What is the importance of mathematics in today's world?

I would say that there are practical, pragmatic, and poetic reasons to study mathematics.

Practical reasons: You will need to be able to compute areas for paint, carpet, sod etc..., compute interest payments and charges, keep a bank account including budgeting and investments, track changes such as your car's mpg, the rate of electric/water/data usage, etc... Statistical knowledge is absolutely necessary to vote for your best interests, to understand when a person or company might be giving you the wrong impression, to understand how to interpret scientific results outside of your field of expertise, etc...

Pragmatic reasons: For better or worse mathematics is used as a gatekeeper. Universities and colleges use math scores to determine who to accept as well as placement. Popular programs use mathematics to screen applicants. In large cities you would take a basic skills test in order to get into a trade union or be trained as a fireman, etc... In today's world math is used determine where you can live, what kind of things you can buy, the most efficient way to move stuff, and many applications of maximizing profits/ minimize costs.

Poetic reasons: Mathematics is a creative activity closely related to poetry and musical expression. There are rules but there are times/places to bend or break the rules. Understanding the beauty of a logical argument, a proof without words, or an organizational construct uplifts the human spirit.

You drop a stone down a well. You hear the splash 2.03 s later. How deep is the well? The speed of sound in air is 343 m/s.

A stone is dropped down a well and the sound of the splash when it strikes the water is heard 2.03 s later. The depth of the well has to be determined.

Let the depth of the well be L meters. The acceleration of the stone due to the gravitational force of attraction is assumed to be constant at 9.8 m/s^2. The speed of sound in air is 343 m/s.

As the stone falls it takes a time t to strike the water below. The equation `L = 0*t + (1/2)*(9.8)*t^2` shows the relation between the depth of the well and the time taken for the stone to fall.

After the stone strikes the water, the time taken by sound to travel upwards and be heard is `L/343` s. This gives us two equations.

`L/343 + t = 2.03` ...(1)

`L = 0*t + (1/2)*(9.8)*t^2` ...(2)

From (1)

`L = (2.03 - t)*343`

Substituting this in (2)

`(2.03 - t)*343 = (1/2)*(9.8)*t^2`

`2.03*343 - 343*t = 4.9*t^2`

Solving this equation gives `t = +-(21*sqrt(310)-350)/10` . The negative root can be ignored.

Substituting `t = +-(21*sqrt(310)-350)/10` in `L = (2.03 - t)*343` gives:

`L = (2.03 -(21*sqrt(310)-350)/10)*343`

L = 19.0998 m

The depth of the well is 19.0998 m.

What does Macbeth mean when he says “We hear our bloody cousins are bestowed in England and Ireland not confessing their parricide, filling...

This quote is from Act III, Scene 1 of Macbeth. Macbeth is speaking here to Banquo, and when he says "bloody cousins," he is referring to Malcolm and Donalbain, the sons of the murdered King Duncan. By using the adjective "bloody," he suggests that it is they who murdered the king. This, he and others have already said in the play, is why they must have fled to England and Ireland, respectively, in the first place. Of course, Malcolm and his younger brother Donalbain, Macbeth tells Banquo, are refusing to confess their involvement in the murder (which, the audience knows, Macbeth was actually responsible for). But even worse, they are involved in "strange invention," or spreading stories about the murder. We can assume, based on context, that they are rallying opposition to Macbeth by telling the truth—that he is Duncan's murderer. Macbeth correctly perceives this as a threat to his power.

What are the major changes Russel went through in Dogsong?

Dogsong is a coming-of-age tale of a 14-year-old young man, Russel. Russel is an Inuit who is caught up in a society that is slowly forgetting its Inuit traditions in favor of new ways of living and beliefs. At the beginning of the book, Russel could best be characterized as angry, restless, and confused. Perhaps confused is the best single word to describe him, because he doesn't understand why he is restless and angry.

Several things happen throughout the book that help Russel's demeanor and outlook change. In many instances, these changes are subtle. However, when the sum of all of these subtle changes is considered, it is obvious that Russel has significantly changed from the beginning of the book to the end.

 Some of the major changes include:

• At the beginning of the book, Russel feels there is something wrong with the way things are now, though he is not able to articulate it. As he comes closer to nature and to his cultural roots, he begins the realization process that he longs to live the way that Inuits used to live. 

• At first, Russel has difficulty trusting the dogs with whom he is traveling. When his lack of trust in them leads to the group getting lost, he instead chooses to trust them. As he does so, he finds that the dogs are able to guide him. Gradually, he and the dogs are on the same wavelength and he is essentially able to be of one mind with them.

• When Russel first leaves Oogruk and finds himself alone with just the dogs and his weapon, he is terrified. However, as Russel begins to trust the his own instincts, he begins to thrive in his surroundings. He finds that he can be the kind of man he wants to be.

The following sentence is ambiguous. Provide two explanations which illustrate the different meaning of the sentence. "They are visiting relatives."

The key word here is “visiting.” Is it used as a verb, or as an adjective?

If “visiting” is used as a verb in combination with the helping verb “are,” then this use is considered a progressive aspect of its form. The action is in progress. The visiting is going on right now. “They” refers to a group of people. This group happens to be in the process of visiting its relatives, somewhere. The implication is that at least a bit of traveling was involved in reaching the destination where these relatives live.

If “visiting” is used as an adjective, then it modifies the plural noun “relatives.” Now the main action switches to the other group mentioned in the sentence. “They” is a pronoun referring to the “visiting relatives.” The word “are” merely indicates a state of being. The sentence describes who these people are. They happen to be relatives who are visiting someone’s home. For how long, we don’t know.

How is a club good different from a common good?

When we discuss types of goods in economics, we have to look at two characteristics of those goods.  We have to ask if the goods are rival in consumption.  This means that, if one person consumes the good, there is less of it for someone else to consume.  We have to ask if people who do not pay can be excluded from consuming the goods.  Based on the answers to these two questions, we categorize the goods.  A club good is one that is excludable but not rival.  A common good is rival but not excludable.

A club good is a good that we can withhold from someone who does not pay for it.  An example of this would be a cell phone network.  Companies have technology that prevents consumers from making calls on their networks if those consumers have not paid for the right to do so.  However, a club good is not rival.  Looking at our cell phone example, if I pay a certain company to use their network, it does not prevent you from paying them for the same service.  I do not use up their network in any significant way, so consumption of this type of good is nonrivalrous.

A common good is just the opposite.  There is rival consumption, but we cannot practically exclude people from using the resource.  The classic example of this is fish in the ocean.  If I catch a certain bunch of fish in my net, you cannot catch those same fish.  If we catch too many, they will eventually run out.  In this way, the consumption of the good is rival.  However, if a company owns part of the ocean, it cannot practically prevent people from fishing there.  Only a government has the means to patrol large areas of ocean and prevent people from fishing there if they have not paid for the right to do so.  In other words, the fish in the ocean are nonexcludable.

The difference between these two, then, is that common goods are rival but not excludable while club goods are excludable but not rival.

What is the role of social media in education?

I would say social media has more than one role in education, and these various roles confer significant advantages and disadvantages. Educators have the responsibility to ensure students benefit from social media in a classroom setting.

Social media can be thought of as another technological tool in a teacher's toolbox. Social media can constructively be used in many ways. For example, a classroom project might be creating a Facebook page for a literary character by choosing favorite quotes, books, and music for the character and uploading photos of what the character might like.  Think about what a page for Jay Gatsby might look like! It might include photos of gangsters and flappers or a clip of people doing the Charleston. Many authors have a Facebook presence today, too, and students could comment on their pages, saying what they like or do not like about their writing. Many authors are happy to accept new Facebook friends. A group can be created on Facebook for group projects or for the entire class. Twitter can be used to summarize a story or to write poems such as haikus. LinkedIn, a form of social media used for professional networking, can be used to construct a student's professional appearance. Social media can also be used as object lessons in how not to bully others or how not to have uncivilized discourse online. These are just a few of the ways a creative teacher can use the resources of social media in education. 

The disadvantages of social media lie in its ability to distract, celebrate the trivial, present negative behaviors to others, and harm mental health and relationships. Students today often cannot seem to put down their devices and focus in their classrooms or on their homework assignments. Some teachers require all electronic devices to be off in the classroom, since it is difficult to control their use. There is little of educational value in learning what a classmate ate for breakfast or looking at more pictures of animals. They have also caused harm in the lives of students foolish enough to post pictures of themselves drinking alcohol or using illegal drugs. Relationships on social media, for teachers and students, can be fraught with risk. Feelings are frequently hurt on Facebook, I am told; people are bullied, and research supports the notion that young people who are users of social media can have feelings of social isolation and depression because of it. While this is not a direct educational harm, it can take away from the feelings of community that every classroom should ideally have. Teachers must be very careful, too, in their social media participation, since all of it can be scrutinized. The faces we present on social media must be above reproach!  

Social media is here to stay, I think. It can be a force for good or evil. It is up to educators to use social media in a way that enhances learning.