How did railroads impact the settlement of the west around 1864?

Under President Abraham Lincoln in 1862, the Pacific Railway Act was approved.  With most of the nation focused on the Civil War, less attention was paid to westward expansion.  In previous decades, westward expansion had been an important focus in the United States.  People traveling west had gone by covered wagon or by horse.  This way of travel was slow and was filled with obstacles.  Wagons were made of wood and could break.  Horses and oxen...

Under President Abraham Lincoln in 1862, the Pacific Railway Act was approved.  With most of the nation focused on the Civil War, less attention was paid to westward expansion.  In previous decades, westward expansion had been an important focus in the United States.  People traveling west had gone by covered wagon or by horse.  This way of travel was slow and was filled with obstacles.  Wagons were made of wood and could break.  Horses and oxen could become ill and die.  The journey could take from four months to over a year.

In 1869, the Transcontinental Railroad was completed.  After this, travel from the East to the West was much quicker and more efficient.  People did not have to worry about feeding livestock or repairing wagons.  Instead, they paid a fare and boarded a train to go west.  By the year 1876, one particularly fast train had a direct route from New York City to San Francisco that took about four days.  This transformed westward expansion in the United States.  New immigrants arriving in New York City could quickly and easily move west, where the population was relatively small and there were more economic opportunities.

How did Renaissance artwork look different from artwork during the Middle Ages?

Because the field of artwork is broad, I will focus on painting. There are many ways in which medieval painting differed from that of the Renaissance, which is what makes the latter period so important as an era of innovation.

First, nearly all medieval painting is religious in its subject matter. A great deal of Renaissance painting is, too, but more liberties are taken in the portrayal of the icons. Renaissance painters individualized the figures,...

Because the field of artwork is broad, I will focus on painting. There are many ways in which medieval painting differed from that of the Renaissance, which is what makes the latter period so important as an era of innovation.

First, nearly all medieval painting is religious in its subject matter. A great deal of Renaissance painting is, too, but more liberties are taken in the portrayal of the icons. Renaissance painters individualized the figures, allowed them to emote, showed the natural movement of the body, and often depicted subjects according to classical standards of beauty. Medieval iconography is less inspired and quite redundant. 

Moreover—and this leads to the second major difference—there is no depiction of perspective in medieval painting. The figures appear flat and two-dimensional. Vanishing point perspective developed during the Renaissance. With this technique, a set of parallel lines are perpendicular to a picture plane, drawing one's eye to a single point in space, then outward. This allowed for the illusion of depth on a flat surface.

The depiction of children is another very important difference. Both medieval and Renaissance painters were interested in the depiction of Madonna and Son. Based on the images, however, one thinks that medieval painters had no concept of what children actually looked like. They are little adults: elongated and earnest in their expressions. Renaissance artists painted babies as they are in real life: chubby, playful, and fidgety.

Why is the work a scalar quantity?

A scalar quantity is a quantity that can be measured by one number. This is in contrast to a vector quantity, that has to be represented by two, or more, numbers. For example, velocity is a vector quantity because it indicates how fastan object is moving, and in what direction.

Force is also a vector quantity, because it represents the strength of the push or pull, and also the direction of that push...

A scalar quantity is a quantity that can be measured by one number. This is in contrast to a vector quantity, that has to be represented by two, or more, numbers. For example, velocity is a vector quantity because it indicates how fast an object is moving, and in what direction.

Force is also a vector quantity, because it represents the strength of the push or pull, and also the direction of that push or pull. Work, however, is a quantity that indicates the result of the force's action: it combines the force with displacement that occurred in the direction of that force. Mathematically, it is expressed as follows:

`W=vecF*Deltavecx`

This is called a scalar product of the two vectors: the force vector and the displacement vector. It is a scalar because the result of it is just one number, calculated as follows:

`W =|vecF||Deltavecx|*cos(theta).`

So, work is the product of the magnitudes of the force and the displacement, and the cosine of the angle between them (`theta` ).

Work can also be thought of as a change in the kinetic energy of and object. The kinetic energy is

`K = 1/2mv^2` , where m is the mass and v is the speed of the object. Since mass and speed are scalar quantities (speed is a magnitude of velocity, and this a scalar), kinetic energy is also a scalar quantity, so the work, which equals the change in kinetic energy, is also a scalar quantity.

In You are Not So Smart, what is McRaney asking you to believe about cognition and decision-making? How does this challenge your assumptions about...

McRaney is primarily asking you to understand the various ways in which your brain limits your perspective and thus your decision-making abilities. The very heuristics and other cognitive processes that allow your brain to function efficiently can also lead to perceptual weaknesses. McRaney hopes to challenge your assumption that you can trust your own thoughts. By understanding the limitations of your thoughts, you can overcome them through more conscious thought efforts and hone your ability to see things objectively.

Heuristics can be very helpful when it comes to processing events and experiences. The human brain can only process so much information at once, and heuristics categorize information and allow the mind to make reasonable predictions that save time and effort. For example, the availability heuristic allows your brain to solve problems more efficiently by calling to mind other events with similar themes. By drawing upon these experiences, you benefit from the accumulation of your life experiences rather than approaching each problem from scratch.

On the other hand, heuristics can also be harmful. Because heuristics simplify things, they make us vulnerable to stereotypes, assumptions, and oversimplification of complex ideas. An example of the harmful effects of the availability heuristic is the fact that most people fear flying more than driving an automobile even though flying is statistically much safer than driving. Because plane crashes are so widely reported in the news, the availability heuristic leads people to believe they are more common and thus that planes are more dangerous.

Past experience, cognitive biases, individual differences, age, the escalation of commitment, and belief in personal relevance are some of the most common factors that influence decision-making. Past experience provides a repository of knowledge your brain uses to draw from in reference to current problems and decisions. Cognitive biases limit your ability to see a decision in a fully objective light. Individual differences allow two people to see the same situation in significantly different ways. Age affects the maturity of the brain which, in turn, affects the way you perceive things and the extent of your decision-making abilities. The escalation of commitment refers to the tendency to hold onto existing ideas and values because you have become invested in them. The degree to which you believe you have personal relevance refers to the fact that you will make decisions with greater frequency and confidence if you perceive yourself as someone who has the authority to make decisions.

Madison argues in this essay that governments must, among their most pressing challenges, channel and control the activities and effects of...

In Federalist Number 10, Madison defines a faction as "...a number of citizens, whether amounting to a majority or a minority of the whole, who are united and actuated by some common impulse of passion, or of interest, adversed to the rights of other citizens, or to the permanent and aggregate interests of the community.”

Clearly, factions, as so defined, constitute a clear and present danger, representing as they do the triumph of sectional interests over those of society as a whole.

So, how should governments go about dealing with this implied threat? Madison makes two proposals. First, we should consider the possibility of removing the causes of factions. One way of doing this would be to restrict liberty. But immediately Madison steps back from the idea as its consequences would clearly be damaging:

But it could not be less folly to abolish liberty, which is essential to political life, because it nourishes faction, than it would be to wish the annihilation of air, which is essential to animal life, because it imparts to fire its destructive agency.

Air is one of the fundamental causes of fire, but we wouldn't try to reduce the danger of fire by restricting the air supply.

What else can be done? Madison considers another possible means of removing the causes of faction. This time, he considers the notion that we could create a society in which everyone has the same opinions and interests.

Aside from the practicalities involved, Madison's second proposal would've been unacceptably democratic to the Founding Fathers, giving too much power to the mob—the "swinish multitude" who lacked sufficient property to possess the disinterested perspective necessary to govern. Having floated the idea, Madison decisively rejects it:

As long as the reason of man continues fallible, and he is at liberty to exercise it, different opinions will be formed.

The problem of faction, then, cannot be controlled by removing its causes.

This leads Madison to conclude that the only way this can be done is to deal with the effects of faction. Why? Because,

The latent causes of faction are thus sown in the nature of man; and we see them everywhere brought into different degrees of activity, according to the different circumstances of civil society.

If faction is an expression of our very nature, then we can only really deal with its effects and try to minimize the harm it can cause.

To this end, Madison places much (some would say too much) faith in the republican nature of the Constitution itself. The system of checks and balances will act as a restraint on the domination of a minority attempting to subvert the interests of the majority:

If a faction consists of less than a majority, relief is supplied by the republican principle, which enables the majority to defeat its sinister views by regular vote. It may clog the Administration, it may convulse the society; but it will be unable to execute and mask its violence under the forms of the Constitution.

But what about a faction of the majority? Madison, in common with the Founding Fathers in general, had a deep suspicion of democracy, fearing that it would lead to a levelling down of society, and of the affairs of the nation being placed in the hands of those without property—people who didn't have a stake in society and who would therefore govern purely in their own interests rather than the good of the country.

Representative government is the solution to this potential problem. This will, it is argued, lead to the election of men

whose wisdom may best discern the true interest of their country, and whose patriotism and love of justice will be least likely to sacrifice it to temporary or partial considerations. 

Under a republican system of government, men of property will take charge. As they are rooted to the land through their property holdings, and as they have sufficient leisure to consider the pressing matter of government, they are most suitably qualified to exercise power.

Madison also argues that having a large, as opposed to a small, republic will act as a check on the growth of faction. The larger the republic, the more "fit characters" there will be to choose from in the governance of the nation. Also the devious wiles of political rhetoric will be much less effective as there will be many more people to convince:

Extend the sphere, and you take in a greater variety of parties and interests; you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens; or if such a common motive exists, it will be more difficult for all who feel it to discover their own strength, and to act in unison with each other.

`y = x/2 + 3 , 1

To find the area of this surface, we rotate the function `y = x/2 + 3 ` about the y-axis (not the x-axis) in the range `1<=x<=5 ` and this way create a finite surface of revolution.

A way to approach this problem is to swap the roles of `x ` and ` ``y `, essentially looking at the page side-on, so that we can use the standard formulae that are usually written in terms of `x ` (ie, that usually refer to the x-axis).

The formula for a surface of revolution A is given by (interchanging the roles of `x ` and `y ` )

`A = int_a^b (2pi x) sqrt(1+(frac(dx)(dy))^2)dy `

Since we are swapping the roles of `x ` and `y `, we need the function `y = x/2 + 3 ` written as `x ` in terms of `y ` as opposed to `y ` in terms of `x `. So we have

`x = 2y - 6 `

To obtain the area required by integration, we are effectively adding together tiny rings (of circumference `2pi x ` at a point `y ` on the y-axis) where each ring takes up length `dy ` on the y-axis. The distance from the circular edge to circular edge of each ring is `sqrt(1+(frac(dx)(dy))^2) dy `

This is the arc length of the function `x = f(y) `in a segment of length `dy ` of the y-axis, which can be thought of as the hypotenuse of a tiny triangle with width `dy ` and height `dx `.

These distances from edge to edge of the tiny rings are then multiplied by the circumference of the surface at that point, `2pi x ` , to give the surface area of each ring. The tiny sloped rings are added up to give the full sloped surface area of revolution.

We have for this function, `x = 2y -6 ` , that  `(dx)/(dy) = 2 `

and since the range (in `y `) over which to take the integral is `1 <=x <=5 `, or equivalently  `7/2 <= y <= 11/2 `  we have `a = 7/2 ` and `b = 11/2 ` .

Therefore, the area required, A, is given by

`A = int_((7)/(2))^((11)/(2)) 2pi (2y -6) sqrt(5) dy `   `= 2sqrt(5)pi int_((7)/(2))^((11)/(2)) 2y - 6 \quad dy `

`= 2sqrt(5)pi y(y - 6)|_((7)/(2))^((11)/(2)) = (sqrt(5))/(2)pi [11(11-6) - 7(7-6)] `

So that the surface area of rotation A is given by

`A = 24(sqrt(5)) pi `

`y = -x + 1` Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

For the region bounded by `y=-x+1` revolve about the x-axis, we can also apply the Shell Method using a horizontal rectangular strip parallel to the axis of revolution (x-axis).We may follow the formula for Shell Method as:

`V = int_a^b 2pi` * radius*height*thickness

where:

radius (r)= distance of the rectangular strip to the axis of revolution

height (h) = length of the rectangular strip

thickness = width  of the rectangular strip  as` dx` or` dy`...

For the region bounded by `y=-x+1` revolve about the x-axis, we can also apply the Shell Method using a horizontal rectangular strip parallel to the axis of revolution (x-axis).We may follow the formula for Shell Method as:

`V = int_a^b 2pi` * radius*height*thickness

where:

radius (r)= distance of the rectangular strip to the axis of revolution

height (h) = length of the rectangular strip

thickness = width  of the rectangular strip  as` dx` or` dy` .

As shown on the attached file,  the rectangular strip has:

`r=y`

`h =f(y) ` or `h = x_2-x_1`

`h = (1-y) -0 = 1-y`

Note: `y =-x+1` can be rearrange into `x=1-y` .

Thickness `= dy`

Boundary values of y: `a=0` to `b =1` .

Plug-in the values on to the formula `V = int_a^b 2pi` * radius*height*thickness, we get:

`V = int_0^1 2pi* y*(1-y)*dy`

Apply basic integration property: `intc*f(x) dx = c int f(x) dx` .

`V = 2pi int_0^1 y*(1-y)*dy`

Simplify: `V = 2pi int_0^1 (y-y^2)dy`

Apply basic integration property:`int (u-v)dy = int (u)dy-int (v)dy` to be able to integrate them separately using Power rule for integration:  `int y^n dy = y^(n+1)/(n+1)` .

`V = 2pi *[ int_0^1 (y) dy -int_0^1 (y^2)dy]`

`V = 2pi *[y^2/2 -y^3/3]|_0^1`

Apply definite integration formula: `int_a^b f(y) dy= F(b)-F(a) ` .

`V = 2pi *[(1)^2/2 -(1)^3/3] -2pi *[(0)^2/2 -(0)^3/3]`

`V = 2pi *[1/2 -1/3] -2pi *[0-0]`

`V = 2pi *[1/6] -2pi *[0]`

`V = (2pi )/6 -0`

`V = (2pi )/6`  or `pi/3`

 We will get the same result whether we use Disk Method or Shell Method for the given bounded region revolve about the x-axis on this problem.