the coelom. The organs of the coelom are supported by the aqueous fluid, which also resists external compression. This compartment is under hydrostatic pressure because of the fluid and supports the other organs of the organism. This type of skeletal system is found in soft-bodied animals such as sea anemones, earthworms, Cnidaria, and other invertebrates.
Movement in a hydrostatic skeleton is provided by muscles that surround the coelom. The muscles in a hydrostatic skeleton contract to change the shape of the coelom; the pressure of the fluid in the coelom produces movement. For example, earthworms move by waves of muscular contractions (peristalsis) of the skeletal muscle of the body wall hydrostatic skeleton, which alternately shorten and lengthen the body. Lengthening the body extends the anterior end of the organism. Most organisms have a mechanism to fix themselves in the substrate. Shortening the muscles then draws the posterior portion of the body forward. Although a hydrostatic skeleton is well-suited to invertebrate organisms such as earthworms and some aquatic organisms, it is not an efficient skeleton for terrestrial animals.
Exoskeleton
An exoskeleton is an external, hard, encasement on the surface of an organism. For example, the shells of crabs and insects are exoskeletons. This skeleton type provides defense against predators, supports the body, and allows for movement through the contraction of attached muscles. As with vertebrates, muscles must cross a joint inside the exoskeleton. Shortening of the muscle changes the relationship of the two segments of the exoskeleton. Arthropods, such as crabs and lobsters, have exoskeletons that consist of 30–50 percent chitin, a polysaccharide derivative of glucose that is a strong-but-flexible material. Chitin is secreted by the epidermal cells. The exoskeleton is further strengthened by the addition of calcium carbonate in organisms such as the lobster. Because the exoskeleton is acellular and does not grow as the organism grows, arthropods must periodically shed their exoskeletons.
Endoskeleton
An endoskeleton consists of hard, mineralized structures located within the soft tissue of organisms. An example of a primitive endoskeletal structure is the spicule of sponges. The bones of vertebrates are composed of tissues, whereas sponges have no true tissues. Endoskeletons provide support for the body, protect internal organs, and allow for movement through contraction of muscles attached to the skeleton.
The human skeleton is an endoskeleton that consists of 206 bones in the adult. It has five main functions: providing support to the body, storing minerals and lipids, producing blood cells, protecting internal organs, and allowing for movement. The skeletal system in vertebrates is divided into the axial skeleton (which consists of the skull, vertebral column, and rib cage), and the appendicular skeleton (which consists of the shoulders, limb bones, the pectoral girdle, and the pelvic girdle).
Key Points
- In organisms with hydrostatic skeletons, the muscles contract to change the shape of the coelom, which then produces movement due to the pressure of the fluid inside the fluid-filled cavity.
- Exoskeletons are external skeletal systems that are made up of chitin and calcium carbonate.
- Organisms with an endoskeleton are supported by a hard, mineralized skeletal system that resides inside the body.
- In vertebrates, the endoskeleton system is further divided into the axial skeleton and appendicular skeleton.
Key Terms
- coelom : a fluid-filled cavity within the body of an animal; the digestive system is suspended within the cavity, which is lined by a tissue called the peritoneum
- peristalsis : the rhythmic, wave-like contraction and relaxation of muscles which propagates in a wave down a muscular tube
- endoskeleton : the internal skeleton of an animal, which in vertebrates is comprised of bone and cartilage
- exoskeleton : a hard outer structure that provides both structure and protection to creatures such as insects, Crustacea, and Nematoda


10.4: PivotTables
By the end of this section, you will be able to:
- Define a PivotTable
- Organize and clean your data prior to designing a PivotTable
- Design/insert a PivotTable
- Understand and use the recommended PivotTable tool
- Add a PivotChart to pictorially present the data
PivotTables help data analysts use raw data and organize it into an interactive tool for summarizing and analyzing the data points. WorldCorp can, for example, gather their manufacturing datasets of each machine’s daily performance and put this information into a table that the PivotTable tool can use to analyze the information. This type of organization would be useful for producing various metrics (such as machine performance) and company Key Performance Indicators (KPIs). The PivotTable tool can also divide data into a set of criteria, which will organize your data by variables you choose. This separation and division of data help you create business intelligence with ease.
Definition and Uses
Excel has various ways to store data and various tools that aid in data intelligence queries. You’ve learned the purpose of using What-If Data Tables, naming ranges, using comparison operators and writing statistical and logical functions—all of which can be combined with arrays (or other tools) to make interactive and automated tables. Excel has an even quicker way to make relational tables and automated business intelligence: PivotTables.
A PivotTable lets users create interactive charts and tables from raw datasets. They can process large, raw datasets and allow the user to choose how the data will be tabulated and summarized using easy drag-and-drop features. PivotTables also help summarize the data for a professional appearance and to show it in a more user-friendly way. Out of thousands of rows and columns of raw data, you can produce a simple, easy-to-read table with different options for displaying your metrics. Some applications of PivotTables might include producing metrics for website views and ad clicks, displaying demographic information for a county or city, or gathering information about small business revenue. In each of these cases, PivotTables can be used to process large, ever-growing datasets into readable tables.
Organizing and Cleaning Your Data
Let’s return to the example of shipped products from WorldCorp’s Portland, Maine, location. PivotTables can be used to aggregate values and create an actionable market analysis and business intelligence for the company.
The first step of creating this PivotTable is to organize and summarize the data. Having a digestible raw data sheet that Excel can read well enough to aggregate the data is critical. To clean the data, you should ensure that:
- There are no merged cells anywhere in the data.
- All filters are inactive.
- There are no SUM functions on the bottom or sides of the worksheets.
- All erroneous, duplicate, and/or blank values are deleted from the data.
- All values in each column are of the same value (e.g., currency, date, text).
Next, convert your data into a top-down table , meaning that all of the data range headers should run along the top row of the sheet. This is the typical format for tables and will make using PivotTables much easier. In general, horizontal tables can be hard to follow. Cross-tabular source data (with headers on top/bottom and left/right) are even more difficult to read for the PivotTable generator. Designate a unique name for each top header; do not repeat any column header names. Figure 10.33 shows a correctly formatted worksheet. You can copy this worksheet into Excel, or you can use the data in the "pivot_table_data" tab of the downloadable Chapter 10 data file workbook.
Inserting a PivotTable and Rearranging Variables
To create a PivotTable, go to the Insert tab, choose the PivotTable icon located on the far left, select the entire dataset, and output to New Worksheet. (Alternatively, you may name your data ranges, as instructed in Defining, Selecting, and Naming a Cell Range, and simply select the named ranges for your data.) The Field List will activate and ask which columns to display, as shown in Figure 10.34. Select the headers (variables) in the order that you want the PivotTable to display them. In the WorldCorp example, to see the quantity each client is buying, select Destination, then Quantity, and then FOB $. As shown in Figure 10.35, the PivotTable neatly displays the data, with interactive features allowing the user to select different pieces of information as variables.
Now, to further delimit the details of the clients, add Date to the variables (Figure 10.36a). Because you chose Destination first, you are separating all the variables and ordering them by client to know what the client is consuming in volume and in dollar amount. Adding the date allows you to see total sales but also breaks down the data by month.
This is only one way to create a PivotTable with this data. If you were to change the order in which you selected the headers in the PivotTable wizard, the whole table would be different. Additionally, you can choose a different set of data to display, which changes how the information is grouped. To demonstrate this, uncheck Date (you have to uncheck Months, too) and instead check Product. As seen in Figure 10.36b, the table gives a totally different perspective. Instead of seeing the data grouped by Date, the grouping has changed to show the products the client ordered and the quantity of each.
You can easily change the arrangement of the PivotTable using the fields in the lower half of the Field List. You can drag and drop the variables from one field to another to change how they appear in the PivotTable. For example, you can change the arrangement of the rows by putting Product first. This automatically changes the PivotTable completely, from being destination oriented (as seen in Figure 10.37a) to being product oriented (Figure 10.37b).
Let’s apply a filter to the results from Figure 10.37b. To filter by date, first check the Date and Month variables on the top half of the Field List. Then, drag and drop the dates from the Rows to the Filter field, as shown in Figure 10.38a. This will allow you to apply a filter to the dates and be able to select only certain dates. To see just the February sales, go to the Month drop-down menu on the PivotTable and select Feb. Now, only the sales from that month will appear (Figure 10.38b).
Using the Recommended PivotTable Tool
If you are unsure how to organize a PivotTable for your data, Excel can recommend one depending on how the data is organized. Let’s see what using a recommended PivotTable might look like.
Go to the Insert tab and select the Recommended PivotTables icon. A dialog box appears with a handful of recommended charts. Scroll and select the one you like best. These recommended tables may be helpful to get you started, but they only compare one variable at a time. However, if you choose one, the new PivotTable will appear in another worksheet, and you can customize it there, adding or reorganizing variables until you have a PivotTable that presents your data in a way that makes sense.
Adding a PivotChart to Visually Present Data
Excel has a feature that allows you to create a PivotChart to visually present the data in your PivotTable. The first step is to select the entire PivotTable, which causes the ribbon tab “ PivotTable Analyze ” to appear. Next, select the PivotChart icon. You will use a small dataset from Figure 10.31 to create a chart of the information. More in-depth coverage of PivotTables is in PivotTables/Charts. Figure 10.39 shows the original data and Figure 10.40 shows the data summarized in a PivotTable by sales agent. To create a chart of the data summarized in the PivotTable, choose PivotChart from the PivotTable Analyze tab.
For the data in Figure 10.40, a bar chart, pie chart, or clustered column chart would be optimal. These chart types are appropriate because they allow you to see the difference in scale between the sales levels of the agents. For this example, choose the column chart. With the column chart, the higher the column, the greater the sales totals for that agent. You can further customize the column chart by adding percentages or labels. The Agent drop-down menu in Figure 10.41 allows you to further filter or change the column chart.
PivotTables can be a powerful tool for businesses in budgeting and creating other financial reports and analyses. You can also use PivotTables for your own personal budgeting. There are many budgeting services available online or through apps, but you can create your own powerful budgets that can help track all personal finance matters, including spending, without paying for an online service or app. As an individual, you might be able to track your budget fairly easily; however, as your financial matters become more complex, or as your family grows, Excel can come in handy to assist you in managing your budgets without the additional expense of a service or application.
To get started, you can download a CSV file of your transactions from your financial accounts. Most institutions will have this function where you review your account and transaction information online. You will want to collect all information, particularly from your spending accounts, such as credit cards. Once these are gathered, be sure to validate the information, then create a column heading named Category. You will then create your categories, such as Food or Entertainment. Some credit cards will auto populate these categories when you download your transactions, but you will want to verify these are accurate. From there, you will complete your PivotTables using the instructions you can find in this textbook. You can opt to create different timelines of your budget, such as daily, monthly, or yearly, depending on your own budget analysis needs.


7.1.3: Nonadjacent Angles
Lesson
Let's look at angles that are not right next to one another.
Exercise \(\PageIndex{1}\): Finding Related Statements
Given \(a\) and \(b\) are numbers, and \(a+b=180\), which statements also must be true?
\(a=180-b\qquad a-180=b\qquad 360=2a+2b\qquad a=90\text{ and }b=90\)
Exercise \(\PageIndex{2}\): Polygon Angles
Use any useful tools in the geometry toolkit to identify any pairs of angles in these figures that are complementary or supplementary.
Exercise \(\PageIndex{3}\): Vertical Angles
Use a straightedge to draw two intersecting lines. Use a protractor to measure all four angles whose vertex is located at the intersection.
Compare your drawing and measurements to the people in your group. Make a conjecture about the relationships between angle measures at an intersection.
Exercise \(\PageIndex{4}\): Row Game: Angles
Find the measure of the angles in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it.
column A
\(P\) is on line \(m\). Find the value of \(a\).
Find the value of \(a\).
column B
Find the value of \(b\).
In right triangle \(LMN\), angles \(L\) and \(M\) are complementary. Find the measure of angle \(L\).
column A
Angle \(C\) and angle \(E\) are supplementary. Find the measure of angle \(E\).
Find the value of \(c\).
Two angles are complementary. One angle measures 37 degrees. Find the measure of the other angle.
column B
\(X\) is on line \(WY\). Find the value of \(b\).
\(B\) is on line \(FW\). Find the measure of angle \(CBW\).
Two angles are supplementary. One angle measures 127 degrees. Find the measure of the other angle.
Summary
When two lines cross, they form two pairs of vertical angles . Vertical angles are across the intersection point from each other.
Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example:
This is always true!
\(a+b=180\) so \(a=180-b\).
\(c+b=180\) so \(c=180-b\).
That means \(a=c\).
Glossary Entries
Definition: Adjacent Angles
Adjacent angles share a side and a vertex.
In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).
Definition: Complementary
Complementary angles have measures that add up to 90 degrees.
For example, a \(15^{\circ}\) angle and a \(75^{\circ}\) angle are complementary.
Definition: Right Angle
A right angle is half of a straight angle. It measures 90 degrees.
Definition: Straight Angle
A straight angle is an angle that forms a straight line. It measures 180 degrees.
Definition: Supplementary
Supplementary angles have measures that add up to 180 degrees.
For example, a \(15^{\circ}\) angle and a \(165^{\circ}\) angle are supplementary.
Definition: Vertical Angles
Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal.
For example, angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^{\circ}\), then angle \(DEB\) must also measure \(120^{\circ}\).
Angles \(AED\) and \(BEC\) are another pair of vertical angles.
Practice
Exercise \(\PageIndex{5}\)
Two lines intersect. Find the value of \(b\) and \(c\).
Exercise \(\PageIndex{6}\)
In this figure, angles \(R\) and \(S\) are complementary. Find the measure of angle \(S\).
Exercise \(\PageIndex{7}\)
If two angles are both vertical and supplementary, can we determine the angles? Is it possible to be both vertical and complementary? If so, can you determine the angles? Explain how you know.
Exercise \(\PageIndex{8}\)
Match each expression in the first list with an equivalent expression from the second list.
- \(5(x+1)-2x+11\)
- \(2x+2+x+5\)
- \(\frac{-3}{8}x-9+\frac{5}{8}x+1\)
- \(2.06x-5.53+4.98-9.02\)
- \(99x+44\)
- \(\frac{1}{4}x-8\)
- \(\frac{1}{2}(6x+14)\)
- \(11(9x+4)\)
- \(3x+16\)
- \(2.06x+(-5.53)+4.98+(-9.02)\)
(From Unit 6.4.5)
Exercise \(\PageIndex{9}\)
Factor each expression.
- \(15a-13a\)
- \(-6x-18y\)
- \(36abc+54ad\)
(From Unit 6.4.2)
Exercise \(\PageIndex{10}\)
The directors of a dance show expect many students to participate but don’t yet know how many students will come. The directors need 7 students to work on the technical crew. The rest of the students work on dance routines in groups of 9. For the show to work, they need at least 6 full groups working on dance routines.
- Write and solve an inequality to represent this situation, and graph the solution on a number line.
- Write a sentence to the directors about the number of students they need.
 Using \(d\) for the number of days and \(f\) for the amount of food in cups, write an equation relating the variables. Use the equation to find how many days a large bag of dog food will last if it contains 210 cups of food.
(From Unit 2.2.2)


2.2.6: Spectrophotometry
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Learning Objectives
Goals:
- Identify the main features on the spectrophotometer and define their functions.
- Use a spectrophotometer to obtain an absorbance spectrum.
Student Learning Outcomes:
Upon completion of this lab, students will be able to:- Identify the parts of a spectrophotometer and their related functions.
- “Blank” a spectrophotometer.
- Obtain an absorbance spectrum for a molecule.
- Use the wavelength absorption scans to determine the dyes in colored skittles.
Introduction
Spectrophotometers are one of the most frequently used tools by scientists to determine both the presence and concentration of dissolved chemicals. As radiant energy (visible light) strikes matter, molecules will absorb certain wavelengths of light and transmit or reflect others based on the nature of their chemical bonds. For example, proteins and nucleic acids absorb wavelengths in the visible light range of 240-300 nanometers (nm), pigments and dyes absorb light in the 400-770-nm range, and other organic molecules absorb wavelengths above 770-nm. Each chemical has a distinctive atomic arrangement and bonding pattern, and thus absorbs or transmits different wavelengths of visible light in a pattern that is unique for that chemical. This unique pattern of light absorption and transmittance creates a “fingerprint” for that chemical. In this exercise you will determine the unique “fingerprint” for a colored molecule and use a spectrophotometer to measure the concentration of a chemical in a given sample.
Spectrophotometers are instruments designed to detect the amount of light energy that is absorbed or transmitted by molecules dissolved in a solution. Since molecules have wavelengths unique to their structure, different chemicals and their concentrations can be identified based on their absorbance or transmittance.
A spectrophotometer is an instrument used for detecting the presence of any light-absorbing particles dissolved in a solution and for measuring the concentration of those particles. A light source inside the spectrophotometer emits a full spectrum of white light towards a compartment where a sample liquid is placed. The samples are prepared in cuvettes that are made using specialized plastics or quartz so that they do not absorb any light and will not affect our measurements. Before the light passes through the sample in the cuvette, an adjustable prism and diffraction grating filters the light so that only a single wavelength of light can be selected and allowed to pass through the sample. All molecules differ in how strongly they absorb each wavelength of light in the visible spectrum because of differences in their molecular structure and composition. This allows us to use a specific wavelength of light to detect the presence of, and quantify, one molecular compound in a simple or complex liquid mixture. Spectrophotometers are also calibrated by using a “blank” solution that we prepare containing all of the components of the solution to be analyzed except for the one compound we are testing for so that the instrument can zero out these background readings and only report values for the compound of interest.
Light passing through a sample solution will partially be absorbed by molecules present in the sample. The amount of light unable to pass through a sample is measured as the absorbance value. Absorbance is directly proportional to the concentration of the molecules and is measured on a logarithmic scale from 0 to infinity. The amount of light that is not absorbed is transmitted or passed through the sample. Compared to the amount of light entering the sample, the amount that exits is measured as a percentage of the light transmitted. Percent transmittance is inversely proportional to the concentration of the molecules in the sample and is measured on a linear scale from 0% to 100%.
A photodetector on the other side of the sample compartment converts the intensity of the light it receives into an electrical signal. The instrument can then calculate and display the absorbance and % transmittance values by measuring the difference between the intensity of light of the selected wavelength entering and exiting the sample.
The absorbance scale reflects the measurement of the amount of light absorbed and converted into absorbance (\(A\)) units by the spectrophotometer. Absorbance units are calculated by using the following equation:
Equation 1
\[\text{Absorbance} (A) = \log_{10} \left(\dfrac{1}{T}\right) \nonumber\]
where “\(T\)” is the decimal form of “\(\%T\)”
\[T = \dfrac{\%T}{100} \nonumber\]
Example
If a solution containing a given dye is found to transmit 10% of the light when placed in a spectrophotometer, its absorbance then would be calculated as follows:
Transmittance (T) = 10% = 10%/100 = 0.10
Absorbance (A) = log 10 (1 / 0.10) = 1.0
Part I: Identifying Food Dyes in Candies
Many foods, drugs and cosmetics are artificially colored with federally approved food dyes (FD & C dyes). These dyes include Red 40, Red 3, Yellow 5, Yellow 6, Blue 1, and Blue 2. Since each dye has an identifiable absorption spectrum and peak, a spectrophotometer may be used to identify the types of FD & C dye used in a product.
Pigments may be extracted from foods and drinks that contain one or more of these dyes. An absorption spectrum of that extract can then determine what dyes are in that food or drink by comparing the peaks of maximum absorbance with information in the table below. If the absorption spectrum of a food extract has a peak at 630 nm and one at 428 nm, you can assume the food contains both Blue #1 and Yellow #5. The following table gives the wavelength of peak absorbance for each of these dyes.
| FD & C Dye | Name | Wavelength (nm) of Maximum Absorbance |
|---|---|---|
| Blue #1 | Brilliant Blue FCF | 630 |
| Green #3 | Solid Green FCF | 625 |
| Blue #2 | Indigo Carmine | 610 |
| Red #3 | Erythrosine | 527 |
| Red #40 | Allura Red AC | 502 |
| Yellow #6 | Sunset Yellow FCF | 484 |
| Yellow #5 | Tartrazine | 428 |
Materials
- Spectrophotometer
- Cuvette
- Skittles
- KimWipes
- Test tubes
Procedure
A. Extracting Dye from Candy (Your Instructor will do this for you)
You will need one test tube and one cuvette for each color to be tested. Measure 4 mL water into one tube. Place 2-4 candies of the same color in a test tube with the water. Gently swirl, and wait one minute. After, pour approximately 1 mL of liquid into a microcentrifuge tube. Spin the microcentrifuge tube at max speed for 60 seconds. Make sure the centrifuge is balanced before spinning. Transfer the clear liquid (supernatant) into a cuvette. Make sure to leave behind the particulates (pellet).
B. Measuring Absorbance with Spectrophotometer
- Turn on the spectrophotometer. Let it warm up for 15 minutes.
- Select wavelength scan.
- Fill a cuvette 2/3 full with DI water to serve as the “BLANK” cuvette.
- Calibrate the Spectrometer
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- Wipe the outside of the BLANK cuvette with a KimWipe
- Place cuvette into the machine so that the clear portions of the cuvette are oriented left to right. (The light needs to pass through clear area on cuvette).
- Press "Zero"
- Remove the blank.
- Determine optimum wavelength absorbance and set up data collection mode.
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- Place a cuvette with a sample into the spectrophotometer
- Press "read"
- Set cursor mode to peak and valley under "Options" > "More" > "Cursor Mode"
- Press the left and right arrows to the right of the graph to select the highest peak.
- Record the wavelength and absorbance in Table 6.2.
- Decide which dyes were used to make each color. Enter the Wavelength of that dye in the last column of table 6.2. Explain your reasoning for each choice in your lab notebook.
Results
|
Color skittle |
Maximum Absorbance |
Wavelength (nm) at Maximum Absorption |
Proposed Dye |
Wavelength (nm) of Maximum Absorption for Proposed Dye |
|---|---|---|---|---|
|
Red |
||||
|
Orange |
||||
|
Yellow |
||||
|
Green |
||||
|
Purple |
Instructions for Cleaning Cuvettes
- Discard solutions to sink
- Rinse with tap water once
- Rinse with DI water two times
- Place in tube rack, allow to air dry
Study Questions
- Name the parts of the spectrophotometer and identify their function.
- What is the difference between % transmittance and absorbance?
- How did you determine which wavelength was absorbed at the highest level? How is this process useful in determining the identity of a molecule?
- How do you clean a cuvette?


7.2: Diseases Common in Elderly Population
More than 37 million “baby boomer” adults born between 1946 and 1964 (60 percent) will manage more than 1 chronic condition by 2030.
Older adults are at high risk for developing chronic illnesses and related disabilities. These chronic conditions include:
- Diabetes mellitus
- Arthritis
- Congestive heart failure
- Dementia
Many experience hospitalizations, nursing home admissions, and low-quality care. They also may lose the ability to live independently at home. Chronic conditions are the leading cause of death among older adults.
Interview an older adult who has a chronic illness and related disability.
Ask about:
- what it is like to deal with the health care system in order to manage the illness
- quality of life and the impact the illness has on daily activities
- caregivers
Why Is the Health of Older Adults Important?
Health Services
- Preventive health services are valuable for maintaining the quality of life and wellness of older adults. In fact, the Patient Protection and Affordable Care Act of 2010 includes provisions related to relevant Medicare services. However, preventive services are underused, especially among certain racial and ethnic groups.
- Ensuring quality health care for older adults is difficult, but the Centers for Medicare & Medicaid Services (CMS) has programs designed to improve physician, hospital, and nursing home care, among others.
- Older adults use many health care services, have complex conditions, and require professional expertise that meets their needs. Most providers receive some type of training on aging, but the percentage of those who actually specialize in this area is small. More certified specialists are needed to meet the needs of this group.
Quality of Life
- Through programs that address chronic illnesses, Federal Government agencies are improving the quality of life for older adults. To combat existing health disparities, many of these programs target minorities and underserved populations.
- The ability to complete basic daily activities may decrease if illness, chronic disease, or injury limit physical or mental abilities of older adults. These limitations make it hard for older adults to remain at home. Early prevention and physical activity can help prevent such declines. Unfortunately, less than 20 percent of older adults engage in enough physical activity, and fewer do strength training. Minority populations often have lower rates of physical activity.
- Most older adults want to remain in their communities as long as possible. Unfortunately, when they acquire disabilities, there is often not enough support available to help them. States that invest in such services show lower rates of growth in long-term care expenditures.
Injury Prevention
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Each year, 1 out of 3 older adults falls. Falls often cause severe disability among survivors. Injuries from falls lead to:
- Fear of falling
- Sedentary behavior
- Impaired function
- Lower quality of life
Falls are the leading cause of death due to unintentional injury among older adults; deaths and injuries can be prevented by addressing risk factors.
Caregivers
- Caregivers for older adults living at home are typically unpaid family members. Caregiver stress often results in unnecessary nursing home placement.
- One to 2 million older adults in the United States are injured or mistreated by a loved one or a caregiver. A measure of elder abuse has been added to encourage data collection on this issue.
The Healthy People 2020 objectives on older adults are designed to promote healthy outcomes for this population. Many factors affect the health, function, and quality of life of older adults.
Individual Behavioral Determinants of Health in Older Adults
Behaviors such as participation in physical activity, self-management of chronic diseases, or use of preventive health services can improve health outcomes.
Social Environment Determinants of Health in Older Adults
Housing and transportation services affect the ability of older adults to access care. People from minority populations tend to be in poorer health and use health care less often than people from nonminority populations.
Health Services-Related Determinants of Health in Older Adults
The quality of the health and social services available to older adults and their caregivers affects their ability to manage chronic conditions and long-term care needs effectively.
Emerging Issues in the Health of Older Adults
Emerging issues for improving the health of older adults include efforts to:
- Coordinate care.
- Help older adults manage their own care.
- Establish quality measures.
- Identify minimum levels of training for people who care for older adults.
- Research and analyze appropriate training to equip providers with the tools they need to meet the needs of older adults.
There is growing recognition that data sources are limited for certain subpopulations of older adults, including the aging lesbian, gay, bisexual, and transgender populations. Research for these groups will inform future health and policy initiatives.
Chronic Diseases among Older Adults
Chronic diseases are long-term illnesses that are rarely cured. Chronic diseases such as heart disease, stroke, cancer, and diabetes are among the most common and costly health conditions. Chronic health conditions negatively affect quality of life, contributing to declines in functioning and the inability to remain in the community. Many chronic conditions can be prevented or modified with behavioral interventions. Six of the seven leading causes of death among older Americans are chronic diseases.
As shown in the bar chart above, the prevalence of certain chronic conditions differs by sex. Women report higher levels of arthritis and hypertension than men. Men report higher levels of heart disease and cancer.
There are differences by race and ethnicity in the prevalence of certain chronic conditions. In 2007–2008, among people age 65 and over, non-Hispanic blacks report higher levels of hypertension and diabetes than non-Hispanic whites (71 percent compared with 54 percent for hypertension and 30 percent compared with 16 percent for diabetes). Hispanics also report higher levels of diabetes than non-Hispanic whites (27 percent compared with 16 percent), but lower levels of arthritis (42 percent compared with 51 percent).
Arthritis, Osteoporosis, and Chronic Back Conditions
Arthritis, osteoporosis, and chronic back conditions all have major effects on quality of life, the ability to work, and basic activities of daily living.
There are more than 100 types of arthritis. Arthritis commonly occurs with other chronic conditions, such as diabetes, heart disease, and obesity. Interventions to treat the pain and reduce the functional limitations from arthritis are important, and may also enable people with these other chronic conditions to be more physically active.
Osteoporosis is a disease marked by reduced bone strength leading to an increased risk of fractures (broken bones). Chronic back pain (CBP) is common, costly, and potentially disabling.
Arthritis
Arthritis affects 1 in 5 adults and continues to be the most common cause of disability. It costs more than $128 billion per year. All of the human and economic costs are projected to increase over time as the population ages.
There are interventions that can reduce arthritis pain and functional limitations, but they remain underused. These include:
- Increased physical activity
- Self-management education
- Weight loss among overweight/obese adults
Osteoporosis
In the United States, an estimated 5.3 million people aged 50 years and older have osteoporosis. Most of these people are women, but about 0.8 million are men. Just over 34 million more people, including 12 million men, have low bone mass, which puts them at increased risk for developing osteoporosis. Half of all women and as many as 1 in 4 men aged 50 years and older will have an osteoporosis-related fracture in their lifetime.
Chronic Back Conditions
About 80 percent of Americans experience low back pain (LBP) in their lifetime. It is estimated that each year:
- 15-20 percent of the population develop protracted back pain.
- 2-8 percent have chronic back pain (pain that lasts more than 3 months).
- 3-4 percent of the population is temporarily disabled due to back pain.
- 1 percent of the working-age population is disabled completely and permanently as a result of LBP.
Americans spend at least $50 billion each year on LBP. LBP is the:
- Second leading cause of lost work time (after the common cold)
- Third most common reason to undergo a surgical procedure
- Fifth most frequent cause of hospitalization
Understanding Arthritis, Osteoporosis, and Chronic Back Conditions
Arthritis
Many factors determine the pain, function, and quality of life of those with arthritis. Greater physical activity can reduce pain and improve function. However, physical activity remains an underused intervention, even though there are a variety of programs to help people with arthritis increase physical activity safely and with little pain. Self-management education can achieve similar positive outcomes by teaching people skills and techniques to deal with the day-to-day issues that result from arthritis. Weight loss among those who are overweight or obese also helps reduce symptoms of arthritis.
Osteoporosis
There are many factors that contribute to osteoporosis and fractures. Nutrition and physical activity are important modifiable (controllable) risk factors. Family history and personal history of fractures are also risk factors for osteoporosis.
Chronic Back Conditions
CBP is often progressive and its cause(s) can be difficult to determine. Most important, previous studies have shown that patients with CBP make up at least 90 percent of total spending on the treatment of lower back pain.
Emerging Issues in Arthritis, Osteoporosis, and Chronic Back Conditions
Several emerging issues may warrant future Healthy People objectives for arthritis, osteoporosis, and chronic back conditions.
- Fatigue is a clinically important symptom of many types of arthritis and other rheumatic conditions. It is often just as problematic as pain.
- Early diagnosis of inflammatory types of arthritis is of growing importance, because the early use of disease-modifying anti-rheumatic drugs has resulted in much more successful treatment of these conditions. There is a continuing effort to develop early biomarkers (both biochemical and imaging markers) of arthritis, osteoporosis, and chronic back conditions to allow adequate and early assessment and treatment of these conditions.
- Worksite accommodation will become a greater issue as the number of working-age people with arthritis and chronic back conditions increases.
- Social participation, an important part of the World Health Organization (WHO) International Classification of Functioning, Disability, and Health (ICF), should be measured for people with arthritis and other chronic conditions.
- Anxiety and depression re frequently observed outcomes associated with chronic conditions such as arthritis, osteoporosis, and chronic back conditions.
- Better measures of arthritis and chronic back pain self-management education will help focus intervention efforts.
- A greater availability of health-related quality-of-life measures will be important in order to monitor nonfatal, but chronic, disabling conditions such as arthritis, osteoporosis, and chronic back conditions.
Dementias, Including Alzheimer’s Disease
Dementia is the loss of cognitive functioning—thinking, remembering, and reasoning—to such an extent that it interferes with a person’s daily life. Dementia is not a disease itself, but rather a set of symptoms. Memory loss is a common symptom of dementia, although memory loss by itself does not mean a person has dementia. Alzheimer’s disease is the most common cause of dementia, accounting for the majority of all diagnosed cases.
Diagnosis of dementia is key to effective treatment and care. It is important to distinguish dementia from temporary, reversible conditions that may cause loss of cognitive functioning. Temporary, reversible conditions include:
- Series of strokes
- Side effects from medication
- Chronic alcoholism
- Some tumors and infections in the brain
- Vitamin B12 deficiency
- Dehydration
These conditions are not dementia, but they can be serious and should be treated by a doctor as soon as possible.
Why Are Dementias, Including Alzheimer’s Disease, Important?
Alzheimer’s disease is the 6th leading cause of death among adults aged 18 years and older. Estimates vary, but experts suggest that up to 5.1 million Americans aged 65 years and older have Alzheimer’s disease. These numbers are predicted to more than double by 2050 unless more effective ways to treat and prevent Alzheimer’s disease are found.
Dementia affects an individual’s health, quality of life, and ability to live independently. It can diminish a person’s ability to effectively:
- Manage medications and medical conditions.
- Maintain a bank account.
- Drive a car or use appliances safely.
- Avoid physical injury.
- Maintain social relationships.
- Carry out activities of daily living, such as bathing or dressing.
People living with dementia are at greater risk for general disability and experience frequent injury from falls. Older adults with dementia are 3 times more likely to have preventable hospitalizations. As their dementia worsens, people need more health services and, oftentimes, long-term care. Many individuals requiring long-term care experience major personal and financial challenges that affect their families, their caregivers, and society.
There are important steps to take to improve the identification of and care for people with dementia. These include:
- Increasing the availability of existing effective diagnostic tools.
- Decreasing the number of people with undiagnosed dementia.
- Reducing the severity of symptoms through better medical management.
- Supporting family caregivers with social, behavioral, and legal resources.
- Encouraging healthy behaviors to reduce the risk of co-occurring conditions.
Understanding Dementias, Including Alzheimer’s Disease
Several factors determine the risk of developing dementia, including age and family history. Other factors affect the management of dementia by families, communities, and the health care system.
Age
Aging is a well-known risk factor for Alzheimer’s disease and other types of dementias. Among adults aged 65 years and older, the prevalence of Alzheimer’s disease doubles every 5 years.
Family History
People with a family history of Alzheimer’s disease are generally considered to be at greater risk of developing the disease. Researchers have identified 3 genes that are linked to early-onset Alzheimer’s disease. Until recently, only 1 gene had been identified that increases the risk of late-onset Alzheimer’s disease. However, during 2009 and 2010, international teams studying the genetics of Alzheimer’s disease have identified and confirmed 3 new genes that are associated with increased risk of late-onset Alzheimer’s disease.
Health Services
Many individuals with Alzheimer’s disease or other dementias are undiagnosed. Primary care providers do not routinely test for Alzheimer’s disease. Alzheimer’s disease and other dementias are more often undiagnosed in rural and minority populations than in urban or white populations.
Some chronic conditions are common in people with Alzheimer’s disease and other dementias. Dementias can greatly complicate the medical management of these conditions; this increases the need for coordination of care among different specialists.
Lack of diagnosis seriously reduces a person’s access to available treatments and valuable information. Active medical management, information and support, and coordination of medical and community services have been shown to improve quality and outcomes of care for people with dementia.
Emerging Issues in Dementias, Including Alzheimer’s Disease
Over the past decade, there has been significant scientific progress in understanding and managing dementia, with most of the research focused on Alzheimer’s disease. During the next decade, it will be important that progress be made in:
- Improving the early diagnosis of Alzheimer’s disease and other dementias.
- Developing interventions to delay or prevent Alzheimer’s disease and other dementias.
- Finding better ways to manage dementia when other chronic conditions are present.
- Understanding the influence of lifestyle factors on a person’s risk of cognitive decline and dementia.
If you had the opportunity to decide how to spend some limited funding to improve the health of older adults, how would spend it? Why?
- Spend it all on a specific condition?
- Which condition?
- Spend it on prevention, treatment, or cure?
- Spend it on national, state, or local efforts?
Sources
Source: Healthy People, healthypeople.gov/2020/topicsobjectives2020/overview.aspx?topicid=31
Chronic Diseases among Older Adults: www.agingstats.gov/agingstatsdotnet/Main_Site/Data/2010_Documents/Docs/OA_2010.pdf
Arthritis, Osteoporosis, and Chronic Back Conditions: healthypeople.gov/2020/topicsobjectives2020/overview.aspx?topicid=3
Dementias, Including Alzheimer’s Disease: Healthy People, healthypeople.gov/2020/topicsobjectives2020/overview.aspx?topicid=7


8: Earth as a Planet
Airless worlds in our solar system seem peppered with craters large and small. Earth, on the other hand, has few craters, but a thick atmosphere and much surface activity. Although impacts occurred on Earth at the same rate, craters have since been erased by forces in the planet’s crust and atmosphere. What can the comparison between the obvious persistent cratering on so many other worlds, and the different appearance of Earth, tell us about the history of our planet?
As our first step in exploring the solar system in more detail, we turn to the most familiar planet, our own Earth. The first humans to see Earth as a blue sphere floating in the blackness of space were the astronauts who made the first voyage around the Moon in 1968. For many people, the historic images showing our world as a small, distant globe represent a pivotal moment in human history, when it became difficult for educated human beings to view our world without a global perspective. In this chapter, we examine the composition and structure of our planet with its envelope of ocean and atmosphere. We ask how our terrestrial environment came to be the way it is today, and how it compares with other planets.
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- 8.1: The Global Perspective
- Earth is the prototype terrestrial planet. Its interior composition and structure are probed using seismic waves. Such studies reveal that Earth has a metal core and a silicate mantle. The outer layer, or crust, consists primarily of oceanic basalt and continental granite. A global magnetic field, generated in the core, produces Earth’s magnetosphere, which can trap charged atomic particles.
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- 8.2: Earth's Crust
- Terrestrial rocks can be classified as igneous, sedimentary, or metamorphic. A fourth type, primitive rock, is not found on Earth. Our planet’s geology is dominated by plate tectonics, in which crustal plates move slowly in response to mantle convection. The surface expression of plate tectonics includes continental drift, recycling of the ocean floor, mountain building, rift zones, subduction zones, faults, earthquakes, and volcanic eruptions of lava from the interior.
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- 8.3: Earth's Atmosphere
- The atmosphere has a surface pressure of 1 bar and is composed primarily of \(N_2\) and \(O_2\), plus such important trace gases as \(H_2O\), \(CO_2\), and \(O_3\). Its structure consists of the troposphere, stratosphere, mesosphere, and ionosphere. Changing the composition of the atmosphere also influences the temperature. Atmospheric circulation (weather) is driven by seasonally changing deposition of sunlight. Many longer term climate variations, such as the ice ages, are related to changes i
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- 8.4: Life, Chemical Evolution, and Climate Change
- Life originated on Earth at a time when the atmosphere lacked \(O_2\) and consisted mostly of \(CO_2\). Later, photosynthesis gave rise to free oxygen and ozone. Modern genomic analysis lets us see how the wide diversity of species on the planet are related to each other. \(CO_2\) and methane in the atmosphere heat the surface through the greenhouse effect; today, increasing amounts of atmospheric \(CO_2\) are leading to the global warming of our planet.
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- 8.5: Cosmic Influences on the Evolution of Earth
- Earth, like the Moon and other planets, has been influenced by the impacts of cosmic debris, including such recent examples as Meteor Crater and the Tunguska explosion. Larger past impacts are implicated in some mass extinctions, including the large impact 65 million years ago at the end of the Cretaceous period that wiped out the dinosaurs and many other species. Today, astronomers are working to predict the next impact in advance, while other scientists are coming to grips with the effect of i
Thumbnail: This image, taken from the International Space Station in 2006, shows a plume of ash coming from the Cleveland Volcano in the Aleutian Islands. Although the plume was only visible for around two hours, such events are a testament to the dynamic nature of Earth’s crust. (credit: modification of work by NASA)


1.7: Add and Subtract Fractions
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Learning Objectives
By the end of this section, you will be able to:
- Add or subtract fractions with a common denominator
- Add or subtract fractions with different denominators
- Use the order of operations to simplify complex fractions
- Evaluate variable expressions with fractions
Note
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions .
Add or Subtract Fractions with a Common Denominator
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.
FRACTION ADDITION AND SUBTRACTION
If \(a,b\), and \(c\) are numbers where \(c\neq 0\), then
\[\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c} \quad \text{and} \quad \dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a - b}{c}\]
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
Manipulative Mathematics
Doing the Manipulative Mathematics activities “Model Fraction Addition” and “Model Fraction Subtraction” will help you develop a better understanding of adding and subtracting fractions.
Example \(\PageIndex{1}\)
Find the sum: \(\dfrac{x}{3} + \dfrac{2}{3}\).
Solution
\[\begin{array} {ll} {} &{\dfrac{x}{3} + \dfrac{2}{3}} \\ {\text{Add the numerators and place the sum over the common denominator}} &{\dfrac{x + 2}{3}} \end{array}\]
Try It \(\PageIndex{2}\)
Find the sum: \(\dfrac{x}{4} + \dfrac{3}{4}\).
- Answer
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\(\dfrac{x + 3}{4}\)
Try It \(\PageIndex{3}\)
Find the sum: \(\dfrac{y}{8} + \dfrac{5}{8}\).
- Answer
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\(\dfrac{y + 5}{8}\)
Example \(\PageIndex{4}\)
Find the difference: \(-\dfrac{23}{24} - \dfrac{13}{24}\)
Solution
\[\begin{array} {ll} {} &{-\dfrac{23}{24} - \dfrac{13}{24}} \\ {\text{Subtract the numerators and place the }} &{\dfrac{-23 - 13}{24}} \\ {\text{difference over the common denominator}} &{} \\ {\text{Simplify.}} &{\dfrac{-36}{24}} \\ {\text{Simplify. Remember, }-\dfrac{a}{b} = \dfrac{-a}{b}} &{-\dfrac{3}{2}} \end{array}\]
Try It \(\PageIndex{5}\)
Find the difference: \(-\dfrac{19}{28} - \dfrac{7}{28}\)
- Answer
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\(-\dfrac{26}{28}\)
Try It \(\PageIndex{6}\)
Find the difference: \(-\dfrac{27}{32} - \dfrac{1}{32}\)
- Answer
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\(-\dfrac{7}{8}\)
Example \(\PageIndex{7}\)
Find the difference: \(-\dfrac{10}{x} - \dfrac{4}{x}\)
Solution
\[\begin{array} {ll} {} &{-\dfrac{10}{x} - \dfrac{4}{x}} \\ {\text{Subtract the numerators and place the }} &{\dfrac{-14}{x}} \\ {\text{difference over the common denominator}} &{} \\ {\text{Rewrite with the sign in front of the fraction.}} &{-\dfrac{14}{x}} \end{array}\]
Try It \(\PageIndex{8}\)
Find the difference: \(-\dfrac{9}{x} - \dfrac{7}{x}\)
- Answer
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\(-\dfrac{16}{x}\)
Try It \(\PageIndex{9}\)
Find the difference: \(-\dfrac{17}{a} - \dfrac{5}{a}\)
- Answer
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\(-\dfrac{22}{a}\)
Now we will do an example that has both addition and subtraction.
Example \(\PageIndex{10}\)
Simplify: \(\dfrac{3}{8} + (-\dfrac{5}{8}) - \dfrac{1}{8}\)
Solution
\[\begin{array} {ll} {\text{Add and Subtract fractions — do they have a }} &{\frac{3}{8} + (-\frac{5}{8}) - \frac{1}{8}} \\ {\text{common denominator? Yes.}} &{} \\ {\text{Add and subtract the numerators and place }} &{\frac{3 + (-5) - 1}{8}} \\ {\text{the result over the common denominator.}} &{} \\ {\text{Simplify left to right.}} &{\frac{-2 - 1}{8}} \\ {\text{Simplify.}} &{-\frac{3}{8}} \end{array}\]
Try It \(\PageIndex{11}\)
Simplify: \(\dfrac{2}{9} + (-\dfrac{4}{9}) - \dfrac{7}{9}\)
- Answer
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\(-1\)
Try It \(\PageIndex{12}\)
Simplify: \(\dfrac{2}{5} + (-\dfrac{4}{9}) - \dfrac{7}{9}\)
- Answer
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\(-\dfrac{2}{3}\)
Add or Subtract Fractions with Different Denominators
As we have seen, to add or subtract fractions, their denominators must be the same. 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.
LEAST COMMON DENOMINATOR
The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.
Note
Doing the Manipulative Mathematics activity “Finding the Least Common Denominator” will help you develop a better understanding of the LCD.
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!
Example \(\PageIndex{13}\)
Add: \(\dfrac{7}{12} + \dfrac{5}{18}\)
Solution
Try It \(\PageIndex{14}\)
Add: \(\dfrac{7}{12} + \dfrac{11}{15}\)
- Answer
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\(\dfrac{79}{60}\)
Try It \(\PageIndex{15}\)
Add: \(\dfrac{13}{15} + \dfrac{17}{20}\)
- Answer
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\(\dfrac{103}{60}\)
ADD OR SUBTRACT FRACTIONS.
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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.
When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.
Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.
In Exercise \(\PageIndex{13}\), the LCD, 36, has two factors of 2 and two factors of 3.
The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.
The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.
We will apply this method as we subtract the fractions in Exercise \(\PageIndex{16}\).
Example \(\PageIndex{16}\)
Subtract: \(\dfrac{7}{15} - \dfrac{19}{24}\)
Solution
Do the fractions have a common denominator? No, so we need to find the LCD.
| Find the LCD. | |
| Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD. | |
| Rewrite as equivalent fractions with the LCD. | |
| Simplify. | |
| Subtract. | \(-\dfrac{39}{120}\) |
| Check to see if the answer can be simplified. | \(-\dfrac{13\cdot3}{40\cdot3}\) |
| Both 39 and 120 have a factor of 3. | |
| Simplify. | \(-\dfrac{13}{40}\) |
Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!
Try It \(\PageIndex{17}\)
Subtract: \(\dfrac{13}{24} - \dfrac{17}{32}\)
- Answer
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\(\dfrac{1}{96}\)
Try It \(\PageIndex{18}\)
Subtract: \(\dfrac{7}{15} - \dfrac{19}{24}\)
- Answer
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\(\dfrac{75}{224}\)
In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.
Example \(\PageIndex{19}\)
Add: \(\dfrac{3}{5} + \dfrac{x}{8}\)
Solution
The fractions have different denominators.
| Find the LCD. | |
| Rewrite as equivalent fractions with the LCD. | |
| Simplify. | |
| Add. |
Remember, we can only add like terms: \(24\) and \(5x\) are not like terms.
Try It \(\PageIndex{20}\)
Add: \(\dfrac{y}{6} + \dfrac{7}{9}\)
- Answer
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\(\dfrac{3y + 14}{18}\)
Try It \(\PageIndex{21}\)
Add: \(\dfrac{x}{6} + \dfrac{7}{15}\)
- Answer
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\(\dfrac{15x + 42}{153}\)
We now have all four operations for fractions. Table \(\PageIndex{1}\) summarizes fraction operations .
| Fraction Multiplication | Fraction Division |
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\(\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}\)
Multiply the numerators and multiply the denominators |
\(\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}\)
Multiply the first fraction by the reciprocal of the second. |
| Fraction Addition | Fraction Subtraction |
|
\(\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}\)
Add the numerators and place the sum over the common denominator. |
\(\dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a - b}{c}\)
Subtract the numerators and place the difference over the common denominator. |
| To multiply or divide fractions, and LCD is NOT needed. To add or subtract fractions, an LCD is needed. |
Example \(\PageIndex{22}\)
Simplify:
- \(\dfrac{5x}{6} - \dfrac{3}{10}\)
- \(\dfrac{5x}{6}\cdot \dfrac{3}{10}\).
Solution
First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.
1. What is the operation? The operation is subtraction.
\[\begin{array} {ll} {\text{Do the fractions have a common denominator? No.}} &{\frac{5x}{6} - \frac{3}{10}} \\ {\text{Rewrite each fractions as an equivalent fraction with the LCD.}} &{\frac{5x\cdot 5}{6\cdot 5} - \frac{3\cdot3}{10\cdot3}} \\ {} &{\frac{25x}{30} - \frac{9}{30}} \\{\text{Subtract the numerators and place the difference over the}} &{\frac{25x - 9}{30}} \\ {\text{common denominators.}} &{} \\ {\text{Simplify, if possible. There are no common factors.}} &{} \\ {\text{The fraction is simplified.}} &{} \end{array}\]
2. What is the operation? Multiplication.
\[\begin{array} {ll} {} &{\frac{5x}{6}\cdot \frac{3}{10}} \\ {\text{To multiply fractions, multiply the numerators and multiply}} &{\frac{5x\cdot 3}{6\cdot 10}} \\ {\text{the denominators}} &{} \\{\text{Rewrite, showing common factors.}} &{\frac{\not 5 x\cdot\not3}{2\cdot\not3\cdot2\cdot\not5}} \\ {\text{common denominators.}} &{} \\ {\text{Simplify.}} &{\frac{x}{4}} \end{array}\]
Try It \(\PageIndex{23}\)
Simplify:
- \(\dfrac{3a}{4} - \dfrac{8}{9}\)
- \(\dfrac{3a}{4}\cdot\dfrac{8}{9}\)
- Answer
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- \(\dfrac{27a - 32}{36}\)
- \(\dfrac{2a}{3}\)
Try It \(\PageIndex{24}\)
Simplify:
- \(\dfrac{4k}{5} - \dfrac{1}{6}\)
- \(\dfrac{4k}{5}\cdot\dfrac{1}{6}\)
- Answer
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- \(\dfrac{24k - 5}{30}\)
- \(\dfrac{2k}{15}\)
Use the Order of Operations to Simplify Complex Fractions
We have seen that a complex fraction is a fraction in which the numerat or or denominator contains a fraction. The fraction bar indicates division . We simplified the complex fraction \(\dfrac{\frac{3}{4}}{\frac{5}{8}}\) by dividing \(\dfrac{3}{4}\) by \(\dfrac{5}{8}\).
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.
Example \(\PageIndex{25}\): How to simplify complex fractions
Simplify: \(\dfrac{(\frac{1}{2})^{2}}{4 + 3^{2}}\)
Solution
Try It \(\PageIndex{26}\)
Simplify: \(\dfrac{(\frac{1}{3})^{2}}{2^{3} + 2}\)
- Answer
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\(\dfrac{1}{90}\)
Try It \(\PageIndex{27}\)
Simplify: \(\dfrac{1 + 4^{2}}{(\frac{1}{4})^{2}}\)
- Answer
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\(272\)
SIMPLIFY COMPLEX FRACTIONS.
- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator. Simplify if possible.
Example \(\PageIndex{28}\)
Simplify: \(\dfrac{\frac{1}{2} + \frac{2}{3}}{\frac{3}{4} - \frac{1}{6}}\)
Solution
\[\begin{array} {ll} {} &{\frac{(\frac{1}{2} + \frac{2}{3})}{(\frac{3}{4} - \frac{1}{6})}} \\ {\text{Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).}} &{\frac{(\frac{3}{6} + \frac{4}{6})}{(\frac{9}{12} - \frac{2}{12})}} \\ {\text{Simplify.}} &{\frac{(\frac{7}{6})}{(\frac{7}{12})}} \\{\text{Divide the numerator by the denominator.}} &{\frac{7}{6}\div\frac{7}{12}} \\ {\text{Simplify.}} &{\frac{7}{6}\cdot\frac{12}{7}} \\ {\text{Divide out common factors.}} &{\frac{7\cdot6\cdot2}{6\cdot7}} \\ {\text{Simplify.}} &{2} \end{array}\]
Try It \(\PageIndex{29}\)
Simplify: \(\dfrac{\frac{1}{3} + \frac{1}{2}}{\frac{3}{4} - \frac{1}{3}}\)
- Answer
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\(2\)
Try It \(\PageIndex{30}\)
Simplify: \(\dfrac{\frac{2}{3} - \frac{1}{2}}{\frac{1}{4} + \frac{1}{3}}\)
- Answer
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\(\dfrac{2}{7}\)
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.
Example \(\PageIndex{31}\)
Evaluate \(x + \dfrac{1}{3}\) when
- \(x = -\dfrac{1}{3}\)
- \(x = -\dfrac{3}{4}\)
Solution
1. To evaluate \(x + \dfrac{1}{3}\) when \(x = -\dfrac{1}{3}\), substitute \(-\dfrac{1}{3}\) for \(x\) in the expression.
| Simplify. | \(0\) |
2. To evaluate \(x + \dfrac{1}{3}\) when \(x = -\dfrac{3}{4}\), substitute \(-\dfrac{3}{4}\) for \(x\) in the expression.
| Rewrite as equivalent fractions with the LCD, 12. | |
| Simplify. | |
| Add. | \(-\dfrac{5}{12}\) |
Try It \(\PageIndex{32}\)
Evaluate \(x + \dfrac{3}{4}\) when
- \(x = -\dfrac{7}{4}\)
- \(x = -\dfrac{5}{4}\)
- Answer
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- \(-1\)
- \(-\dfrac{1}{2}\)
Try It \(\PageIndex{33}\)
Evaluate \(y + \dfrac{1}{2}\) when
- \(y = \dfrac{2}{3}\)
- \(y = -\dfrac{3}{4}\)
- Answer
-
- \(\dfrac{7}{6}\)
- \(-\dfrac{1}{12}\)
Example \(\PageIndex{34}\)
Evaluate \(-\dfrac{5}{6} - y\) when \(y = -\dfrac{2}{3}\)
Solution
| Rewrite as equivalent fractions with the LCD, \(6\). | |
| Subtract. | |
| Simplify. | \(-\dfrac{1}{6}\) |
Try It \(\PageIndex{35}\)
Evaluate \(y + \dfrac{1}{2}\) when \(y = \dfrac{2}{3}\)
- Answer
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\(-\dfrac{1}{4}\)
Try It \(\PageIndex{36}\)
Evaluate \(y + \dfrac{1}{2}\) when \(y = \dfrac{2}{3}\)
- Answer
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\(-\dfrac{17}{8}\)
Example \(\PageIndex{37}\)
Evaluate \(2x^{2}y\) when \(x = \dfrac{1}{4}\) and \(y = -\dfrac{2}{3}\).
Solution
Substitute the values into the expression.
| \(2x^{2}y\) | |
| Simplify exponents first. | \(2(\frac{1}{16})(-\frac{2}{3})\) |
| Multiply. Divide out the common factors. Notice we write \(16\) as \(2\cdot2\cdot4\) to make it easy to remove | \(-\frac{\not2\cdot1\cdot\not2}{\not2\cdot\not2\cdot4\cdot3}\) |
| Simplify. | \(-\frac{1}{12}\) |
Try It \(\PageIndex{38}\)
Evaluate \(3ab^{2}\) when \(a = -\dfrac{2}{3}\) and \(b = -\dfrac{1}{2}\).
- Answer
-
\(-\dfrac{1}{2}\)
Try It \(\PageIndex{39}\)
Evaluate \(4c^{3}d\) when \(c = -\dfrac{1}{2}\) and \(d = -\dfrac{4}{3}\).
- Answer
-
\(\dfrac{2}{3}\)
The next example will have only variables, no constants.
Example \(\PageIndex{40}\)
Evaluate \(\dfrac{p + q}{r}\) when \(p = -4, q = -2\), and \(r = 8\).
Solution
To evaluate \(\dfrac{p + q}{r}\) when \(p = -4, q = -2\), and \(r = 8\), we substitute the values into the expression.
| \(\dfrac{p + q}{r}\) | |
| Add in the numerator first. | \(\dfrac{-6}{8}\) |
| Simplify. | \(-\dfrac{3}{4}\) |
Try It \(\PageIndex{41}\)
Evaluate \(\dfrac{a+b}{c}\) when \(a = -8, b = -7\), and \(c = 6\).
- Answer
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\(-\dfrac{5}{2}\)
Try It \(\PageIndex{42}\)
Evaluate \(\dfrac{x+y}{z}\) when \(x = 9, y = -18\), and \(z = -6\).
- Answer
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\(\dfrac{3}{2}\)
Key Concepts
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Fraction Addition and Subtraction:
If \(a, b\), and \(c\) are numbers where \(c\neq 0\), then
\(\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c}\) 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. -
Strategy for Adding or Subtracting 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. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
-
Do they have a common denominator?
-
Simplify Complex Fractions
- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator. Simplify if possible.


7.16: Putting It Together- Revision and Proofreading
Now that you have learned to recognize strategies for revision, make them an integral part of your writing process. For every writing project, budget time to consider the following:
- Content: Does the content of this project fulfill the purpose of the assignment? Will the audience be persuaded by the ideas and evidence? What strategies could I use to improve the content?
- Structure: Is the organization of the project appropriate for the purpose of the assignment? Will the audience find the project well-organized and logical? What strategies might I use to improve the structure?
- Style: Does the style of the project align with the purpose of the assignment? Will the audience recognize the style as appropriate and effective? What strategies might I use to improve the style?
- Grammar and mechanics: Will readers be hindered and irritated by errors in grammar and mechanics? Will errors in grammar and mechanics damage my ethos as a writer? What strategies might I use to edit and proofread the project?
Contributors and Attributions
CC licensed content, Shared previously
- Image of woman at desk. Authored by : Andrea Piacquadio. Provided by : Pexels. Located at : www.pexels.com/photo/woman-in-pink-long-sleeve-shirt-using-laptop-3785408/. License : Other . License Terms : www.pexels.com/photo-license/


6.4: Characteristics of Suitable Site Index Trees
6.4 Characteristics of Suitable Site Index Trees
“Site trees” are used to assess a site’s inherent ability to grow trees, and therefore should be the best trees on the site. We select trees expressing the full potential of the site, not those that have been subjected to damage, injury or disease. A tree with a broken top obviously is not as tall as it would have been without the breakage on any given site, and using such a tree would falsely indicate a lower site index than the site is capable of producing. Therefore, in selecting which trees to measure for obtaining site index, care must be taken to avoid those trees that misrepresent the true quality of a site. A site tree must meet all three of the following criteria:
1. It must be in the Dominant or Codominant crown class.
2. It must be free from past disturbance, injury or damage . A site index tree cannot have broken tops, scarred trunks, damaged or compacted root systems, insect injury and so on. These occurrences reduce a tree’s health and vigor so that it does not express the site’s full potential for growth.
3. It must be free from past suppression . Some trees can tolerate heavy shade when they are young, and then put on rapid growth when the canopy opens up, allowing full sunlight to shine on them. This pattern is not acceptable for a site index tree, since light availability dominated the tree’s ability to grow. The tree was not expressing the site’s full potential when shaded.
Dominant and codominant trees should be examined carefully for outward signs of injury or defect before measuring for site index. In addition, each increment core sample used to estimate age should be checked for evidence of past suppression or injury. Dramatic shifts in the ring sizes, presence of rot, charcoal, and other abnormalities can indicate previous impacts (Figure 6.6).
The following are illustrations of trees not suitable for site index determination, due to defect, injury or past suppression (Figure 6.7).


Chapter 5: The Production of Extras in a Precarious Creative Economy
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Over the past twenty years, regional governments around the world and global film industry corporations have collaborated, if not colluded, to provide a steady stream of workers for film location shooting through legislated incentives. Seeking to reduce labor costs in relation to other fixed expenses, industry executives have successfully used incentives to reduce budgets. Meanwhile, regional policymakers have looked to film and television production as a panacea for anemic economic growth and declining employment indices.¹ Together, governments and industry have made labor into one of the primary fault lines in the political economy of film production.
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5.1: Introduction
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Introduction to the focus of this chapter: the film industry strategy of creating a moral economy that calls for low-paying or unpaid jobs, as a form of boosterism for local economy and culture, through the case study of the production of the HBO drama Treme in post-Katrina New Orleans.
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5.2: The Treme Moral Economy
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The production of the HBO drama Treme in post-Katrina New Orleans. Includes discussion of the choice of the production to hire local residents as extras and media labor, as simultaneously an economically efficient and a moral choice; the application of this case to Mark Banks' concept of a moral economy.
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5.3: A Little Extra Work for a Lot of Extras
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The role played by wanting to be a part of something larger in attracting would-be extras to a production; contrasts between the low-paid extras of most major productions and the high standards set by Treme in a precarious local economy.
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5.4: To Be (an) Extra
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Discussion of some of the reasons why so many New Orleanians sought to be extras on Treme, even at the cost of giving up free time and working at the bottom of the production hierarchy: a sense of emotional solidarity with the show as a means of post-Katrina recovery, memorialization of personal experiences, or seeking to record themselves as belonging to the city. Contrasts between some extras' lived experiences and the sense of authenticity that the show sought to create.
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5.5: Concluding Futures?
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The broader labor implications of so many New Orleanians volunteering for low-paid or unpaid labor on Treme, including the exclusion of people who lack the free time or the social connections to work on the show, the undermining of improvement of public services, and the promotion of the idea that private companies are the best managers of the public good.
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5.6: Notes
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15.12: When to Make Capital Budgeting Decisions
Learning Outcomes
- Identify situations that require capital budgeting decision making
Mary walks into your office and announces, “Our machinery is 20 years old, and has become inefficient. We should think about replacing it with one of the new, faster pieces of equipment.” She has several pieces of paper in her hand showing beautiful and shiny machines will really high price tags!
Your company has a capital expenditure budget of $100,000 available this year and each of these machines has a price that approaches that amount. They also have payment and lease options. But you are thinking “will these new machines create enough in higher efficiency and costs savings to recoup this large investment, or will the ones we have continue to work just fine? Or which of the three would make the most financial sense for us right now?”
Watch It
Capital budgeting can help you to make these decisions.
Capital budgeting decisions are for the big stuff. When might we need to make such a decision?
- Cost reduction: Should we buy new equipment to reduce manufacturing costs? Might a newer machine save us money even considering the initial capital outlay?
- Expansion: Would a new plant, warehouse or other building help to increase sales and capacity? Are we not making the income we could, by not expanding?
- Picking the equipment: There may be three or more options for equipment at various prices with different options. Which one will make the most sense? We know we need new stuff, but with all of the various options out there, evaluation of all of the options is important.
- Leasing or buying: Should we purchase a new piece of equipment or lease it? Sometimes leasing may be a better option!
- Replacement: Should we replace a machine now, or wait? The machine we are currently using to make widgets still works, but it is slow and frequently breaking down. A new one is a huge investment, but how long will it take to pay for itself?
All of these situations require analysis to figure out which is going to be the best decision in the long run. We make these kinds of decisions personally when we evaluate buying a new car, dishwasher or house. Businesses go through a similar process to determine what will work best, and ultimately create the best financial outcome.
Practice Question


9.13: Logic of Hypothesis Testing (Exercises)
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You may want to use the Binomial Calculator for some of these exercises.
General Questions
Q1
An experiment is conducted to test the claim that James Bond can taste the difference between a Martini that is shaken and one that is stirred. What is the null hypothesis? (relevant section)
Q2
The following explanation is incorrect. What three words should be added to make it correct? (relevant section)
The probability value is the probability of obtaining a statistic as different from the parameter specified in the null hypothesis as the statistic obtained in the experiment. The probability value is computed assuming that the null hypothesis is true.
Q3
Why do experimenters test hypotheses they think are false? (relevant section)
Q4
State the null hypothesis for:
- An experiment testing whether echinacea decreases the length of colds.
- A correlational study on the relationship between brain size and intelligence.
- An investigation of whether a self-proclaimed psychic can predict the outcome of a coin flip.
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A study comparing a drug with a placebo on the amount of pain relief. (A one-tailed test was used.)
(relevant section & relevant section)
Q5
Assume the null hypothesis is that \(\mu =50\) and that the graph shown below is the sampling distribution of the mean (\(M\)). Would a sample value of \(M= 60\) be significant in a two-tailed test at the \(0.05\) level? Roughly what value of \(M\) would be needed to be significant? (relevant section & relevant section)
Q6
A researcher develops a new theory that predicts that vegetarians will have more of a particular vitamin in their blood than non-vegetarians. An experiment is conducted and vegetarians do have more of the vitamin, but the difference is not significant. The probability value is \(0.13\). Should the experimenter's confidence in the theory increase, decrease, or stay the same? (relevant section)
Q7
A researcher hypothesizes that the lowering in cholesterol associated with weight loss is really due to exercise. To test this, the researcher carefully controls for exercise while comparing the cholesterol levels of a group of subjects who lose weight by dieting with a control group that does not diet. The difference between groups in cholesterol is not significant. Can the researcher claim that weight loss has no effect? (relevant section)
Q8
A significance test is performed and \(p = 0.20\). Why can't the experimenter claim that the probability that the null hypothesis is true is \(0.20\)? (relevant section, relevant section & relevant section)
Q9
For a drug to be approved by the FDA, the drug must be shown to be safe and effective. If the drug is significantly more effective than a placebo, then the drug is deemed effective. What do you know about the effectiveness of a drug once it has been approved by the FDA (assuming that there has not been a Type I error)? (relevant section)
Q10
When is it valid to use a one-tailed test? What is the advantage of a one-tailed test? Give an example of a null hypothesis that would be tested by a one-tailed test. (relevant section)
Q11
Distinguish between probability value and significance level. (relevant section)
Q12
Suppose a study was conducted on the effectiveness of a class on "How to take tests." The SAT scores of an experimental group and a control group were compared. (There were \(100\) subjects in each group.) The mean score of the experimental group was \(503\) and the mean score of the control group was \(499\). The difference between means was found to be significant, \(p = 0.037\). What do you conclude about the effectiveness of the class? (relevant section & relevant section)
Q13
Is it more conservative to use an alpha level of \(0.01\) or an alpha level of \(0.05\)? Would beta be higher for an alpha of \(0.05\) or for an alpha of \(0.01\)? (relevant section)
Q14
Why is \(H_o: M_1 = M_2\) not a proper null hypothesis? (relevant section)
Q15
An experimenter expects an effect to come out in a certain direction. Is this sufficient basis for using a one-tailed test? Why or why not? (relevant section)
Q16
How do the Type I and Type II error rates of one-tailed and two-tailed tests differ? (relevant section & relevant section)
Q17
A two-tailed probability is \(0.03\). What is the one-tailed probability if the effect were in the specified direction? What would it be if the effect were in the other direction? (relevant section)
Q18
You choose an alpha level of \(0.01\) and then analyze your data.
- What is the probability that you will make a Type I error given that the null hypothesis is true?
- What is the probability that you will make a Type I error given that the null hypothesis is false? (relevant section)
Q19
Why doesn't it make sense to test the hypothesis that the sample mean is \(42\)? (relevant section & relevant section)
Q20
True/false: It is easier to reject the null hypothesis if the researcher uses a smaller alpha (\(\alpha\)) level. (relevant section & relevant section)
Q21
True/false: You are more likely to make a Type I error when using a small sample than when using a large sample. (relevant section)
Q22
True/false: You accept the alternative hypothesis when you reject the null hypothesis. (relevant section)
Q23
True/false: You do not accept the null hypothesis when you fail to reject it. (relevant section)
Q24
True/false: A researcher risks making a Type I error any time the null hypothesis is rejected. (relevant section)


Skip to main content
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Why is the United States a leader in entrepreneurship?
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What commonalities do the top five countries have that put them in the lead for entrepreneurial ventures?
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One of Roxanne Quimby’s motivations in starting businesses included helping employ people in her community. How does or doesn’t this motivation fit in the definition of an entrepreneur as someone who identifies a problem and solves that problem?
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What are the similarities and differences between the business ventures of Wander Girls and Roxanne Quimby?
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Why is knowing your vision for the future important?
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Why should we create goals using the SMART method?
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What is the value in using divergent thinking when trying to solve a problem?
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How would you explain the concept of an entrepreneurial vision to your college roommate who is a history major?
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What is the primary difference between divergent thinking and brainstorming?
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Identify five disruptive technologies and the industries that these technologies impact.
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In thinking about the future, and your own experiences, what opportunities are there for disruptive technologies in one or two familiar industries?
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What is the difference between an entrepreneurial mindset and entrepreneurial spirit?
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Describe your own level of entrepreneurial spirit.


22.2: Force between two current-carrying wires
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Consider two infinite parallel straight wires, a distance \(h\) apart, carrying upwards currents, \(I_{1}\) and \(I_{2}\), respectively, as illustrated in Figure \(\PageIndex{1}\).
The first wire will create a magnetic field, \(\vec B_{1}\), in the shape of circles concentric with the wire. At the position of the second wire, the magnetic field \(B_{1}\) is into the page, and has a magnitude:
\[\begin{aligned} B_{1}=\frac{\mu_{0}I_{1}}{2\pi h} \end{aligned}\]
Since the second wire carries a current, \(I_{2}\), upwards, it will experience a magnetic force, \(\vec F_{2}\), from the magnetic field, \(B_{1}\), that is towards the left (as illustrated in Figure \(\PageIndex{1}\) and determined from the right-hand rule). The magnetic force, \(\vec F_{2}\), exerted on a section of length, \(l\), on the second wire has a magnitude given by:
\[\begin{aligned} F_{2}=I_{2}||\vec l\times\vec B_{1}||=I_{2}lB_{1}\frac{\mu_{0}I_{2}I_{1}l}{2\pi h} \end{aligned}\]
where we used the fact that the angle between \(\vec l\) and \(\vec B\) is \(90^{◦}\). We expect, from Newton’s Third Law, that an equal and opposite force should be exerted on the first wire. Indeed, the second wire will create a magnetic field, \(\vec B_{2}\), that is out of the page at the location of the first wire, with magnitude:
\[\begin{aligned} B_{2}=\frac{\mu_{0}I_{2}}{2\pi h} \end{aligned}\]
This leads to a magnetic force, \(\vec F_{1}\), exerted on the first wire, that points to the right (from the right-hand rule). On a section of length, \(l\), of the first wire, the magnetic force from the magnetic field, \(\vec B_{2}\), has magnitude:
\[\begin{aligned} F_{1}=I_{1}||\vec l\times\vec B_{2}||=I_{1}lB_{2}\frac{\mu_{0}I_{1}I_{2}}{2\pi h} \end{aligned}\]
which does indeed have the same magnitude as the force exerted on the second wire. Thus, when two parallel wires carry current in the same direction, they exert equal and opposite attractive forces on each other.
Exercise \(\PageIndex{1}\)
Two parallel wires carry current in opposite directions, as shown in Figure \(\PageIndex{2}\). What force do they exert on each other?
- There will be no force, since the currents cancel.
- There will be an attractive force between the wires.
- There will be a repulsive force between the wires.
- Answer
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The attractive force between two wires used to be the basis for defining the Ampere, the S.I. (base) unit for electric current. Before 2019, the Ampere was defined to be “that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in vacuum, would produce between these conductors a force equal to \(2 × 10^{−17}\text{N}\) per meter of length”. Recently, the definition was updated to be based on defining the Coulomb in such a way that the elementary charge has a numerical value of \(e = 1.602 176 634 × 10^{−19}\text{C}\), and the Ampere corresponds to one Coulomb per second.
The force between two wires is a good system to understand how any physical quantity cannot depend on our choice of the right-hand to define cross-products. As mentioned in the previous chapter, any physical quantity, such as the direction of the force exerted on a wire, will always depend on two successive uses of the right hand. In this system, we first used the right-hand rule for axial vectors to determine the direction of the magnetic field from one of the wires. We then used the right-hand rule to determine the direction of the cross-product to determine the direction of the force on the other wire. You can verify that you get the same answer if you, instead, use your left-hand to define the direction of the magnetic field (which will be in the opposite direction), and then again for the cross-product. This also highlights that the magnetic field (and the electric field) is just a mathematical tool that we use to, ultimately, describe the motion of charges or compass needles.
Exercise \(\PageIndex{2}\)
When current is flowing in a straight cable, how to you expect the charges to be distributed radially through the cross-section of the cable?
- Uniformly in radius (current density does not depend on \(r\)).
- There will be an excess of positive charges on the outside of the cable.
- There will be an excess of negative charges on the outside of the cable.
- Answer
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