3.3: Examples
You have escaped from your cell in a dungeon underneath the castle shown in the figure. If you can run through the tunnels and find an exit on the other side of the river (30 m wide and 205 m north, as shown), you'll escape from the evil wizard!
- First you take a tunnel 180 m long traveling 30\(^\circ\) north of east.
- Then you make a right angle turn to the left, and travel another 180 m.
At the end of this tunnel, you see a hatch in the ceiling. Will you escape to the other side of the river if you take this hatch?
Solve the following two problems using vector components.
- Sandra drives her car directly North for 2.0 km, and then turns 20 degrees to the East and drives another 4.0 km. Find the vector (magnitude and direction) from her initial location to her final location.
- Joe and Max shake hands and say goodbye. Joe walks east 1.00 km to a coffee shop. Max flags a cab and rides 5.00 km in a direction 30 degrees north of east to a bookstore. Find the vector (magnitude and direction) from the coffee shop (Joe) to the bookstore (Max).
Example \(\PageIndex{3}\): Analytical Computation of a Resultant
Three displacement vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) in a plane are specified by their magnitudes A = 10.0, B = 7.0, and C = 8.0, respectively, and by their respective direction angles with the horizontal direction \(\alpha\) = 35°, \(\beta\) = −110°, and \(\gamma\) = 30°. The physical units of the magnitudes are centimeters. Resolve the vectors to their scalar components and find the following vector sums:
- \(\vec{R}\) = \(\vec{A}\) + \(\vec{B}\) + \(\vec{C}\),
- \(\vec{D}\) = \(\vec{A}\) − \(\vec{B}\), and
- \(\vec{S}\) = \(\vec{A}\) − 3 \(\vec{B}\) + \(\vec{C}\).
First, we use Equation 3.2.13 to find the scalar components of each vector and then we express each vector in its vector component form given by \(\overrightarrow{\mathbf{A}}=A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}\). Then, we use analytical methods of vector algebra to find the resultants.
SolutionWe resolve the given vectors to their scalar components:
\[ \begin{cases} A_{x} = A \cos \alpha = (10.0\; cm) \cos {35^{o}} = 8.19\; cm \\ A_{y} = A \sin \alpha = (10.0\; cm) \sin{35^{o}} = 5.73\; cm \end{cases}\]
\[ \begin{cases} B_{x} = B \cos \beta = (7.0\; cm) \cos (-110^{o}) = -2.39\; cm \\ B_{y} = B \sin \beta= (7.0\; cm) \sin (-110^{o}) = -6.58\; cm \end{cases}\]
\[ \begin{cases} C_{x} = C \cos \gamma= (8.0\; cm) \cos (30^{o}) = 6.93\; cm \\ C_{y} = C \sin \gamma= (8.0\; cm) \sin(30^{o}) = 4.00\; cm \end{cases}\]
For (a) we may substitute directly into the given expression to find the scalar components of the resultant:
\[ \begin{cases} R_{x} = A_{x} + B_{x} + C_{x} = 8.19\; cm - 2.39\; cm + 6.93\; cm = 12.73\; cm \\ R_{y} = A_{y} + B_{y} + C_{y} = 5.73\; cm - 6.58\; cm + 4.00\; cm = 3.15\; cm \end{cases}\]
Therefore, the resultant vector is \(\vec{R} = R_{x} \hat{i} + R_{y} \hat{j} = (12.7 \hat{i} + 3.1 \hat{j})\)cm. For (b), we may want to write the vector difference as
\[\vec{D} = \vec{A} - \vec{B} = (A_{x} \hat{i} + A_{y} \hat{j}) - (B_{x} \hat{i} + B_{y} \hat{j}) = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j} \ldotp\]
Hence the difference vector is \(\vec{D} = D_{x} \hat{i} + D_{y} \hat{j} = (10.6 \hat{i} + 12.3 \hat{j})\)cm.
For (c), we can write vector \(\vec{S}\) in the following explicit form:
\[ \vec{S} = \vec{A} - 3 \vec{B} + \vec{C} = (A_{x} \hat{i} + A_{y} \hat{j}) - 3(B_{x} \hat{i} + B_{y} \hat{j}) + (C_{x} \hat{i} + C_{y} \hat{j}) = (A_{x} - 3 B_{x} + C_{x}) \hat{i} + (A_{y} - 3 B_{y} + C_{y}) \hat{j} \ldotp\]
Then, the scalar components of \(\vec{S}\) are
\[ \begin{cases} S_{x} = A_{x} - 3B_{x} + C_{x} = 8.19\; cm - 3(-2.39\; cm) + 6.93\; cm = 22.29\; cm \\ S_{y} = A_{y} - 3B_{y} + C_{y} = 5.73\; cm -3(-6.58\; cm) + 4.00\; cm = 29.47\; cm \end{cases}\]
The vector is \(\vec{S} = S_{x} \hat{i} + S_{y} \hat{j} = (22.3 \hat{i} + 29.5 \hat{j})\)cm.
SignificanceHaving found the vector components, we can illustrate the vectors by graphing or we can compute magnitudes and direction angles, as shown in Figure \(\PageIndex{1}\).
Exercise \(\PageIndex{4}\)
Three displacement vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{F}\) are specified by their magnitudes A = 10.00, B = 7.00, and F = 20.00, respectively, and by their respective direction angles with the horizontal direction \(\alpha\) = 35°, \(\beta\) = −110°, and \(\varphi\) = 110°. The physical units of the magnitudes are centimeters. Use the analytical method to find vector \(\vec{F}\) = \(\vec{A}\) + 2 \(\vec{B}\) − \(\vec{F}\). Verify that G = 28.15 cm and that \(\theta_{G}\) = −68.65°.
Example \(\PageIndex{5}\): The Tug-of-War Game
Four dogs named Astro, Balto, Clifford, and Dug play a tug-of-war game with a toy (Figure \(\PageIndex{2}\)). Astro pulls on the toy in direction \(\alpha\) = 55° south of east, Balto pulls in direction \(\beta\) = 60° east of north, and Clifford pulls in direction \(\gamma\) = 55° west of north. Astro pulls strongly with 160.0 units of force (N), which we abbreviate as A = 160.0 N. Balto pulls even stronger than Astro with a force of magnitude B = 200.0 N, and Clifford pulls with a force of magnitude C = 140.0 N. When Dug pulls on the toy in such a way that his force balances out the resultant of the other three forces, the toy does not move in any direction. With how big a force and in what direction must Dug pull on the toy for this to happen?
Strategy
We assume that east is the direction of the positive x-axis and north is the direction of the positive y-axis. As in Example \(\PageIndex{1}\), we have to resolve the three given forces — \(\vec{A}\) (the pull from Astro), \(\vec{B}\) (the pull from Balto), and \(\vec{C}\) (the pull from Clifford)—into their scalar components and then find the scalar components of the resultant vector \(\vec{R}\) = \(\vec{A}\) + \(\vec{B}\) + \(\vec{C}\). When the pulling force \(\vec{D}\) from Dug balances out this resultant, the sum of \(\vec{D}\) and \(\vec{R}\) must give the null vector \(\vec{D}\) + \(\vec{R}\) = \(\vec{0}\). This means that \(\vec{D}\) = \(- \vec{R}\) so the pull from Dug must be antiparallel to \(\vec{R}\).
Solution
The direction angles are \(\theta_{A}\) = \(− \alpha\) = −55°, \(\theta_{B}\) = 90° − \(\beta\) = 30°, and \(\theta_{C}\) = 90° + \(\gamma\) = 145°, and substituting them into Equation 2.4.13 gives the scalar components of the three given forces:
\[ \begin{cases} A_{x} = A \cos \theta_{A} = (160.0\; N) \cos (-55^{o}) = + 91.8\; N \\ A_{y} = A \sin \theta_{A} = (160.0\; N) \sin (-55^{o}) = -131.1\; N \end{cases}\]
\[ \begin{cases} B_{x} = B \cos \theta_{B} = (200.0\; N) \cos 30^{o} = + 173.2\; N \\ B_{y} = B \sin \theta_{B} = (200.0\; N) \sin 30^{o} = + 100.0\; N \end{cases}\]
\[ \begin{cases} C_{x} = C \cos \theta_{C} = (140.0\; N) \cos 145^{o} = -114.7\; N \\ C_{y} = C \sin \theta_{C} = (140.0\; N) \sin 145^{o} = + 80.3\; N \end{cases}\]
Now we compute scalar components of the resultant vector \(\vec{R} = \vec{A} + \vec{B} + \vec{C}\):
\[ \begin{cases} R_{x} = A_{x} + B_{x} + C_{x} = + 91.8\; N+ 173.2\; N- 114.7\; N = +150.3 \; N\\ R_{y} = A_{y} + B_{y} + C_{y} = -131.1\; N + 100.0\; N + 80.3\; N= +49.2\; N\end{cases}\]
The antiparallel vector to the resultant \(\vec{R}\) is
\[\vec{D} = -\vec{R} = -R_{x} \hat{i} - R_{y} \hat{j} = (-150.3 \hat{i} - 49.2 \hat{j}) N \ldotp\]
The magnitude of Dug's pulling force is
\[D = \sqrt{D_{x}^{2} + D_{y}^{2}} = \sqrt{(-150.3)^{2} + (-49.2)^{2}} N = 158.1\; N \ldotp\]
The direction of Dug's pulling force is
\[\theta = \tan^{-1} \left(\dfrac{D_{y}}{D_{x}}\right) = \tan^{-1} \left(\dfrac{-49.2\; N}{-150.3\; N}\right) = \tan^{-1} \left(\dfrac{49.2}{150.3}\right) = 18.1^{o}\ldotp\]
Dug pulls in the direction 18.1° south of west because both components are negative, which means the pull vector lies in the third quadrant.
Exercise \(\PageIndex{6}\)
Suppose that Balto in Example \(\PageIndex{2}\) leaves the game to attend to more important matters, but Astro, Clifford, and Dug continue playing. Astro and Clifford’s pull on the toy does not change, but Dug runs around and bites on the toy in a different place. With how big a force and in what direction must Dug pull on the toy now to balance out the combined pulls from Clifford and Astro? Illustrate this situation by drawing a vector diagram indicating all forces involved.
Example \(\PageIndex{7}\): Vector Algebra
Find the magnitude of the vector \(\vec{C}\) that satisfies the equation 2 \(\vec{A}\) − 6 \(\vec{B}\) + 3 \(\vec{C}\) = 2 \(\hat{j}\), \(\vec{A}\) = \(\hat{i}\) − 2\(\hat{k}\) and \(\vec{B}\) = − \(\hat{j}\) + \(\frac{\hat{k}}{2}\) .
StrategyWe first solve the given equation for the unknown vector \(\vec{C}\). Then we substitute \(\vec{A}\) and \(\vec{B}\); group the terms along each of the three directions \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\); and identify the scalar components C x , C y , and C z . Finally, we substitute into Equation 3.3.6 to find magnitude C.
Solution\[\begin{split} 2 \vec{A} - 6 \vec{B} +& 3 \vec{C} = 2 \hat{j}\\ & 3 \vec{C} = 2 \hat{j} - 2 \vec{A} + 6 \vec{B} \\ &\vec{C} = \frac{2}{3} \hat{j} - \frac{2}{3} \vec{A} + 2 \vec{B}\\ & \quad = \frac{2}{3} \hat{j} - \frac{2}{3} (\hat{i} - 2\hat{k}) + 2 \big(- \hat{j} + \frac{\hat{k}}{2}\big)\\ & \quad = \frac{2}{3} \hat{j} - \frac{2}{3} \hat{i} + \frac{4}{3} \hat{k} - 2 \hat{j} + \hat{k}\\ & \quad = -\frac{2}{3} \hat{i} + \big(\frac{2}{3} - 2 \big)\hat{j} + \big(\frac{4}{3}\ + 1 \big)\hat{k}\\ & \quad = -\frac{2}{3} \hat{i} - \frac{4}{3} \hat{j} + \frac{7}{3} \hat{k} \end{split}\]
The components are C x = \(-\frac{2}{3}\), C y = \(-\frac{4}{3}\), and C z = \(\frac{7}{3}\), and substituting into Equation 3.3.6 gives
\[C = \sqrt{C_{x}^{2} + C_{y}^{2} + C_{z}^{2}} = \sqrt{\left(-\dfrac{2}{3}\right)^{2} + \left(-\dfrac{4}{3}\right)^{2} + \left(\dfrac{7}{3}\right)^{2}} = \sqrt{\frac{23}{3}} \ldotp\]
Example \(\PageIndex{8}\): Displacement of a Skier
Starting at a ski lodge, a cross-country skier goes 5.0 km north, then 3.0 km west, and finally 4.0 km southwest before taking a rest. Find his total displacement vector relative to the lodge when he is at the rest point. How far and in what direction must he ski from the rest point to return directly to the lodge?
StrategyWe assume a rectangular coordinate system with the origin at the ski lodge and with the unit vector \(\hat{i}\) pointing east and the unit vector \(\hat{j}\) pointing north. There are three displacements: \(\vec{D}_{1}\), \(\vec{D}_{2}\), and \(\vec{D}_{3}\). We identify their magnitudes as D 1 = 5.0 km , D 2 = 3.0 km , and D 3 = 4.0 km . We identify their directions are the angles \(\theta_{1}\) = 90°, \(\theta_{2}\) = 180°, and \(\theta_{3}\) = 180° + 45° = 225°. We resolve each displacement vector to its scalar components and substitute the components into Equation 3.2.13 to obtain the scalar components of the resultant displacement \(\vec{D}\) from the lodge to the rest point. On the way back from the rest point to the lodge, the displacement is \(\vec{B}\) = − \(\vec{D}\). Finally, we find the magnitude and direction of \(\vec{B}\).
SolutionScalar components of the displacement vectors are
\[ \begin{cases} D_{1x} = D_{1} \cos \theta_{1} = (5.0\; km) \cos 90^{o} = 0 \\ D_{1y} = D_{1} \sin \theta_{1} = (5.0\; km) \sin 90^{o} = 5.0\; km \end{cases}\]
\[ \begin{cases} D_{2x} = D_{2} \cos \theta_{2} = (3.0\; km) \cos 180^{o} = -3.0 \;km\\ D_{2y} = D_{2} \sin \theta_{2} = (3.0\; km) \sin 180^{o} = 0 \end{cases}\]
\[ \begin{cases} D_{3x} = D_{3} \cos \theta_{3} = (4.0\; km) \cos 225^{o} = -2.8\; km \\ D_{3y} = D_{3} \sin \theta_{3} = (4.0\; km) \sin 225^{o} = -2.8\; km \end{cases}\]
Scalar components of the net displacement vector are
\[ \begin{cases} D_{x} = D_{1x} + D_{2x} + D_{3x} = (0 - 3.0 - 2.8)km = -5.8\; km \\ D_{y} = D_{1y} + D_{2y} + D_{3y} = (5.0 + 0 - 2.8)km = + 2.2\; km \end{cases}\]
Hence, the skier’s net displacement vector is \(\vec{D}\) = D x \(\hat{i}\) + D y \(\hat{j}\) = (−5.8 \(\hat{i}\) + 2.2 \(\hat{j}\))km . On the way back to the lodge, his displacement is \(\vec{B}\) = − \(\vec{D}\) = −(−5.8 \(\hat{i}\)+ 2.2 \(\hat{j}\))km = (5.8 \(\hat{i}\) − 2.2 \(\hat{j}\))km. Its magnitude is B = \(\sqrt{B_{x}^{2} + B_{y}^{2}}\) = \(\sqrt{(5.8)^{2} + (−2.2)^{2}}\) km = 6.2 km and its direction angle is \(\theta\)= tan −1 \(\left(\dfrac{−2.2}{5.8}\right)\) = −20.8°. Therefore, to return to the lodge, he must go 6.2 km in a direction about 21° south of east.
Significance
Notice that no figure is needed to solve this problem by the analytical method. Figures are required when using a graphical method; however, we can check if our solution makes sense by sketching it, which is a useful final step in solving any vector problem.
Example \(\PageIndex{9}\): Displacement of a Jogger
A jogger runs up a flight of 200 identical steps to the top of a hill and then runs along the top of the hill 50.0 m before he stops at a drinking fountain (Figure \(\PageIndex{3}\)). His displacement vector from point A at the bottom of the steps to point B at the fountain is \(\vec{D}_{AB}\) = (−90.0 \(\hat{i}\) + 30.0 \(\hat{j}\))m. What is the height and width of each step in the flight? What is the actual distance the jogger covers? If he makes a loop and returns to point A, what is his net displacement vector?
Strategy
The displacement vector \(\vec{D}_{AB}\) is the vector sum of the jogger’s displacement vector \(\vec{D}_{AT}\) along the stairs (from point A at the bottom of the stairs to point T at the top of the stairs) and his displacement vector \(\vec{D}_{RB}\) on the top of the hill (from point T at the top of the stairs to the fountain at point B). We must find the horizontal and the vertical components of \(\vec{D}_{TB}\). If each step has width w and height h, the horizontal component of \(\vec{D}_{TB}\) must have a length of 200w and the vertical component must have a length of 200h. The actual distance the jogger covers is the sum of the distance he runs up the stairs and the distance of 50.0 m that he runs along the top of the hill.
Solution
In the coordinate system indicated in Figure \(\PageIndex{3}\), the jogger’s displacement vector on the top of the hill is \(\vec{D}_{RB}\) = (−50.0 m) \(\hat{i}\). His net displacement vector is
\[\vec{D}_{AB} = \vec{D}_{AT} + \vec{D}_{TB} \ldotp \nonumber\]
Therefore, his displacement vector \(\vec{D}_{TB}\) along the stairs is
\[\begin{split} \vec{D}_{AT}& = \vec{D}_{AB} - \vec{D}_{TB} = (-90.0 \hat{i} + 30.0 \hat{j})m - (-50.0 m)\hat{i}) = [(-90.0 50.0) \hat{i} + 30.0 \hat{j})]m \\ & = (-40.0 \hat{i} + 30.0 \hat{j})m \ldotp \end{split}\]
Its scalar components are D ATx = −40.0 m and D ATy = 30.0 m. Therefore, we must have
\[200 w=|-40.0| \mathrm{m} \text { and } 200 h=30.0 \: \mathrm{m} \ldotp \nonumber\]
Hence, the step width is w = \(\frac{40.0\; m}{200}\) = 0.2 m = 20 cm, and the step height is w = \(\frac{30.0\; m}{200}\) = 0.15 m = 15 cm. The distance that the jogger covers along the stairs is
\[\vec{D}_{AT} = \sqrt{\vec{D}_{ATx}^{2} + \vec{D}_{ATy}^{2}} = \sqrt{(-40.0)^{2} + (30.0)^{2}}m = 50.0\; m \ldotp \nonumber\]
Thus, the actual distance he runs is D AT + D TB = 50.0 m + 50.0 m = 100.0 m. When he makes a loop and comes back from the fountain to his initial position at point A, the total distance he covers is twice this distance,or 200.0 m. However, his net displacement vector is zero, because when his final position is the same as his initial position, the scalar components of his net displacement vector are zero.
In many physical situations, we often need to know the direction of a vector. For example, we may want to know the direction of a magnetic field vector at some point or the direction of motion of an object. We have already said direction is given by a unit vector, which is a dimensionless entity—that is, it has no physical units associated with it. When the vector in question lies along one of the axes in a Cartesian system of coordinates, the answer is simple, because then its unit vector of direction is either parallel or antiparallel to the direction of the unit vector of an axis. For example, the direction of vector \(\vec{d}\) = -5 m \(\hat{i}\) is unit vector \(\vec{d}\) = -\(\hat{i}\). The general rule of finding the unit vector \(\vec{V}\) of direction for any vector \(\vec{V}\) is to divide it by its magnitude V :
\[\widehat{\mathbf{V}}=\frac{\overrightarrow{\mathbf{V}}}{V} \label{2.26} \cdot \]
We see from this expression that the unit vector of direction is indeed dimensionless because the numerator and the denominator in Equation \ref{2.26} have the same physical unit. In this way, Equation \ref{2.26} allows us to express the unit vector of direction in terms of unit vectors of the axes. The following example illustrates this principle.
Example \(\PageIndex{10}\): The Unit Vector of Direction
If the velocity vector of a military convoy is \(\vec{v}\) = (4.000 \(\hat{i}\) + 3.000 \(\hat{j}\) + 0.100 \(\hat{k}\))km/h, what is the unit vector of its direction of motion.
StrategyThe unit vector of the convoy’s direction of motion is the unit vector \(\hat{v}\) that is parallel to the velocity vector. The unit vector is obtained by dividing a vector by its magnitude, in accordance with Equation \ref{2.26}.
SolutionThe magnitude of the vector \(\vec{v}\) is
\[v = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}} = \sqrt{4.000^{2} + 3.000^{2} + 0.100^{2}}km/h = 5.001\; km/h \ldotp \nonumber\]
To obtain the unit vector \(\hat{v}\), divide \(\vec{v}\) by its magnitude:
\[\begin{split} \hat{v}& = \frac{\vec{v}}{v} = \frac{(4.000 \hat{i} + 3.00 \hat{j} + 0.100 \hat{k})km/h}{5.001\; km/h} \\ & = \frac{(4.000 \hat{i} + 3.000 \hat{j} + 0.1100 \hat{k})}{5.001} \\ & = \frac{4.000}{5.001} \hat{i} + \frac{3.000}{5.001} \hat{j} + \frac{0.100}{5.001} \hat{k} \\ & = (79.98 \hat{i} + 59.99 \hat{j} + 2.00 \hat{k}) \times 10^{-2} \ldotp \end{split}\]
SignificanceNote that when using the analytical method with a calculator, it is advisable to carry out your calculations to at least three decimal places and then round off the final answer to the required number of significant figures, which is the way we performed calculations in this example. If you round off your partial answer too early, you risk your final answer having a huge numerical error, and it may be far off from the exact answer or from a value measured in an experiment.
Exercise \(\PageIndex{11}\)
Verify that vector \(\hat{v}\) obtained in Example \(\PageIndex{3}\) is indeed a unit vector by computing its magnitude. If the convoy in Example 2.6.1 was moving across a desert flatland—that is, if the third component of its velocity was zero—what is the unit vector of its direction of motion? Which geographic direction does it represent?


13.4: Stream Types
Stream channels can be straight or curved, deep and slow, or rapid and choked with coarse sediments. The cycle of erosion has some influence on the nature of a stream, but there are several other factors that are important.
Youthful streams that are actively down-cutting their channels tend to be relatively straight and are typically ungraded (meaning that rapids and falls are common). As shown in Figures 13.0.1 and 13.4.1, youthful streams commonly have a step-pool morphology, meaning that the stream consists of a series of pools connected by rapids and waterfalls. They also have steep gradients and steep and narrow V-shaped valleys—in some cases steep enough to be called canyons.
In mountainous terrain, such as that in western Alberta and B.C., steep youthful streams typically flow into wide and relatively low-gradient U-shaped glaciated valleys. The youthful streams have high sediment loads, and when they flow into the lower-gradient glacial valleys where the velocity isn’t high enough to carry all of the sediment braided patterns develop, characterized by a series of narrow channels separated by gravel bars (Figure \(\PageIndex{2}\)).
Braided streams can develop anywhere there is more sediment than a stream is able to transport. One such environment is in volcanic regions, where explosive eruptions produce large amounts of unconsolidated material that gets washed into streams. Streams in the volcanic Mt. Meager area of southwestern British Columbia are good examples of this (Figure \(\PageIndex{3}\)).
A stream that occupies a wide, flat flood plain with a low gradient typically carries only sand-sized and finer sediments and develops a sinuous flow pattern. As you saw in Figure \(\PageIndex{1}\), when a stream flows around a corner, the water on the outside has farther to go and tends to flow faster. This leads to erosion of the banks on the outside of the curve, deposition on the inside, and formation of a point bar (Figure \(\PageIndex{4}\)). Over time, the sinuosity of the stream becomes increasingly exaggerated, and the channel migrates around within its flood plain, forming a meandering pattern.
A well-developed meandering river is shown in Figure \(\PageIndex{5}\). The meander in the middle of the photo has reached the point where the thin neck of land between two parts of the channel is about to be eroded through. When this happens, another oxbow lake will form like the others in the photo.
Gradient is the key factor controlling stream velocity, and of course, velocity controls sediment erosion and deposition. This map shows the elevations of Priest Creek in the Kelowna area. The length of the creek between 1,600 meters and 1,300 meters elevation is 2.4 kilometers, so the gradient is 300/2.4 = 125 meters per kilometre.
- Use the scale bar to estimate the distance between 1,300 meters and 600 meters and then calculate that gradient.
- Estimate the gradient between 600 and 400 meters.
- Estimate the gradient between 400 meters on Priest Creek and the point where Mission Creek enters Okanagan Lake.
See Appendix 3 for Exercise 13.4 answers .
At the point where a stream enters a still body of water—a lake or the ocean—sediment is deposited and a delta forms. The Fraser River has created a large delta, which extends out into the Strait of Georgia (Figure \(\PageIndex{7}\)). Much of the Fraser delta is very young in geological terms. Shortly after the end of the last glaciation (10,000 years ago), the delta did not extend past New Westminster. Since that time, all of the land that makes up Richmond, Delta, and parts of New Westminster and south Surrey has formed from sediment from the Fraser River. (You can see this in more detail at Geoscape Vancouver .)
Image Descriptions
Figure \(\PageIndex{5}\) image description: A part of the Nowitna River has curved around so sharply that it almost forms a circle before curving the other way again. Eventually, as the barrier between these two parts of the channel erodes, they will be joined and form an oxbow lake. [Return to Figure \(\PageIndex{5}\)]
Media Attributions
- Figures 13.4.1, 13.4.2, 13.4.4, 13.4.6: © Steven Earle. CC BY.
- Figure \(\PageIndex{3}\) : “Meager Creek” © Isaac Earle. CC BY.
- Figure \(\PageIndex{5}\): “ Nowitna river ” by Oliver Kumis. CC BY-SA.
- Figure \(\PageIndex{7}\): Delta of the Fraser River by NASA. Taken September 2011. Adapted by Steven Earle. Public domain.


6.1: Introduction
The nurse is working in the private practice of a health-care provider who specializes in obstetrics and gynecology. After greeting the patient, a 38-year-old person assigned female at birth (AFAB) who identifies as female, the nurse asks why she is seeking care. She replies that she has always had menstrual cramps, but they’ve been significantly worse lately, even causing her to frequently miss work and social outings. These cramps often last at least 2 to 3 days every menstrual cycle, often requiring ibuprofen (Motrin), 600 mg three to four times a day, to manage. She also uses hot water bottles and gentle exercise, neither of which provides much relief. Upon further assessment, the patient appears very distressed about the pain and how it affects her life. The nurse’s thorough assessment of the patient’s symptoms is important in determining if the symptoms fall within the expected range or if another cause like a structural or tissue disorder is present.


14: Cosmic Samples and the Origin of the Solar System
Imagine you are a scientist examining a sample of rock that had fallen from space a few days earlier and you find within it some of the chemical building blocks of life. How could you determine whether those “organic” materials came from space or were merely the result of earthly contamination?
We conclude our survey of the solar system with a discussion of its origin and evolution. Some of these ideas were introduced in Other Worlds: An Introduction to the Solar System; we now return to them, using the information we have learned about individual planets and smaller members of the solar system. In addition, astronomers have recently discovered several thousand planets around other stars, including numerous multiplanet systems. This is an important new source of data, providing us a perspective that extends beyond our own particular (and perhaps atypical) solar system.
But first, we want to look at another crucial way that astronomers learn about the ancient history of the solar system: by examining samples of primitive matter , the debris of the processes that formed the solar system some 4.5 billion years ago. Unlike the Apollo Moon rocks, these samples of cosmic material come to us free of charge—they literally fall from the sky. We call this material cosmic dust and meteorites.
-
- 14.1: Meteors
- When a fragment of interplanetary dust strikes Earth’s atmosphere, it burns up to create a meteor. Streams of dust particles traveling through space together produce meteor showers, in which we see meteors diverging from a spot in the sky called the radiant of the shower. Many meteor showers recur each year and are associated with particular comets that have left dust behind as they come close to the Sun and their ices evaporate (or have broken up into smaller pieces).
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- 14.2: Meteorites - Stones from Heaven
- Meteorites are the debris from space that survive to reach the surface of Earth. Meteorites are called finds or falls according to how they are discovered; the most productive source today is the Antarctic ice cap. Meteorites are classified as irons, stony-irons, or stones accordingly to their composition. Most stones are primitive objects, dated to the origin of the solar system. The most primitive are the carbonaceous meteorites that can contain a number of organic (carbon-rich) molecules.
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- 14.3: Formation of the Solar System
- Meteorites, comets, and asteroids are survivors of the solar nebula out of which the solar system formed. This nebula was the result of the collapse of an interstellar cloud of gas and dust, which contracted (conserving its angular momentum) to form our star, the Sun, surrounded by a thin, spinning disk of dust and vapor. Condensation in the disk led to the formation of planetesimals, which became the building blocks of the planets.
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- 14.4: Comparison with Other Planetary Systems
- The first planet circling a distant solar-type star was announced in 1995. Twenty years later, thousands of exoplanets have been identified, including planets with sizes and masses between Earth’s and Neptune’s, which we don’t have in our own solar system. A few percent of exoplanet systems have “hot Jupiters,” massive planets that orbit close to their stars, and many exoplanets are also in eccentric orbits.
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- 14.5: Planetary Evolution
- fter their common beginning, each of the planets evolved on its own path. Different possible outcomes are illustrated by comparison of the terrestrial planets (Earth, Venus, Mars, Mercury, and the Moon). All are rocky, differentiated objects. The level of geological activity is proportional to mass: greatest for Earth and Venus, less for Mars, and absent for the Moon and Mercury. However, tides from another nearby world can also generate heat to drive geological activity.
Thumbnail: This illustration depicts a disk of dust and gas around a new star. Material in this disk comes together to form planetesimals. (credit: modification of work by University of Copenhagen/Lars Buchhave, NASA).


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abduction
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rotating out and away from the body
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abductor pillow
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a pillow that keeps the hips in a neutral position to avoid external rotation and adduction
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active ROM
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range of movement performed independently by the patient
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active-assisted ROM
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a procedure performed with partial assistance from an external force; the patient is able to perform the ROM exercise but may need additional assistance when they encounter pain or weakness
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activity of daily living (ADL)
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a routine task of everyday life necessary for independent self-care
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adduction
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movement close to the midline of the body
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adipose tissue
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body fat
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aerobic exercise
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any cardiovascular activity that increases a patient’s heart rate and respiration rate
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alternating pressure mattress
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a mattress that is powered with air and can be programmed to inflate and deflate certain sections in a programmable cycle
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balance
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state of equilibrium
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balance exercise
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a movement to reduce the risk of falls among patients by focusing on body alignment and control
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bandage
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a strip of material that can be used for slings, to hold splints in place, provide support for the limb or joint, or restrict movement
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basic ADL
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an activity to manage basic needs such as toileting, dressing, bathing, eating, and ambulating
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cane
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a singular assistive device that aids in balance when walking or helps compensate for an injury or disability
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cast
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a splint that is set around the limb
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chemical restraint
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any drug that is used to manage a patient’s behavior in order to reduce the safety risk to the patient or others
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congenital abnormality
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an impairment of body function or structure present at birth
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coordination
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organization of different elements of the complex body, including muscular, skeletal, and sensory functions, to enable the elements to work together effectively
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crutches
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an aid used for standing, walking, and mobility for patients who cannot support the weight of their bodies with their leg, knee, or ankle
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external factor
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an environmental or social factor that contributes to immobility
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external fixation
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fixation of a fracture where pins and wires are inserted into the bone percutaneously and held together externally
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fibrous connective tissue
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a type of tissue that protects, supports, and holds bones, muscles, organs, and other tissues within the body in place
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foam mattresses
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a mattress made of crosscut foam that allows parts of the mattress to shift along with the patient’s body
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foot drop syndrome
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the inability to raise the front part of the foot due to weakness and paralysis of the muscles that allow the foot to lift and maintain itself in a perpendicular position
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functional brace
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a casting that is applied to areas of the limb, leaving the joint free
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functional status
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a measurement of a patient’s ability to perform physical movements (e.g., walking, standing) and higher-level activities (e.g., ADLs, filling occupational and societal roles)
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gait
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a manner of walking or moving on foot
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gait belt
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a strap placed around a patient’s waist to help nurses assist with mobility and transfers
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gel overlay mattress
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a mattress made with a combination of foam and gel; it conforms to the patient’s body and retains less body heat
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immobilization device
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any device, including casts, braces, slings, traction, and external fixation, that keeps a part of the patient’s body in a fixed position
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instrumental ADL
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a complex activity that keeps patients independent in the community, such as managing medications, preparing food, and managing household chores
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internal factor
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a factor that affects mobility that is specific to each individual and may impact an individual’s mobility in different ways
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isokinetic exercise
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a resistance-based exercises that provide variable resistance to a movement
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isometric exercise
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a movement related to contractions of specific muscles or group of muscles and then their release
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isotonic exercises
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a movement that require muscles to resist constant weight over a range of motion
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low air loss mattress
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a mattress that delivers active therapy by pushing air through tiny holes on the surface of the mattress to keep the skin dry, wick moisture, and accommodate pressure points. It is attached to an air pump or compressor that maintains inflation
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mobility
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the ability of a patient to change and control their body position
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non-weight-bearing
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a situation in which an affected limb or limbs cannot support any weight
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orthostatic hypotension
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a temporary drop in blood pressure when moving from a supine position to a sitting or standing position
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passive ROM
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range of motion performed when a patient is unable to move or not permitted to move a body part
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peristalsis
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the wavelike movement of the gastrointestinal system to move contents forward
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physical restraint
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any manual, physical, or mechanical device, material, or equipment attached to or adjacent to the patient’s body that the individual cannot remove easily and restricts movement or normal access to one’s body
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podus boot
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a device used in foot drop syndrome that covers the ankle and foot to support the muscles, immobilize joints, and correct positioning
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posture
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a fundamental mechanism to ensure balance against gravity, aligning the body, and assisting in external environmental perception and action
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proprioception
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the body’s ability to sense movement, action, and location of parts of the body
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range of motion (ROM)
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the extent to which a part of the body can be moved around a joint or a fixed point
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repositioning
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changing the position of a patient
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restraint
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any measure, physical or pharmacological, that prevents or reduces the full movement of the patient during hospital care
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skeletal traction
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the use of pins wires, or screws inserted distally from the fracture of the bone to create traction by attaching a string and weight
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sliding board
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a piece of equipment that is flat and rigid, and creates a bridge for transfers; used for patients who cannot complete a transfer between two surfaces or if a standing transfer is not safe
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sling
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a device used to support an upper limb by limiting movement or immobilizing the arm to prevent movement in the wrist, elbow, or shoulder
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sling transfer
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a patient transfer from a bed, wheelchair, toilet, or shower using a lift machine and flat piece of material placed under the patient
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specialty mattress
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a mattress designed to relieve pressure over bony prominences to prevent injury
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splint
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a firm support that can be made of metal, plastic, or plaster
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stand and pivot
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a technique for patients, who can partially weight-bear, by moving from sitting to standing and then just turning with one leg to sitting again
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strength
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the quality or state of being strong
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strength exercise
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a movement focused on increasing the maximum amount of force a muscle can exert
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stretching
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movements to lengthen muscles
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therapeutic exercise
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an intentional movement to develop strength, endurance, and increased range of motion and flexibility
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traction
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the process of pulling on a broken bone or dislocated part of the body, in a slow and steady manner, to realign and stabilize it into the proper position
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venous stasis
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a decrease in venous return to the heart
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vestibular function
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the body’s ability to compensate in response to self and external forces and keep balance
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walker
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an aide for ambulation and transfers that has four points of contact with the ground
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wedge pillow
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a large triangular pillow made of foam that elevates different parts of the body
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weight-bearing
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the ability of a patient to put all their weight on a specific limb that is recovering from injury


11.11: Trends in Social Media
What is Social Media, and How Has it Changed Promotion?
Advances in technology continue to change the marketing landscape. As you will see in the following sections, marketers are harnessing new technology to hone their marketing message and reach more customers.
The business world now relies on the internet for much of its communications, marketing or otherwise. Almost all companies have Facebook accounts, and individual leaders of companies have separate individual accounts on Linked In, Twitter, Instagram, and other social media sites. New social media sites are popping up almost every week. The phenomenon of social media has created a business climate in which thousands of impressions can be made with one social media post. That impression could be positive or negative. Within social media, there are “stars” of social media—individuals who have developed audiences in the millions who follow their posts every day. Social media is a hugely powerful tool for marketers. It has it challenges, though, because the platforms are constantly changing and evolving. Also, the audiences being reached often read (or view) and believe the messages seen on various social media platforms without understanding the context of the message. A social media post that goes viral can close down a business, even if the post is not true. That’s what makes social media the newest challenge/opportunity for marketers. Companies that want to retain market share and build their image must develop tactics for the use of social media and for defending against problems created by the use of this powerful marketing tool.
Promotion Through Blogs
Blogs provide marketers with a real-time dialogue with customers and an avenue to promote their products or services. A blog is an online journal with regularly updated content. This content is pushed to subscribers by RSS (really simple syndication) or e-mail and allows for response and discussion from site visitors. RSS enables users to automatically gather updates from various websites, especially news sites and blogs, and display headlines and a brief summary of those updates in a single location. Blogs can be considered to be offerings of social media unless the site is actually part of the company’s main web page.
Well-run marketing blogs usually focus tightly on one niche area, product line, or vertical market segment. The aim is to provide the blog’s readers with a constantly renewing source of news and insight about that topic. About 366 million blogs are registered on Tumblr, and more than 23 million blog entries are posted daily.
Many companies have set up their own blogs, including General Motors, Apple, the American Cancer Society, and Microsoft, to name a few. These companies blog because they: (1) get real-time input from customers and prospects; (2) create and maintain relationships; (3) can have a continuing dialogue with loyal customers and prospective clients; and (4) can zero in on specific marketing goals. For example, Disney uses a blog called Disney Baby to cater to the needs of new mothers. Each of their bloggers has a personal bio that helps provide a connection for the new mother to the blogger and provides a deeper connection to the Disney brand.
Firms can also use emerging search tools such as BlogPulse, Feedster, PubSub, and Technorati to monitor conversations about their company and brands. A public relations department might then decide to feed new-product information to bloggers who are evangelists for their brand.
Advertisers Jump on Podcasts and Videos
Podcasts are basically blogs with a multimedia file. The trend developed when a new version of iTunes software made it easy for people to create their own podcasts and post them on a website. There are more than 8,000 podcasters in the United States. Besides individuals, companies are beginning to do their own podcasts as well as posting videos from the company on YouTube as another marketing channel. For listeners, the advantage of a podcast is convenience. Companies now have the ability to use streaming video, which potential customers can download to their mobile devices; for example, ABC News offering a digital version of its programming. The customers’ favorite programs download automatically from the internet, usually free of charge, and they can listen to the programs any time they wish. They can also listen wherever they wish, if they have a mobile device to receive the downloads.
Gimlet Media is one of the nation’s largest podcasters, offering material from nearly 40 different stations as podcasts. At first ad-free, Gimlet’s podcasts are done for direct-to-consumer companies like Blue Apron, as well as for traditional advertisers like Pepsi and Ford. Gimlet now includes a short advertisement before the programming—short enough that people won’t fast-forward through it. Gimlet also received a $5 million investment from advertising giant WPP, a clear sign that the business community sees a bright future in podcasts.
Pet owners can go to http://www.purina.com and opt in to receive Purina’s podcasts. The products will offer advice ranging from animal training to pet insurance to nutrition for older pets. Weekly tips will also be sent on things such as how to help your dog lose weight. Owners spend close to $25 billion a year on pet food. The aim of the podcasts is to build brand loyalty with a soft sell.
Videos have become another important promotions channel. Literally hundreds of thousands of videos can be viewed on YouTube, the top video-hosting site on the internet. Many people now log in to YouTube to watch videos on a particular product and how the product can be used. Entrepreneurs and other small-business owners have made extensive use of YouTube to provide value to their customers by creating and uploading informational videos that highlight their products.
CONCEPT CHECK
- How are companies embracing social media as a way to connect with customers?
- What has been the effect of social media on traditional advertising?


9.15: Influences on Organizational Culture
- Discuss the sources of culture in an organization.
- Explain the methods for maintaining a culture.
- List the visible signs of culture.
No matter where you work, you will experience organizational culture. Employees can usually tell from the get-go if their place of employment is serious or fun, people-oriented or results-oriented. But how did the culture get started? How do employees get to know about the corporate culture and help to keep it in place?
Sources of Culture in an Organization
Founders
When a company is founded, there is usually a single individual or group of individuals involved. The founder or founders have a vision for their new company—and that vision helps to form the corporate culture. In some cases, the founder is very intentional about creating a particular culture; he or she may actually want to create a business in which, for example, innovation or teamwork is valued. in other cases, the founder’s personality unintentionally forms the culture.
Some individual founders have such strong personalities and values that the company continues to reflect their goals even as it grows—and even after the founder dies. Walt Disney, for example, modeled leadership, teamwork, and innovation so that, even today, the Disney Corporation is built around the values and assumptions of its founder.
Another good example of the way that founders’ values create corporate culture is Ben and Jerry’s Ice Cream. Founders Ben Cohen and Jerry Greenfield started out to create a company with strong social values—and they succeeded. The company started in 1978, but even today the company continues to focus on sustainability, environmental activism, social activism, and charity.
Industry
It’s one thing to be creative, innovative, and fun in the hospitality or entertainment business. But that type of culture won’t work well in an industry that’s built around regulations and policies that cannot be changed or bent. Industries such as pharmaceuticals and nuclear power require attention to detail and cannot tolerate a “creative” approach to following rules. True, a pharmaceutical company can be people-oriented to a degree, but its willingness to support the individual needs of employees must be secondary to its absolute compliance with regulations and the law.
Methods for Maintaining Corporate Culture
Why do some companies maintain their culture whereas others see it fall apart? The answer lies in how the company goes about recruiting, hiring, onboarding, and training its employees.
- Recruiting . To find employees who will fit into the corporate culture, recruiters must look in the right places. When looking for upper-level managers, recruiters should look at corporations with a similar culture to their own. When looking for entry-level employees, recruiters should tap college programs or websites that reflect their corporate culture.
- Hiring . When interviewing job candidates, managers and human resources managers must spend some time assessing the candidate’s assumptions and values. Is this person a collaborator or a competitor? Are candidates detail-oriented or innovative? Candidates whose personal assumptions and values match the corporate culture are much more likely to help maintain that culture over time.
- Onboarding . The “onboarding” process is really a new employee orientation process. During onboarding, human resources personnel help the new employee get to know company policies and practices. It’s during onboarding, for example, that a new employee may learn that each team puts together a skit for the company holiday party or that bonuses can be earned as a result of exceeding sales goals.
- Training . Training can be both formal and informal. Whereas formal training may teach new employees how to use company software or systems, informal training may involve one-on-one conversations with peers and managers. During those conversations, new employees learn how the company culture manifests itself in the workplace. For example, they may learn that everyone—even bad players—take part in departmental softball games, or that recycling is a “must” in the lunchroom. All these subtle bits of information add up to an understanding of corporate culture.
Once an employee is hired, he or she may feel comfortable or uncomfortable in the new workplace. Typically, those employees who feel “at home” in the corporate culture tend to stick around, whereas those who feel like outsiders tend to leave at the first good opportunity.
Visible Signs of Culture
When you walk into a business setting, you should be able to see visible signs of the business’s organizational culture. If you spend a few weeks on the job, you should see even more. Here are just a few things to look out for:
- How employees dress. Most employers have some kind of dress code. Some are quite casual whereas others require more formal clothing. In a minority of businesses there is no dress code at all, and employees are welcome to wear short, sleeveless tops, and even flip-flops to work.
- Snack areas and candy jars. How people-oriented is your business? In some workplaces, employees have all-day access to snack bars, free soft drinks, coffee, and other goodies. Candy jars are also a sign that a business is open to people-oriented treats.
- Plaques and awards. In some businesses, plaques and awards honoring employees are placed front and center. Some honor individuals as “employee of the month,” whereas others recognize specific achievements such as “most sales made in a month.”
- Mission statement. In many corporations, the organizational mission is posted on every floor. The mission is intended to remind employees why they are there and what their goals and attitudes should be.
- Events and rituals. Many workplaces have customs or rituals that become obvious within just a couple of weeks on the job. Whether it’s “happy hour Friday,” “dress down Tuesday,” or weekly update meetings, employees quickly learn the routines.
- Physical layout and décor. When you walk into a workplace, what do you see? In some offices, gray cubicles dominate; in others, the workspace is wide open. In some businesses, individuals are encouraged to bring in and show off photos, posters, and plants; in others employees are asked to keep personal items to a minimum.
Recognizing the culture is the beginning of adapting to it and, perhaps, using it to persuade management to your ideas.
Contributors and Attributions
- Influences on Organizational Culture. Authored by : Lisa Jo Rudy and Lumen Learning. License : CC BY: Attribution
- Image: Office space. Authored by : MagicDesk. Located at : https://pixabay.com/en/office-pantry-cup-coffee-lunch-730680/ . License : CC0: No Rights Reserved


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2.7: Cases and Problems
Chapter Summary
- The selection process refers to the steps involved in choosing someone who has the right qualifications to fill a current or future job opening.
- There are five main steps in the selection process. First, criteria should be developed to determine how the person will be chosen. Second, a review of the applications and résumés is conducted, often via a computer program that can find keywords. Next, interview the employee. The last steps involve administering tests, such as a personality test or drug test, and making the offer to the right candidate.
- The first step in selection is to review résumés. Even before you do this, though, it is important to develop criteria against which each candidate will be measured. Criteria can come from the job description as well as the job qualifications.
- Other tools, such as cognitive ability tests, credit checks, or personality tests, can be used to determine qualifications. When developing your criteria for interviewing, determine the level the applicant needs to meet to meet the minimum criteria—for example, a minimum score for a personality test.
- We should be concerned with validity and reliability of measurement tools. Validity refers to how valid the test is—that is, how well a test measures a candidate’s abilities to do a job. Reliability refers to which selection techniques yield similar data or results over time. It is important to choose the right measurement tool used to determine whether the candidate meets the criteria.
- Use of criteria before the interview process starts is also important to make sure disparate impact or disparate treatment do not occur in the interview process.
- When hiring, there is the option of internal and external candidates. Each has its own set of advantages and disadvantages. Internal candidates may be able to “hit the ground running” but external candidates may come in with new perspectives. Even if an internal candidate seems to be the best hire, it is best to still perform the process of posting the job and interviewing, since other less vocal employees might be qualified internal candidates as well. In other words, don’t assume one person is the obvious choice for the promotion.
- Traditional, telephone, panel, informational, meal, group, and video are types of interviews. A combination of several of these may be used to determine the best candidate for the job. A structured interview format means the questions are determined ahead of time, and unstructured means the questions are based on the individual applicant. The advantage of a structured interview is that all candidates are rated on the same criteria. Before interviewing occurs, criteria and questions for a structured interview should be developed.
- Interview questions can revolve around situational questions or behavioral questions . Situational questions focus on asking someone what they would do in a given situation, while behavioral questions ask candidates what they would have done in certain situations.
- Interview questions about national origin, marital status, age, religion, and disabilities are illegal. To avoid any legal issues, it is important for interviewers to be trained on which questions cannot be asked. The halo effect , which assumes that one desirable trait means all traits are desirable, should also be avoided.
- The process involved in interviewing a person includes the following steps: recruit new candidates; establish criteria for which candidates will be rated; develop interview questions based on the analysis; set a time line for interviewing and decision making; connect schedules with others involved in the interview process; set up interviews with candidates and set up any testing procedures; interview the candidates and perform any necessary testing; and once all results are back, meet with the hiring team to discuss each candidate and make a decision based on the established criteria. Finally, put together an offer for the candidate.
- Developing a rapport, being honest, and managing the interview process are tips to having a successful interview.
- Once the interview process is complete, some companies use other means of measuring candidates. For example, work samples are an excellent way of seeing how someone might perform at your company.
- An aptitude test or achievement test can be given. An aptitude test measures how well someone might be able to do something, while an achievement test measures what the candidate already knows. Tests that measure cognitive ability and personality are examples.
- Some organizations also perform drug tests and physical tests. A physical test might consist of being able to lift a certain amount of weight, if required for the job. Honesty tests are also given, which measure the honesty level of the candidate. However, these tests may not be reliable, since someone can guess the “right” answer.
- Facebook, Twitter, and other social networking websites are used to gather information about a candidate. Calling references is another option.
- Every person interviewing the candidate should have a selection model; this method utilizes a statistical approach as opposed to a clinical approach . The selection table lists the criteria on the left and asks interviewers to provide a rating for each. This method can allow for a more consistent way of measuring candidates.
- The job of the HR professional isn’t finished once the selection is made. The next step is to make the offer. This step is important, because if it isn’t done properly, you could lose the candidate or have ill feelings at the onset of the employment relationship.
- Once you have made the decision to hire someone, make the offer to the candidate right away. Normally this is done through a phone call and a follow-up e-mail, outlining the details of the offer.
- It is not unusual for someone to negotiate salary or benefits. Know how far you can negotiate, and also be aware of how your current employees will be affected if you offer this person a higher salary.
- If you are having trouble coming to an agreement, be creative in what you can offer; for example, offer flextime instead of higher pay.
Chapter Case
The Four-Fifths Rule
The four-fifths rule is a way of measuring adverse impact in selection processes of organizations. It works like this: assume your organization requires a cognitive test for employment. You set a test score of 70 as the required pass rate for the candidate to be considered for an interview. Based on our numbers, if 50 percent of men passed this test with a score of 70, then four-fifths or 40 percent of women should also be able to pass the test. You might calculate it like this:
| Gender | Total who scored 70 or above | Total who took the test | Percent |
|---|---|---|---|
| Male | 52 | 62 | 83.8 or 84% passed |
| Female | 36 | 58 | 62.07 or 62% |
If you divide the total of who scored above 70 by the total number who took the test, it shows the percentage of 84 percent passed the test. If you divide the number of women who passed by the total number of women who took the test, you come up with 62 percent. Then divide 62 percent by 84 percent (62/84 = 73.8%). The resulting 74 percent means that it is below the 80 percent or the four-fifths rule, and this test could be considered to have disparate impact.
52/62 = 84% of men who took the test passed the test
36/58 = 62% of women who took the test passed the test
62/84 = 73.8%, less than 80%, which could show disparate impact
This is only an indicator as to how the selection process works for the organization, and other factors, such as sample size, can impact the reliability of this test. Using the tables below, please calculate possible disparate impact and then answer the questions that follow.
| National Origin | Passing Test Score | Total Number Taking the Test | Percent |
|---|---|---|---|
| Caucasians | 56 | 89 | |
| Minority groups | 48 | 62 |
| Age | Passing Test Score | Total Number Taking the Test | Percent |
|---|---|---|---|
| People under 40 | 28 | 52 | |
| People over 40 | 23 | 61 |
| Gender | Passing Test Score | Total Number Taking the Test | Percent |
|---|---|---|---|
| Male | 71 | 82 | |
| Female | 64 | 85 |
-
Please calculate the above numbers using the four-fifths rule. Based on your calculation:
- Which group or groups might be affected negatively by this test?
- What would be your considerations before changing any selection tools based on this data?
- How might you change your selection process to ensure disparate impact isn’t occurring at your organization?
- In a team of two, take the Big Five personality test online ( http://www.outofservice.com/bigfive/ ) and compare scores.
- Assume you are hiring a retail salesperson and plan to administer the same Big Five personality test you took above. In your team, develop minimum percentile scores for each of the five areas that would be acceptable for your new hire.


7.2.34: KewAr Code (Requires Alternative Activity)
Overall Usability
Requires Alternative Activity:
- This activity is not usable by some users.
Important Reminders
The activity may be usable if:
- An optional description for each image is not included
- A brief alt-text description (150 characters maximum) and hover text for each image are included
- Labels for the images are set to not be displayed.
However, many images would require a longer description that cannot be added to the activity without making the reading order of the activity content unusable with screen reading software. Provide an alternative version of the activity that includes detailed text descriptions for meaningful visual information and accurate content reading order.
Users who require magnification up to 400% may need to be provided with an alternative activity.
Visible Indication of Focus
The slider button has a visible indication of focus.
Color Contrast
The visible indication of focus for the slider button has good color contrast.
Keyboard Usability
The activity is usable with the keyboard. The slider button works with arrow keys.
No Keyboard Traps
There are no keyboard traps.
Magnification
Content magnifies up to 200% using browser settings without loss of content or functionality. The image content does not magnify up to 400%.
An alternative version of the content should be included that supports magnification up to 400% without requiring horizontal scrolling to view content and interact with content.
Usability with Screen Reading Software
In popup content, H5P creates visual headings from the Content type that are not headings that screen reading software can recognize. When the popup content is open, H5P does not keep the keyboard focus inside the popup content.


5.1: Bone Tissue and the Skeletal System
Introduction
Bones make good fossils. While the soft tissue of a once living organism will decay and fall away over time, bone tissue will, under the right conditions, undergo a process of mineralization, effectively turning the bone to stone. A well-preserved fossil skeleton can give us a good sense of the size and shape of an organism, just as your skeleton helps to define your size and shape. Unlike a fossil skeleton, however, your skeleton is a structure of living tissue that grows, repairs, and renews itself. The bones within it are dynamic and complex organs that serve a number of important functions, including some necessary to maintain homeostasis.
The Functions of the Skeletal System
Bone , or osseous tissue , is a hard, dense connective tissue that forms most of the adult skeleton, the support structure of the body. In the areas of the skeleton where bones move (for example, the ribcage and joints), cartilage , a semi-rigid form of connective tissue, provides flexibility and smooth surfaces for movement. The skeletal system is the body system composed of bones and cartilage and performs the following critical functions for the human body:
- supports the body
- facilitates movement
- protects internal organs
- produces blood cells
- stores and releases minerals and fat
Support, Movement, and Protection
The most apparent functions of the skeletal system are the gross functions—those visible by observation. Simply by looking at a person, you can see how the bones support, facilitate movement, and protect the human body.
Just as the steel beams of a building provide a scaffold to support its weight, the bones and cartilage of your skeletal system compose the scaffold that supports the rest of your body. Without the skeletal system, you would be a limp mass of organs, muscle, and skin.
Bones also facilitate movement by serving as points of attachment for your muscles. While some bones only serve as a support for the muscles, others also transmit the forces produced when your muscles contract. From a mechanical point of view, bones act as levers and joints serve as fulcrums (Figure 5.2). Unless a muscle spans a joint and contracts, a bone is not going to move. For information on the interaction of the skeletal and muscular systems, that is, the musculoskeletal system, seek additional content.
Bones also protect internal organs from injury by covering or surrounding them.
Mineral Storage, Energy Storage, and Hematopoiesis
On a metabolic level, bone tissue performs several critical functions. For one, the bone matrix acts as a reservoir for a number of minerals important to the functioning of the body, especially calcium, and potassium. These minerals, incorporated into bone tissue, can be released back into the bloodstream to maintain levels needed to support physiological processes. Calcium ions, for example, are essential for muscle contractions and controlling the flow of other ions involved in the transmission of nerve impulses.
 The softer connective tissue that fills the interior of most bone is referred to as bone marrow (Figure 5.5). There are two types of bone marrow: yellow marrow and red marrow. Yellow marrow contains adipose tissue; the triglycerides stored in the adipocytes of the tissue can serve as a source of energy. Red marrow is where hematopoiesis —the production of blood cells—takes place. Red blood cells, white blood cells, and platelets are all produced in the red marrow.
Contributors and Attributions
- Chapter 6. Authored by : OpenStax College. Provided by : Rice University. Located at : http://cnx.org/contents/14fb4ad7-39a1-4eee-ab6e-3ef2482e3e22<EMAIL_ADDRESS>. Project : Anatomy & Physiology. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/content/col11496/latest/ .


2.7: Algebra Tips and Tricks Part III (Factoring)
Factoring
When factoring an expression like this:
\(x^2 - 8x + 15\)
The goal is to write this like \((x + a)(x + b)\) for some numbers \(a\) and \(b\), where \(a\) and \(b\) could be positive, negative, or zero. Since \((x + a)(x + b) = x^2 + (a + b)x + (ab)\) we see we need \(a + b = -8\) and \(ab = 15\). That way, when you foil it back out, you have \(x^2 - 8x + 15\). We see if \(a = -3\) and \(b = -5\), this works for both \(a + b = -8\) and \(ab = 15\). Thus,
\(x^2 - 8x + 15 = (x - 3)(x - 5)\)
Let’s do a couple more examples.
-
Factor \(x^2 + 3x + 2\).
In this case we want \(a + b = 3\) and \(ab = 2\). \(a = 1\) and \(b = 2\) works, so \(x^2 + 3x + 2 = (x + 1)(x + 2)\).
-
Factor \(x^2 + 5x - 84\).
This is a bit harder because the numbers are bigger, but we can still do it. We want \(a + b = 5\), and \(ab = -84\). We can see that \(84\) is \(12\) times \(7\). So if we have \(a = 12\) and \(b = -7\), then \(a + b = 5\) and \(ab = -84\). Hence \(x^2 + 5x - 84 = (x + 12)(x - 7)\).
-
Factor \(x^2 - 64\).
In this case, we want \(a + b = 0\) and \(ab = -64\). But notice that this means \(a = -b\), and hence \(-a^2 = -64\), which means \(a^2 = 64\). That means \(a = 8\), so \(b = -8\) (or vice versa). Hence \(x^2 - 64 = (x + 8)(x - 8)\).