4.6A: Ribosomes
- Compare and contrast ribosome structure and function in prokaryotes and eukaryotes
Ribosomes are tiny spherical organelles that make proteins by joining amino acids together. Many ribosomes are found free in the cytosol, while others are attached to the rough endoplasmic reticulum. The purpose of the ribosome is to translate messenger RNA (mRNA) to proteins with the aid of tRNA. In eukaryotes, ribosomes can commonly be found in the cytosol of a cell, the endoplasmic reticulum or mRNA, as well as the matrix of the mitochondria. Proteins synthesized in each of these locations serve a different role in the cell. In prokaryotes, ribosomes can be found in the cytosol as well. This protein-synthesizing organelle is the only organelle found in both prokaryotes and eukaryotes, asserting the fact that the ribosome is a trait that evolved early on, most likely present in the common ancestor of eukaryotes and prokaryotes. Ribosomes are not membrane bound.
Ribosomes are composed of two subunits, one large and one small, that only bind together during protein synthesis. The purpose of the ribosome is to take the actual message and the charged aminoacyl-tRNA complex to generate the protein. To do so, they have three binding sites. One is for the mRNA; the other two are for the tRNA. The binding sites for tRNA are the A site, which holds the aminoacyl-tRNA complex, and the P site, which binds to the tRNA attached to the growing polypeptide chain.
In most bacteria, the most numerous intracellular structure is the ribosome which is the site of protein synthesis in all living organisms. All prokaryotes have 70S (where S=Svedberg units) ribosomes while eukaryotes contain larger 80S ribosomes in their cytosol. The 70S ribosome is made up of a 50S and 30S subunits. The 50S subunit contains the 23S and 5S rRNA while the 30S subunit contains the 16S rRNA. These rRNA molecules differ in size in eukaryotes and are complexed with a large number of ribosomal proteins, the number and type of which can vary slightly between organisms. The ribosome is the most commonly observed intracellular multiprotein complex in bacteria.
Ribosome assembly consists of transcription, translation, the folding of rRNA and ribosomal proteins, the binding of ribosomal proteins, and the binding and release of the assembly components to make the ribosome. In vivo assembly of the 30S subunit has two intermediates (p130S and p230S) and the 50S subunit has three intermediates (p150S, p250S, and p350S). However, the reconstitution intermediates are not the same as in vitro. The intermediates of the 30S subunit yield 21S and 30S particles while the intermediates of the 50S subunit yield 32S, 43S, and 50S particles. The intermediates in the in vivo assembly are precursor rRNA which is different from in vitro which uses matured rRNA. To complete the mechanism of ribosome assembly, these precursor rRNA gets transformed in the polysomes.
Key Points
- All prokaryotes have 70S (where S=Svedberg units) ribosomes while eukaryotes contain larger 80S ribosomes in their cytosol. The 70S ribosome is made up of a 50S and 30S subunits.
- Ribosomes play a key role in the catalysis of two important and crucial biological processes. peptidyl transfer and peptidyl hydrolysis.
- Ribosomes are tiny spherical organelles that make proteins by joining amino acids together. Many ribosomes are found free in the cytosol, while others are attached to the rough endoplasmic reticulum.
Key Terms
- ribosome : Small organelles found in all cells; involved in the production of proteins by translating messenger RNA.
- translation : A process occurring in the ribosome, in which a strand of messenger RNA (mRNA) guides assembly of a sequence of amino acids to make a protein.
- Svedberg : The Svedberg unit (S) offers a measure of particle size based on its rate of travel in a tube subjected to high g-force.


10.6: Calculating Moments of Inertia
- Calculate the moment of inertia for uniformly shaped, rigid bodies
- Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one already known
- Calculate the moment of inertia for compound objects
In the preceding subsection, we defined the moment of inertia but did not show how to calculate it. In this subsection, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. This section is very useful for seeing how to apply a general equation to complex objects (a skill that is critical for more advanced physics and engineering courses).
Moment of Inertia
We defined the moment of inertia I of an object to be
\[I = \sum_{i} m_i r_i^2 \]
for all the point masses that make up the object. Because \(r\) is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. To see this, let’s take a simple example of two masses at the end of a massless (negligibly small mass) rod (Figure \(\PageIndex{1}\)) and calculate the moment of inertia about two different axes. In this case, the summation over the masses is simple because the two masses at the end of the barbell can be approximated as point masses, and the sum therefore has only two terms.
In the case with the axis in the center of the barbell, each of the two masses m is a distance \(R\) away from the axis, giving a moment of inertia of
\[I_{1} = mR^{2} + mR^{2} = 2mR^{2} \ldotp\]
In the case with the axis at the end of the barbell—passing through one of the masses—the moment of inertia is
\[I_{2} = m(0)^{2} + m(2R)^{2} = 4mR^{2} \ldotp\]
From this result, we can conclude that it is twice as hard to rotate the barbell about the end than about its center.
In this example, we had two point masses and the sum was simple to calculate. However, to deal with objects that are not point-like, we need to think carefully about each of the terms in the equation. The equation asks us to sum over each ‘piece of mass’ a certain distance from the axis of rotation. But what exactly does each ‘piece of mass’ mean? Recall that in our derivation of this equation, each piece of mass had the same magnitude of velocity, which means the whole piece had to have a single distance r to the axis of rotation. However, this is not possible unless we take an infinitesimally small piece of mass dm, as shown in Figure \(\PageIndex{2}\).
The need to use an infinitesimally small piece of mass dm suggests that we can write the moment of inertia by evaluating an integral over infinitesimal masses rather than doing a discrete sum over finite masses:
\[I = \sum_{i} m_{i} r_{i}^{2}\]
becomes
\[I = \int r^{2} dm \ldotp \label{10.19}\]
This, in fact, is the form we need to generalize the equation for complex shapes. It is best to work out specific examples in detail to get a feel for how to calculate the moment of inertia for specific shapes. This is the focus of most of the rest of this section.
A uniform thin rod with an axis through the center
Consider a uniform (density and shape) thin rod of mass M and length L as shown in Figure \(\PageIndex{3}\). We want a thin rod so that we can assume the cross-sectional area of the rod is small and the rod can be thought of as a string of masses along a one-dimensional straight line. In this example, the axis of rotation is perpendicular to the rod and passes through the midpoint for simplicity. Our task is to calculate the moment of inertia about this axis. We orient the axes so that the z-axis is the axis of rotation and the x-axis passes through the length of the rod, as shown in the figure. This is a convenient choice because we can then integrate along the x-axis.
We define dm to be a small element of mass making up the rod. The moment of inertia integral is an integral over the mass distribution. However, we know how to integrate over space, not over mass. We therefore need to find a way to relate mass to spatial variables. We do this using the linear mass density \(\lambda\) of the object, which is the mass per unit length. Since the mass density of this object is uniform, we can write
\[\lambda = \frac{m}{l}\; or\; m = \lambda l \ldotp\]
If we take the differential of each side of this equation, we find
\[dm = d(\lambda l) = \lambda (dl)\]
since \(\lambda\) is constant. We chose to orient the rod along the x-axis for convenience—this is where that choice becomes very helpful. Note that a piece of the rod dl lies completely along the x-axis and has a length dx; in fact, dl = dx in this situation. We can therefore write dm = \(\lambda\)(dx), giving us an integration variable that we know how to deal with. The distance of each piece of mass dm from the axis is given by the variable x, as shown in the figure. Putting this all together, we obtain
\[I = \int r^{2} dm = \int x^{2} dm = \int x^{2} \lambda dx \ldotp\]
The last step is to be careful about our limits of integration. The rod extends from x = \(− \frac{L}{2}\) to x = \(\frac{L}{2}\), since the axis is in the middle of the rod at x = 0. This gives us
\[\begin{split} I & = \int_{- \frac{L}{2}}^{\frac{L}{2}} x^{2} \lambda dx = \lambda \frac{x^{3}}{3} \Bigg|_{- \frac{L}{2}}^{\frac{L}{2}} \\ & = \lambda \left(\dfrac{1}{3}\right) \Bigg[ \left(\dfrac{L}{2}\right)^{3} - \left(- \dfrac{L}{2}\right)^{3} \Bigg] = \lambda \left(\dfrac{1}{3}\right) \left(\dfrac{L^{3}}{8}\right) (2) = \left(\dfrac{M}{L}\right) \left(\dfrac{1}{3}\right) \left(\dfrac{L^{3}}{8}\right) (2) \\ & = \frac{1}{12} ML^{2} \ldotp \end{split}\]
Next, we calculate the moment of inertia for the same uniform thin rod but with a different axis choice so we can compare the results. We would expect the moment of inertia to be smaller about an axis through the center of mass than the endpoint axis, just as it was for the barbell example at the start of this section. This happens because more mass is distributed farther from the axis of rotation.
A Uniform Thin Rod with Axis at the End
Now consider the same uniform thin rod of mass \(M\) and length \(L\), but this time we move the axis of rotation to the end of the rod. We wish to find the moment of inertia about this new axis (Figure \(\PageIndex{4}\)).
The quantity \(dm\) is again defined to be a small element of mass making up the rod. Just as before, we obtain
\[I = \int r^{2} dm = \int x^{2} dm = \int x^{2} \lambda dx \ldotp\]
However, this time we have different limits of integration. The rod extends from \(x = 0\) to \(x = L\), since the axis is at the end of the rod at \(x = 0\). Therefore we find
\[\begin{align} I & = \int_{0}^{L} x^{2} \lambda\, dx \\[4pt] &= \lambda \frac{x^{3}}{3} \Bigg|_{0}^{L} \\[4pt] &=\lambda \left(\dfrac{1}{3}\right) \Big[(L)^{3} - (0)^{3} \Big] \\[4pt] & = \lambda \left(\dfrac{1}{3}\right) L^{3} = \left(\dfrac{M}{L}\right) \left(\dfrac{1}{3}\right) L^{3} \\[4pt] &= \frac{1}{3} ML^{2} \ldotp \label{ThinRod} \end{align} \]
Note the rotational inertia of the rod about its endpoint is larger than the rotational inertia about its center (consistent with the barbell example) by a factor of four.
The Parallel-Axis Theorem
The similarity between the process of finding the moment of inertia of a rod about an axis through its middle and about an axis through its end is striking, and suggests that there might be a simpler method for determining the moment of inertia for a rod about any axis parallel to the axis through the center of mass. Such an axis is called a parallel axis . There is a theorem for this, called the parallel-axis theorem , which we state here but do not derive in this text.
Let m be the mass of an object and let d be the distance from an axis through the object’s center of mass to a new axis. Then we have
\[I_{\text{parallel-axis}} = I_{\text{center of mass}} + md^{2} \ldotp \label{10.20}\]
Let’s apply this to the uniform thin rod with axis example solved above:
\[I_{parallel-axis} = I_{center\; of\; mass} + md^{2} = \frac{1}{12} mL^{2} + m \left(\dfrac{L}{2}\right)^{2} = \left(\dfrac{1}{12} + \dfrac{1}{4}\right) mL^{2} = \frac{1}{3} mL^{2} \ldotp\]
This result agrees with our more lengthy calculation (Equation \ref{ThinRod}). Equation \ref{10.20} is a useful equation that we apply in some of the examples and problems.
What is the moment of inertia of a cylinder of radius \(R\) and mass \(m\) about an axis through a point on the surface, as shown below?
- Answer
-
\[I_{parallel-axis} = I_{center\; of\; mass} + md^{2} = mR^{2} + mR^{2} = 2mR^{2} \nonumber \]
A Uniform Thin Disk about an Axis through the Center
Integrating to find the moment of inertia of a two-dimensional object is a little bit trickier, but one shape is commonly done at this level of study—a uniform thin disk about an axis through its center (Figure \(\PageIndex{5}\)).
Since the disk is thin, we can take the mass as distributed entirely in the xy-plane. We again start with the relationship for the surface mass density , which is the mass per unit surface area. Since it is uniform, the surface mass density \(\sigma\) is constant:
\[\sigma = \frac{m}{A}\] or \[\sigma A = m\] so \[dm = \sigma (dA)\]
Now we use a simplification for the area. The area can be thought of as made up of a series of thin rings, where each ring is a mass increment dm of radius \(r\) equidistant from the axis, as shown in part (b) of the figure. The infinitesimal area of each ring \(dA\) is therefore given by the length of each ring (\(2 \pi r\)) times the infinitesimmal width of each ring \(dr\):
\[A = \pi r^{2},\; dA = d(\pi r^{2}) = \pi dr^{2} = 2 \pi rdr \ldotp\]
The full area of the disk is then made up from adding all the thin rings with a radius range from \(0\) to \(R\). This radius range then becomes our limits of integration for \(dr\), that is, we integrate from \(r = 0\) to \(r = R\). Putting this all together, we have
\[\begin{split} I & = \int_{0}^{R} r^{2} \sigma (2 \pi r) dr = 2 \pi \sigma \int_{0}^{R} r^{3} dr = 2 \pi \sigma \frac{r^{4}}{4} \Big|_{0}^{R} \\ & = 2 \pi \sigma \left(\dfrac{R^{4}}{4} - 0 \right) = 2 \pi \left(\dfrac{m}{A}\right) \left(\dfrac{R^{4}}{4}\right) = 2 \pi \left(\dfrac{m}{\pi R^{2}}\right) \left(\dfrac{R^{4}}{4}\right) = \frac{1}{2} mR^{2} \ldotp \end{split}\]
Note that this agrees with the value given in Figure 10.5.4 .
Calculating the Moment of Inertia for Compound Objects
Now consider a compound object such as that in Figure \(\PageIndex{6}\), which depicts a thin disk at the end of a thin rod. This cannot be easily integrated to find the moment of inertia because it is not a uniformly shaped object. However, if we go back to the initial definition of moment of inertia as a summation, we can reason that a compound object’s moment of inertia can be found from the sum of each part of the object:
\[I_{total} = \sum_{i} I_{i} \ldotp \label{10.21}\]
It is important to note that the moments of inertia of the objects in Equation \(\PageIndex{6}\) are about a common axis . In the case of this object, that would be a rod of length L rotating about its end, and a thin disk of radius \(R\) rotating about an axis shifted off of the center by a distance \(L + R\), where \(R\) is the radius of the disk. Let’s define the mass of the rod to be m r and the mass of the disk to be \(m_d\).
The moment of inertia of the rod is simply \(\frac{1}{3} m_rL^2\), but we have to use the parallel-axis theorem to find the moment of inertia of the disk about the axis shown. The moment of inertia of the disk about its center is \(\frac{1}{2} m_dR^2\) and we apply the parallel-axis theorem (Equation \ref{10.20}) to find
\[I_{parallel-axis} = \frac{1}{2} m_{d} R^{2} + m_{d} (L + R)^{2} \ldotp\]
Adding the moment of inertia of the rod plus the moment of inertia of the disk with a shifted axis of rotation, we find the moment of inertia for the compound object to be
\[I_{total} = \frac{1}{3} m_{r} L^{2} + \frac{1}{2} m_{d} R^{2} + m_{d} (L + R)^{2} \ldotp\]
Applying moment of inertia calculations to solve problems
Now let’s examine some practical applications of moment of inertia calculations.
A 25-kg child stands at a distance \(r = 1.0\, m\) from the axis of a rotating merry-go-round (Figure \(\PageIndex{7}\)). The merry-go-round can be approximated as a uniform solid disk with a mass of 500 kg and a radius of 2.0 m. Find the moment of inertia of this system.
Strategy
This problem involves the calculation of a moment of inertia. We are given the mass and distance to the axis of rotation of the child as well as the mass and radius of the merry-go-round. Since the mass and size of the child are much smaller than the merry-go-round, we can approximate the child as a point mass. The notation we use is m c = 25 kg, r c = 1.0 m, m m = 500 kg, r m = 2.0 m. Our goal is to find \(I_{total} = \sum_{i} I_{i}\) (Equation \ref{10.21}).
Solution
For the child, \(I_c = m_cr^2\), and for the merry-go-round, \(I_m = \frac{1}{2}m_m r^2\). Therefore
\[I_{total} = 25(1)^{2} + \frac{1}{2} (500)(2)^{2} = 25 + 1000 = 1025\; kg\; \cdotp m^{2} \ldotp \nonumber \]
Significance
The value should be close to the moment of inertia of the merry-go-round by itself because it has much more mass distributed away from the axis than the child does.
Find the moment of inertia of the rod and solid sphere combination about the two axes as shown below. The rod has length 0.5 m and mass 2.0 kg. The radius of the sphere is 20.0 cm and has mass 1.0 kg.
Strategy
Since we have a compound object in both cases, we can use the parallel-axis theorem to find the moment of inertia about each axis. In (a), the center of mass of the sphere is located at a distance \(L + R\) from the axis of rotation. In (b), the center of mass of the sphere is located a distance \(R\) from the axis of rotation. In both cases, the moment of inertia of the rod is about an axis at one end. Refer to Table 10.4 for the moments of inertia for the individual objects.
a.
\[\begin{split} I_{total} & = \sum_{i} I_{i} = I_{Rod} + I_{Sphere}; \\ I_{Sphere} & = I_{center\; of\; mass} + m_{Sphere} (L + R)^{2} = \frac{2}{5} m_{Sphere} R^{2} + m_{Sphere} (L + R)^{2}; \\ I_{total} & = I_{Rod} + I_{Sphere} = \frac{1}{3} m_{Rod} L^{2} + \frac{2}{5} m_{Sphere} R^{2} + m_{Sphere} (L + R)^{2}; \\ I_{total} & = \frac{1}{3} (20\; kg)(0.5\; m)^{2} + \frac{2}{5} (1.0\; kg)(0.2\; m)^{2} + (1.0\; kg)(0.5\; m + 0.2\; m)^{2}; \\ I_{total} & = (0.167 + 0.016 + 0.490)\; kg\; \cdotp m^{2} = 0.673\; kg\; \cdotp m^{2} \ldotp \end{split}\]
b.
\[\begin{split} I_{Sphere} & = \frac{2}{5} m_{Sphere} R^{2} + m_{Sphere} R^{2}; \\ I_{total} & = I_{Rod} + I_{Sphere} = \frac{1}{3} m_{Rod} L^{2} + \frac{2}{5} (1.0\; kg)(0.2\; m)^{2} + (1.0\; kg)(0.2\; m)^{2}; \\ I_{total} & = (0.167 + 0.016 + 0.04)\; kg\; \cdotp m^{2} = 0.223\; kg\; \cdotp m^{2} \ldotp \end{split}\]
Significance
Using the parallel-axis theorem eases the computation of the moment of inertia of compound objects. We see that the moment of inertia is greater in (a) than (b). This is because the axis of rotation is closer to the center of mass of the system in (b). The simple analogy is that of a rod. The moment of inertia about one end is \(\frac{1}{3}\)mL 2 , but the moment of inertia through the center of mass along its length is \(\frac{1}{12}\)mL 2 .
A pendulum in the shape of a rod (Figure \(\PageIndex{8}\)) is released from rest at an angle of 30°. It has a length 30 cm and mass 300 g. What is its angular velocity at its lowest point?
Strategy
Use conservation of energy to solve the problem. At the point of release, the pendulum has gravitational potential energy, which is determined from the height of the center of mass above its lowest point in the swing. At the bottom of the swing, all of the gravitational potential energy is converted into rotational kinetic energy.
Solution
The change in potential energy is equal to the change in rotational kinetic energy, \(\Delta U + \Delta K = 0\).
At the top of the swing:
\[U = mgh_{cm} = mgL^2 (\cos \theta). \nonumber \]
At the bottom of the swing,
\[U = mg \frac{L}{2}. \nonumber\]
At the top of the swing, the rotational kinetic energy is K = 0. At the bottom of the swing, K = \(\frac{1}{2} I \omega^{2}\). Therefore:
\[\Delta U + \Delta K = 0 \Rightarrow (mg \frac{L}{2} (1 - \cos \theta) - 0) + (0 - \frac{1}{2} I \omega^{2}) = 0 \nonumber\]
or
\[\frac{1}{2} I \omega^{2} = mg \frac{L}{2} (1 - \cos \theta) \ldotp \nonumber\]
Solving for \(\omega\), we have
\[\omega = \sqrt{mg \frac{L}{I} (1 - \cos \theta)} = \sqrt{mg \frac{L}{\frac{1}{3} mL^{2}} (1 - \cos \theta)} = \sqrt{g \frac{3}{L} (1 - \cos \theta)} \ldotp \nonumber\]
Inserting numerical values, we have
\[\omega = \sqrt{(9.8\; m/s^{2}) \left(\dfrac{3}{0.3\; m}\right) (1 - \cos 30)} = 3.6\; rad/s \ldotp \nonumber\]
Significance
Note that the angular velocity of the pendulum does not depend on its mass.


6.4: Addition and Subtraction of Decimals
- understand the method used for adding and subtracting decimals
- be able to add and subtract decimals
- be able to use the calculator to add and subtract decimals
The Logic Behind the Method
Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have
\(4 \dfrac{37}{100} + 3 \dfrac{22}{100}\) Performing the addition, we get
\(\begin{array} {rcl} {4.37 + 3.22 = 4 \dfrac{37}{100} + 3 \dfrac{22}{100}} & = & {\dfrac{4 \cdot 100 + 37}{100} + \dfrac{3 \cdot 100 + 22}{100}} \\ {} & = & {\dfrac{437}{100} + \dfrac{322}{100}} \\ {} & = & {\dfrac{437 + 322}{100}} \\ {} & = & {\dfrac{759}{100}} \\ {} & = & {7 \dfrac{59}{100}} \\ {} & = & {\text{seven and fifty-nine hundredths}} \\ {} & = & {7.59} \end{array}\)
Thus, \(4.37 + 3.22 = 7.59\).
The Method of Adding and Subtracting Decimals
When writing the previous addition, we could have written the numbers in columns.
\(\begin{array} {r} {4.37} \\ {\underline{+3.22}} \\ {7.59} \end{array}\)
This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.
Method of Adding and Subtracting
Decimals
To add or subtract decimals:
Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
Add or subtract the numbers as if they were whole numbers.
Place a decimal point in the resulting sum or difference directly under the other decimal points.
Find the following sums and differences.
\(9.813 + 2.140\)
Solution
\(\begin{array} {r} {9.813} \\ {\underline{+2.140}} \\ {11.953} \end{array}\) The decimal points are aligned in the same column.
\(841.0056 + 47.016 + 19.058\)
Solution
\(\begin{array} {r} {841.0056} \\ {47.016\ \ } \\ {\underline{+19.058\ \ }} \end{array}\)
To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.
\(1.314 - 0.58\)
Solution
\(\begin{array} {r} {1.314} \\ {\underline{-0.58\ \ }} \end{array}\) Write a 0 in the thousandths position.
\(16.01 - 7.053\)
Solution
\(\begin{array} {r} {16.01\ \ } \\ {\underline{-7.053}} \end{array}\) Write a 0 in the thousandths position.
Find the sum of 6.88106 and 3.5219 and round it to three decimal places.
Solution
\(\begin{array} {r} {6.88106} \\ {\underline{+3.5219\ \ }} \end{array}\) Write a 0 in the ten thousandths position.
We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get
10.403
Wendy has $643.12 in her checking account. She writes a check for $16.92. How much is her new account balance?
Solution
To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.
After writing a check for $16.92, Wendy now has a balance of $626.20 in her checking account.
Pracitce Set A
Find the following sums and differences.
\(3.187 + 2.992\)
- Answer
-
6.179
Pracitce Set A
\(14.987 - 5.341\)
- Answer
-
9.646
Pracitce Set A
\(0.5261 + 1.0783\)
- Answer
-
1.6044
Pracitce Set A
\(1.06 - 1.0535\)
- Answer
-
0.0065
Pracitce Set A
\(16,521.07 + 9,256.15\)
- Answer
-
25,777.22
Pracitce Set A
Find the sum of 11.6128 and 14.07353, and round it to two decimal places.
- Answer
-
25.69
Calculators
The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encountered. There are some inexpensive calculators that can handle 13 decimal places.
Use a calculator to find each sum or difference.
42.0638 + 126.551
Solution
| Display Reads | ||
| Type | 42.0638 | 42.0638 |
| Press | + | 42.0638 |
| Type | 126.551 | 126.551 |
| Press | = | 168.6148 |
The sum is 168.6148.
Find the difference between 305.0627 and 14.29667.
Solution
| Display Reads | ||
| Type | 305.0627 | 305.0627 |
| Press | - | 305.0627 |
| Type | 14.29667 | 14.29667 |
| Press | = | 290.76603 |
The difference is 290.76603
51.07 + 3,891.001786
Solution
Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.
\(\begin{array} {r} {51.070000} \\ {\underline{3891.001786}} \\ {3942.071786} \end{array}\)
The sum is 3,942.071786.
Practice Set B
Use a calculator to perform each operation.
\(4.286 + 8.97\)
- Answer
-
13.256
Practice Set B
\(452.0092 - 392.558\)
- Answer
-
59.4512
Practice Set B
Find the sum of 0.095 and 0.001862
- Answer
-
0.096862
Practice Set B
Find the difference between 0.5 and 0.025
- Answer
-
0.475
Practice Set B
Find the sum of 2,776.00019 and 2,009.00012.
- Answer
-
Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031
Exercises
For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.
Exercise \(\PageIndex{1}\)
\(1.84 + 7.11\)
- Answer
-
8.95
Exercise \(\PageIndex{2}\)
\(15.015 - 6.527\)
Exercise \(\PageIndex{3}\)
\(11.842 + 28.004\)
- Answer
-
39.846
Exercise \(\PageIndex{4}\)
\(3.16 - 2.52\)
Exercise \(\PageIndex{5}\)
\(3.55267 + 8.19664\)
- Answer
-
11.74931
Exercise \(\PageIndex{6}\)
\(0.9162 - 0.0872\)
Exercise \(\PageIndex{7}\)
\(65.512 - 8.3005\)
- Answer
-
57.2115
Exercise \(\PageIndex{8}\)
\(761.0808 - 53.198\)
Exercise \(\PageIndex{9}\)
\(4.305 + 2.119 - 3.817\)
- Answer
-
2.607
Exercise \(\PageIndex{10}\)
\(19.1161 + 27.8014 + 39.3161\)
Exercise \(\PageIndex{11}\)
\(0.41276 - 0.0018 - 0.00011\)
- Answer
-
0.41085
Exercise \(\PageIndex{12}\)
\(2.181 + 6.05 + 1.167 + 8.101\)
Exercise \(\PageIndex{13}\)
\(1.0031 + 6.013106 + 0.00018 + 0.0092 + 2.11\)
- Answer
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9.135586
Exercise \(\PageIndex{14}\)
\(27 + 42 + 9.16 - 0.1761 + 81.6\)
Exercise \(\PageIndex{15}\)
\(10.28 + 11.111 + 0.86 + 5.1\)
- Answer
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27.351
For the following 10 problems, solve as directed. A calculator may be useful.
Exercise \(\PageIndex{16}\)
Add 6.1121 and 4.916 and round to 2 decimal places.
Exercise \(\PageIndex{17}\)
Add 21.66418 and 18.00184 and round to 4 decimal places.
- Answer
-
39.6660
Exercise \(\PageIndex{18}\)
Subtract 5.2121 from 9.6341 and round to 1 decimal place.
Exercise \(\PageIndex{19}\)
Subtract 0.918 from 12.006 and round to 2 decimal places.
- Answer
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11.09
Exercise \(\PageIndex{20}\)
Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.
Exercise \(\PageIndex{21}\)
A checking account has a balance of $42.51. A check is written for $19.28. What is the new balance?
- Answer
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$23.23
Exercise \(\PageIndex{22}\)
A checking account has a balance of $82.97. One check is written for $6.49 and another for $39.95. What is the new balance?
Exercise \(\PageIndex{23}\)
A person buys $4.29 worth of hamburger and pays for it with a $10 bill. How much change does this person get?
- Answer
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$5.71
Exercise \(\PageIndex{24}\)
A man buys $6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is $2.12, and he buys it, how much of the $20 bill is left?
Exercise \(\PageIndex{25}\)
A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading?
- Answer
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869.37
Exercises for Review
Exercise \(\PageIndex{26}\)
Find the difference between 11,206 and 10,884.
Exercise \(\PageIndex{27}\)
Find the product, \(820 \cdot 10,000\).
- Answer
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8,200,000
Exercise \(\PageIndex{28}\)
Find the value of \(\sqrt{121} - \sqrt{25} + 8^2 + 16 \div 2^2\).
Exercise \(\PageIndex{29}\)
Find the value of \(8 \dfrac{1}{3} \cdot \dfrac{36}{75} \div 2 \dfrac{2}{5}\).
- Answer
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\(\dfrac{20}{9} = \dfrac{5}{3}\) or \(2 \dfrac{2}{9}\)
Exercise \(\PageIndex{30}\)
Round 1.08196 to the nearest hundredth.


12.3: Preparing Unit-Dose Packaged Medications
By the end of this section, you will be able to:
- Discuss the guidelines for withdrawing medications from a vial
- Identify how to withdraw medications from a glass ampule
- Recognize how to use prefilled cartridges for medication administration
- Discuss the guidelines for safely reconstituting powdered medication
- Explain the guidelines for mixing medications in one syringe
Parenteral medications may be supplied in a variety of packages, such as vials, glass ampules, and prefilled cartridges. Sometimes, the medication will come prepared, ready for the nurse to administer. Other times, it may be necessary for the nurse to withdraw a dose of medication from a vial, add a liquid to a powdered medication to form a solution that may be suitable for the parenteral route, or mix multiple medications in the same syringe prior to administration. Each of these preparations requires specific guidelines to be followed when preparing the medication. This section will provide you with the knowledge needed to demonstrate appropriate preparation of unit-dosed medications for parenteral administration.
Withdrawing Medications from a Vial
Withdrawing medications from a vial is a common procedure for the nurse to perform. A vial is a glass or plastic container with a rubber stopper on top that contains a liquid medication (Figure 12.18). To prepare for the task, the nurse should gather the medication vial, syringe , needle (if not already packaged with the syringe), and an alcohol swab. Perform hand hygiene and ensure the work surface is clean. Examine the vial to verify the correct medication, concentration (if applicable), and the expiration date of the vial. If the vial is out of date, discard the vial according to the agency’s policy and obtain a new vial.
To prepare the vial, remove the plastic cap from the vial if it is the first time the vial is being accessed. Use an alcohol swab to clean the rubber top. While the rubber top is drying, prepare the syringe. Remove the syringe and needle from the package, using caution to touch only the nonsterile surfaces. Attach the needle to the end of the syringe using aseptic technique. Once assembled, pull the plunger back to equal the amount of the medication to be administered, leaving the cap of the needle on during the process. Remove the cap, insert the needle in the center of the rubber top, and insert the air into the vial . Inserting air into the vial will reduce the vacuum effect and allow the medication to be removed from the vial more easily.
Invert the vial and needle, so the needle is pointing upward. Ensure the bevel is below the level of the solution. Pull back the plunger to obtain the correct dose. Remove any air bubbles by gently tapping the syringe. Gently press the plunger upward to expel the air bubbles from the syringe. If many bubbles are present, inject all of the medication back into the vial and slowly withdraw the medication again. Ensure all bubbles are removed and the correct dose is obtained, then invert the vial and syringe so the needle is pointing downward. Remove the needle from the vial and engage the needle safety device.
While recapping needles is generally discouraged because of the risk of needlestick injuries and potential exposure to bloodborne pathogens, there may be times when recapping the needle is necessary, such as if the needle is built into the syringe. If recapping is necessary, there are several guidelines to follow to minimize the risk of an accidental needlestick injury:
- Use a one-handed technique: Keep one hand behind your back or at your side away from the needle during the recapping process.
- Maintain visual contact: Keep your eyes on the needle while recapping to ensure you can control the process.
- Use a scoop technique: Place the cap on the countertop. Hold the barrel of the syringe with your dominant hand and use a scooping motion to insert the needle into the cap. When the needle is covered by the cap, tip the needle and cap upward and use your nondominant hand to click the cap securely on the needle. Another option is to press the needle and cap on a hard surface such as wall to click the cap onto the needle. During the process, use aseptic technique to avoid contaminating the needle and exposed plunger.
- Use mechanical devices: If available, use mechanical devices designed for recapping needles safely. These devices often have mechanisms to hold and secure the needle cap without requiring direct manual contact.
See the competency checklist for Withdrawing Medication from a Vial. You can find the checklists on the Student resources tab of your book page on openstax.org.
Single-Dose Vials
Single-dose vial s are approved to be used one time for one patient. They do not contain antimicrobial preservatives; hence, harmful bacteria may grow in the vial and infect the patient if it is reused. Therefore, it is important to discard the vial after every use, even if there is unused medication left in the vial after removing the required dose. Single-dose vials are preferred over multidose vial s, when possible, to reduce the risk of contamination.
Single-dose vials may come in a variety of sizes; therefore, it is important to always check the medication label to determine if the vial is single-dose, rather than relying on the volume of the vial. If the vial says single-dose and it has already been accessed or the medication has expired, dispose of the vial according to the agency’s policy and obtain a new vial. In general, the rule of “when in doubt, throw it out” is a good one to follow here.
Multidose Vials
Multidose vials contain more than one dose of medication. Vials are labeled for multiuse or single use. The manufacturer adds antimicrobial preservation to multidose vials. This helps prevent bacterial growth if safe injection practices are followed. This preservative, however, may not be effective against viruses. Multidose vials should be dedicated to a single patient, when possible. If the multidose vial enters the patient’s room, it should be used only for that patient to prevent inadvertent contamination. Otherwise, the multidose vial should be kept in the medication preparation area at all times. New needles and syringes should be used for every injection, even if the multidose vial is used for only one patient. Do not leave needles in vials for subsequent medication withdrawals. If barcode administration is used, the pharmacy will usually supply a sheet of barcodes that may be applied to the syringe and later scanned in the patient’s room, identifying the medication.
When using multidose vials, it is important to note the date when the vial was first accessed by placing a label on the vial with the date, time, and nurse’s initials. This beyond-use date should be calculated according to the agency’s policy, usually twenty-eight days unless the manufacturer specifies differently, and written on the vial. The beyond-use date should never extend past the manufacturer’s expiration date. For expiration dates that display only the month and year, the unopened product is usable until the end of the month unless otherwise noted by the manufacturer. The vial should be discarded if sterility is compromised, the manufacturer’s expiration date has passed, the vial is unlabeled with the date it was first accessed, the beyond-use date has passed, or the solution appears abnormal (e.g., discoloration, floating particles, or cloudy when it should be clear).
Withdrawing Medications from a Glass Ampule
An ampule (Figure 12.19) is a small glass container that holds a single dose of a medication prepared in liquid form. Prior to opening the ampule, gently tap the ampule to move all of the liquid contents to the bottom of the ampule and wipe the scored line with an alcohol swab. To open the glass ampule, place a gauze around the neck of the ampule and snap the neck of the ampule off, at the scored line, away from your body. To withdraw the medication from the ampule, use a blunt fill needle with a filter to prevent glass particles from being drawn up into the syringe with the medication. Replace the filtered needle with a regular needle before administering the medication to the patient. After an ampule is opened, the contents must be used right away and not stored for later use.
See the competency checklist for Withdrawing Medication from a Glass Ampule. You can find the checklists on the Student resources tab of your book page on openstax.org.
Special Considerations
When removing medications from an ampule, it is important to remember that the ampule is made of glass. Therefore, precautions should be taken to avoid cutting yourself. As mentioned in the clinical procedures, it is important to wrap a gauze pad around the neck of the ampule to protect your hands from being cut by the breaking glass. In addition, the glass ampule should always be placed in the sharps container instead of a regular trash can. It is important to be aware of any glass shards that may result from opening the ampule and to ensure all glass is cleaned up properly. If the ampule breaks or does not result in a clean break around the scored line, causing shards of glass, do not use the medication. Carefully clean up, dispose of the glass, and obtain a new ampule. By following these protocols and using a filtered needle, the nurse may avoid drawing glass up into the syringe with the medication. Injecting glass into the patient may cause harmful side effects, including pulmonary thrombi, embolisms, phlebitis, granuloma formation, and inflammation.
Using Prefilled Cartridges
Some medications are packaged in prefilled cartridges. These prefilled cartridges may fall into two categories: injector device s or prefilled syringe s. Prefilled cartridges have many benefits, including enhanced convenience, accuracy, sterility, and safety.
Prefilled cartridges come prepared with the medication premeasured and ready to use, which is much more convenient than having to draw up the medication from a vial or ampule . When a medication is already drawn up and ready to go, injections may be administered more quickly. Prefilled cartridges may be particularly helpful during emergent situations.
Prefilled cartridges may also enhance accuracy, because the manufacturer ensures the cartridges contain the correct amount of medication. It removes the potential for the individual to draw up the wrong amount of medication. This may be particularly helpful when patients need to self-administer the medication and have not been trained to calculate and measure doses.
Sterility is also enhanced when using prefilled cartridges. Prefilled cartridges have a shelf life of approximately two to three years, whereas medications that are drawn up from a vial or ampule typically have a shelf life of approximately twelve hours. Not to mention, there is less risk of contamination if the cartridge is prefilled and ready for the user to administer.
See the competency checklist for Preparing Medication Using a Prefilled Sterile Cartridge. You can find the checklists on the Student resources tab of your book page on openstax.org.
Injector Devices
An injector device (Figure 12.20) is a spring-loaded syringe that delivers a single, preloaded dose of a drug when the device is firmly pressed against the body. These devices can deliver medications through subcutaneous or intramuscular routes. They are easy to self-administer, which may improve patient compliance and reduce anxiety. Some examples of injector devices include epinephrine injections (EPIPEN), insulin pens, diazepam (Valium) auto-injectors, and naloxone ( Narcan ) auto-injectors.
Prefilled Syringes
A prefilled syringe (Figure 12.21) is a disposable syringe that is supplied with the necessary amount of the medication to be injected in a single dose. It is designed to simplify the process of medication administration by eliminating the need for healthcare professionals or patients to manually draw up the medication from a vial into a syringe before injection. By using a prefilled syringe, the risk of vial contamination or improper technique of preparing the medication is reduced, as well as reducing medication waste. In many cases, the prefilled syringe contains a Luer-lock that allows the needle to be twisted in place and then the medication is ready to administer immediately. If there is an air bubble present in the prefilled syringe, gently tap the syringe until the bubbles rise to the top of the syringe and slowly push the plunger up to expel the bubbles, using caution not to expel any of the medication. However, do not remove the air bubble found in enoxaparin (Lovenox) prefilled syringes, as this bubble is needed to help push the medication into the body. After administration, both the needle and the syringe must be discarded because reusing either component may transmit diseases. Some examples of medications that come in prefilled syringes include enoxaparin, propofol (Diprivan), fentanyl, hydromorphone (Dilaudid), and labetalol.
Prefilled syringes may also come packaged for use in a single-dose cartridge system, which loads a specific prefilled cartridge into a plastic holder for easy administration. This system reduces waste by reusing the plastic holder instead of requiring the medication to be withdrawn with a syringe and needle. When using the reusable plastic holder, it is important to wipe the holder down between uses with an appropriate antiseptic cleanser. If withdrawing medication from the single-dose cartridge system with a needle and syringe, do not inject air into the prefilled syringe because the pressure in the prefilled syringe will increase, forcing the rubber stopper and medication to be expelled from the prefilled syringe.
Guidelines for Reconstituting Powdered Medication
Some medications are unstable when supplied in a liquid form; therefore, they are packaged in a powdered form. Prior to administering these medications via the parenteral route, the nurse must reconstitute the powder (solute), or make it into liquid again, with a diluent (solvent) to create a liquid solution. It is important for the nurse to know how to properly reconstitute these medications to ensure that safe medication administration practices are followed.
Selection of Diluent
A diluent is the liquid used to dilute a powdered medication into a liquid form. Sterile solutions must be used when reconstituting parenteral medications. The type of diluent used varies according to the medication. Typically, parenteral medications are diluted with sterile water or sterile normal saline. The type and amount of diluent to be used should be identified in the medication package insert, in the medication label, or the electronic medication administration record (eMAR). If in doubt, consult the pharmacy or a medication reference guide to confirm.
Safety Considerations
When reconstituting powdered medications, it is important to locate the manufacturer’s directions for reconstituting, typically found on the vial or package insert. Within the manufacturer’s directions, there should be clear instructions regarding which diluent to use, as well as the volume of liquid needed to reconstitute the medication. The label should also identify proper storage for the medication after mixing, including the length of time the medication can be stored after reconstitution. To correctly calculate the dose, it is important to note the strength or concentration of the medication after it has been reconstituted. If in doubt or if any of the necessary information cannot be found, consult the pharmacy or a medication reference guide. Reconstitution instructions must always be verified and never be assumed.
The nurse is preparing to administer IV acyclovir to a patient. After retrieving the vial, the nurse notes that the medication is in a powdered form. Recognizing that the powdered medication must be turned into a solution before administering the medication via the IV route, the nurse analyzes the cues to determine whether the medication needs to be reconstituted. The nurse considers common diluents used for reconstituting medications and hypothesizes that sterile water or normal saline may be required; however, the nurse examines the medication label to determine which diluent to use. After reading the medication label, the nurse notes that the instructions read, “Inject 20 mL sterile water for injection into vial.” The nurse takes action by reconstituting the powdered medication with 20 mL of sterile water.
After a medication has been reconstituted, it is best practice for the individual who prepared the medication to administer the medication. Should the individual preparing the medication not use the appropriate type or amount of diluent , the nurse who administered the medication may be held responsible should a medication error occur because of improper reconstitution . If the individual who prepared the medication is not the person administering the medication, best practice is to label the syringe with the type and amount of diluent used, as well as the expiration date of the diluent, and the time in which the medication must be used after being reconstituted.
Special consideration should be given to hazardous medications, such as chemotherapeutics or other cytotoxic drugs. When reconstituting hazardous medications, the procedure must take place inside a chemical fume hood or other containment device to provide safety in the event the powdered medication is released into the air. In addition to using a containment device, personal protective equipment (PPE) should also be worn to protect the individual preparing the medication from accidental exposure. Due to the required safety precautions, hazardous medications must always be reconstituted by trained professionals in the pharmacy.
Procedural Steps for Reconstituting Powdered Medication
Gather the necessary supplies: diluent, medication vial , needle , alcohol swab, and syringe large enough to hold the volume of the diluent. Check expiration dates on the diluent and medication vials. If expired, discard vials according to agency policy and obtain new vials. After performing hand hygiene, remove the plastic caps from the vials, clean the rubber stoppers with an alcohol swab, and allow the alcohol to dry. Then the nurse assembles the needle and syringe.
Inject air into the diluent vial, equivalent to the volume of diluent required. Invert the vial and needle so the needle is pointing up with the bevel below the level of solution inside the vial. Then withdraw the required amount of diluent and remove the syringe and needle from the vial. If there is more diluent in the vial than required, only remove the required volume, and dispose of the remaining diluent according to the agency’s policy. Invert the vial and needle so the needle is facing upward, tap the syringe so air bubbles move to the top. Gently depress the plunger to remove all air bubbles. Inject the diluent into the medication vial. Gently roll the medication vial between your hands until the powder is dissolved, inspecting the vial to ensure all particles are dissolved. Invert the vial and needle so the needle is facing upward and withdraw the required amount of solution. If there is more solution in the vial than required, only remove the required volume, and dispose of the remaining solution according to the agency’s policy. Invert the vial and needle so the needle is facing downward and remove the needle and syringe from the vial. Gently remove all air bubbles as described and engage the needle safety device.
See the competency checklist for Reconstituting Powdered Medication. You can find the checklists on the Student resources tab of your book page on openstax.org.
Dual-Compartment Vial System
Some medications come prepared with the powdered medication and the diluent in the same package; however, they are separated in different compartments until they are ready for use (Figure 12.22). One example of a medication that comes prepared in a dual-compartment vial system is methylprednisolone sodium succinate (Solu-Medrol). To reconstitute the medication, simply press down on the plastic activator. This will displace the rubber stopper and allow the diluent into the lower compartment with the powdered medication. Gently swirl the vial (do not shake) to mix the solution. Ensure no particles are remaining before drawing up the medication. Remove the circular plastic covering on top of the plastic activator to reveal the rubber stopper and withdraw the medication according to the procedure for withdrawing medications from a vial.
Guidelines for Mixing Medications in One Syringe
Some medications must be mixed from two vials or from a vial and an ampule . Other times, the nurse may mix medications to avoid the need for multiple injections. When mixing medications in one syringe , it is essential to first ensure compatibility of the medications. Consideration should also be given to ensure the volume of the mixed medications is appropriate for the injection site. Special consideration must be given when mixing insulins. It is important to follow the guidelines for mixing medications into one syringe to ensure patient safety, comply with aseptic technique, and prevent contamination of the contents.
See the competency checklist for Mixing Medications from Two Vials in One Syringe. You can find the checklists on the Student resources tab of your book page on openstax.org.
Compatibility
It is important to ensure the medications are compatible before mixing medications into one syringe. Compatibility is the ability to administer two or more medications together without them interacting with one another. Incompatible medications drawn up or mixed together may alter the chemical makeup or effects of either medication. After being combined, the medications may have reduced effects or become inactive, harmful chemical reactions may occur, toxicity of the medications may be increased, or microparticles may be created and injected into the body. Compatibility of the medications must be assessed before administering the medications. To do so, the nurse may use an electronic medication interaction checker, look up the medications in a medication reference guide, or use a compatibility chart often provided by the organization (Figure 12.23) for quick access to commonly used medications. If uncertain about whether medications are compatible, consult a pharmacist.
Nurse:
Mike, RN
Clinical setting:
Inpatient psychiatric unit
Years in practice:
5
Facility location:
Southeastern United States
I was recently asked to float to an inpatient psychiatric unit. During the float assignment, a patient on the unit became very aggressive. My first action was to implement nonpharmacological and verbal de-escalation techniques but they were not effective. The prescribing provider had ordered diphenhydramine (Benadryl) 50 mg, haloperidol (Haldol) 5 mg, and lorazepam (Ativan) 2 mg IM times one dose. Reading the order, I understood that the three different medications were ordered via the IM route. I was concerned about giving all three medications at one time. After speaking with other professional personnel on the unit, I learned this combination of medications is widely used in the behavioral health setting and is known as a B-52 cocktail. So, I considered whether the medications could be combined into one syringe to avoid having to give the patient three injections. Therefore, I opened the electronic medication interaction checker, entered all three medications into the system, and concluded that all three medications were compatible. My next step was to calculate the total amount of the medications and ensure that the total volume did not exceed the maximum volume for an IM injection. Finally, I drew up the medications into one syringe and administered them to the patient in a single injection.
Volume Limits for Different Routes
When mixing medications into one syringe, it is important to consider the overall amount of fluid to be injected. This is particularly important when administering medications via the intradermal (ID) , subcutaneous (SQ) , and intramuscular (IM) routes. Injecting too much volume may be difficult for the body to absorb, cause increased pain for the patient, and result in adverse events at the injection site. The maximum volume for ID injections is 0.5 mL or less. Typically, the maximum volume for SQ injections is about 1.5 mL; however, volumes of 2 mL to 3 mL may be tolerated when injected into the abdomen. The maximum volume for IM injections is between 1 mL and 3 mL, though this can depend on the injection site. Children can withstand less volume via the IM route, with no more than 1 mL of fluid injected into the ventrogluteal muscles and no more than 3 mL injected into the vastus lateralis (1 mL in infants) (Table 12.1).
| Route | Typical Maximum Volume |
|---|---|
| Intradermal |
|
| Subcutaneous |
|
| Intramuscular |
|
Mixing Medications from Vial and Ampule
When mixing medications from a vial and an ampule , withdraw the medication from the vial first, followed by the ampule. Determine the amount of medication to be removed from the vial, the amount to be removed from the ampule, and the total volume that should be administered. Follow the guidelines for removing medication from a vial, remove the needle from the syringe using the scoop or hands-free method, add a filtered needle, and then proceed with removing the medication from the ampule. Use caution not to remove more than the exact required dose from the ampule. Remove the filtered needle and replace it with an appropriate-sized needle according to the route of administration.
Mixing Insulins
Some insulins, such as intermediate-acting insulin and short- or rapid-acting insulins, may be mixed together in a single syringe so the individual will require only one injection. Mixing insulins is sometimes needed to help control an individual’s blood sugar throughout the day, because the insulins have varying onsets. Insulin glargine (Lantus) and insulin detemir (Levemir) may never be mixed in the same syringe with each other or other insulins because doing so may alter the onset of the insulins, making it difficult to control the patient’s blood sugar. It is important to follow proper sequencing for mixing insulins to prevent contaminating the insulin vials, which may adversely affect the onset of the medications.
Procedural Sequence for Mixing Insulins
When mixing insulins, obtain the two types of insulin, an insulin syringe large enough to hold the volume of both insulins combined, and alcohol swabs. To prepare the vials, gently roll the intermediate-acting insulin (cloudy vial) between your hands. Wipe the top of each insulin vial with an alcohol swab and allow the alcohol to dry. Inject air equal to the required units of the intermediate-acting insulin into the cloudy insulin vial. Remove the syringe and needle from the intermediate-acting vial. Inject air equal to the required units of short- or rapid-acting insulin into the clear vial of insulin. With the needle in the clear vial, invert the vial and syringe so the needle is pointed up. Ensure that the bevel of the needle is below the level of solution and slowly pull down on the plunger until the desired units of clear insulin are withdrawn into the syringe. Gently tap on the syringe to remove any air bubbles. If there is a large quantity of air bubbles, the insulin may be inserted back into the vial and the process repeated. Gently push up on the plunger to remove the air bubbles from the syringe. If any air bubbles are left in the syringe, the patient will receive less than the intended dose of insulin.
After the correct dose of clear insulin has been withdrawn and there are no air bubbles present, remove the needle from the clear insulin vial . Push the needle into the cloudy insulin vial and invert the bottle upside down with the needle pointed up. Slowly pull down the plunger until the total number of cloudy units are obtained (Figure 12.24). Be cautious that you have the correct number of units because the cloudy insulin has mixed with the clear insulin upon withdrawing the medication from the vial and cannot be pushed back into the vial. If too much cloudy insulin is withdrawn from the vial, discard the dose and start the procedure over.
See the competency checklist for Procedural Sequence for Mixing Insulins. You can find the checklists on the Student resources tab of your book page on openstax.org.


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16.8: Critical Thinking Questions
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How do people learn to create art and music in modern Western societies? Describe your own experiences becoming enculturated in the American style of creating art and music.
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Pick a favorite image, one that was created specifically as an art object. This might be a painting, a photograph, a depiction of a sculpture, or something similar. Analyze it in an anthropological fashion. Who created it? Why was it created? Who was the audience or market for this object? What message(s) do you perceive when looking at it?
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How far back can you trace the history of your favorite style of music? In your analysis, include instrumentation, rhythms, vocalizations (if any), and sites of performance.
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What role(s) do sports play in your own culture? Address both informal sports (e.g., pickup games between friends) and professional/national teams.
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How would global and racial inclusion in the United States differ without the contributions of art, music, and sports professionals?


11.4: Enthalpy- Heat of Combustion
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- Scott Van Bramer
- Widener University
Learning Objectives
- State the first law of thermodynamics
- Define enthalpy and explain its classification as a state function
- Write and balance thermochemical equations
- Calculate enthalpy changes for various chemical reactions
- Explain Hess’s law and use it to compute reaction enthalpies
Standard enthalpy of combustio n (\(ΔH_C^\circ\)) is the enthalpy change when 1 mole of a substance burns (combines vigorously with oxygen) under standard state conditions; it is sometimes called “heat of combustion.” For example, the enthalpy of combustion of ethanol, −1366.8 kJ/mol, is the amount of heat produced when one mole of ethanol undergoes complete combustion at 25 °C and 1 atmosphere pressure, yielding products also at 25 °C and 1 atm.
\[\ce{C2H5OH}(l)+\ce{3O2}(g)⟶\ce{2CO2}+\ce{3H2O}(l)\hspace{20px}ΔH_{298}^\circ=\mathrm{−1366.8\: kJ} \label{5.4.8}\]
Enthalpies of combustion for many substances have been measured; a few of these are listed in Table \(\PageIndex{1}\). Many readily available substances with large enthalpies of combustion are used as fuels. These include hydrogen; carbon as coal or charcoal; and hydrocarbons like methane, propane, and the major components of gasoline.
| Substance | Combustion Reaction | Enthalpy of Combustion \(ΔH_c^\circ \left(\mathrm{\dfrac{kJ}{mol} \:at\:25°C}\right)\) |
|---|---|---|
| carbon | \(\ce{C}(s)+\ce{O2}(g)⟶\ce{CO2}(g)\) | −393.5 |
| hydrogen | \(\ce{H2}(g)+\frac{1}{2}\ce{O2}(g)⟶\ce{H2O}(l)\) | −285.8 |
| magnesium | \(\ce{Mg}(s)+\frac{1}{2}\ce{O2}(g)⟶\ce{MgO}(s)\) | −601.6 |
| methane | \(\ce{CH4}(g)+\ce{2O2}(g)⟶\ce{CO2}(g)+\ce{2H2O}(l)\) | −890.8 |
| ethanol | \(\ce{C2H5OH}(l)+\ce{3O2}(g)⟶\ce{CO2}(g)+\ce{3H2O}(l)\) | −1366.8 |
| isooctane | \(\ce{C8H18}(l)+\dfrac{25}{2}\ce{O2}(g)⟶\ce{8CO2}(g)+\ce{9H2O}(l)\) | −5461 |
| glucose | \(\ce{C6H12O6}(s)+\dfrac{6}{2}\ce{O2}(g)⟶\ce{6CO2}(g)+\ce{6H2O}(l)\) | -2813 |
Example \(\PageIndex{1}\): Using Enthalpy of Combustion
As Figure \(\PageIndex{1}\) suggests, the combustion of gasoline is a highly exothermic process. Let us determine the approximate amount of heat produced by burning 1.00 L of gasoline, assuming the enthalpy of combustion of gasoline is the same as that of isooctane, a common component of gasoline. The density of isooctane is 0.692 g/mL.
Solution
Starting with a known amount (1.00 L of isooctane), we can perform conversions between units until we arrive at the desired amount of heat or energy. The enthalpy of combustion of isooctane provides one of the necessary conversions. Table \(\PageIndex{1}\) gives this value as −5460 kJ per 1 mole of isooctane (C 8 H 18 ).
Using these data,
The combustion of 1.00 L of isooctane produces 33,100 kJ of heat. (This amount of energy is enough to melt 99.2 kg, or about 218 lbs, of ice.)
Note: If you do this calculation one step at a time, you would find:
\(\begin {align*}
&\mathrm{1.00\:L\:\ce{C8H18}⟶1.00×10^3\:mL\:\ce{C8H18}}\\
&\mathrm{1.00×10^3\:mL\:\ce{C8H18}⟶692\:g\:\ce{C8H18}}\\
&\mathrm{692\:g\:\ce{C8H18}⟶6.07\:mol\:\ce{C8H18}}\\
&\mathrm{692\:g\:\ce{C8H18}⟶−3.31×10^4\:kJ}
\end {align*}\)
Exercise \(\PageIndex{1a}\)
How much heat is produced by the combustion of 125 g of acetylene?
- Answer
-
6.25 × 10 3 kJ
Exercise \(\PageIndex{1b}\)
How much heat is produced by the combustion of 125 g of glucose?
- Answer
-
1.95 × 10 3 kJ
Emerging Algae-Based Energy Technologies (Biofuels)
As reserves of fossil fuels diminish and become more costly to extract, the search is ongoing for replacement fuel sources for the future. Among the most promising biofuels are those derived from algae (Figure \(\PageIndex{2}\)). The species of algae used are nontoxic, biodegradable, and among the world’s fastest growing organisms. About 50% of algal weight is oil, which can be readily converted into fuel such as biodiesel. Algae can yield 26,000 gallons of biofuel per hectare—much more energy per acre than other crops. Some strains of algae can flourish in brackish water that is not usable for growing other crops. Algae can produce biodiesel, biogasoline, ethanol, butanol, methane, and even jet fuel.
According to the US Department of Energy, only 39,000 square kilometers (about 0.4% of the land mass of the US or less than \(\dfrac{1}{7}\) of the area used to grow corn) can produce enough algal fuel to replace all the petroleum-based fuel used in the US. The cost of algal fuels is becoming more competitive—for instance, the US Air Force is producing jet fuel from algae at a total cost of under $5 per gallon. The process used to produce algal fuel is as follows: grow the algae (which use sunlight as their energy source and CO 2 as a raw material); harvest the algae; extract the fuel compounds (or precursor compounds); process as necessary (e.g., perform a transesterification reaction to make biodiesel); purify; and distribute (Figure \(\PageIndex{3}\)).
Footnotes
- 1 For more on algal fuel, see www.theguardian.com/environme...n-fuel-problem.
Summary
If a chemical change is carried out at constant pressure and the only work done is caused by expansion or contraction, the heat produced or required for the change is called the enthalpy change with the symbol Δ H , or \(ΔH^\circ\) for reactions occurring under standard state conditions. The value of Δ H for a reaction in one direction is equal in magnitude, but opposite in sign, to Δ H for the reaction in the opposite direction, and Δ H is directly proportional to the quantity of reactants and products. Examples of enthalpy changes include enthalpy of combustion, enthalpy of fusion, enthalpy of vaporization, and standard enthalpy of formation. The standard enthalpy of formation, \(ΔH^\circ_\ce{f}\), is the enthalpy change accompanying the formation of 1 mole of a substance from the elements in their most stable states at 1 bar (standard state). Many of the processes are carried out at 298.15 K. If the enthalpies of formation are available for the reactants and products of a reaction, the enthalpy change can be calculated using Hess’s law: If a process can be written as the sum of several stepwise processes, the enthalpy change of the total process equals the sum of the enthalpy changes of the various steps.
Key Equations
- \(ΔU=q+w\)
- \(ΔH^\circ_\ce{reaction}=∑n×ΔH^\circ_\ce{f}\ce{(products)}−∑n×ΔH^\circ_\ce{f}\ce{(reactants)}\)
Glossary
- enthalpy ( H )
- sum of a system’s internal energy and the mathematical product of its pressure and volume
- Hess’s law
- if a process can be represented as the sum of several steps, the enthalpy change of the process equals the sum of the enthalpy changes of the steps
- hydrocarbon
- compound composed only of hydrogen and carbon; the major component of fossil fuels
- standard enthalpy of combustion (\(ΔH^\circ_\ce{c}\))
- heat released when one mole of a compound undergoes complete combustion under standard conditions
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


23.7B: Absorptive State
When the gastrointestinal tract is full, anabolism exceeds catabolism; this is the absorptive state.
- Differentiate among the nutrients in the absorptive state
Key Points
- During the absorptive state, anabolic processes use glucose in a variety of ways.
- In the liver, glucose is converted to glycogen or fat, which store energy for future use. Fat is also stored in adipose tissue and glycogen in muscle tissue.
- Glucose is also carried in the bloodstream to cells where it will be used to provide energy for cellular processes.
- Also during the absorptive state, chylomicrons, the product of fat digestion, are reconstituted to fat and stored in adipose tissue or, in a low carb environment, are used as an energy source.
- The liver deaminates amino acids to keto acids which can be used in the krebs cycle to produce ATP, or can be converted to fat, or can be used by other body cells to create proteins.
Key Terms
- absorptive state : The period during digestion when anabolism exceeds catabolism.
- Kreb’s cycle : The Kreb’s cycle is a series of chemical reactions used by all aerobic organisms to generate energy through the oxidization of acetate derived from carbohydrates, fats and proteins into carbon dioxide.
- glycogen : A polysaccharide that is the main form of carbohydrate storage in animals; converted to glucose as needed.
EXAMPLES
The baby who has finished nursing has a full tummy and now will probably fall asleep. During this sleep period, anabolic processes are busy building up stores of fats and glycogen that will be needed in the future to provide energy for the growing baby.
Absorptive state is the period in which the gastrointestinal tract is full and the anabolic processes exceed catabolism. The fuel used for this process is glucose.
Carbohydrates
Simple sugars are sent to the liver where they are converted to glucose. The glucose then travels to the blood or is converted to glycogen and fat (triglyceride) for energy storage. The glycogen and fat will be stored in the liver and adipose tissue, respectively, as reserves for the post-absorptive state. The remaining glucose is taken in for use by body cells or stored in skeletal muscle as glycogen.
Triglycerides
Chylomicrons are lipoprotein particles that consist of triglycerides (85-92%), phospholipids (6-12%), cholesterol (1-3%) and proteins (1–2%). This main product of fat digestion is first broken down to fatty acids and glycerol through hydrolysis using lipoprotein lipase. This allows them to pass freely through capillary walls. Most of this will be reconstituted as triglycerides and stored in adipose tissue. The rest will be used for energy in adipose cells, skeletal muscle and hepatocytes. In a low carb environment, other cells of the body will also begin to use triglycerides as energy sources.
Amino acids
The liver deaminates amino acids to keto acids to be used in the Kreb’s cycle in order to generate energy in the form of ATP. They may also be converted to fat for energy storage. Some are used to make plasma proteins, but most leave through liver sinusoids to be used by body cells to construct proteins.
Glucose Metabolism : Glucose metabolism and various forms of it in the process.


5.15: Colons- The Signposts
- Demonstrate the standard uses of colons
The colon is like a sign on the highway, announcing that something important is coming. It acts as an arrow pointing forward, telling you to read on for important information. A common analogy used to explain the colon is that it acts as a flare in the road, signaling that something meaningful lies ahead.
The colon is not just used to introduce a list; it is far more flexible. The colon can appear after the first word(s) of a sentence, just before the final word(s) of a sentence, or even between two independent clauses. It can be used to provide emphasis, to explain, or to summarize. Thus, it is one of the most powerful punctuation marks.
Use the colon when you wish to provide emphasis.
- To address this problem, we must turn to one of the biologist’s most basic tools: the petri dish.
- My grandfather, a research scientist, gave me some critical advice about petri dishes: don’t drop them.
Use the colon to introduce material that explains or summarizes what has preceded it.
- The petri dish: one of the biologist’s most basic tools.
- In low-carbon steels, banding tends to affect two properties in particular: tensile ductility and yield strength.
- The research is conclusive: global warming is really happening.
Use the colon to present a list or series, particularly when there is a lot of similar material to join or when the items in the list include commas. When you are using a colon in this way, the colon usually gives the idea of “as follows” or “which is/are.” That means you should not include a colon after words like “for example,” or when you may be separating a verb from its object.
- A compost facility may not be located in the following: within 300 feet of an exceptional-value wetland, within 100 feet of a perennial stream, or within 50 feet of a property line.
- A backyard compost pile can process many items: fruit and vegetable scraps; crushed egg shells; spoiled soy, rice, almond, or coconut milk; loose leaf tea.
- Incorrect sentence: The three primary colors are: red, blue, and yellow. (The colon is not necessary to introduce this list because the first part of the sentence is grammatically incomplete without including the colors.)
Is the colon used correctly in the following sentences? Select yes or no.
https://assessments.lumenlearning.co...sessments/5171
Understanding the functions of the semicolon and colon will help you use them effectively. Remember, a semicolon connects complete ideas or items in a list that have internal commas. Colons act as signposts to alert readers to important information.
https://h5p.cwr.olemiss.edu/h5p/embed/29
Review the distinctions between semicolons and colons in the following interactive.
https://lumenlearning.h5p.com/content/1290906746630560368/embed
Contributors and Attributions
- Revision and Adaptation.. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Semicolons, Colons, and Dashes. Authored by : Joe Schall. Provided by : The Pennsylvania State University. Located at : The%20Pennsylvania%20State%20University. Project : Penn State's College of Earth and Mineral Sciences' OER Initiative. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Colons. Authored by : Julie Sevastopoulos. Provided by : Grammar-Quizzes.. Located at : Grammar-Quizzes.. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Semicolons and Colons. Authored by : Excelsior Online Writing Lab. Located at : https://owl.excelsior.edu/writing-refresher/grammar-refresher/semicolons-colons/ . License : CC BY: Attribution
- Image of road sign. Authored by : Eric Deschaintre. Provided by : Unsplash. Located at : https://unsplash.com/photos/xtnUQQwwnXA . License : Other . License Terms : https://unsplash.com/license


11.10: References
American College of Obstetricians and Gynecologists. (2014). Obstetrics data definitions (Version 1.0). https://www.acog.org/practice-manage...ta-definitions
American College of Obstetricians and Gynecologists. (2017, Reaffirmed 2022). Methods for estimating the due date. ACOG Committee Opinion, 700 . https://www.acog.org/clinical/clinic...g-the-due-date
American College of Obstetricians and Gynecologists. (2018). Air travel during pregnancy . https://www.acog.org/clinical/clinic...ring-pregnancy
American College of Obstetricians and Gynecologists. (2019). Prenatal genetic testing chart . https://www.acog.org/womens-health/i...-testing-chart
American College of Obstetricians and Gynecologists. (2020a). Gestational hypertension and preeclampsia. ACOG Practice Bulletin, 222. https://www.acog.org/clinical/clinic...d-preeclampsia
American College of Obstetricians and Gynecologists. (2020b). Prevention of group B streptococcal early-onset disease in newborns. ACOG Committee Opinion, 797. https://www.acog.org/clinical/clinic...se-in-newborns
American College of Obstetricians and Gynecologists. (2020c). Gestational hypertension and preeclampsia. ACOG Practice Bulletin, 222. https://www.acog.org/clinical/clinic...d-preeclampsia
American College of Obstetricians and Gynecologists. (2021a). Updated cervical cancer screening guidelines . https://www.acog.org/clinical/clinic...ing-guidelines
American College of Obstetricians and Gynecologists. (2021b). ACOG response to consensus statement on paracetamol use during pregnancy . https://www.acog.org/news/news-artic...ring-pregnancy
American College of Obstetricians and Gynecologists. (2021c). What medicine can I take for allergies while I’m pregnant? https://www.acog.org/womens-health/e...le-im-pregnant
American College of Obstetricians and Gynecologists. (2021d). Anemia in pregnancy. ACOG Practice Bulletin, 233. https://files.medelement.com/uploads...6949ba8e0c.pdf
American College of Obstetricians and Gynecologists. (2022). Nutrition during pregnancy . https://www.acog.org/store/products/...ring-pregnancy
Brough, L., Rees, G. A., Crawford, M. A., Morton, R. H., & Dorman, E. K. (2010). Effect of multiple-micronutrient supplementation on maternal nutrient status, infant birth weight and gestational age at birth in a low-income, multi-ethnic population. British Journal of Nutrition, 104 (3), 437–445. https://doi.org/10.1017/S0007114510000747
Cano Panlagua, A., & Amariles Muñoz, P. (2017). A structured review of hepatotoxic medicines during pregnancy. Revista Colombiana de Gastroenterología, 32 (1), 38–46. https://doi.org/10.22516/25007440.128
Chen, H., Qian, N., Yan, L., & Jiang, H. (2018). Role of serum vitamin A and E in pregnancy. Experimental and Therapeutic Medicine, 16 (6), 5185–5189. https://doi.org/10.3892/etm.2018.6830
Demarco, R. F., & Healey-Walsh, J. (2020). Community and public health nursing: Evidence for practice (3rd ed.). Wolters Kluwer.
Dibley, M. J. & Jeacocke, D. A. (2001). Safety and toxicity of vitamin A supplements in pregnancy. Food and Nutrition Bulletin, 22 (3), 248–266. journals.sagepub.com/doi/pdf...82650102200304
Ecker, J., Abuhamad, A., Hill, W., Bailit, J., Bateman, B. T., Berghelia, V., Blake-Lamb, T., Guille, C., Landau, R., Minkoff, H., Pranhu, M., Rosenthal, E., Terplan, M., Wright, T. E., & Yonkers, K. A. (2019, March 27). Substance use disorders in pregnancy: Clinical, ethical, and research imperatives of the opioid epidemic: A report of a joint workshop of the Society for Maternal-Fetal Medicine, American College of Obstetricians and Gynecologists, and American Society of Addiction Medicine. https://doi.org/10.1016/j.ajog.2019.03.022
Erchick D. J., Khatry S. K., Agrawal, N. K., Katz, J., LeClerq, S. C., Rai, B., Reynolds, M. A., & Mullary. L. C. (2020). Risk of preterm birth associated with maternal gingival inflammation and oral hygiene behaviours in rural Nepal: A community-based, prospective cohort study. BMJ Open, 10, e036515. https://doi.org/10.1136/bmjopen-2019-036515
Georgieff, M. K., Krebs N. F., & Cusick S. E. (2019). The benefits and risks of iron supplementation in pregnancy and childhood. Annual Review of Nutrition, 21 (39), 121-–146. doi.org/10.1146/annurev-nutr-082018-124213
Guest, H. (2021). A concept analysis of trauma-informed care. Nursing Forum, 56 (4), 1000-1007. doi.org/10.1111/nuf.12626
Kellie, F. J. (2017). Vitamin K supplementation during pregnancy for improving outcomes. Cochrane Database of Systematic Reviews (6), CD010920. doi.org/10.1002/14651858.CD010920.pub2
King, T. L., Brucker, M. C., Osborne, K., Jevitt, C. M. (2019). Varney’s midwifery (6th ed.) Jones & Bartlett.
Moretti, R., Morelli, M. E., & Caruso, P. (2019). Vitamin D in neurological diseases: A rationale for a pathogenic impact. International Journal of Molecular Sciences, 19 (8), 2245. https://doi.org/10.3390/ijms19082245
National Institute of Drug Abuse. (2020). Substance use while pregnant and breastfeeding . https://nida.nih.gov/publications/re...-breastfeeding
Osterman, M. J. K. & Martin, J. A. (2018). Timing and adequacy of prenatal care in the United States, 2016. National Vital Statistics Reports, 67 (3). https://stacks.cdc.gov/view/cdc/55174
Rodríguez-Blanque, R., Sanchez-Garcia, J. C., Sanchez-Lopez, A. M., & Aguilar-Cordero, M. J. (2019). Physical activity during pregnancy and its influence on delivery time: A randomized clinical trial. PeerJ, 7 (e6370). http://doi.org/10.7717/peerj.6370
Royal College of Obstetricians and Gynecologists. (2015). Female genital mutilation and its management. Green Top Guideline No. 53. https://www.rcog.org.uk/media/au0jn5of/gtg-53-fgm.pdf
Sarecka-Hujar, B., & Szulc-Musioł, B. (2022). Herbal medicines—Are they effective and safe during pregnancy? Pharmaceutics , 14 (1), 171. https://doi.org/10.3390/pharmaceutics14010171
Smaill, F. M., & Vazquez, J. C. (2019). Antibiotics for asymptomatic bacteriuria in pregnancy. The Cochrane database of systematic reviews , 2019 (11), CD000490. doi.org/10.1002/14651858.CD000490.pub4
Substance Abuse and Mental Health Services Administration. (2023). Drug testing resources . https://www.samhsa.gov/workplace/dru...ting-resources
U.S. Department of Health and Human Services. (n.d.). Healthy People 2030: Pregnancy and childbirth. https://health.gov/healthypeople/obj...and-childbirth
U.S. Food & Drug Administration. (2014, updated March 5, 2021). Pregnancy and lactation labeling (drugs) final rule . https://www.fda.gov/drugs/labeling-i...ugs-final-rule
U.S. Food and Drug Administration. (2020). FDA recommends avoiding use of NSAIDs in pregnancy at 20 weeks or later because they can result in low amniotic fluid . https://www.fda.gov/media/142967/download
World Health Organization. (2023, January 31). Female genital mutilation . https://www.who.int/news-room/fact-s...tal-mutilation


Dimensional Analysis
Skills to Develop
- Use dimensional analysis to calculate solutions with the correct units
A dimension is any measurable extent, such as length, time, and mass. Units help describe the measurement according to certain standards. In the metric system for example, a one-dimensional (1-D) length is measured in meters (m) a two-dimensional (2-D) area is measured in meters squared (m 2 ), and a three-dimensional (3-D) volume is measured in meters cubed (m 3 ). Other types of quantities (time, mass, temperature) are measured using different units because they have different dimensions. Analysis means to think about something, often focusing on one part at a time. Putting it all together, dimensional analysis means thinking about units piece by piece. Dimensional analysis can by to correctly go between different types of units, to catch mistakes in one's calculations, and to make many useful calculations in real life.
Essentially, dimensional analysis means multiplying by one. You collect a set of "conversion factors" or ratios that equal one, and then multiply a quantity that you are interested in by those "ones." For example, if you want to know how many seconds it would take to get from New York to Philadelphia, you'd do it like this:
First, using the express train it takes 2.5 hours to get to Philadelphia from a station in New York. Then, we know that 1 hour = 60 minutes and 1 minute = 60 seconds, so (1h / 60 min) = 1, and (1 min / 60 s) = 1. Now, all we have to do it multiply our starting number (2.5 h) by "one" twice, making sure that the units cancel correctly so that we have only seconds at the end.
\[(2.5\; \cancel{ h}) \left(\dfrac{60\; \cancel{min}}{1\; \cancel{ h}}\right)\left(\dfrac{60 s}{1 \; \cancel{min}}\right) = 9.0 \times 10^3 s\]
If each part is not put in the right place, the units will come out wrong. For example:
\[\left(\dfrac{1}{2.5\;\cancel{ h}}\right)\left(\dfrac{1 \,\cancel{h}}{60\,m \cancel{ min}}\right)\left(\dfrac{1 \, \cancel{min}}{60 \,s}\right) = 1.1 \times 10^{-4} s^{-1}\]
In this case, we put the starting quantity on the bottom, so we got s -1 when the units are canceled out. Here is an example of not being able to cancel out the units correctly:
\[(2.5 h)\left(\dfrac{1 h}{60 min}\right)\left(\dfrac{60 s}{1 min}\right) = 2.5 s\cdot h^2\cdot min^{-2}$$
The important part is that if you check the units to make sure that they come out right, you can be pretty sure you set the calculation up right!
Here is an example of how dimensional analysis can help. A student was calculating initial velocity (v 0 ) from this equation:
\[d = (v_0)t + \dfrac{at^2}{2} \nonumber\]
But the student had derived the equation incorrectly, and used this equation instead:
\[v_0 = \dfrac{d}{t} - \dfrac{at^2}{2} \nonumber \]
So the student had the wrong answer, but didn't know that because he just put the numbers for d, t, and a into his calculator using the wrong equation. If he had checked the units, he would have seen that (d/t) has units of meters per second (m/s) while (at 2 )/2 has units of meters (m).
Dimensional analysis is often useful when you want to estimate some quantity in the real world. For instance, maybe you want to know how much money you spend on coffee each month. If you spend $5 per cup and have 2 cups per day, and there are approximately 30 days in a month, than you can set up a calculation just like those above to calculate dollars per month spent on coffee. This works for many important, less obvious situations, for instance in business, to get an approximate idea of some quantity.


11.1: The Cerebrum
The iconic gray mantle of the human brain, which appears to make up most of the mass of the brain, is the cerebrum ( Figure 11.1 ). The wrinkled portion is the cerebral cortex , and the rest of the structure is beneath that outer covering. There is a large separation between the two sides of the cerebrum called the longitudinal fissure . It separates the cerebrum into two distinct halves, a right and left cerebral hemisphere . Deep within the cerebrum, the white matter of the corpus callosum provides the major pathway for communication between the two hemispheres of the cerebral cortex.
Cerebral Cortex
The cerebrum is covered by a continuous layer of gray matter that wraps around either side of the forebrain—the cerebral cortex. This thin, extensive region of wrinkled gray matter is responsible for the higher functions of the nervous system. A gyrus (plural = gyri) is the ridge of one of those wrinkles, and a sulcus (plural = sulci) is the groove between two gyri. The pattern of these folds of tissue indicates specific regions of the cerebral cortex.
The head is limited by the size of the birth canal, and the brain must fit inside the cranial cavity of the skull. Extensive folding in the cerebral cortex enables more gray matter to fit into this limited space. If the gray matter of the cortex were peeled off of the cerebrum and laid out flat, its surface area would be roughly equal to one square meter.
The folding of the cortex maximizes the amount of gray matter in the cranial cavity. During embryonic development, as the telencephalon expands within the skull, the brain goes through a regular course of growth that results in everyone’s brain having a similar pattern of folds. The surface of the brain can be mapped on the basis of the locations of large gyri and sulci. Using these landmarks, the cortex can be separated into four major regions, or lobes ( Figure 11.2 ). The lateral sulcus that separates the temporal lobe from the other regions is one such landmark. Superior to the lateral sulcus are the parietal lobe and frontal lobe , which are separated from each other by the central sulcus . The posterior region of the cortex is the occipital lobe , which has no obvious anatomical border between it and the parietal or temporal lobes on the lateral surface of the brain. From the medial surface, an obvious landmark separating the parietal and occipital lobes is called the parieto- occipital sulcus .
The fact that there is no obvious anatomical border between these lobes is consistent with the functions of these regions being interrelated.


10.4: Delinquency
[1] The juvenile court oversees cases for youth between the ages of 7 and 17. Seven is considered the lower limit of the reaches or protections of the juvenile justice system, while 17 is the upper limit. At 18, youth are considered adults and are tried under the laws of the adult criminal justice system. However, some states have differing upper age limits. For example, in Oregon, the Oregon Youth Authority houses youth until the age of 25. Other states have similar provisions and although the lower limit is seven years of age, most states do not intervene in cases under nine.
Callstack:
at (Bookshelves/Corrections/Introduction_to_the_American_Criminal_Justice_System_(Burke_et_al.)/10:_Juvenile_Justice/10.04:_Delinquency), /content/body/p[2]/@function, line 1, column 1
Podcast: Caught
Callstack:
at (Bookshelves/Corrections/Introduction_to_the_American_Criminal_Justice_System_(Burke_et_al.)/10:_Juvenile_Justice/10.04:_Delinquency), /content/body/p[3]/@function, line 1, column 1
-
Feld, B.C. (1999).
Bad Kids: Race and the Transformation of the Juvenile Court.
New York: Oxford University Press.
↵
-
Feld, B.C. (1999).
Bad Kids: Race and the Transformation of the Juvenile Court.
New York: Oxford University Press.
↵
-
Feld, B.C. (1999).
Bad Kids: Race and the Transformation of the Juvenile Court.
New York: Oxford University Press.
↵


9: Math Models and Geometry
We are surrounded by all sorts of geometry. Architects use geometry to design buildings. Artists create vivid images out of colorful geometric shapes. Street signs, automobiles, and product packaging all take advantage of geometric properties. In this chapter, we will begin by considering a formal approach to solving problems and use it to solve a variety of common problems, including making decisions about money. Then we will explore geometry and relate it to everyday situations, using the problem-solving strategy we develop.
-
- 9.1: Use a Problem Solving Strategy (Part 1)
- In earlier chapters, you translated word phrases into algebraic expressions and word sentences into algebraic equations and solved some word problems that applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated. Now we'll develop a strategy you can use to solve any word problem.
-
- 9.2: Use a Problem Solving Strategy (Part 2)
- In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Some number word problems ask you to find two or more numbers. Be sure to read the problem carefully to discover how all the numbers relate to each other.
-
- 9.3: Solve Money Applications
- Solving coin word problems is much like solving any other word problem. However, what makes them unique is that you have to find the total value of the coins instead of just the total number of coins. For coins of the same type, the total value can be found by multiplying the number of coins by the value of an individual coin. You may find it helpful to put all the numbers into a table to make sure they check.
-
- 9.4: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 1)
- An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. If the sum of the measures of two angles is 180°, then they are supplementary angles. But if their sum is 90°, then they are complementary angles. We will adapt our Problem Solving Strategy for Geometry Applications. Since these applications will involve geometric shapes, it will help to draw a figure and label it with the information from the problem.
-
- 9.5: Use Properties of Angles, Triangles, and the Pythagorean Theorem (Part 2)
- Triangles are named by their vertices. For any triangle, the sum of the measures of the angles is 180°. Some triangles have special names such as the right triangle which has one 90° angle. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. To solve problems that use the Pythagorean Theorem, we will need to find square roots.
-
- 9.6: Use Properties of Rectangles, Triangles, and Trapezoids (Part 1)
- Many geometry applications will involve finding the perimeter or the area of a figure. The perimeter is a measure of the distance around a figure. The area is a measure of the surface covered by a figure. The volume is a measure of the amount of space occupied by a figure. A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and the adjacent side as the width, W.
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- 9.7: Use Properties of Rectangles, Triangles, and Trapezoids (Part 2)
- Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of a triangle is one-half the base times the height. An isosceles triangle is a triangle with two sides of equal length is while a triangle that has three sides of equal length is an equilateral triangle. A trapezoid is four-sided figure with two sides that are parallel, the bases, and two sides that are not. The area of a trapezoid is one-half the height times the sum of the bases.
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- 9.8: Solve Geometry Applications- Circles and Irregular Figures
- In this section, we will work on geometry applications for circles and irregular figures. To solve applications with circles, we use the properties of circles from Decimals and Fractions. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. To find the area of one of these irregular figures, we split it into figures whose formulas we know and then add the areas of the figures.
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- 9.9: Solve Geometry Applications- Volume and Surface Area (Part 1)
- The surface area is a square measure of the total area of all the sides of a rectangular solid. The amount of space inside the rectangular solid is the volume, a cubic measure. The volume, V, of any rectangular solid is the product of the length, width, and height. To find the surface area of a rectangular solid, find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.
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- 9.10: Solve Geometry Applications- Volume and Surface Area (Part 2)
- A sphere is the shape of a basketball, like a three-dimensional circle. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height h of a cylinder is the distance between the two bases. In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.
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- 9.11: Solve a Formula for a Specific Variable
- For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula d = rt where d = distance, r = rate, and t = time. To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all the other variables and constants on the other side. The result is another formula, made up only of variables.
Figure 9.1 - Note the many individual shapes in this building. (credit: Bert Kaufmann, Flickr)
Contributors and Attributions
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Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


9: Viruses
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- 9.7: Viral Diversity
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- 9.7A: Overview of Bacterial Viruses
- 9.7B: RNA Bacteriophages
- 9.7C: Single-Stranded DNA Bacteriophages
- 9.7D: Double-Stranded DNA Bacteriophages
- 9.7E: Mu- A Double-Stranded Transposable DNA Bacteriophage
- 9.7F: Virulent Bacteriophages and T4
- 9.7G: Temperate Bacteriophages - Lambda and P1
- 9.7H: Viruses of Archaea
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- 9.11: DNA Viruses in Eukaryotes
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- 9.11A: Plant DNA Viruses
- 9.11B: Replication of Double-Stranded DNA Viruses of Animals
- 9.11C: Double-Stranded DNA Viruses - Herpesviruses
- 9.11E: Attachment and Entry of Herpes Simplex
- 9.11E: Replication of Herpes Simplex Virus
- 9.11F: Immunodeficiency
- 9.11G: Double-Stranded DNA Viruses - Pox Viruses
- 9.11H: Double-Stranded DNA Viruses- Adenoviruses
- 9.11I: Retroviruses and Hepadnavirus
- 9.11J: Treatment of Animal Viral Infections
Thumbnail: This colorized transmission electron microscopic (TEM) image revealed some of the ultrastructural morphology displayed by an Ebola virus virion. (Public Domain; Frederick A. Murphy via CDC).


1.4: Preventing Injuries and Violence
By the end of this chapter you will be able to:
- Recognize the common types of unintentional and intentional injuries
- Identify ways to reduce chances of unintentional and intentional injuries
- Explain how risk from injury changes throughout life stages.
What steps can you take to reduce the chances of accidental injuries?
How can you reduce chances of injuries from violence?
There are many choices you make in life that impact your health and well-being and some of those choices, like smoking cigarettes, have a direct connection to a negative health implication, like lung cancer. Although the choices you make may or may not directly impact the possibility of suffering intentional and unintentional injuries, understanding the most common reasons for injury can help you increase your safety.