College Algebra, Chapter 8, 8.1, Section 8.1, Problem 16

Determine the focus, directions and focal diameter of the parabola $y = -2x^2$. Then, sketch its graph.



The equation $\displaystyle y = -2x^2; x^2 = \frac{-1}{2}y$ is a parabola that opens downward. The parabola has the form $x^2 = 4py$. So


$
\begin{equation}
\begin{aligned}

4p =& \frac{-1}{2}
\\
\\
p =& \frac{-1}{8}

\end{aligned}
\end{equation}
$


So, the focus is at $\displaystyle (0,p) = \left(0, \frac{-1}{8} \right)$ and directrix $\displaystyle y = -p = \frac{-1}{8}$. Also, $\displaystyle 2p = 2 \left( \frac{-1}{8} \right) = \frac{-1}{4}$, thus the endpoints of the latus rectum are at $\displaystyle \left( \frac{-1}{4}, \frac{-1}{8} \right)$ and $\displaystyle \left( \frac{1}{4}, \frac{-1}{8} \right)$. The focal diameter is $\displaystyle |4p| = \left| 4 \left( \frac{-1}{8} \right) \right| = \frac{1}{2} $ units. Therefore, the graph is

How does one describe the U.S. court system? What are the types of courts? How does the federal court system compare with the state court system? Who are the participants in a lawsuit? What are the initial steps in a lawsuit and the steps in trial as well as post-trial procedures? What are two alternative dispute resolution procedures?

Under the US Constitution, the highest court is the Supreme Court. The role of the Supreme Court is to hear the following: lawsuits between two or more states, cases concerning "ambassadors and other public ministers," cases of appeal, and any cases questioning a point of "constitutional and/or federal law" (United States Courts, "About the Supreme Court"). The Constitution grants Congress the power to oversee the Supreme Court, and over the years, the number of seats on the court has changed. The Judiciary Act of 1837 established the nine seats held today, including the chief justice and eight associate justices (Federal Judicial Center, "The Supreme Court of the United States and the Federal Judiciary").
The 13 U.S. Courts of Appeal count as the second type of U.S. courts. If a defendant does not think the decision handed by a district court was correctly guided by the law, the defendant has the right to appeal to the appellate court of the defendant's district. The role of the appellate courts is to decide "whether or not the law was applied correctly in the trial court" (United States Courts, "Court Role and Structure").The third type of court is the district court, also called the trial court. Each state has "at least one district court," and there are a total of 94 U.S. District Courts across the nation ("Court Role and Structure"). Nearly all criminal and civil cases are heard by the district courts before being appealed to higher courts, should a case be appealed. In district courts, judges have the right oversee jury trials and sometimes to try cases without a jury. District court judges decide on the following "questions of law":

... the admissibility of certain kinds of evidence, the scope of a search warrant, or the legality of an arrest. (Georgia Tech, "Federal District Courts")

District courts will decide if, beyond reasonable doubt, a defendant committed a crime.The primary difference between federal district courts and state district courts concerns the types of cases the courts can hear. State courts can hear almost any case concerning state laws and state citizens. Cases include "robberies, traffic violations, broken contracts, and family disputes" ("Federal vs. State Courts--Key Differences"). The only kinds of cases state district courts cannot hear concern federal crimes, "antitrust, bankruptcy, patent, copyright, and some maritime cases," which are all instead heard by federal district courts, in addition to any cases concerning the United States ("Federal vs. State Courts").
https://www.uscourts.gov/about-federal-courts/court-role-and-structure

Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 33

Predator - prey models are often used to study the interaction between species, according to the study of ecosystems. Consider populations of Tundra Wolves, given y $w(t)$, and Caribou, given by $c(t)$, in Northern Canada. The interaction has been modeled by the equations.

$\displaystyle \frac{dc}{dt} = ac - bcw \qquad \frac{dw}{dt} = -cw + dcw$

a.) What values of $\displaystyle \frac{dc}{dt}$ and $\displaystyle \frac{dw}{dt}$ correspond to stable populations?
b.) How would the statement "The Caribou go extinct" be represented mathematically?
c.) Suppose that $a = 0.05$, $b = 0.001$, $c = 0.05$, and $d = 0.0001$. Find all population pairs $(c,w)$ that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

a.) If the populations are stable, the rate of change are neither increasing nor dercreasing, that is, $\displaystyle \frac{dc}{dt} = 0$ and $\displaystyle \frac{dw}{dt} = 0$.

b.) "The Caribou go extinct" can be represented mathematically as $c(t) = 0$

c.) For the satbility of the Caribou,

$
\begin{equation}
\begin{aligned}
\frac{dc}{dt} = 0 &= 0.05 c - 0.001 cw\\
\\
0 & = c ( 50 - w )\\
\\
\text{so, } (c,w) &= (0,50)
\end{aligned}
\end{equation}
$

For the stability of the Tundra Wolves,


$
\begin{equation}
\begin{aligned}
\frac{dw}{dt} = 0 &= -0.05 c + 0.0001 cw\\
\\
0 & = w(c-500)\\
\\
\text{so, } (c,w) &= (500,0)
\end{aligned}
\end{equation}
$


We can conclude that the two populations can live in balance if $(c,w) = (500,50)$ or the population of the Wolves and Caribou can be represented in the ratio of 1:10.

What is the main idea of the poem "Ozymandias"?

In the poem, Shelley describes the ruins of a once great statue of a sphinx intended to represent the almighty reign of Ramses II, also known as Ozymandias. However, instead of witnessing the powerful image of an omnipotent ruler, all that remains of Ozymandias's statue is a "Half sunk," broken image of a domineering man that is decaying in the sand. Ironically, Ramses's original intentions of his statue have the opposite effect on travelers, who only witness how time impacts one's legacy and accomplishments. Shelley's poem examines the transitory nature of life, legacy, power, and government institutions. The decaying, broken image of Ozymandias's visage portrays how time destroys every human accomplishment. The inscription on the bottom of the statue is also ironic and symbolically represents how one's pursuit of power and glory are illusory and fleeting.

The main idea of this poem is that all tyrants are eventually defeated and reduced to nothing. Although Ozymandias thought he was a great and terrifying monarch, ruling over a mighty kingdom, all that is left of him now is a broken statue on an empty desert where his "works" once flourished. 
 Ozymandias had the following inscription put on his statue:


My name is Ozymandias, King of Kings; 
Look on my Works, ye Mighty, and despair!

 

The inscription is ironic; while Ozymandias means for other rulers to be frightened of his displays of power, what now would frighten them would be the complete and utter destruction of Ozymandias's power and glory.
Shelley was a radical. He was a supporter of the French Revolution. Shelley is making the point that no king is all-powerful, no matter what he may think. 

How can I write an essay on racial tension/conflict by comparing Langston Hughes's "Theme for English B" and Rudyard Kipling's "The White Man's Burden"?

One way to write a comparison essay is to note the similarities between both poems while acknowledging a main difference between them. Of course, we can make this comparison clear in the thesis statement of the first paragraph. Here are two suggestions for a thesis statement:
1) While both poems acknowledge the consequences of imperialism, Langston Hughes's poem is the only one that explores the question of personal identity.
2) While both poems acknowledge the racial divide that separates the conquered from the colonizing force, Kipling's poem is the only one that characterizes imperialism as a potentially moral enterprise.
From here, we can conceivably use five or more paragraphs to address one of the thesis statements above. In this answer, I will address the first statement: While both poems acknowledge the consequences of imperialism, Langston Hughes's poem is the only one that explores the question of personal identity.
Additionally, whether we choose thesis statement 1 or thesis statement 2, a discussion of imperialism would serve our purposes. Before we discuss the consequences of imperialism (as evidenced in the poems), it would be a good idea to discuss the basis for American imperialism.
 
Kipling's poem was published at the beginning of the Philippine-American War, while Hughes's poem explores African American integration into white society after the Civil War and Reconstruction. The speaker of Hughes's poem was presumably born in Winston-Salem, North Carolina, and he now attends college in Harlem, New York. The concepts of Manifest Destiny and American exceptionalism (where America deems itself a superior force for good because of its distinctive institutions and freedoms) can be seen in the ideas behind Reconstruction and the Philippine-American war ("To seek another’s profit / And work another’s gain / Take up the White Man’s burden...").

During the period of what was known as Radical Reconstruction, the South debuted its first state-funded integrated public school, saw African American participation in public office rise, and passed laws against racial discrimination in public accommodations. 
 
For information on the Philippine-American War, American exceptionalism and Manifest Destiny, please refer to the three links below.
 
Both poems do acknowledge the consequences of imperialism. For example, Hughes's poem highlights the divide between two different worldviews. The speaker's white teacher tells the speaker (a black student) to write in his authentic voice. However, the speaker questions what his authentic voice really is; he wonders whether race impacts personal preferences and whether it colors one's perception about life. Here, the conquered (speaker of the poem) questions his place within a society dominated by the progeny of past imperialists. ("It’s not easy to know what is true for you or me / at twenty-two, my age.")
Meanwhile, Kipling acknowledges that conquered natives will not submit readily to their oppression ("Your new-caught, sullen peoples, / Half devil and half child..."). So, both poems acknowledge that there are very real consequences to imperialism. First, it is unlikely that the conquered will thank their new masters for their inferior status in society. Second, the question of identity comes into play: how will the conquered align their native assumptions with the unfamiliar and offensive presuppositions of their imperial masters?
Next, we can discuss how Hughes's poem explores the question of personal identity, an element that is missing from Kipling's poem. First, Hughes humanizes the speaker by giving us some important information about him. Here, we can include information from the poem to support this statement. For example, the speaker is twenty-two; he was born in Winston-Salem, etc. The speaker comes to an interesting conclusion about life: both white and black people can like the same things even though they don't belong to the same race. Second, we can make the point that Hughes's poem acknowledges the humanity that connects all races ("You are white— / yet a part of me, as I am a part of you"). Third, Hughes makes the point that this belief about shared humanity is also very American in nature: 

You are white—
yet a part of me, as I am a part of you.
That’s American.
Sometimes perhaps you don’t want to be a part of me.   
Nor do I often want to be a part of you.
But we are, that’s true!

Although both poems acknowledge the validity of racial tensions, Hughes's poem is a departure from Kipling's in that the former acknowledges the shared humanity between the races while the latter views the imperial force as a superior civilizing power. 
https://www.newworldencyclopedia.org/entry/American_exceptionalism

https://www.newworldencyclopedia.org/entry/Manifest_Destiny

https://www.newworldencyclopedia.org/entry/Philippine-American_War

How much did Jimmy make from all his robberies?

The text does not say exactly how much Jimmy makes from all the robberies he does before his arrest. However, after his release from prison, Jimmy makes about eight hundred dollars from a safe burglary in Richmond, Indiana. Two weeks after the Richmond affair, he makes about fifteen hundred dollars from another burglary in Logansport. Next, he makes five thousand dollars from a burglary in Jefferson City. This amounts to a total of about seven thousand and three hundred dollars in currency.
After the last of the three burglaries, Jimmy relocates to Elmore in Arkansas. While there, he meets and falls in love with the beautiful Annabel Adams, whose father owns the Elmore Bank. He also starts a shoe store, which he initially intends to use as a front for his safe-cracking activities. However, his love for Annabel encourages him to quit his old ways for the more honest shoe-selling business. He becomes successful in his line of trade and wants to give away his safe-cracking tools to an old friend so that he can settle down with the love of his life.

What behavior of Mr. Wickham’s (in regard to Miss King) does Mrs. Gardiner find odd?

Mrs. Gardiner finds it odd, not to say sinister, of Mr. Wickham to be paying court to Miss King after she's recently come into a large inheritance. This confirms Mrs. Gardiner in her suspicion that Wickham is an incorrigible gold-digger who simply cannot be trusted around young, eligible ladies.
At this stage in the story, Elizabeth has yet to be apprised of Wickham's villainy by Mr. Darcy who, in due course, will tell her all about his caddish and disreputable behavior towards his sister. But for now, Elizabeth defends what she sees as Wickham's prudence against Mrs. Gardiner's dire warnings of his far from pure intentions. It's only later on, when she reads Mr. Darcy's shocking revelations about Wickham's true character, that Elizabeth will finally see this importunate rake for what he really is.

Elizabeth likes Mr. Wickham so much, and Mrs. Gardiner trusts the judgment of her sensible niece. So when Mr. Wickham abandons his attentions to Elizabeth and begins to pay more attention to Miss King, a young woman who has just inherited a large sum of money, Mrs. Gardiner says that she "'should be sorry to think [their] friend mercenary.'" She fears that, perhaps, Mr. Wickham is now only paying attention to Miss King because of her inheritance of ten thousand pounds. Elizabeth, however, seems to find this concern somewhat problematic. Her aunt would call such an interest a mercenary one, but Elizabeth suggests that it might actually be more "'prudent.'" In the past, her aunt cautioned Elizabeth against marrying him when neither of them have any money, but now she criticizes Wickham for his interest in Miss King because she does have money. Elizabeth is, of course, keen to defend the man she believes to be her friend, but there is also some truth to what she says.

In chapter 27 of Jane Austen's Pride and Prejudice, Mrs. Gardiner asks Elizabeth about Miss King, who is now Wickham's love interest. Wickham, who was presumably courting Elizabeth before, is now seen courting Miss King. Coincidentally, Miss King just inherited a large sum of money, which makes the intentions of Mr. Wickham all the more suspicious. Already, Mrs. Gardiner has an opinion of Wickham. She thinks him a "mercenary."
As such, Mrs. Gardiner wants to know what kind of girl Miss King is in order to determine the type of man that Mr. Wickham really is. The result is that, according to Mrs. Gardiner:

[Wickham] paid [Miss King] not the smallest attention till her grandfather’s death made her mistress of this fortune.

Therefore, Mrs. Gardiner's assumptions that Mr. Wickham is only interested in people for what he can get from them are correct. Wickham is attracted to the ten thousand pounds that Miss King brings to the relationship, while caring very little about the woman herself. He is quick enough to drop Elizabeth, even when they were getting along well. Soon enough, Elizabeth will see the extent of Wickham's wickedness when he diverts his attention to someone else: her sister Lydia, whom he disgraces.