Does Mary have a guilty conscience?

We don't know for sure if Mary does have a guilty conscience. However, if she does, then she doesn't really show it. Although her killing of Patrick is carried out in the heat of the moment, it is still a crime all the same. And Mary's subsequent behavior doesn't suggest that she feels any remorse for what she's done. She meticulously sets about covering up her crime and establishing an alibi. There's the coldly methodical way she concocts a story for the police. Then, of course, there's her novel method for disposing of the evidence. Again, one would have to say that this doesn't appear to be the behavior of someone with a guilty conscience.
Besides, if Mary really did feel any guilt or remorse for what she's done, then she has ample opportunity to confess to the police when they arrive. It's clear from all the available evidence that this is not a premeditated act; no charge of murder would've been filed against Mary as a consequence. Yet she chooses not to confess, and this, in conjunction with all the other evidence, indicates that Mary doesn't really have a guilty conscience at all.

y=sqrt(4-x^2) , -1

To calculate the surface area generated by curve y=f(x) revolving about x-axis between a and b, we use the following formula
S_x=2pi int_a^b y sqrt(1+y'^2)dx
Let us therefore first find the derivative y'.
y'=1/(2sqrt(4-x^2))cdot(-2x)=-x/sqrt(4-x^2)
y'^2=x^2/(4-x^2)
We can now calculate the surface.
S_x=2pi int_-1^1sqrt(4-x^2)sqrt(1+x^2/(4-x^2))dx=
2pi int_-1^1sqrt(4-x^2)sqrt(4-x^2+x^2)/sqrt(4-x^2)dx=
2pi int_-1^1 2dx=4pi x|_-1^1=4pi(1+1)=8pi
The area of surface generated by revolving the given curve about x-axis between -1 and 1 is 8pi.
Graphs of the curve and the surface generated by curve's revolution can be seen in the images below.

College Algebra, Chapter 4, 4.6, Section 4.6, Problem 72

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function $\displaystyle r(x) = \frac{2x^3 + 2x}{x^2 - 1}$.

By applying Long Division,







By factoring,

$\displaystyle r(x) = \frac{2x^3 + 2x}{x^2 - 1} = \frac{2x (x^2 + 1)}{(x - 1)(x + 1)}$

Thus,

$\displaystyle r(x) = \frac{2x^3 + 2x}{x^2 - 1} = 2x + \frac{4x}{x^2 - 1}$

Therefore, the line $y = 2x$ is the slant asymptote.

The vertical asymptotes occur where the denominator is , that is, where the function is undefined. Hence the lines $\displaystyle x = 1$ and $x = -1$ are the vertical asymptotes.

To sketch the graph of the function, we must first determine the intercepts.

$x$-intercepts: The $x$-intercepts are the zeros of the numerator, in our case, $x = 0$ is the only real $x$-intercept.

$y$-intercept: To find $y$-intercept, we set $x = 0$ into the original form of the function

$\displaystyle r(0) = \frac{2(0) (0^2 + 1)}{(0 -1)(0 + 1)} = 0$

The $y$-intercept is .

Next, we must determine the end behavior of the function near the vertical asymptote. By using test values, we found out that $y \to \infty$ as $\displaystyle x \to 1^+$ and $x \to - 1^+$. On the other hand as $y \to -\infty$ as $x \to -1^-$ and $\displaystyle x \to 1^-$. So the graph is

A 200-gallon tank is half full of distilled water. At time t = 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 3 gallons per minute. At what time will the tank be full?

Total volume of the tank=200 gallon
Given that, tank is half full,
so remaining volume of the tank to be filled=200-100=100 gallon.
Since 5 gallons of solution comes into the tank per minute and 3 gallons of solution goes out each minute,
Net flow of solution in the tank=5-3= 2 gallons per minute.
Time required=100/2=50
So, the time required for the tank be full = 50 minutes 

Is Sara a flat or dynamic character?

Sarah Ruth is a flat character as she remains religious, unattractive and uninterested in frivolities throughout the story. When Parker first meets her, he observes that he wouldn’t want anything to do with her, because of her “dry hot and rough hands” and her plain looks. Also, she is not attracted to his tattoos, something that quite surprises Parker as he had never before met a woman who did not like his tattoos. She refers to the tattoos as “vanity of vanities." It is her rejection of his tattoos that torments Parker.
Later on, after he marries her, Parker observes that she has not changed much. She is plain-looking and, to make matters worse, pregnant—something that makes her even less attractive to him. He thinks that she obsesses over righteousness. This is why she does not imbibe, smoke, use foul language, or even use body makeup.
However, Parker forgets that Sarah comes from a religious family and lives her life the only way she knows how to. Her father was a preacher who must have taught her the scriptures well. Parker may have thought that marriage would finally change Sarah Ruth into a doting, more agreeable wife. However, she remains unchanged and this makes him “gloomier than ever." The text states that each morning, Parker would leave determined not to return, yet came back in spite of everything.

In Cranes by Hwang Sun-won, what was Songsam's relationship with Tokchae like when they were children?

Although now torn apart by war and the separation of North Korea and South Korea, Songsam and Tokchae were quite close when they were children, as indicated by a few flashbacks throughout “Cranes.” In one particularly touching flashback, Songsam and Tokchae are stealing chestnuts from a tree when Songsam falls and suffers being pricked by chestnut needles all over his bottom. Tokchae patiently removes the needles and then hands his friend a bunch of chestnuts to enjoy.
The pivotal flashback is to when the boys were twelve and decided to secretly catch and keep a crane, only to let it go free when they feared it would be captured or killed by someone else. It is this memory that triggers Songsam’s drastic act of compassion; he forms an idea, using this memory of crane-catching, to let his old friend Tokchae escape capture and execution.

When Tokchae and Songsam were children, they were childhood friends with each other. 
The story gives some small details from the characters' pasts that indicate that the two men were fairly close friends.  For example, Songsam initially thinks that offering Tokchae a cigarette is bad timing, but then he remembers that Tokchae and he shared stuff with each other when they were children.

Then suddenly he thought that Tok-chae, too, must want a puff. He thought of the days when they used to share dried gourd leaves behind walls, hidden from the adults.

Near the story's conclusion, Tokchae and Songsam arrive at a field together.  The reader is told that this is the same field where the two men captured and freed a crane together as young boys.  

Once, when Song-sam and Tok-chae were about twelve, they had set a trap here, without the knowledge of the adults, and had caught a crane, a Tanjong crane.

Tokchae believes that he is about to be executed in this very same field, but because of their childhood bond, Songsam decides to let Tokchae run away.  Just as the two men let the crane go free as boys, Songsam is now letting Tokchae go free as well.  

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 6

a.) Find all zeros of $P(x) = x^5 + 9x^3 $ of $P$, real and complex

b.) Factor $P$ completely.



a.) We first factor $P$ as follows.


$
\begin{equation}
\begin{aligned}

P(x) =& x^5 + 9x^3
&& \text{Given}
\\
\\
=& x^3 (x^2 + 9)
&& \text{Factor out } x^3

\end{aligned}
\end{equation}
$


We find the zeros of $P$ by setting each factor equal to :

Setting $x^3 = 0$, we see that $x = 0$ is a zero. More over, setting $x^2 + 9 = 0$, we get $x^2 = -9$, so $x = \pm 3i$. So the zeros of $P$ are $0, 3i$ and $-3i$.

b.) Since the zeros are $0, 3i$ and $-3i$, by the complex Factorization Theorem $P$ factors as


$
\begin{equation}
\begin{aligned}

P(x) =& x (x - 3i) [x - (-3i)]
\\
\\
=& x (x - 3i)(x + 3i)

\end{aligned}
\end{equation}
$