College Algebra, Chapter 2, 2.1, Section 2.1, Problem 38

Plot the points $A(5,1), B(0,6)$ and $C(-5,1)$ on a coordinate plane. Where must the point $D$ be located so that the quadrilateral $ABCD$ is a square? Find the area of this square.

If the Quadrilateral $ABCD$ is a square, then $d_{AB} = d_{BC} = d_{CD} = d_{AD}$

By using distance formula,

$
\begin{equation}
\begin{aligned}
d_{AB} &= \sqrt{(6-1)^2 + (0-5)^2}\\
\\
&= \sqrt{5^2 + (-5)^2}\\
\\
&= \sqrt{25+25}\\
\\
&= \sqrt{50}\\
\\
&= 5\sqrt{2} \text{ units}\\
\\
d_{AB} &= \sqrt{(y-1)^2 + (x-5)^2} && \text{Distance of point } A(5,1) \text{ and } B(x,y)\\
\\
(d_{AD})^2 &= (y-1)^2 + (x-5)^2 && \text{Square both sides}\\
\\
(5\sqrt{2})^2 &= (y-1)^2 + (x-5)^2 && \text{Substitute } d_{AB} \text{ to } d_{AD}\\
\\
(y-1)^2 &= 50 - (x-5)^2\\
\\
d_{CD} &= \sqrt{(x-(-5))^2 + (y-1)^2} && \text{Distance of point } C(-5,1) \text{ and } D(x,y)\\
\\
(d_{CD})^2 &= (x+5)^2 + (y-1)^2 && \text{Square both sides}\\
\\
(5\sqrt{2})^2 &= (x+5)^2 + (y-1)^2 && \text{Substitute } d_{AB} \text{ to } d_{CD}\\
\\
50 &= (x+5)^2 + \left[50 - (x-5)^2 \right] && \text{Substitute } (y-1)^2 \text{ from } d_{AD}\\
\\
0 &= (x+5)^2 - (x-5)^2 && \text{Subtract to } 50\\
\\
0 &= x^2 + 10x + 25 - x^2 + 10x - 25 && \text{Expand} \\
\\
0 &= 20x && \text{Combine like terms}\\
\\
0 &= x && \text{Solve for } x
\end{aligned}
\end{equation}
$


If $x=0$, then

$
\begin{equation}
\begin{aligned}
(y-1)^2 &= 50-(0-5)^2\\
\\
(y-1)^2 &= 50-25\\
\\
(y-1)^2 &= 25\\
\\
y -1 &= \pm 5\\
\\
y &= \pm 5 + 1\\
\\
y &= 6 \text{ and } y = -4
\end{aligned}
\end{equation}
$

The point $(0,6)$ is already $B$. Therefore, point $D$ is $(0,-4)$
Thus, the area of the square $ABCD$ is $A = \left( d_{AB} \right)^2 = \left( 5\sqrt{2} \right)^2 = 50$ square units.

What is the role of women in society nowadays?

Today, gender roles are changing as a result of modernity and its effects on life. While in the past, women the world over were mainly caregivers with roles centered around the home, today they even take on those roles that are considered traditionally male. Roles in areas such as caregiving are, however, still dominated largely by women. For instance, the United States Department of Labor states that the top ten occupation types dominated by women today are speech-language pathologists, preschool and kindergarten teachers, dental hygienists, medical transcriptionists, dental assistants, childcare workers, hairdressers and cosmetologists, secretaries and administrative assistants, medical assistants, and nutritionists. It appears that women still dominate occupations that were traditionally female, such as the beauty industry, caregiving, and teaching. They have also made strides in some traditionally male fields, such as architecture, production work, farming, and computer occupations. However, fields such as oil and gas mining and mechanic work are still predominantly male.
Also, the number of women taking up formal employment away from home has increased. According to the US Department of Labor women comprise 46.9% of the total national workforce. This means that many women are no longer just stay at home mothers. However, most employed women also double up as workers in their homes, doing household chores and taking care of their families.
The increasing number of female single parents has meant that women must sometimes be the sole breadwinners of their families – a role assigned to men in the past. This has led to more women seeking formal education as a means towards better employment opportunities. 

What check does the president have on Congress?

Checks and balances, the idea of making sure no one branch of government has complete or unchecked power, was an idea written by James Madison in the Federalist Papers, essay number 51.  Checks and balances also call for a separation of powers within government.
The president, head of the executive branch of government, can check the legislature, just as the legislative branch can check the president.  When a bill comes up in Congress, it needs to be passed by both houses in the Capitol.  If legislation is passed through both the House of Representatives and the Senate, the president has the power to veto this legislation, essentially shutting it down.  Congress can override the presidential veto with a two-thirds vote.
Another check the president has on Congress is the ability to call a special session, propose executive orders, and reject portions of bills (instead of the whole bill); while some of these are considered formal or informal presidential powers, they still count as a mechanism that can check the power of Congress.

Does the main character in the novel reflect tensions of the Civil Rights movement?

Although To Kill a Mockingbird takes place nearly seventy years after the legal end to slavery, Harper Lee’s coming of age novel clearly depicts the racial inequity that necessitated the emergence of The Civil Rights Movement. Many of Maycomb County’s characters confirmed the ubiquity of the deeply ingrained racist views of the American people in the early 1930s. These characters, along with Tom Robinson’s conviction for a crime he clearly did not commit, illustrate the truth about America in the decades between the ratification of The Thirteenth Amendment and The Civil Rights Movement: that true freedom did not exist for African Americans. To Kill a Mockingbird reminds us that African Americans in the 1930’s were not free to receive an equal education, to obtain equal housing, to avoid discrimination in America’s every institution, or --as was the case for Tom Robinson, as well as countless other black men living during this time -- to move about the world without the very real risk of death as the result of a wrongful, unjust conviction.
Scout, the main character in To Kill a Mockingbird, embodies many of the tensions of The Civil Rights movement, particularly in her point of view as a young child growing up in a racist, hypocritical world. Although Scout’s father, Atticus, insists, through his own example and in his direct teachings, that Scout learn to treat all people equally, regardless of race, Scout still, inevitably, is indoctrinated into America’s tradition of unfair and unequal treatment of black people. Although Scout does not exhibit overt racism, her narration often illustrates how deeply she has internalized people of color as “the other,” such as when she refers to Tom as a “good negro” during the trial, espousing a clear double standard of morality for black people. Scout does have moments of mature revelation, such as when she famously says to Atticus that it would be “like killing a mockingbird” to reveal the truth about Boo Radley’s killing of Mr. Ewell; still, she clearly struggles to completely overcome the racist views of the society she belongs to despite the careful teachings of her father, thus elucidating the immense challenge of achieving the Civil Rights Movement's goal for just world, free of racial prejudice.

Although To Kill a Mockingbird is set during the Great Depression, the racial tension in the novel is equally applicable to the climate of the civil rights movement; this tension highlights the absence of and need for racial equality in America. Tom Robinson's legal troubles were a common occurrence during the Jim Crow era, as many black males were unjustly accused of, and then harshly punished for, inappropriate conduct with white females. Beatings, lynchings, and worse were often inflicted upon black men at the hands of white men in the volatile racial climate of the rural south; Tom's legal predicament in the novel is similarly grave. The likelihood of a black man being acquitted by a white jury was small, so the accusations against Tom in To Kill A Mockingbird accurately reflect life both during the Great Depression and the civil rights movement.
Atticus Finch is respected by many people as an example of how a white ally should fight for equality; his courageous representation of Tom in court provides a moral example of how one should fight for what is right, even when the decision is unpopular. Although the civil rights movement was largely championed by people of color, white supporters also marched, boycotted, and contributed financially in many cases, and Atticus was undoubtedly a hero to white Americans who fought for racial equality. Atticus's respect and support for Calpurnia, the family's black housekeeper, demonstrates the importance of mutually respectful relationships across color lines; in many ways she is considered a part of the family more than an employee. The above character relationship provides a small but accurate snapshot of the issues and tension present both during the Great Depression and the American civil rights movement.

How did mass culture contribute to American global dominance?

One of the most significant and visible features of America's expanding global presence has been the spread of American mass culture around the world. Beginning in the 1950s, when American consumer goods flooded postwar Europe, American culture has been ubiquitous around the world. One observer in the late 1990s commented that 

...[I]mages of America are so pervasive in this global village that it is almost as if instead of the world immigrating to America, America has emigrated to the world, allowing people to aspire to be Americans even in distant countries.

This process has been largely due to the power and reach of American corporations, which became "multinational" in the late twentieth century. McDonald's, Coca-Cola, and other corporations have expanded their reach to the far corners of the globe. Not just American goods, but film, music, fashion, and other aspects of American mass culture have flooded the world as well. American movies enjoy runs in European and Asian theaters, teenagers around the world wear American fashions, and people everywhere listen to American pop stars. This process has never been completely hegemonic, and it has always flowed in both directions—the emergence of football (soccer) as a popular pastime in the United States is one prominent example—but there is no doubt that, for better or for worse, American mass culture has exerted a profound influence around the world. 
http://www.globalization101.org/pop-culture/

What is the best way to determine tone in the short stories "A Rock Trying to Be a Stone" by Sergio Troncoso and "Guests of the Nation" by Frank O'Connor?

The best way to determine tone in "A Rock Trying to Be a Stone" and "Guests of the Nation" is to examine diction and syntax. For example, in "A Rock," the diction reflects the socioeconomic status of the barely educated youths dealing with a brutal lack of opportunity: the narrator holds most emotion in check, stating facts coldly, objectively. In "Guests," the diction reflects the narrator's generally pleasant emotion as he describes people and events, beginning his story with "Well, chums...?" Syntax elements are different in each, the first having short, choppy sentences, and the second having longer, more flowing sentences.
Tone is the attitude, perspective, and emotion that the speaker has toward the subject matter being written about. Tone is detected in the diction a speaker uses and elements of syntax. To avoid a mistake often made, remember that tone is not detected in setting. Setting reveals something very different from tone: mood. Setting establishes mood within the story; diction and syntax establish author tone external to the story. Tone descriptors identify attitude (a way of thinking reflected in behavior), perspective (point of view regarding something), and emotion (feeling occurring toward someone, something, or some occurrence). Tone may be critical and subjective or critical and objective; it may be solemn or cynical. Tone may also be humorous or straightforward (think of "Garfield" compared with an economics textbook).
The clearest connection of diction to conflict and theme appears in the titles of both short stories. In "A Rock Trying to Be a Stone," the difference between a rock and stone highlights both conflict and theme as Turi tries to understand and do right in an environment that doesn't offer much opportunity for either. He is a rock trying to be a [gem] stone in a rocky environment. In "Guest of the Nation," the difference between guest and hostage dramatizes the conflict and theme as "chums" have to choose between duty to friends and duty to country. The bitterly ironic contrast between a guest protected by the nation and a hostage shot by the nation highlights the struggle the men are going through.
https://sites.google.com/site/makingsenseoftext/text/author-style/syntax

In Janus, what are Andrea's traits, strengths, or weaknesses? As the story progresses, to what degree do her traits become more or less prominent, and how does that work out for the character?

Andrea is a real estate agent, and she is a cunning real estate agent.  This single trait is both a strength and a weakness for Andrea.  It's a strength because she knows how to cunningly display a property for prospective buyers.  She will light a fire in the fireplace to make the house seem more welcoming, she will put special fragrances in the air, and she will even bring her dog if a possible buyer is a dog lover.  Andrea refers to these tactics as her "tricks."  She's a shrewd real estate agent, and it has made her successful.  I see that as a strength of hers.  
Unfortunately, Andrea's cunning and shrewd practices have started to consume her.  She becomes obsessed with the bowl's ability to help sell a house.  She reveres the bowl with superstitious fanaticism.  Even beyond that, Andrea begins to think of the bowl as a person that she has a relationship with.  

Could it be that she had some deeper connection with the bowl--a relationship of some kind?

As the story progresses, readers come to realize that Andrea is a broken, hollow, and empty woman.  Her marriage is intact in only legal terms.  She feels a closer kinship with the bowl than she does with her own husband.  As the story closes, readers finally learn why this bowl means so much to her.  

She had first seen the bowl several years earlier, at a crafts fair she had visited half in secret, with her lover.

The bowl reminds Andrea of her former lover.  He left Andrea when she refused to leave her husband for him.  The bowl is more than just a realtor's trick to her.  It's Andrea's way of trying to stay emotionally close to a part of her life that has long passed her by.  Her marriage is dead.  Her affair is broken, yet in her eyes, the bowl is still perfect. 

In its way, it was perfect: the world cut in half, deep and smoothly empty.

Andrea's devotion to bowl is perhaps her way of trying to hang on to something from a life that she no longer is able to have.  Andrea is very much like the bowl.  She appears perfect; however, she and the bowl are both ultimately hollow and empty.