College Algebra, Chapter 9, 9.1, Section 9.1, Problem 78

Suppose that a biologist is trying to find the optimal salt concentration for the growth of a certain species of mollusk.
She begins with a brine solution that has $\displaystyle 4 \frac{\text{g}}{\text{L}}$ of salt and increases the concentration $10\%$ every day.
Let $C_0$ denote the initial concentration and $C_n$ the concentration after $n$ days.
a.) Find a recursive definition of $C_n$
b.) Find the salt concentration after 8 days.

a.) If the initial concentration is 4, then $C_0 = 4$. So, $C_n = 0.10 C_{(n-1)} + 4$

b.) After 8 days, the salt concentration will be

$
\begin{equation}
\begin{aligned}
C_1 &= 0.10 C_{(1-1)} + 4 &&& C_2 &= 0.10 C_{(2-1)} + 4\\
\\
&= 0.10 C_0 + 4 &&& &= 0.10 C_1 + 4\\
\\
&= 0.10 (4) +4 &&& &= 4.44\\
\\
&= 4.40 &&& C_3 &= 0.10 C_{(3-1)} + 4\\
\\
C_4 &= 0.10 C_{(4-1)} + 4 &&& &= 0.10 C_2 + 4\\
\\
&= 0.10 C_3 + 4 &&& &= 4.444\\
\\
&= 4.4444 &&& C_5 &= 0.10 C_{(5-1)} + 4\\
\\
C_6 &= 0.10 C_{(6-1)} + 4 &&& &= 0.10 C_4 + 4\\
\\
&= 0.10 C_5 + 4 &&& &= 4.44444\\
\\
&= 4.444444 &&& C_7 &= 0.10 C_{(7-1)} + 4\\
\\
C_8 &= 0.10 C_{(8-1)} + 4 &&& &= 0.10C_6 + 4\\
\\
&= 0.10 C_7 + 4 &&& &=4.4444444\\
\\
&= 4.44444444
\end{aligned}
\end{equation}
$

Therefore, the salt concentration after 8 days is $\displaystyle 4.44444444 \frac{\text{g}}{\text{L}}$

How does a democratic government work?

To answer this question, I will assume you are asking about how a democratic republic works? In a country like the United States, which has a democratic republic, we elect leaders to make the laws for us. The people vote for whom they want to represent them in government. These people are then expected to carry out the wishes of the majority of the people they represent. In theory, the power lies with the people since they have the ability to choose who will represent them.
If you are asking about a direct democracy, then all of the people who are citizens would be involved in the process of making the laws. This would be very difficult today if a country expected all of its citizens to be involved in the process. Ancient Athens is often used to describe a direct democracy, but even in ancient Athens, there were limits on who could participate. Only male citizens were allowed to participate in ancient Athens. Many who could participate in ancient Athens didn't participate because they had military or work commitments.
A democratic republic or representative democracy is an example of a democracy used in some countries today.
https://www.history.com/topics/ancient-greece/ancient-greece-democracy

https://www.ushistory.org/gov/1c.asp

College Algebra, Chapter 8, 8.2, Section 8.2, Problem 10

Determine the vertices, foci and eccentricity of the ellipse $\displaystyle \frac{x^2}{16} + \frac{y^2}{25} = 1$. Determine the lengths of the major and minor
axes, and sketch the graph.
If we rewrite the given ellipse to $\displaystyle \frac{x^2}{4^2} + \frac{y^2}{5^2} = 1$, then it will have a form of
$\displaystyle \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$. So, the vertices of the ellipse is $(0,\pm a)$ which is $(0,\pm 5)$ and the foci of the ellipse
$c = \sqrt{a^2-b^2} = \sqrt{5^2 - 4^2} = 1$ is $(0,\pm 1)$. To determine the eccentricity of the ellipse we use the formula
$\displaystyle e = \frac{c}{a} = \frac{1}{5} = 0.20$. Next the lengths of the major and minor axes is determined to be $2a$ and $2b$ respectively.
So, the length of the major axis is $2(5) = 10$ while the length of the minor axis is $2(4) = 8$. Therefore, the graph is

y' = sqrt(x)y Solve the differential equation

An ordinary differential equation (ODE) has differential equation for a function with single variable. A first order ODE follows (dy)/(dx)= f(x,y) .
It can also be in a form of N(y) dy= M(x) dx as variable separable differential equation..
 To be able to set-up the problem as N(y) dy= M(x) dx , we let y' = (dy)/(dx).
 The problem: y'=sqrt(x)y becomes:
(dy)/(dx)=sqrt(x)y
Rearrange by cross-multiplication, we get:
(dy)/y=sqrt(x)dx
Apply direct integration on both sides: int (dy)/y=int sqrt(x)dx  to solve for the general solution of a differential equation.
 
For the left side, we applying basic integration formula for logarithm:
int(dy)/y= ln|y|
For the right side, we apply the Law of Exponent: sqrt(x)=x^(1/2) then follow the Power Rule of integration: int x^n dx = x^(n+1)/(n+1)+C
int sqrt(x)* dx= int x^(1/2)* dx
                   = x^(1/2+1)/(1/2+1)+C
                  = x^(3/2)/(3/2)+C
                  = x^(3/2)*(2/3)+C
                  = (2x^(3/2))/3+C
Combining the results from both sides, we get the general solution of the differential equation as:
ln|y|=(2x^(3/2))/3+C
or
y =e^(((2x^(3/2))/3+C))

How do contemporary audiences react to one of the scenes in the play Romeo and Juliet?

Let’s take the balcony scene, Act II, Scene 2, as an example for examining contemporary audience reactions to Romeo and Juliet. This scene is one of the most famous scenes in all of Shakespeare. There is no mention of an actual balcony, but it is traditionally portrayed that way to give Juliet more room to move around. She could just as easily be at a window. Either way, Juliet is up high and Romeo is down below in the Capulet's orchard.
On the one hand, a modern audience might like the scene because people know what to expect. Even people who don’t know much Shakespeare know some of the lines, and everyone knows the story of Romeo and Juliet. This could be a good or bad thing. Either way, this scene is definitely iconic.
Act II, Scene 2 takes place after Romeo and Juliet have seen each other and spoken briefly at the ball. They fell in love at first sight. In this scene, Romeo and Juliet speak to each other without speaking directly to each other, as neither knows the other is there.
Juliet is frustrated that she has fallen in love with a Montague, the sworn enemies of the Capulets.

JULIET
O Romeo, Romeo! wherefore art thou Romeo?Deny thy father and refuse thy name;Or, if thou wilt not, be but sworn my love,And I'll no longer be a Capulet (Act II, Scene 2).

One of the reasons this scene is so powerful is the contemplation of the meaning of name and family. In the famous line about the importance of a name, Juliet reminds us that our names are not a fundamental part of who we are. Romeo, likewise, is unconcerned with his name.  
Name is also metaphorical for family. These two young people are willing to cut ties with their families to be together. It’s powerful and dramatic. This scene is about the transformative nature of love, aa well as its tendency to blind us to practicality, and perhaps reality. 
This scene might also stand out to a contemporary audience because two young people agree to get married before they really know each other. In Shakespeare’s day, that may have seemed romantic.  Today, modern audiences might just say Juliet is too young and she and Romeo are getting married too quickly.

ROMEO
O, wilt thou leave me so unsatisfied?
JULIET
What satisfaction canst thou have to-night?
ROMEO
The exchange of thy love's faithful vow for mine.
JULIET
I gave thee mine before thou didst request it:And yet I would it were to give again (Act II, Scene 2).

Modern audiences know how Romeo and Juliet's relationship ends. It is hard to watch this scene and not think about how the two young people will end up dead in a matter of days due to a combination of fate, poor judgement, and hasty decision making.
Modern audiences will likely still enjoy Romeo and Juliet’s story, as it is considered a love for the ages. Romeo and Juliet meet and have a whirlwind romance, never really getting to know one another before their deaths. The balcony scene exemplifies the challenges of their relationship.

Regarding the ongoing Syrian Civil War, who are the important actors and what are their preferences over possible outcomes? What factors help or hinder states in reaching a cooperative solution in this war? And has any international institution been helpful in trying to resolve this civil war?

There are actually three key groups at play in the Syrian Civil War, which has now been underway for over six years. These groups are:
Islamic State, or Daesh, a terrorist organization who are opposed to the current president, Bashar Al-Assad.
Bashar Al-Assad as a representative of the legal government of Syria, and his supporters. Importantly, he is backed by the national army.
"Rebel" fighters, who are not allied to Islamic State but oppose the government.
Unrest began in Syria in 2011 because it was felt that the government was not sufficiently democratic: protests led to the murder of five people by the government, which drove the rest of the country to action. While the protesters originally only wanted greater transparency in government, they soon came to call for the removal of Al-Assad and his government from power. Al-Assad continues to refuse. So, the government's goal is to reassert its power over the country. The rebels' goal is to dissolve the current government. Meanwhile, Islamic State also wants the government to be dissolved, but for its own reasons: it has been taking over lands in Iraq and expanding into Syria. Its aim is to spread its hateful extremist message as far and widely as possible. In actual fact, the civil war in Syria is three-sided, in that both the government and the rebel fighters are attempting to beat IS, which is not "on the side of" the rebels, as they are opposed to IS as well as to Al-Assad. They want democracy, which IS most certainly does not represent.
In attempting to reach a solution, this is certainly presenting a problem. Civil wars are always more complicated when there are more than two groups fighting; it is suggested that there are thousands of different factions with differing goals fighting against the Syrian government. Meanwhile, both the US and the UK have intervened with air-strikes against IS, but these have also had the side-effect of destroying many civilians' houses. However, an agreement between the US and Russia in 2013, which resulted in the destruction of chemical weapons in Syria, has lessened some tensions.
Many European countries have taken in significant numbers of refugees fleeing the war in Syria, but this has been hotly contested in some areas, with the US and UK governments, in particular, anxious about the possible spread of Muslim extremists.
https://www.bbc.co.uk/newsround/16979186

In Casablanca, from what does Rick need redemption and how is his redemption achieved?

To understand why Rick needs redemption, let's look at what Rick and his rival for Isla's love, Victor Lazlo, each represent. Victor represents the idealistic vision of a cause. He is willing to give everything to fight against tyranny. He has spent time as a political prisoner and appears even stronger for it. Even after achieving his own freedom, he puts himself back into danger. Rick, on the other hand, ostensibly cares for nobody but himself. Before the war, he and Ilsa lived carefree in Paris, not concerning themselves with the coming conflict. He represents the notion that everything will be fine if one does not get involved with the struggles of nations at war.
We do, however, see cracks in that veneer throughout the film. For example, he helps a young refugee couple purchase their travel papers; although, he appears reluctant to do so. This proves that he does know justice from injustice. However, he has cultivated the image of a callous businessman who takes no sides in the conflict surrounding him. He unemotionally serves Germans and French partisans alike in his bar. It is from this self-imposed mask of selfish indifference he wears that he must redeem himself.
Rick finally does receive redemption during the final scene of the film. He has made a deal with Ilsa, that he will help Lazlo escape, but only if she agrees to stay with Rick. Once again, Rick has shown his callous selfishness by crafting this deal. It appears that he has even convinced Ilsa that this is the satisfactory course of action. Ilsa has confessed that she still loves Rick but is committed to helping Lazlo with his cause. By staying with Rick to help Lazlo escape, perhaps everyone is getting what they want. Rick's moment of redemption comes when he tells Ilsa that she needs to leave with Lazlo after all. In fact, he already has her travel documents prepared. By letting her go, he has finally committed a selfless act to forward the cause of the resistance against fascism. He has made a stand and taken a side. This is further cemented when he shoots Major Heinrich Strasser, the German officer who tries to stop Lazlo's escape. There is no going back for Rick after this. He has given up everything: his love, his bar, even his safety for the cause.