College Algebra, Chapter 4, 4.1, Section 4.1, Problem 68

Suppose a a certain drug is taken orally, the concentration of the drug in the patient's blood stream after $t$ minutes is given by $C(t) = 0.06t - 0.0002t^2$, where $0 \leq t \leq 240$ and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?

The function $C$ is a quadratic function with $a = -0.0002$ and $b = 0.06$. Thus, its maximum value occurs when

$\displaystyle t = - \frac{b}{2a} = - \frac{0.06}{2 (-0.0002)} = 150$ min.

The maximum concentration is $C(150) = 0.06(150) - 0.0002 (150)^2 = 4.5 $ mg/L.

So the serum will reached maximum concentration is $150$ min. and the maximum concentration of serum is $4.5$ mg/L.

Why is Robbie in disgrace?

Robbie Turner is in disgrace because of the lie of an upper-class child named Briony Tallis. She witnesses Robbie, the lower-class son of a maid, flirting with her sister, Cecilia. Briony, only thirteen years old, misinterprets Robbie's actions, as he and Cecilia, long friends, are now in love with each other. When Briony sees her cousin Lola being attacked by an assailant, Briony thinks that Robbie is Lola's attacker and accuses him. As a lower-class person, Robbie has few rights in the justice system, and he is sent to prison. Briony later reveals to the reader that she lied about having seen Robbie attacking Lola (and that it was actually a sexual encounter) and that the real attacker was someone else—a man who wound up later marrying Lola. However, Briony is not able to reveal the truth in his lifetime, and Robbie dies in the war in a state of disgrace.

What are some quotes about the fire in the novel Lord of the Flies?

At first, the boys, under the direction of Ralph, attempt to light a fire as a signal to passing ships. Here is the description of their attempt from Chapter 2:


"On one side the air was cool, but on the other the fire thrust out a savage arm of heat that crinkled hair on the instant. Boys who felt the evening wind on their damp faces paused to enjoy the freshness of it and then found they were exhausted. They flung themselves down in the shadows that lay among the shattered rocks. The beard of flame diminished quickly; then the pile fell inwards with a soft, cindery sound, and sent a great tree of sparks upwards that leaned away and drifted downwind. The boys lay, panting like dogs."

Their attempt to keep the fire going and to generate smoke is fruitless, as they find it's impossible to keep the fire burning at that level of intensity. They are quickly exhausted from the effort, but they try to keep the fire going in the increasingly vain hope of rescue. 
Piggy notices that the fire has burnt out of control. The author describes the raging fire in the following way:



"Smoke was rising here and there among the creepers that festooned the dead or dying trees. As they watched, a flash of fire appeared at the root of one wisp, and then the smoke thickened. Small flames stirred at the trunk of a tree and crawled away through leaves and brushwood, dividing and increasing. One patch touched a tree trunk and scrambled up like a bright squirrel. The smoke increased, sifted, rolled outwards. The squirrel leapt on the wings of the wind and clung to another standing tree, eating downwards. Beneath the dark canopy of leaves and smoke the fire laid hold on the forest and began to gnaw. Acres of black and yellow smoke rolled steadily toward the sea. At the sight of the flames and the irresistible course of the fire, the boys broke into shrill, excited cheering."

The fire at first seems to rally the boys to action, but, as Piggy notes, the fire is out of control and is consuming all their firewood. They have spent themselves with the first effort to try to be rescued, and their misspent energies do not bode well for their chances at rescue.
Ralph continues to try to keep the fire burning, even as Jack and his followers branch off into a rival band of hunters. Ralph tells the little boys who still follow him:

"'The fire's the most important thing. Without the fire we can't be rescued. I'd like to put on war-paint and be a savage. But we must keep the fire burning. The fire's the most important thing on the island, because, because--'He paused again and the silence became full of doubt and wonder. Piggy whispered urgently. 'Rescue.'"

Notably, Ralph can't even remember the reason he is keeping the fire burning, and the boys eventually let the fire burn out when a ship is passing. Ralph says, "If I blow the conch and they don't come back; then we've had it. We shan't keep the fire going. We'll be like animals. We'll never be rescued." The fire and the conch are the boys' connection to rescue and to civilization. When the fire burns out, it's a symbol that they no longer care about being rescued and have descended into brutality and chaos.

Early in the novel, Ralph comments that they must make a signal fire in order for them to be rescued. He hopes that a passing ship will notice the smoke and stop to pick them up. As soon as they successfully light their first fire, Ralph yells for the boys to grab more wood to feed the flames. Golding writes,

"Life became a race with the fire and the boys scattered through the upper forest. To keep a clean flag of flame flying on the mountain was the immediate end and no one looked further" (56).

The signal fire is a symbol of hope, rescue, and civilization throughout the novel. The boys' focus on and enthusiasm for maintaining the signal fire illustrates their initial affinity for civility.
In Chapter 5, Ralph holds a meeting to discuss how the boys are not completing the tasks agreed upon during the assemblies. After commenting on the shabby huts, lack of water in the coconuts, and the location of where the boys are relieving themselves, he says,

"The fire is the most important thing on the island. How can we ever be rescued except by luck, if we don’t keep a fire going? Is a fire too much for us to make?...Look at us! How many are we? And yet we can’t keep a fire going to make smoke. Don’t you understand? Can’t you see we ought to—ought to die before we let the fire out?" (Golding 31).

Ralph is pleading with the boys to focus on the importance of having a signal fire, but they do not share the same feelings. The inability to maintain a signal fire correlates with the boys' inability to create a civil society on the island. Their indifferent feelings about maintaining a signal fire reflect their gradual descent into savagery.

The fire is an important symbol in The Lord of the Flies. It represents civility and the boys' hopes for rescue, but, in the case of the forest fire, it also demonstrates how the boys can quickly lose control over a powerful and deadly force.

When Nag first meets Rikki-Tikki, why does he try to keep Rikki-Tikki engaged in conversation?

When Rikki-tikki-tavi initially meets Nag for the first time in the garden, he comes face-to-face with an opposing, intimating figure. Nag understands that a mongoose's presence means sure death for himself and Nagaina as he stares down Rikki-tikki. Nag attempts to keep Rikki-tikki's attention by speaking to him and asks Rikki-tikki why it is okay for him to eat eggs if he is not allowed to eat birds. As Rikki-tikki is about to respond, Darzee warns him to look out as Nagaina strikes. Rikki-tikki quickly pounces in the air as high as he could jump and narrowly avoids Nagaina's venomous bite. Essentially, Nag attempts to distract Rikki-tikki's attention so that his wife can ambush him from behind. Nagaina is quietly biding her time to strike and attacks Rikki-tikki before he could respond to her husband. Nag and Nagaina's initial ambush attempt emphasizes their clever, sneaky nature and establishes them as the main antagonists in the story.

At first Nag introduces himself to Rikki-Tikki in a way designed to strike fear into the mongoose's heart and to make Rikki feel like a young thing. But Rikki's mongoose nature won't allow him to remain frightened for long, and Nag knows what Rikki knows: that mongooses in the garden mean death for snakes. But in this scene, Nag knows something that Rikki does not at first, and that is that Nagaina, Nag's wife, is slithering up in the grass behind Rikki. Therefore Nag tries to get Rikki into a philosophical discussion about whether it's right for Rikki to eat eggs if it's wrong for Nag to eat birds. He is hoping to distract Rikki long enough to allow Nagaina to strike him from behind. But Darzee sings out with a warning, and Rikki jumps high in the air as Nagaina's strike misses. The next time Nag and Rikki meet, there is no discussion. It's a fight to the death--Nag's death. 

What were some challenges of Mesopotamia?

Mesopotamian civilization faced many challenges. The first was that the neolithic transition led to increasing urbanization and population density. This meant that areas immediately surrounding cities needed to feed increasing numbers of people while, at the same time, houses and temples were being built on land that might have been used for agriculture. Continuous intensive agriculture eventually depletes and salinizes the soil, meaning that new areas, more distant from the urban center, need to be farmed. To transport food, therefore, one faces the challenge of building transportation networks.
Next, in urban areas, human waste disposal became a problem. Human waste can contaminate drinking water supplies. Also, crowded living conditions can lead to epidemic diseases.
Because the "fertile crescent" was an area rich in arable land and natural resources, nomadic tribes were constantly trying to pillage from the cities and other groups were trying to conquer them.

Was Shinji's suicide beautiful?

Lieutenant Shinji commits suicide in "Patriotism" using the Japanese method of seppuku, which is a ritual form of suicide by way of disembowelment. One could consider his act to be beautiful since he voluntarily chooses death over the option of following military orders and betraying his friends, who are guilty of plotting to overthrow the government.
Both Lieutenant Shinji and his wife, Reiko, choose suicide and immediately make peace with the decision, carefully preparing their home for death and even making passionate love before committing that final act. Reiko calmly watches her husband's violent finale, then unlocks the door so their bodies will be soon found, before slitting her own throat with the dagger she has held onto since their wedding night.
One could argue that beauty is demonstrated in the unbreakable commitment between husband and wife, in the calm resolve with which they undertake the most grim task of all, and in their choice of an honorable death out of loyalty to conscience.

Prove that the time period of oscillations of a pendulum is independent of the mass or material of the bob.

First we will treat the pendulum as a point mass (m) a distance L from the axis of rotation. Use newtons second law with the moment of inertia I about the axis of rotation, and alpha is the angular acceleration.
tau=I*alpha
F_g*L*sin(theta)=I*(d^2(theta))/dt^2
-m*g*L*sin(theta)=(m*L^2)(d^2(theta))/(dt^2)
0=L/g*sin(theta)+(d^2(theta))/dt^2
If we assume the angular displacements are small, we can use the small angle approximation.
0=L/g*theta+(d^2(theta))/dt^2
This is a differential equation that has a solution of the form
theta(t)=theta_0*cos(omega*t+phi)
Where the angular frequency is
omega^2=g/L
The period of oscillation is
T=2*pi*omega=2*pi*(g/L)^(1/2)
The period of oscillation is indeed independent of the mass of the pendulum.
https://courses.lumenlearning.com/boundless-physics/chapter/periodic-motion/

https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendula.html