11.4: Apportionment Methods
- Describe and interpret the apportionment problem.
- Apply Hamilton’s Method.
- Describe and interpret the quota rule.
- Apply Jefferson’s Method.
- Apply Adams’s Method.
- Apply Webster’s Method.
- Compare and contrast apportionment methods.
- Identify and contrast flaws in various apportionment methods.
A Closer Look at the Apportionment Problem
In Standard Divisors, Standard Quotas, and the Apportionment Problem we calculated the standard divisor and the standard quotas in various apportionment scenarios. The results of those calculations routinely led to fractions and decimals of units. However, the seats in the House of Representatives, laptops in a classroom, or a variety of other resources, are indivisible, meaning they cannot be divided up into fractional parts. This leaves a decision to be made. For example, if the standard quota for the number of laptops to be distributed to a classroom is 12.44 units, how do we deal with the fractional part of 0.44? It is unclear if the classroom should receive 12 units, 13 units, or some other value. Let’s try traditional rounding to the nearest whole number value.
The board of trustees of a college has recently approved the installation of 70 new emergency blue lights in three parking lots. The number of lights in each lot will be proportionate to the size of the parking lot, which isto be measured in acres. The total number of acres is 34; so the standard divisor is . The standard quota for each lot is listed in the table below. Use this information to answer each question.
| Lot | Acres | Lot’s Standard Quota |
|---|---|---|
| A | 15 | emergency blue lights |
| B | 9 | emergency blue lights |
| C | 10 | emergency blue lights |
- Use traditional rounding to determine the number of lights assigned to each lot.
- Find the sum of the values from part 1.
- Does the sum found in part 2 equal the number of lights available?
- Answer
-
- If traditional rounding is used, there will be 31, 19, and 21 lights distributed to each lot, respectively.
- The total of these values is 71.
- No, the total from part 2 is one more than the number of lights available. In other words, one of the parking lots must get 1 fewer light than apportioned.
The science department of a high school has received a grant for 34 laptops. They plan to apportion them among their six classrooms based on each classroom’s student capacity. Use the standard quotas in the table below to answer each question.
| Room | Room Capacity | Room’s Standard Quota |
|---|---|---|
| A | 30 | \(30 \div 4.88 \approx 6.15\) laptops |
| B | 25 | \(25 \div 4.88 \approx 5.12\) laptops |
| C | 28 | \(28 \div 4.88 \approx 5.74\) laptops |
| D | 32 | \(32 \div 4.88 \approx 6.56\) laptops |
| E | 24 | \(24 \div 4.88 \approx 4.92\) laptops |
| F | 27 | \(27 \div 4.88 \approx 5.53\) laptops |
Use traditional rounding to determine the number of laptops assigned to each classroom.
Find the sum of the values from part 1.
Does the sum found in part 2 equal the number of laptops available?
Example
demonstrates that we cannot successfully apportion indivisible resources by rounding off each standard quota using traditional rounding. This leaves us with a problem. What is a fair way to distribute the fractional parts of the standard quotas? We will refer to this as the apportionment problem . Several methods for making this decision will be discussed.In 2015, the U.S. Air Force had a fleet of approximately 281 A-10C Thunderbolt II aircraft. Suppose that the Air Force administration wanted to distribute 27 aircrafts across six bases based on the number of qualified pilots stationed at those bases. Use the information in the table below to answer each question.
The reading coach at an elementary school has 13 gift cards to distribute to their three students as a reward for time spent reading. When they calculated the standard quota for each student based on the number of minutes they student had read, the results were: 4.49 gift cards, 4.03 gift cards, and 4.48 gift cards. How does this demonstrate the apportionment problem?
Hamilton's Method of Apportionment
One of the problems encountered when standard quotas are transformed into whole numbers using traditional rounding is that it is possible for the sum of the values to be greater than the number of seats available. A reasonable way to avoid this is to always round down, even when the first decimal place is five or greater. For example, a standard quota of 12.33 and a standard quota of 12.99 would both round down to 12. This is called the lower quota .
The Air Force administration wants to distribute 27 aircrafts across six bases based on the number of qualified pilots stationed at those bases. The standard quotas for each base are listed in the table below. Use this information to answer the questions.
| Base | Standard Quota |
|---|---|
| (A) Alpha | aircraft |
| (B) Bravo | aircraft |
| (C) Charlie | aircraft |
| (D) Delta | aircraft |
| (E) Echo | aircraft |
| (F) Foxtrot | aircraft |
- Give the lower quota for each Air Force base.
- Find the sum of the lower quotas. By how much does this sum fall short of the actual number of aircraft?
- Answer
-
- Round down. The lower quota for each Air Force base is 5, 5, 2, 6, 3, 3, respectively.
- The sum is 24. This is 3 fewer than the actual number of aircraft.
The apportionment of 70 new emergency blue lights in three parking lots is based on acreage. The standard quota for each lot is listed in the table below. Use this information to answer each question.
| Lot | Acres | Lot’s Standard Quota |
|---|---|---|
| A | 15 | \(15 \div 0.4857 \approx 30.88\) emergency blue lights |
| B | 9 | \(9 \div 0.4857 \approx 18.53\) emergency blue lights |
| C | 10 | \(10 \div 0.4857 = 20.59\) emergency blue lights |
Give the lower quota for each parking lot.
Find the sum of the lower quotas.
By how much does this sum fall short of the actual number of emergency lights?
If the standard quotas are all rounded down, their sum will always be less than or equal to the house size. Then, it would only remain to find a fair way to distribute any remaining seats. Alexander Hamilton, who was a general in the American Revolution, author of the Federalist Papers, and the first secretary of the treasury, took this approach to apportionment.
Steps for Hamilton’s Method of Apportionment
There are five steps we follow when applying Hamilton’s Method of apportionment:
- Find the standard divisor.
- Find each state’s standard quota.
- Give each state the state’s lower quota (with each state receiving at least 1 seat).
- Give each remaining seat one at a time to the states with the largest fractional parts of their standard quotas until no seats remain.
- Check the solution by confirming that the sum of the modified quotas equals the house size.
Hamilton Method of Apportionment
Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of the five counties based on the residents under 19 years old, This data is provided in the table below. Using the Hamilton method, calculate how many schools would be funded in each state.
| Hawaii | Honolulu | Kalawao | Kauai | Maui | Total | |
|---|---|---|---|---|---|---|
| Residents under age 19 | 46,310 | 224,230 | 20 | 16,560 | 38,450 | 325,570 |
- Answer
-
Step 1: Calculate the standard divisor. Divide the total population, 325,570, by the house size, 616 seats. The standard divisor is 528.52.
Step 2: Find each state’s standard quota:
Hawaii Honolulu Kalawao Kauai Maui Total Standard Quota 616 Step 3: Find each state’s lower quota and their sum:
Hawaii Honolulu Kalawao Kauai Maui Total Lower Quota 87 424 1 31 72 615 Step 4: Compare the sum of the states’ lower quotas, 615, to the house size, 616. One seat remains to be apportioned and must be given to the state with the largest fractional part: Maui with 0.75. So, the final Hamilton quotas are as follows: Hawaii 87, Honolulu 424, Kalawao 1, Kauai 31, and Maui 73.
Step 5: Find the total to confirm the sum of the quotas equals the house size, 616. Then . The apportionment is complete.
In the country of Imaginaria, there will be four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Suppose there will be 35 seats in the legislature of Imaginaria. Use Hamilton’s method of apportionment to determine the number of seats in each state based on the populations in the following table.
| Fictionville | Pretendstead | Illusionham | Mythbury | Total | |
|---|---|---|---|---|---|
| Population | 71,000 | 117,000 | 211,000 | 1,194,000 | 1,593,000 |
Check out websites such as Ms. Hearn Math for a free Hamilton apportionment calculator.
This can be a useful tool to confirm your results!
The Quota Rule
A characteristic of an apportionment that is considered favorable is when the final quota values all either result from rounding down or rounding up from the standard quotas. The value that results from rounding down is called the lower quota, and the value that results from rounding up is called the upper quota .
As we explore more methods of apportionment, we will consider whether they satisfy the quota rule. If a scenario exists in which a particular apportionment allocates a value greater than the upper quota or less than the lower quota, then that apportionment violates the quota rule and the apportionment method that was used violates the quota rule.
Several apportionment methods have been used to allocate 125 seats to ten states and the results are shown in the table below. Determine which apportionments do not satisfy the quota rule and justify your answer.
| State A | State B | State C | State D | State E | State F | State G | |
|---|---|---|---|---|---|---|---|
| Standard Quota | 41.26 | 16.00 | 5.77 | 2.64 | 7.82 | 10.47 | 0.21 |
| Lower Quota | 41 | 16 | 5 | 2 | 7 | 10 | 0 |
| Upper Quota | 42 | 17 | 6 | 3 | 8 | 11 | 1 |
| Method X | 43 | 16 | 5 | 2 | 7 | 10 | 1 |
| Method Y | 41 | 16 | 6 | 2 | 8 | 10 | 1 |
| Method Z | 42 | 16 | 7 | 3 | 7 | 9 | 1 |
- Answer
-
Look for states such that the number of seats allocated differs from the lower or upper quota. Method X violates the quota rule because State A receives 43 seats instead of 41 or 42. Method Z violates the quota rule because State C receives 7 seats instead of 5 or 6 and State F receives 9 instead of 10 or 11.
Apportionment Method V has been used to allocate 125 seats to ten states as shown in the table below. Determine whether the apportionment satisfies the quota rule and justify your answer.
| State A | State B | State C | State D | State E | State F | State G | |
|---|---|---|---|---|---|---|---|
| Standard Quota | 41.26 | 10.70 | 16.00 | 13.11 | 17.00 | 5.77 | 2.64 |
| Lower Quota | 41 | 10 | 16 | 13 | 17 | 5 | 2 |
| Upper Quota | 42 | 11 | 17 | 14 | 18 | 6 | 3 |
| Method V | 42 | 11 | 17 | 13 | 18 | 4 | 2 |
It is possible for an apportionment method to satisfy the quota rule in some scenarios but violate it in others. However, because the Hamilton method always begins with the lower quota and either adds one to it or keeps it the same, the final Hamilton quota will always consist of values that are either lower quota values or upper quota values. When an apportionment method has this characteristic, it is said to satisfy the quota rule. So, we can say:
The Hamilton method of apportionment satisfies the quota rule.
Although the Hamilton method of apportionment satisfies the quota rule, it can result in some unexpected outcomes, which has caused it to pass in and out of favor of the U.S. government over the years. There are several apportionment methods that have been popular alternatives, such as Jefferson’s method of apportionment that the founders of Imaginaria should consider.
Jefferson’s Method of Apportionment
Another approach to dealing with the fractional parts of the standard quotas is to modify the standard divisor so that the total of the resulting modified lower quotas is the necessary number of seats. This is the approach used by Jefferson.
In Jefferson’s method, the change to the standard divisor is made so that the total of the modified lower quotas equals the house size. The change in the standard divisor to get the modified divisor is relatively small. There is not a formula for this. The modified divisor is found by “guess and check.” It is important to remember that increasing the divisor decreases the quotas, but decreasing the divisor increases the quotas. So, if you need a larger quota, try reducing the divisor, and if you need a smaller quota, try increasing the divisor.
Suppose the population of a state is 50 and the standard divisor is 12.5.
- Find the state’s standard quota.
- Increase the standard divisor by 2 units and use the modified divisor to determine the modified quota for the state.
- Decrease the modified divisor from part 2 by 1.5 units and use the new modified divisor to determine the modified quota for the state.
- Choose any value of divisor between the value of the modified divisor from part 2 and the value of the modified divisor from part 3 and use it to determine the modified quota for the state.
- Which modified quota was the largest, the modified quota from part 2, from part 3, or from part 4? Explain why.
- Answer
-
1. The state's standard quota is \(\frac{50}{12.5}=4\).
2. The modified divisor is 14.5 . The modified quota is \(\frac{50}{14.5} \approx 3.45\).
3. The modified divisor is 13 . The modified quota is \(\frac{50}{13} \approx 3.85\).
4. One value between 13 and 14.5 is 13.5 . With a modified divisor of 13.5 , the modified quota is \(\frac{50}{13.5} \approx 3.70\).
5. The modified quota from part 3 was the largest because the divisor was the smallest of the three. Dividing the same number by a smaller value gives a larger result.
Suppose the population of a state is 12 and the standard divisor is 0.225.
- Find the state’s standard quota.
- Decrease the standard divisor by 0.200 units and use the modified divisor to determine the modified quota for the state.
- Increase the modified divisor from part 2 by 0.100 units and use the new modified divisor to determine the modified quota for the state.
- Choose any value of divisor between the value of the modified divisor from part 2 and the value of the modified divisor from part 3 and use it to determine the modified quota for the state.
- Which modified quota was the smallest, the modified quota from part 2, from part 3, or from part 4? Explain why.
When you use Jefferson’s method, you might have to adjust the divisor several times find modified lower quotas that sum to the house size. First, guess what the divisor should be based on the sum of the lower quotas and then increase or decrease it from there based on whether the sum needs to be smaller or larger respectively. If the result still does not produce lower quotas that sum to the house size, adjust again. Keep a record of the values that didn't work to help you narrow your search.
Steps for Jefferson’s Method of Apportionment
We take four steps to apply Jefferson’s Method of apportionment:
- Step 1: Find the standard divisor.
- Step 2: Find each state’s quota. This will be the standard quota the first time Step 2 is completed and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas.
- Step 3: Find the states’ lower quotas (with each state receiving at least one seat), and their sum.
- Step 4: If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the lower quotas is less than the number of seats, reduce the standard divisor. If the sum of the lower quotas is greater than the number of seats, increase the standard divisor. Return to Step 2 using the modified divisor.
Suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of five counties based on the residents under the age of 19. With the data in the table below, apply Jefferson’s method to apportion the schools to the counties.
| Hawaii | Honolulu | Kalawao | Kauai | Maui | Total | |
|---|---|---|---|---|---|---|
| Residents under Age 19 | 46,310 | 224,230 | 20 | 16,560 | 38,450 | 325,570 |
- Answer
-
Step 1: The process for finding the standard divisor, standard quotas, and lower quotas is the same in the Hamilton and Jefferson methods of apportionment. We walked through the Hamilton Method in Example 11.34, and following these steps resulted in lower quotas as shown in the table below.
Hawaii Honolulu Kalawao Kauai Maui Total Standard Quota 616 Lower Quota 87 424 1 31 72 615 Step 2: Compare the sum of the states’ lower quotas, 615, to the house size, 616. Since 615 is less than 616, use a modified divisor that is less than the standard divisor of 528.52. Try 526.00.
Step 3: Find each state’s modified quota, lower quota, and the sum of the lower quotas based on the modified divisor of 526:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Lower Quota 88 426 1 31 72 618 Step 4: The new sum of the lower quotas is 2 units greater than 616. We have overshot the goal. So, increase the divisor to a value between 526.00 and 528.52. Try 527.00.
Step 5: Repeat the process of finding the quotas. Find each state’s modified quota, lower quota, and the sum of the lower quotas based on the modified divisor of 526.00:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Lower Quota 87 425 1 31 72 616 Step 6: The new sum of the lower quotas equals the house size. The apportionment is complete.
The apportionment is: Hawaii County 87, Honolulu County 425, Kalawao County 1, Kauai 31, and Maui 72 schools.
When using Jefferson’s method, the modified divisors you use may be different from what another person chooses, but final apportionment values will be the same.
Let’s return to the Imaginarian states of Fictionville, Pretendstead, Illusionham, and Mythbury. Suppose that there are going to be 35 seats in the legislature. This time use Jefferson’s method of apportionment to determine the number of seats in each state based on the populations in the table below. How many seats would each state receive?
| Fictionville | Pretendstead | Illusionham | Mythbury | Total | |
|---|---|---|---|---|---|
| Population | 71,000 | 117,000 | 211,000 | 1,194,000 | 1,593,000 |
Notice that, in this apportionment, Mythbury received more than the upper quota. Since this apportionment of representatives to Imaginarian states by Jefferson’s method does not satisfy the quota rule, we say that:
Jefferson’s method violates the quota rule.
We have discussed two apportionment methods: one that satisfies the quota rule and one that does not. Before you decide which method to use in Imaginaria, there are a couple more options to consider.
Jefferson Apportionment Method
It is possible to create Excel spreadsheets that complete the calculations necessary to complete a Jefferson Apportionment. In some cases, this work has already been done and posted online. Check out websites such Ms. Hearn Math for a free Jefferson apportionment calculator.
This can be a useful tool to confirm your results!
Adams’s Method of Apportionment
Adams’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did, Adams used the upper quotas.
To apply Adams’s Method of apportionment, there are four steps we follow:
- Find the standard divisor.
- Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas.
- Find the states’ upper quotas and their sum.
- If the sum from Step 3 equals the number of seats, the apportionment is complete. If the sum of the upper quotas is less than the number of seats, reduce the standard divisor. If the sum of the upper quotas is greater than the number of seats, increase the standard divisor. Return to Step 2 using the modified divisor.
As in earlier examples, suppose that the Hawaii State Department of Education has a budget for 616 schools and is doing a research study to determine the equitable number of schools to have in each of the five counties based on the residents under the age of 19. Use the data in the following table and the Adams method to apportion the schools to the counties.
| Hawaii | Honolulu | Kalawao | Kauai | Maui | Total | |
|---|---|---|---|---|---|---|
| Residents under Age 19 | 46,310 | 224,230 | 20 | 16,560 | 38,450 | 325,570 |
- Answer
-
Step 1: The steps of finding the standard divisor and each state’s quota are the same in the Jefferson and Adams methods. As in Example 11.37, the standard divisor is 528.52.
Step 2: Find each state’s upper quota and their sum:
Hawaii Honolulu Kalawao Kauai Maui Total Standard Quota 616 Upper Quota 88 425 1 32 73 619 Step 3: Compare the sum of the states’ upper quotas, 619, to the house size, 616. Since 619 is greater than 616, we need to reduce the size of the quotas. Use a modified divisor that is greater than the standard divisor of 528.52. Try 534.00.
Step 4: Find each state’s modified quota, upper quota, and the sum of the upper quotas based on the modified divisor of 534:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 88 420 1 32 72 613 Step 5: The new sum of the upper quotas is 3 units less than 616. Larger quotas are needed. So, decrease the divisor to a value between 534.00 and 528.52. Try 532.00.
Step 6: Find each state’s modified quota, upper quota, and the sum of the upper quotas based on the modified divisor of 532.00:
County Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 88 422 1 32 73 616 Step 7: The new sum of the upper quotas equals the house size. The apportionment is complete.
The apportionment is Hawaii County 88, Honolulu County 422, Kalawao County 1, Kauai 32, and Maui 73 schools.
When using Adams’s method, just as with Jefferson’s method, the modified divisors you use may be different from what another person chooses, but final apportionment values will be the same.
There are four states in Imaginaria: Fictionville, Pretendstead, Illusionham, and Mythbury. Assume there will be 35 seats in the legislature of Imaginaria. Use Adams's method of apportionment to determine the number of seats in each state based on the populations in the table below. How many seats would each state receive?
| Fictionville | Pretendstead | Illusionham | Mythbury | Total | |
|---|---|---|---|---|---|
| Population | 71,000 | 117,000 | 211,000 | 1,194,000 | 1,593,000 |
In this apportionment, Mythbury received less than the state’s lower quota. So, this apportionment is an example of a scenario in which the Adams’s method violates the quota rule.
Adams’s method of apportionment violates the quota rule.
So far, only Hamilton’s method satisfies the quota rule, but there is one more apportionment method you should consider for Imaginaria.
Adams Method Apportionment Calculator
Check out websites such as Ms. Hearn Math for a free Adams Method apportionment calculator.
This can be a useful tool to confirm your results!
Webster’s Method of Apportionment
Webster’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did or the sum of the upper quotas as Adams did, Webster used traditional rounding.
To apply Webster’s method of apportionment, there are four steps we take:
- Find the standard divisor.
- Find each state’s quota. This will be the standard quota the first time Step 2 is completed, and the standard divisor is used, but Step 2 may be repeated as needed using a modified divisor and resulting in modified quotas.
- Round each state’s quota to the nearest whole number and find the sum of these values.
- If the sum of the rounded quotas equals the number of seats, the apportionment is complete. If the sum of the rounded quotas is less than the number of seats, reduce the divisor. If the sum of the rounded quotas is greater than the number of seats, increase the divisor. Return to Step 2 using the modified divisor.
When using Webster’s method, just as with Jefferson’s method, the modified divisors you use may be different from what another person chooses, but final apportionment values will be the same.
Use the data in the table below to apportion 616 schools to Hawaiian counties. This time, use Webster’s method.
| Hawaii | Honolulu | Kalawao | Kauai | Maui | Total | |
|---|---|---|---|---|---|---|
| Residents under Age 19 | 46,310 | 224,230 | 20 | 16,560 | 38,450 | 325,570 |
- Answer
-
To apply Webster’s method of apportionment, there are four steps we take:
Step 1: The processes of finding the standard divisor and standard quota are the same in the Jefferson, Adams, and Webster’s methods. As in the previous examples, the standard divisor is 528.52.
Step 2: Find each state’s rounded quota and their sum:
Hawaii Honolulu Kalawao Kauai Maui Total Standard Quota 616 Rounded Quota 88 424 1 31 73 617 Step 3: Compare the sum of the states’ rounded quotas, 617, to the house size, 616. Since 617 is greater than 616, we need to reduce the size of the quotas. Use a modified divisor that is greater than the standard divisor of 528.52. Try 534.00.
Step 4: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified divisor of 534:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 87 420 1 31 72 612 Step 5: The new sum of the rounded quotas is 4 units less than 616. Larger quotas are needed. So, decrease the divisor to a value between 534.00 and 528.52. Try 530.00.
Step 6: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified divisor of 530.00:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 87 423 1 31 73 615 Step 7: The new sum of the rounded quotas is 1 unit less than 616. Larger quotas are needed. So, decrease the divisor to a value between 528.52 and 530.00. Try 529.50.
Step 8: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified divisor of 529.50:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 87 423 1 31 73 615 Step 9: The new sum is still only 1 unit less than 616. Larger quotas are needed, but not much larger. So, decrease the divisor to a value between 528.52 and 529.50. Try 529.30.
Step 10: Find each state’s modified quota, rounded quota, and the sum of the rounded quotas based on the modified divisor of 529.30:
Hawaii Honolulu Kalawao Kauai Maui Total Modified Quota 616 Upper Quota 87 424 1 31 73 616 Step 11: The new sum of the rounded quotas equals the house size. The apportionment is complete.
The apportionment is Hawaii County 87, Honolulu County 424, Kalawao County 1, Kauai 31, and Maui 73 schools.
If you use Webster’s method to apportion 35 legislative seats to the 4 states of Imaginaria, Fictionville, Pretendstead, Illusionham, and Mythbury, with the populations given in the table below, what is the resulting apportionment?
| Fictionville | Pretendstead | Illusionham | Mythbury | Total | |
|---|---|---|---|---|---|
| Population | 71,000 | 117,000 | 211,000 | 1,194,000 | 1,593,000 |
So far, we know that the Hamilton method satisfies the quota rule, while the Jefferson and Adams methods do not. The apportionments in the Example and Your Turn above are both scenarios in which the Webster method satisfies the quota rule. Does it always? We have a little more work to do to find out. However, one thing is clear. Not all apportionment methods have the same results. Before you make such an important decision for Imaginaria, it’s important to think about the differences in the apportionments that result from these four methods. How will the differences affect the citizens of Imaginaria?
Check out websites such as Ms. Hearn Math for a free Webster Method apportionment calculator.
This can be a useful tool to confirm your results!
Comparing Apportionment Methods
Recall that the four apportionment methods discussed in this chapter differ in two main ways:
- Whether or not a modified divisor is used
- The type of rounding of the quotas that is used
How might these differences affect Imaginarians? In the next two examples, we will compare the results when different apportionment methods are applied to the same scenario.
Let’s use the results from Example
, Example , Example , and Example to compare the four apportionment methods we have discussed. The following table summarizes the results of the results of the Hamilton, Jefferson, Adams and Webster methods when applied to the apportionment of 616 schools to Hawaiian counties.| Hawaii | Honolulu | Kalawao | Kauai | Maui | |
|---|---|---|---|---|---|
| Under 19 years old | 46,310 | 224,230 | 20 | 16,560 | 38,450 |
| Hamilton | 87 | 424 | 1 | 31 | 73 |
| Jefferson | 87 | 425 | 1 | 31 | 72 |
| Adams | 88 | 422 | 1 | 32 | 73 |
| Webster | 87 | 424 | 1 | 31 | 73 |
- Do any of the apportionment methods result in the same apportionment? If so, which ones?
- Which apportionment method would the citizens of the largest county likely favor most and least? Justify your answer.
- As a group, which apportionment method would the citizens of the other four counties likely favor most and least? Justify your answer.
- Answer
-
- Yes, the Hamilton and Webster methods result in the same apportionment.
- The largest county is Honolulu. The citizens would likely favor the Jefferson method of apportionment most since they received the most seats by that method. They would likely favor the Adams method of apportionment least because they received the least number of seats by that method.
- As a group, the other four counties received 192 seats by either the Hamilton or Webster method, 194 seats by the Adams method, and 191 seats by the Jefferson method. They would likely favor the Adams method the most and favor the Jefferson methods the least.
In Your Turn
, Example , Example , and Example , you apportioned 35 legislative seats among the four states of Imaginaria using the Hamilton, Jefferson, Adams, and Webster methods of apportionment. To understand how the differences in the apportionments might affect Imaginarians, answer these questions.Which apportionment method would the citizens of the largest state likely favor most and least? Justify your answer.
As a group, which apportionment method would the citizens of the other three states likely favor most and least? Justify your answer.
The Adams method favored the smaller states and the Jefferson method favored the larger states in the previous example, but is this the case in general?
Since the Jefferson method begins with the lower quotas, any adjustment to the quotas will be an increase. As you have seen, this is accomplished by using a modified divisor that is smaller than the standard divisor. The next example compares the impact of a decreasing divisor on the modified quotas of large states to the impact of the same size decrease on small states.
The following table displays the effect of reducing the size of the divisor. Observe the effect this has on the modified quotas of smaller states versus larger states and use the table answer each question.
| Modified Quotas | ||||
|---|---|---|---|---|
| State | Population | Divisor: 10,500 | Divisor: 10,000 | Divisor: 9,500 |
| A | 10,000 | 0.95 | 1 | 1.05 |
| B | 100,000 | 9.52 | 10 | 10.53 |
| C | 1,000,000 | 95.24 | 100 | 105.26 |
- When the divisor decreases from 10,500 to 10,000, how many representatives are gained by each state based on the lower quota?
- When the divisor decreases from 10,000 to 9,500, how many representatives are gained by each state based on the lower quota?
- Which state gains the most representatives each time the divisor is decreased?
- Answer
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- Since a state must have at least one seat, State A begins with 1 seat and still has one seat. State B begins with 9 seats and increases to 10 seats. State C begins with 95 seats and increases to 100 seats. So, State A gains 0, B gains 1, and C gains 5 seats.
- State A begins with 1 and still has 1. State B begins with 10 and still has 10. State C begins with 100 and increases to 105. So, State A gains 0, State B gains 0, and State C gains 5.
- State C, the largest state, gains the most representatives each time the divisor is decreased.
This example demonstrates that the Jefferson method is biased toward states with larger populations because the modified divisor is smaller than the standard divisor. On the other hand, the Adams’s method, which begins with the upper quotas, must increase the standard divisor in order to reduce the quotas. Once again, the effect on the number of seats is greater for the larger states, but this time they are decreased. This means that the Adams’s method favors states with smaller populations.
The following table displays the effect of increasing the size of the divisor. Observe the effect this has on the modified quotas of smaller states versus larger states and use the table to answer each question.
| Modified Quotas | ||||
|---|---|---|---|---|
| State | Population | Divisor: 11,500 | Divisor: 12,000 | Divisor: 12,500 |
| A | 10,000 | 0.87 | 0.83 | 0.8 |
| B | 100,000 | 8.7 | 8.33 | 8 |
| C | 1,000,000 | 86.96 | 83.33 | 80 |
When the divisor increases from 11,500 to 12,000, how many representatives are lost by each state based on the upper quota?
When the divisor increases from 12,000 to 12,500, how many representatives are lost by each state based on the upper quota?
Which state loses the most representatives each time the divisor is increased?
Flaws in Apportionment Methods
As we have seen, different apportionment methods can have the same results in some scenarios but different results in others. Citizens of states which receive fewer seats with a particular apportionment method will view the apportionment method as flawed and argue in favor of a different method. This inevitably creates debates regarding the use of one method over another. Methods that favor larger states are likely to be challenged by smaller states, methods that favor smaller states are likely to be challenged by larger states, and methods that violate the quota rule are likely to be challenged by states of any size depending on the circumstances.
Suppose that the State of Hawaii House of Representatives had 51 representatives, each with their own district. Imagine that redistricting were underway, and the representative districts were to be apportioned to each of five counties based on population. The following table shows the apportionment that would result from the use of the Jefferson, Adams, and Webster methods of apportionment.
| Hawaii | Honolulu | Kalawao | Kauai | Maui | |
|---|---|---|---|---|---|
| Population | 201,500 | 974,600 | 100 | 72,300 | 167,400 |
| Lower Quota | 7 | 35 | 0 | 2 | 6 |
| Upper Quota | 8 | 36 | 1 | 3 | 7 |
| Jefferson | 7 | 35 | 1 | 2 | 6 |
| Adams | 7 | 34 | 1 | 3 | 6 |
| Webster | 7 | 34 | 1 | 3 | 6 |
From the table, you can see that Hawaii, Kalawao, and Maui receive the same number of seats regardless of the method used. However, citizens of Honolulu would likely reject the Adams and Webster methods arguing that they violate the quota rule. Similarly, citizens of Kauai would probably reject the Jefferson method based on the argument that it unfairly favors the larger states. This scenario demonstrates that the Adams and Webster methods violate the quota rule, but the Jefferson method also violates the quota rule at times. The Hamilton method is the only method that satisfies the quota rule in all scenarios. It also consistently favors neither larger nor smaller states. Unfortunately, it can have some strange and results in certain circumstances, which you will see in the next section.
In addition to your choice of voting method and your choice of apportionment method, there is another important decision to make which could potentially have a huge impact on the fairness of elections in Imaginaria—the creation of electoral districts. In example above, we imagined that there were 51 state legislators in Hawaii, each representing their own district. But how did the legislators decide on the boundaries for these districts? Typically, boundaries are drawn so that each district has approximately the same number of residents, but the percentage of residents in each district with a particular political affiliation can swing the power from one group to another. When the districts are drawn to impact the power of a political party, ethnic or racial group, or other group, this is called gerrymandering. For example, districts can be drawn so that a particular group is spread thinly across districts, increasing the likelihood that they will not have strong representation.
"The Gerry-mander" first appeared in this cartoon-map in the Boston Gazette, March 26, 1812, and was soon reproduced in several other Massachusetts newspapers in response to election district changes initiated by governor Eldridge Gerry. Note that while the practice is named after him, Gerry was not the first to employ it.
Jonathan Mattingly is a mathematician who was featured in a Nature article titled “The Mathematicians Who Want to Save Democracy” . Mattingly is a mathematician at Duke University in North Carolina and he runs election simulations based on alternate versions of electoral districts in order to analyze the effects of gerrymandering. He has even been asked to testify as an expert witness in court. Mattingly and other mathematicians who are working on the problem will potentially have an impact on the redistricting that will occur as a result of the 2020 census. (Carrie Arnold, “The Mathematicians Who Want to Save Democracy,” Nature 546 , 200–202, 2017.)
Check Your Understanding
- Which of the four apportionment methods discussed in this section does not use a modified divisor?
- Which of the four apportionment methods discussed in this section satisfies the quota rule?
- Which of the four apportionment methods discussed in this section is biased toward states with larger populations?
- Which of the four apportionment methods discussed in this section is biased toward states with smaller populations?
- Which of the four apportionment methods discussed in this section begin the apportionment with a state’s upper quota and adjust down?
- Which of the four apportionment methods discussed in this section begin the apportionment with a state’s lower quota and adjust up?
- Which of the four apportionment methods discussed in this section use traditional rounding?
- Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or smaller states more?


3: Acid-Base Equilibria
This chapter will illustrate the chemistry of acid-base reactions and equilibria, and provide you with tools for quantifying the concentrations of acids and bases in solutions.
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- 3.1: Brønsted-Lowry Acids and Bases
- Compounds that donate a proton (a hydrogen ion) to another compound is called a Brønsted-Lowry acid. The compound that accepts the proton is called a Brønsted-Lowry base. The species remaining after a Brønsted-Lowry acid has lost a proton is the conjugate base of the acid. The species formed when a Brønsted-Lowry base gains a proton is the conjugate acid of the base. Amphiprotic species can act as both proton donors and proton acceptors. Water is the most important amphiprotic species.
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- 3.2: pH and pOH
- The concentration of hydronium ion in a solution of an acid in water is greater than \( 1.0 \times 10^{-7}\; M\) at 25 °C. The concentration of hydroxide ion in a solution of a base in water is greater than \( 1.0 \times 10^{-7}\; M\) at 25 °C. The concentration of H3O+ in a solution can be expressed as the pH of the solution; \(\ce{pH} = -\log \ce{H3O+}\). The concentration of OH− can be expressed as the pOH of the solution: \(\ce{pOH} = -\log[\ce{OH-}]\). In pure water, pH = 7 and pOH = 7.
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- 3.3: Relative Strengths of Acids and Bases
- The strengths of Brønsted-Lowry acids and bases in aqueous solutions can be determined by their acid or base ionization constants. Stronger acids form weaker conjugate bases, and weaker acids form stronger conjugate bases. Thus strong acids are completely ionized in aqueous solution because their conjugate bases are weaker bases than water. Weak acids are only partially ionized because their conjugate bases are compete successfully with water for possession of protons.
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- 3.4: Hydrolysis of Salt Solutions
- The characteristic properties of aqueous solutions of Brønsted-Lowry acids are due to the presence of hydronium ions; those of aqueous solutions of Brønsted-Lowry bases are due to the presence of hydroxide ions. The neutralization that occurs when aqueous solutions of acids and bases are combined results from the reaction of the hydronium and hydroxide ions to form water. Some salts formed in neutralization reactions may make the product solutions slightly acidic or slightly basic.
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- 3.5: Polyprotic Acids
- An acid that contains more than one ionizable proton is a polyprotic acid. The protons of these acids ionize in steps. The differences in the acid ionization constants for the successive ionizations of the protons in a polyprotic acid usually vary by roughly five orders of magnitude. As long as the difference between the successive values of Ka of the acid is greater than about a factor of 20, it is appropriate to break down the calculations of the concentrations sequentially.
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- 3.6: Buffers
- A solution containing a mixture of an acid and its conjugate base, or of a base and its conjugate acid, is called a buffer solution. Unlike in the case of an acid, base, or salt solution, the hydronium ion concentration of a buffer solution does not change greatly when a small amount of acid or base is added to the buffer solution. The base (or acid) in the buffer reacts with the added acid (or base).
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- 3.7: Acid-Base Titrations
- A titration curve is a graph that relates the change in pH of an acidic or basic solution to the volume of added titrant. The characteristics of the titration curve are dependent on the specific solutions being titrated. The pH of the solution at the equivalence point may be greater than, equal to, or less than 7.00. The choice of an indicator for a given titration depends on the expected pH at the equivalence point of the titration, and the range of the color change of the indicator.
Contributors
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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>).


5.5A: The Nucleus
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- Last updated
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- Ying Liu
- City College of San Francisco
Describe the structure and the function of the nucleus in eukaryotic cells.
Define the following:
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nuclear envelope
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nucleolus
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nucleosome
- nuclear envelope
- nucleolus
- nucleosome
- Electron micrograph of a nucleus courtesy of Dennis Kunkel's Microscopy .
Inside the nucleus is a fluid called nucleoplasm, a nucleolus (see Figure \(\PageIndex{31}\)) , and linear chromosomes composed of negatively charged DNA associated with positively charged basic proteins called histones to form structures known as nucleosomes. The nucleosomes are part of what is called chromatin , the DNA and proteins that make up the chromosomes. The nucleolus is an area within the nucleus that is involved in the assembly of ribosomal subunits. An area of DNA called the nucleolar organizer directs the synthesis of ribosomal RNA (rRNA) that subsequently combines with ribosomal proteins to form immature ribosomal subunits that mature after they leave the nucleus by way of the pores in the nuclear envelope and mature in the cytoplasm. Genes located along the DNA are transcribed into RNA molecules, primarily messenger RNA (mRNA), transfer RNA (tRNA, and ribosomal RNA (rRNA). Messenger RNA is then translated into protein at the ribosomes. In general then, DNA determines what proteins and enzymes an organism can synthesize and, therefore, what chemical reactions it is able to carry out.
The DNA in eukaryotic cells is packaged in a highly organized way. It consists of a basic unit called a nucleosome , a beadlike structure 11 nm in diameter that consists of 146 base pairs of DNA wrapped around eight histone molecules. The nucleosomes are linked to one another by a segment of DNA approximately 60 base pairs long called linker DNA (see Figure \(\PageIndex{27}\)A) . Another histone associated with the linker DNA then packages adjacent nucleotides together to form a nucleosome thread 30nm in diameter. Finally, these packaged nucleosome threads form large coiled loops that are held together by nonhistone scaffolding proteins. These coiled loops on the scaffolding proteins interact to form the condensed chromatin seen in chromosomes during mitosis.
When the cell is not replicating, the DNA and proteins appear as a threadlike mass called chromatin. During mitosis , the chromatin coils into thick rodlike bodies called chromosomes (see Figure \(\PageIndex{31}\)A) and a spindle apparatus guides the separation and movement of the chromosomes for cell division so each cell winds up with a full complement of chromosomes. During sexual reproduction the nuclei of sex cells divide by meiosis producing cells with half the normal number of chromosomes (one from each homologous pair).
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For More Information: DNA from Unit 6
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For More Information: DNA Replication from Unit 6
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For More Information: Mitosis from Unit 6
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| Concept map for Eukaryotic Cell Structure |
Summary
- Eukaryotic cells contain much more DNA than do bacteria, and this DNA is organized as multiple chromosomes located within a nucleus.
- The nucleus in eukaryotic cells is separated from the cytoplasm by a nuclear envelope.
- The nucleolus is an area within the nucleus that is involved in the assembly of ribosomal subunits.
- Genes located along the DNA are transcribed into RNA molecules, primarily messenger RNA (mRNA), transfer RNA (tRNA, and ribosomal RNA (rRNA). Messenger RNA is then translated into protein at the ribosomes.
- In general then, DNA determines what proteins and enzymes an organism can synthesize and, therefore, what chemical reactions it is able to carry out.
Questions
Study the material in this section and then write out the answers to these questions. Do not just click on the answers and write them out. This will not test your understanding of this tutorial.
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_____ Separates the chromosomes from the cytoplasm.
(ans)
_____ A basic unit of eukaryotic DNA appearing as beadlike structures consisting of DNA wrapped around histone molecules. (ans)
- nuclear envelope
- nucleolus
- nucleosome
- chromosomes


I wept and said, "Oh, pardon me!  Why am I obliged to live? To drag hour after hour, to see the trees wave their branches restlessly, to feel the air, & to suffer in all I feel keenest agony. My frame is strong, but my soul sinks beneath this endurance of living anguish. Death is the goal that I would attain, but, alas! I do not even see the end of the course.
But Woodville's words had magic in them, when beginning with the sweetest pity, he would raise me by degrees out of myself and my sorrows until I wondered at my own selfishness: but he left me and despair returned; the work of consolation was ever to begin anew. I often desired his entire absence; for I found that I was grown out of the ways of life and that by long seclusion, although I could support my accustomed grief, and drink the bitter daily draught with some degree of patience, yet I had become unfit for the slightest novelty of feeling. Expectation, and hopes, and affection were all too much for me. I knew this, but at other times I was unreasonable and laid the blame upon him, who was most blameless, and pevishly thought that if his gentle soul were more gentle, if his intense sympathy were more intense, he could drive the fiend from my soul and make me more human. I am, I thought, a tragedy; a character that he comes to see act: now and then he gives me my cue that I may make a speech more to his purpose: perhaps he is already planning a poem in which I am to figure. I am a farce and play to him, but to me this is all dreary reality: he takes all the profit and I bear all the burthen.
Chapter XIIt is a strange circumstance but it often occurs that blessings by their use turn to curses; and that I who in solitude had desired sympathy as the only relief I could enjoy should now find it an additional torture to me. During my father's life time I had always been of an affectionate and forbearing disposition, but since those days of joy alas! I was much changed. I had become arrogant, peevish, and above all suspicious. Although the real interest of my narration is now ended and I ought quickly to wind up its melancholy catastrophe, yet I will relate one instance of my sad suspicion and despair and how Woodville with the goodness and almost the power of an angel, softened my rugged feelings and led me back to gentleness.
He had promised to spend some hours with me one afternoon but a violent and continual rain prevented him. I was alone the whole evening. I had passed two whole years alone unrepining, but now I was miserable. He could not really care for me, I thought, for if he did the storm would rather have made him come even if I had not expected him, than, as it did, prevent a promised visit. He would well know that this drear sky and gloomy rain would load my spirit almost to madness: if the weather had been fine I should not have regretted his absence as heavily as I necessarily must shut up in this miserable cottage with no companions but my own wretched thoughts. If he were truly my friend he would have calculated all this; and let me now calculate this boasted friendship, and discover its real worth. He got over his grief for Elinor, and the country became dull to him, so he was glad to find even me for amusement; and when he does not know what else to do he passes his lazy hours here, and calls this friendship—It is true that his presence is a consolation to me, and that his words are sweet, and, when he will he can pour forth thoughts that win me from despair. His words are sweet,—and so, truly, is the honey of the bee, but the bee has a sting, and unkindness is a worse smart that that received from an insect's venom. I will put him to the proof. He says all hope is dead to him, and I know that it is dead to me, so we are both equally fitted for death. Let me try if he will die with me; and as I fear to die alone, if he will accompany [me] to cheer me, and thus he can shew himself my friend in the only manner my misery will permit.
It was madness I believe, but I so worked myself up to this idea that I could think of nothing else. If he dies with me it is well, and there will be an end of two miserable beings; and if he will not, then will I scoff at his friendship and drink the poison before him to shame his cowardice. I planned the whole scene with an earnest heart and franticly set my soul on this project. I procured Laudanum and placing it in two glasses on the table, filled my room with flowers and decorated the last scene of my tragedy with the nicest care. As the hour for his coming approached my heart softened and I wept; not that I gave up my plan, but even when resolved the mind must undergo several revolutions of feeling before it can drink its death.
Now all was ready and Woodville came. I received him at the door of my cottage and leading him solemnly into the room, I said: "My friend, I wish to die.  What slave will not, if he may, escape from his chains? Look, I weep: for more than two years I have never enjoyed one moment free from anguish. I have often desired to die; but I am a very coward. It is hard for one so young who was once so happy as I was; [ sic ] voluntarily to divest themselves of all sensation and to go alone to the dreary grave; I dare not. I must die, yet my fear chills me; I pause and shudder and then for months I endure my excess of wretchedness.  Then we shall find Elinor and what I have lost. Look, I am prepared; there is the death draught, let us drink it together and willingly & joyfully quit this hated round of daily life[.]
"You turn from me; yet before you deny me reflect, Woodville, how sweet it were to cast off the load of tears and misery under which we now labour: and surely we shall find light after we have passed the dark valley. That drink will plunge us in a sweet slumber, and when we awaken what joy will be ours to find all our sorrows and fears past. A little patience, and all will be over ; aye, a very little patience; for, look, there is the key of our prison; we hold it in our own hands, and are we more debased than slaves to cast it away and give ourselves up to voluntary bondage? Even now if we had courage we might be free. Behold, my cheek is flushed with pleasure at the imagination of death; all that we love are dead. Come, give me your hand, one look of joyous sympathy and we will go together and seek them; a lulling journey; where our arrival will bring bliss and our waking be that of angels. Do you delay? Are you a coward, Woodville? Oh fie! Cast off this blank look of human melancholy. Oh! that I had words to express the luxury of death that I might win you. I tell you we are no longer miserable mortals; we are about to become Gods; spirits free and happy as gods. What fool on a bleak shore, seeing a flowery isle on the other side with his lost love beckoning to him from it would pause because the wave is dark and turbid?
"Do you mark my words; I have learned the language of despair: I have it all by heart, for I am Despair; and a strange being am I, joyous, triumphant Despair. But those words are false, for the wave may be dark but it is not bitter. We lie down, and close our eyes with a gentle good night, and when we wake, we are free. Come then, no more delay, thou tardy one! Behold the pleasant potion! Look, I am a spirit of good, and not a human maid that invites thee, and with winning accents, (oh, that they would win thee!) says, Come and drink."
As I spoke I fixed my eyes upon his countenance, and his exquisite beauty, the heavenly compassion that beamed from his eyes, his gentle yet earnest look of deprecation and wonder even before he spoke wrought a change in my high strained feelings taking from me all the sterness of despair and filling me only with the softest grief. I saw his eyes humid also as he took both my hands in his; and sitting down near me, he said:
"This is a sad deed to which you would lead me, dearest friend, and your woe must indeed be deep that could fill you with these unhappy thoughts. You long for death and yet you fear it and wish me to be your companion. But I have less courage than you and even thus accompanied I dare not die. Listen to me, and then reflect if you ought to win me to your project, even if with the over-bearing eloquence of despair you could make black death so inviting that the fair heaven should appear darkness. Listen I entreat you to the words of one who has himself nurtured desperate thoughts, and longed with impatient desire for death, but who has at length trampled the phantom under foot, and crushed his sting. Come, as you have played Despair with me I will play the part of Una with you and bring you hurtless from his dark cavern. Listen to me, and let yourself be softened by words in which no selfish passion lingers.
"We know not what all this wide world means; its strange mixture of good and evil. But we have been placed here and bid live and hope. I know not what we are to hope; but there is some good beyond us that we must seek; and that is our earthly task. If misfortune come against us we must fight with her; we must cast her aside, and still go on to find out that which it is our nature to desire. Whether this prospect of future good be the preparation for another existence I know not; or whether that it is merely that we, as workmen in God's vineyard, must lend a hand to smooth the way for our posterity. If it indeed be that; if the efforts of the virtuous now, are to make the future inhabitants of this fair world more happy; if the labours of those who cast aside selfishness, and try to know the truth of things, are to free the men of ages, now far distant but which will one day come, from the burthen under which those who now live groan, and like you weep bitterly; if they free them but from one of what are now the necessary evils of life, truly I will not fail but will with my whole soul aid the work. From my youth I have said, I will be virtuous; I will dedicate my life for the good of others; I will do my best to extirpate evil and if the spirit who protects ill should so influence circumstances that I should suffer through my endeavour, yet while there is hope and hope there ever must be, of success, cheerfully do I gird myself to my task.
"I have powers; my countrymen think well of them. Do you think I sow my seed in the barren air, & have no end in what I do? Believe me, I will never desert life untill this last hope is torn from my bosom, that in some way my labours may form a link in the chain of gold with which we ought all to strive to drag Happiness from where she sits enthroned above the clouds, now far beyond our reach, to inhabit the earth with us. Let us suppose that Socrates, or Shakespear, or Rousseau had been seized with despair and died in youth when they were as young as I am; do you think that we and all the world should not have lost incalculable improvement in our good feelings and our happiness thro' their destruction. I am not like one of these; they influenced millions: but if I can influence but a hundred, but ten, but one solitary individual, so as in any way to lead him from ill to good, that will be a joy to repay me for all my sufferings, though they were a million times multiplied; and that hope will support me to bear them[.]
"And those who do not work for posterity; or working, as may be my case, will not be known by it; yet they, believe me, have also their duties. You grieve because you are unhappy[;] it is happiness you seek but you despair of obtaining it. But if you can bestow happiness on another; if you can give one other person only one hour of joy ought you not to live to do it? And every one has it in their power to do that. The inhabitants of this world suffer so much pain. In crowded cities, among cultivated plains, or on the desart mountains, pain is thickly sown, and if we can tear up but one of these noxious weeds, or more, if in its stead we can sow one seed of corn, or plant one fair flower, let that be motive sufficient against suicide. Let us not desert our task while there is the slightest hope that we may in a future day do this.
 I have a mother whose support and hope I am. I have a friend who loves me as his life, and in whose breast I should infix a mortal sting if I ungratefully left him.  Nor shall you, my friend; cheer up; cease to weep, I entreat you. Are you not young, and fair, and good? Why should you despair? Or if you must for yourself, why for others? If you can never be happy, can you never bestow happiness[?] Oh! believe me, if you beheld on lips pale with grief one smile of joy and gratitude, and knew that you were parent of that smile, and that without you it had never been, you would feel so pure and warm a happiness that you would wish to live for ever again and again to enjoy the same pleasure[.]
"Come, I see that you have already cast aside the sad thoughts you before franticly indulged. Look in that mirror; when I came your brow was contracted, your eyes deep sunk in your head, your lips quivering; your hands trembled violently when I took them; but now all is tranquil and soft. You are grieved and there is grief in the expression of your countenance but it is gentle and sweet.
These words are shadowy as I repeat them but they were indeed words of fire and produced a warm hope in me (I, miserable wretch, to hope!) that tingled like pleasure in my veins. He did not leave me for many hours; not until he had improved the spark that he had kindled, and with an angelic hand fostered the return of somthing that seemed like joy. He left me but I still was calm, and after I had saluted the starry sky and dewy earth with eyes of love and a contented good night, I slept sweetly, visited by dreams, the first of pleasure I had had for many long months.
But this was only a momentary relief and my old habits of feeling returned; for I was doomed while in life to grieve, and to the natural sorrow of my father's death and its most terrific cause, immagination added a tenfold weight of woe. I believed myself to be polluted by the unnatural love I had inspired, and that I was a creature cursed and set apart by nature. I thought that like another Cain, I had a mark set on my forehead to shew mankind that there was a barrier between me and they [ sic ]. Woodville had told me that there was in my countenance an expression as if I belonged to another world; so he had seen that sign: and there it lay a gloomy mark to tell the world that there was that within my soul that no silence could render sufficiently obscure. Why when fate drove me to become this outcast from human feeling; this monster with whom none might mingle in converse and love; why had she not from that fatal and most accursed moment, shrouded me in thick mists and placed real darkness between me and my fellows so that I might never more be seen?, [ sic ] and as I passed, like a murky cloud loaded with blight, they might only perceive me by the cold chill I should cast upon them; telling them, how truly, that something unholy was near? Then I should have lived upon this dreary heath unvisited, and blasting none by my unhallowed gaze. Alas! I verily believe that if the near prospect of death did not dull and soften my bitter [fe]elings, if for a few months longer I had continued to live as I then lived, strong in body, but my soul corrupted to its core by a deadly cancer[,] if day after day I had dwelt on these dreadful sentiments I should have become mad, and should have fancied myself a living pestilence: so horrible to my own solitary thoughts did this form, this voice, and all this wretched self appear; for had it not been the source of guilt that wants a name?
This was superstition. I did not feel thus franticly when first I knew that the holy name of father was become a curse to me: but my lonely life inspired me with wild thoughts; and then when I saw Woodville & day after day he tried to win my confidence and I never dared give words to my dark tale, I was impressed more strongly with the withering fear that I was in truth a marked creature, a pariah, only fit for death.
Chapter XIIAs I was perpetually haunted by these ideas, you may imagine that the influence of Woodville's words was very temporary; and that although I did not again accuse him of unkindness, yet I soon became as unhappy as before. Soon after this incident we parted. He heard that his mother was ill, and he hastened to her. He came to take leave of me, and we walked together on the heath for the last time. He promised that he would come and see me again; and bade me take cheer, and to encourage what happy thoughts I could, untill time and fortitude should overcome my misery, and I could again mingle in society.
"Above all other admonition on my part," he said, "cherish and follow this one: do not despair. That is the most dangerous gulph on which you perpetually totter; but you must reassure your steps, and take hope to guide you. Hope, and your wounds will be already half healed: but if you obstinately despair, there never more will be comfort for you. Believe me, my dearest friend, that there is a joy that the sun and earth and all its beauties can bestow that you will one day feel. The refreshing bliss of Love will again visit your heart, and undo the spell that binds you to woe, untill you wonder how your eyes could be closed in the long night that burthens you. I dare not hope that I have inspired you with sufficient interest that the thought of me, and the affection that I shall ever bear you, will soften your melancholy and decrease the bitterness of your tears. But if my friendship can make you look on life with less disgust, beware how you injure it with suspicion. Love is a delicate sprite and easily hurt by rough jealousy. Guard, I entreat you, a firm persuasion of my sincerity in the inmost recesses of your heart out of the reach of the casual winds that may disturb its surface. Your temper is made unequal by suffering, and the tenor of your mind is, I fear, sometimes shaken by unworthy causes; but let your confidence in my sympathy and love be deeper far, and incapable of being reached by these agitations that come and go, and if they touch not your affections leave you uninjured."
These were some of Woodville's last lessons. I wept as I listened to him; and after we had taken an affectionate farewell, I followed him far with my eyes until they saw the last of my earthly comforter. I had insisted on accompanying him across the heath towards the town where he dwelt: the sun was yet high when he left me, and I turned my steps towards my cottage. It was at the latter end of the month of September when the nights have become chill. But the weather was serene, and as I walked on I fell into no unpleasing reveries. I thought of Woodville with gratitude and kindness and did not, I know not why, regret his departure with any bitterness. It seemed that after one great shock all other change was trivial to me; and I walked on wondering when the time would come when we should all four, my dearest father restored to me, meet in some sweet Paradise[.] I pictured to myself a lovely river such as that on whose banks Dante describes Mathilda gathering flowers, which ever flows
And then I repeated to myself all that lovely passage that relates the entrance of Dante into the terrestrial Paradise; and thought it would be sweet when I wandered on those lovely banks to see the car of light descend with my long lost parent to be restored to me. As I waited there in expectation of that moment, I thought how, of the lovely flowers that grew there, I would wind myself a chaplet and crown myself for joy: I would sing sul margine d'un rio , my father's favourite song, and that my voice gliding through the windless air would announce to him in whatever bower he sat expecting the moment of our union, that his daughter was come. Then the mark of misery would have faded from my brow, and I should raise my eyes fearlessly to meet his, which ever beamed with the soft lustre of innocent love. When I reflected on the magic look of those deep eyes I wept, but gently, lest my sobs should disturb the fairy scene.
I was so entirely wrapt in this reverie that I wandered on, taking no heed of my steps until I actually stooped down to gather a flower for my wreath on that bleak plain where no flower grew, when I awoke from my day dream and found myself I knew not where.
The sun had set and the roseate hue which the clouds had caught from him in his descent had nearly died away. A wind swept across the plain, I looked around me and saw no object that told me where I was; I had lost myself, and in vain attempted to find my path. I wandered on, and the coming darkness made every trace indistinct by which I might be guided. At length all was veiled in the deep obscurity of blackest night; I became weary and knowing that my servant was to sleep that night at the neighbouring village, so that my absence would alarm no one; and that I was safe in this wild spot from every intruder, I resolved to spend the night where I was. Indeed I was too weary to walk further: the air was chill but I was careless of bodily inconvenience, and I thought that I was well inured to the weather during my two years of solitude, when no change of seasons prevented my perpetual wanderings.
I lay upon the grass surrounded by a darkness which not the slightest beam of light penetrated—There was no sound for the deep night had laid to sleep the insects, the only creatures that lived on the lone spot where no tree or shrub could afford shelter to aught else—There was a wondrous silence in the air that calmed my senses yet which enlivened my soul, my mind hurried from image to image and seemed to grasp an eternity. All in my heart was shadowy yet calm, untill my ideas became confused and at length died away in sleep.
When I awoke it rained: I was already quite wet, and my limbs were stiff and my head giddy with the chill of night. It was a drizzling, penetrating shower; as my dank hair clung to my neck and partly covered my face, I had hardly strength to part with my fingers, the long strait locks that fell before my eyes. The darkness was much dissipated and in the east where the clouds were least dense the moon was visible behind the thin grey cloud—
Its presence gave me a hope that by its means I might find my home. But I was languid and many hours passed before I could reach the cottage, dragging as I did my slow steps, and often resting on the wet earth unable to proceed.
I particularly mark this night, for it was that which has hurried on the last scene of my tragedy, which else might have dwindled on through long years of listless sorrow. I was very ill when I arrived and quite incapable of taking off my wet clothes that clung about me. In the morning, on her return, my servant found me almost lifeless, while possessed by a high fever I was lying on the floor of my room.
I was very ill for a long time, and when I recovered from the immediate danger of fever, every symptom of a rapid consumption declared itself. I was for some time ignorant of this and thought that my excessive weakness was the consequence of the fever; [ sic ] But my strength became less and less; as winter came on I had a cough; and my sunken cheek, before pale, burned with a hectic fever. One by one these symptoms struck me; & I became convinced that the moment I had so much desired was about to arrive and that I was dying. I was sitting by my fire, the physician who had attended me ever since my fever had just left me, and I looked over his prescription in which digitalis was the prominent medecine. "Yes," I said, "I see how this is, and it is strange that I should have deceived myself so long; I am about to die an innocent death, and it will be sweeter even than that which the opium promised."
I rose and walked slowly to the window; the wide heath was covered by snow which sparkled under the beams of the sun that shone brightly thro' the pure, frosty air: a few birds were pecking some crumbs under my window. I smiled with quiet joy; and in my thoughts, which through long habit would for ever connect themselves into one train, as if I shaped them into words, I thus addressed the scene before me:
"I salute thee, beautiful Sun, and thou, white Earth, fair and cold! Perhaps I shall never see thee again covered with green, and the sweet flowers of the coming spring will blossom on my grave. I am about to leave thee; soon this living spirit which is ever busy among strange shapes and ideas, which belong not to thee, soon it will have flown to other regions and this emaciated body will rest insensate on thy bosom
"For it will be the same with thee, who art called our Universal Mother, when I am gone. I have loved thee; and in my days both of happiness and sorrow I have peopled your solitudes with wild fancies of my own creation. The woods, and lakes, and mountains which I have loved, have for me a thousand associations; and thou, oh, Sun! hast smiled upon, and borne your part in many imaginations that sprung to life in my soul alone, and which will die with me. Your solitudes, sweet land, your trees and waters will still exist, moved by your winds, or still beneath the eye of noon, though [w]hat I have felt about ye, and all my dreams which have often strangely deformed thee, will die with me. You will exist to reflect other images in other minds, and ever will remain the same, although your reflected semblance vary in a thousand ways, changeable as the hearts of those who view thee. One of these fragile mirrors, that ever doted on thine image, is about to be broken, crumbled to dust. But everteeming Nature will create another and another, and thou wilt loose nought by my destruction.
"Thou wilt ever be the same. Recieve then the grateful farewell of a fleeting shadow who is about to disappear, who joyfully leaves thee, yet with a last look of affectionate thankfulness. Farewell! Sky, and fields and woods; the lovely flowers that grow on thee; thy mountains & thy rivers; to the balmy air and the strong wind of the north, to all, a last farewell. I shall shed no more tears for my task is almost fulfilled, and I am about to be rewarded for long and most burthensome suffering. Bless thy child even even [ sic ] in death, as I bless thee; and let me sleep at peace in my quiet grave."
I feel death to be near at hand and I am calm. I no longer despair, but look on all around me with placid affection. I find it sweet to watch the progressive decay of my strength, and to repeat to myself, another day and yet another, but again I shall not see the red leaves of autumn; before that time I shall be with my father. I am glad Woodville is not with me for perhaps he would grieve, and I desire to see smiles alone during the last scene of my life; when I last wrote to him I told him of my ill health but not of its mortal tendency, lest he should conceive it to be his duty to come to me for I fear lest the tears of friendship should destroy the blessed calm of my mind. I take pleasure in arranging all the little details which will occur when I shall no longer be. In truth I am in love with death; no maiden ever took more pleasure in the contemplation of her bridal attire than I in fancying my limbs already enwrapt in their shroud: is it not my marriage dress? Alone it will unite me to my father when in an eternal mental union we shall never part.
I will not dwell on the last changes that I feel in the final decay of nature. It is rapid but without pain: I feel a strange pleasure in it. For long years these are the first days of peace that have visited me. I no longer exhaust my miserable heart by bitter tears and frantic complaints; I no longer the [ sic ] reproach the sun, the earth, the air, for pain and wretchedness. I wait in quiet expectation for the closing hours of a life which has been to me most sweet & bitter. I do not die not having enjoyed life; for sixteen years I was happy: during the first months of my father's return I had enjoyed ages of pleasure: now indeed I am grown old in grief; my steps are feeble like those of age; I have become peevish and unfit for life; so having passed little more than twenty years upon the earth I am more fit for my narrow grave than many are when they reach the natural term of their lives.
Again and again I have passed over in my remembrance the different scenes of my short life: if the world is a stage and I merely an actor on it my part has been strange, and, alas! tragical. Almost from infancy I was deprived of all the testimonies of affection which children generally receive; I was thrown entirely upon my own resources, and I enjoyed what I may almost call unnatural pleasures, for they were dreams and not realities. The earth was to me a magic lantern and I [a] gazer, and a listener but no actor; but then came the transporting and soul-reviving era of my existence: my father returned and I could pour my warm affections on a human heart; there was a new sun and a new earth created to me; the waters of existence sparkled: joy! joy! but, alas! what grief! My bliss was more rapid than the progress of a sunbeam on a mountain, which discloses its glades & woods, and then leaves it dark & blank; to my happiness followed madness and agony, closed by despair.
This was the drama of my life which I have now depicted upon paper. During three months I have been employed in this task. The memory of sorrow has brought tears; the memory of happiness a warm glow the lively shadow of that joy. Now my tears are dried; the glow has faded from my cheeks, and with a few words of farewell to you, Woodville, I close my work: the last that I shall perform.
Farewell, my only living friend; you are the sole tie that binds me to existence, and now I break it[.] It gives me no pain to leave you; nor can our seperation give you much. You never regarded me as one of this world, but rather as a being, who for some penance was sent from the Kingdom of Shadows; and she passed a few days weeping on the earth and longing to return to her native soil. You will weep but they will be tears of gentleness. I would, if I thought that it would lessen your regret, tell you to smile and congratulate me on my departure from the misery you beheld me endure. I would say; Woodville, rejoice with your friend, I triumph now and am most happy. But I check these expressions; these may not be the consolations of the living; they weep for their own misery, and not for that of the being they have lost. No; shed a few natural tears due to my memory: and if you ever visit my grave, pluck from thence a flower, and lay it to your heart; for your heart is the only tomb in which my memory will be enterred.
My death is rapidly approaching and you are not near to watch the flitting and vanishing of my spirit. Do no[t] regret this; for death is a too terrible an [ sic ] object for the living. It is one of those adversities which hurt instead of purifying the heart; for it is so intense a misery that it hardens & dulls the feelings. Dreadful as the time was when I pursued my father towards the ocean, & found their [ sic ] only his lifeless corpse; yet for my own sake I should prefer that to the watching one by one his senses fade; his pulse weaken—and sleeplessly as it were devour his life in gazing. To see life in his limbs & to know that soon life would no longer be there; to see the warm breath issue from his lips and to know they would soon be chill—I will not continue to trace this frightful picture; you suffered this torture once; I never did. And the remembrance fills your heart sometimes with bitter despair when otherwise your feelings would have melted into soft sorrow.
So day by day I become weaker, and life flickers in my wasting form, as a lamp about to loose it vivifying oil. I now behold the glad sun of May. It was May, four years ago, that I first saw my beloved father; it was in May, three years ago that my folly destroyed the only being I was doomed to love. May is returned, and I die. Three days ago, the anniversary of our meeting; and, alas! of our eternal seperation, after a day of killing emotion, I caused myself to be led once more to behold the face of nature. I caused myself to be carried to some meadows some miles distant from my cottage; the grass was being mowed, and there was the scent of hay in the fields; all the earth look[ed] fresh and its inhabitants happy. Evening approached and I beheld the sun set. Three years ago and on that day and hour it shone through the branches and leaves of the beech wood and its beams flickered upon the countenance of him whom I then beheld for the last time. I now saw that divine orb, gilding all the clouds with unwonted splendour, sink behind the horizon; it disappeared from a world where he whom I would seek exists not; it approached a world where he exists not[.] Why do I weep so bitterly? Why my [ sic ] does my heart heave with vain endeavour to cast aside the bitter anguish that covers it "as the waters cover the sea." I go from this world where he is no longer and soon I shall meet him in another.
Farewell, Woodville, the turf will soon be green on my grave; and the violets will bloom on it. There is my hope and my expectation; your's are in this world; may they be fulfilled.


11.2.1: What is Fair Use?
Fair Use and Educational Use
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Are you incorporating any materials in your research final product that were created by someone else, such as images or text from other works? These materials could be protected by copyright. For example, content you find online, text, books, movies, songs, email, images, and videos are most likely copyrighted. Fortunately, U.S. copyright law includes an exception that allows you to use copyrighted work in your assignments for class.
However, if you would like to share your research product outside of the classroom (such as on a webpage or blog or in your portfolio), you will need permission from the copyright owner(s) unless your use is covered under another statutory exception. Fair use is one such exception, and it can apply to a wide variety of uses.
Fair Use and Educational Use
Fair use plays an important role in education. Although educational use receives several protections in copyright law, not all educational use is automatically fair use. It’s important to know that there are limits to how you can use others’ creative works even as a student or teacher in the classroom.
In this section, you will learn about fair use and strategies to help determine whether or not a proposed use of someone else’s copyrighted works falls under the fair use exception. Understanding how to properly perform a fair use analysis and assert your fair use rights can help you to build upon others’ works with confidence.
Copyright in the U.S. is intended to promote the creation of new works by providing an incentive for creators. However, recognizing that new works often build on or incorporate existing works, the law strikes a balance between the rights of creators and the rights of users via exceptions to the exclusive rights of the creator.
The fair use exception is detailed in Section 107 of the U.S. Copyright Act. Unlike other copyright exceptions, fair use is flexible and can apply to a broad array of uses. It is designed to be adaptable to new uses and technologies so that Congress doesn’t have to create new exceptions before a new technology can be utilized.
Watch a short introduction to fair use from the OSU Libraries’ Copyright Resources Center .
A YouTube element has been excluded from this version of the text. You can view it online here .


1.5: Emotional Intelligence
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Most of you have heard of IQ or intelligence quotient. But have you heard of EI or EQ - emotional intelligence? Daniel Goleman is an American psychologist who studied emotional intelligence. His book, Emotional Intelligence: Why It Can Matter More Than IQ 8 , takes a look at the extreme importance of emotional intelligence. He found that there are five key components that comprise EI or EQ:
You could be the smartest person in the world, but if you are not able to manage your emotions, you probably will have a difficult time in school and in the workplace. You could excel in all academic situations, but if you do not work well with other people, you may have difficulty doing well in any workplace environment. Many people believe that EI or EQ is just as important or perhaps more important than IQ. What do you think?
8 Goleman, D. (1995). Emotional intelligence: Why it can matter more than IQ . New York: Bantam Books.


13.1: Reproductive System
In biological terms sexual reproduction involves the union of gametes - the sperm and the ovum - produced by two parents. Each gamete is formed by meiosis . This means each contains only half the chromosomes of the body cells ( haploid ). Fertilization results in the joining of the male and female gametes to form a zygote which contains the full number of chromosomes ( diploid ). The zygote then starts to divide by mitosis to form a new animal with all its body cells containing chromosomes that are identical to those of the original zygote (see diagram 13.1).
Diagram 13.1 - Sexual reproduction
The offspring formed by sexual reproduction contain genes from both parents and show considerable variation. For example, kittens in a litter are all different although they (usually) have the same mother and father. In the wild this variation is important because it means that when the environment changes some individuals may be better adapted to survive than others. These survivors pass their “superior” genes on to their offspring. In this way the characteristics of a group of animals can gradually change over time to keep pace with the changing environment. This “survival of the fittest” or “ natural selection ” is the mechanism behind the theory of evolution .


8.5.1: Dot Product (Exercise)
section 8.5 exercises
Two vectors are described by their magnitude and direction in standard position. Find the dot product of the vectors.
1. Magnitude: 6, Direction: \(45^{\circ}\); Magnitude: 10, Direction: \(120^{\circ}\)
2. Magnitude: 8, Direction: \(220^{\circ}\); Magnitude: 7, Direction: \(305^{\circ}\)
Find the dot product of each pair of vectors.
3. \(\langle 0, 4 \rangle; \langle -3, 0 \rangle\)
4. \(\langle 6, 5 \rangle; \langle 3, 7 \rangle\)
5. \(\langle -2, 1 \rangle; \langle -10, 13 \rangle\)
6. \(\langle 2, -5 \rangle; \langle 8, -4 \rangle\)
Find the angle between the vectors
7. \(\langle 0, 4 \rangle; \langle -3, 0 \rangle\)
8. \(\langle 6, 5 \rangle; \langle 3, 7 \rangle\)
9. \(\langle 2, 4 \rangle; \langle 1, -3 \rangle\)
10. \(\langle -4, 1 \rangle; \langle 8, -2 \rangle\)
11. \(\langle 4, 2 \rangle; \langle 8, 4 \rangle\)
12. \(\langle 5, 3 \rangle; \langle -6, 10 \rangle\)
13. Find a value for \(k\) so that \(\langle 2, 7 \rangle\) and \(\langle k, 4 \rangle\) will be orthogonal.
14. Find a value for \(k\) so that \(\langle -3, 5 \rangle\) and \(\langle 2, k \rangle\) will be orthogonal.
15. Find the magnitude of the projection of \(\langle 8, -4 \rangle\) onto \(\langle 1, -3 \rangle\).
16. Find the magnitude of the projection of \(\langle 2, 7 \rangle\) onto \(\langle 4, 5 \rangle\).
17. Find the projection of \(\langle -6, 10 \rangle\) onto \(\langle 1, -3 \rangle\).
18. Find the projection of \(\langle 0, 4 \rangle\) onto \(\langle 3, 7 \rangle\).
19. A scientist needs to determine the angle of reflection when a laser hits a mirror. The picture shows the vector representing the laser beam, and a vector that is orthogonal to the mirror. Find the acute angle between these, the angle of reflection.
20. A triangle has coordinates at \(A\): (1,4), \(B\): (2,7), and \(C\): (4,2). Find the angle at point \(B\).
21. A boat is trapped behind a log lying parallel to the dock. It only requires 10 pounds of force to pull th e boat directly towards you, but because of the log, you'll have to pull at a \(45^{\circ}\) angle. How much force will you have to pull with? (We're going to assume that the log is very slimy and doesn't contribute any additional resistance.)
22. A large boulder needs to be dragged to a new position. If pulled directly horizontally, the boulder would require 400 pounds of pulling force to move.  How much force will the truck need to pull with?
23. Find the work done against gravity by pushing a 20 pound cart 10 feet up a ramp that is \(10^{\circ}\) above horizontal. Assume there is no friction, so the only force is 20 pounds downwards due to gravity.
24. Find the work done against gravity by pushing a 30 pound cart 15 feet up a ramp that is \(8^{\circ}\) above horizontal. Assume there is no friction, so the only force is 30 pounds downwards due to gravity.
25. An object is pulled to the top of a 40 foot ramp that forms a \(10^{\circ}\) angle with the ground. It is pulled by rope exerting a force of 120 pounds at a \(35^{\circ}\) angle relative to the ground. Find the work done.
26. An object is pulled to the top of a 30 foot ramp that forms a \(20^{\circ}\) angle with the ground. It is pulled by rope exerting a force of 80 pounds at a \(30^{\circ}\) angle relative to the ground. Find the work done.
- Answer
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1. \(6 \cdot 10 \cdot \cos(75^{\circ}) = 15.529\)
3. (0)(-3) + (4)(0) = 0
5. (-2)(-10) + (1)(13) = 33\)
7. \(\cos^{-1} (\dfrac{0}{\sqrt{4}\sqrt{3}}) = 90^{\circ}\)
9. \(\cos^(-1)(\dfrac{(2)(1) + (4)(-3)}{\sqrt{2^2 + 4^2}\sqrt{1^2 + (-3)^2}}) = 135^{\circ}\)
11. \(\cos^(-1)(\dfrac{(4)(8) + (2)(4)}{\sqrt{4^2 + 8^2}\sqrt{2^2 + 4^2}}) = 0^{\circ}\)
13. \((2)(k) + (7)(4) = 0, k = -14\)
15. \(\dfrac{(8)(1) + (-4)(-3)}{\sqrt{1^2 + (-3)^2}} = 6.325\)
17. \((\dfrac{(-6)(1) + (10)(-3)}{\sqrt{1^2 + (-3)^2}}^2) \langle 1, -3 \rangle = \langle -3.6, 10.8 \rangle\)
19. The vectors are \(\langle 2, 3 \rangle\) and \(\langle -5, -2 \rangle\). The acute angle between the vectors is \(34.509^{\circ}\)
21. 14.142 pounds
23. \(\langle 10\cos(10^{\circ}), 10\sin(10^{\circ}) \rangle \cdot \langle 0, -20 \rangle\), so 34.7296 ft-lbs
25. \(40 \cdot 120 \cdot \cos(25^{\circ}) = 4350.277\) ft-lbs