Basically, the U.S. Constitution grants Congress the power to determine how many justices sit on SCOTUS. This number has ranged between 5 and 10, but since 1869 the number has been set at 9.
What do you mean by diversity in court?The 116 individuals who have been nominated and confirmed as justices to the Supreme Court are represented by the gender, ethnicity, and backgrounds in terms of religion, geography, and economics. Since the court was created in 1789, some of these qualities have been brought up as concerns. For the first 180 years of the court's existence, justices were primarily white male Protestants of Anglo or Northwestern European ancestry.
Prior to the 20th century, a small number of Catholics were appointed, but concerns about diversity on the court focused more on geographic variety—representing all regions of the nation—than on ethnic, religious, or gender diversity.
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What does Mrs. Dubose's struggle say about humanity?
Answer:
Dubose struggles against old age; she has ailments which are so grave that there is little that can be done for her but alleviate the pain caused by these ailments through the use of morphine. But, as Atticus later informs Jem and Scout, Mrs. Dubose decided to die without being on any drugs.
Explanation:
Mrs. Dubose's struggle say about humanity is you never really know what someone is going through or how why they act the way they do.
What is To kill a mockingbird?To Kill a Mockingbird explores how good and evil may coexist inside a single community or individual while also telling the story of a young girl coming of age and a darker drama about the causes and effects of racism and prejudice.
Dubose fights against aging, but due to her severe illnesses, there is little that can be done for her other than to utilize morphine to treat the pain they create. But as Atticus goes on to tell Jem and Scout, Mrs. Dubose made the decision to pass away without any medications.
Therefore, you never really know what someone is going through or why they act the way they do, according to Mrs. Dubose's fight with humanity.
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If you depoit 4000 in to an account paying 9% annual interet compounded monthly how long until there i 10000 in the account
Answer:
To find out how long it will take for the balance in the account to reach $10,000, you can use the following formula:
n = log(10000 / 4000) / log(1 + (0.09 / 12))
This formula calculates the number of periods (in this case, months) required for the balance to grow from $4,000 to $10,000 at a 9% annual interest rate compounded monthly.
Plugging in the values, we get:
n = log(10000 / 4000) / log(1 + (0.09 / 12))
= log(2.5) / log(1.0075)
= 0.964 / 0.0075
= 128.53
So it would take approximately 128 months, or just over 10 years, for the balance in the account to reach $10,000.
Note: This is just an approximation and the actual time required may vary slightly due to the way that the interest is compounded.
The mean price of computers sold at a particular store is $949 with a standard deviation of $75. The z-score of a certain computer is -0.8. What is the price of the computer?
Answer:
the price of the computer is $889
Explanation:
The calculation of the price of the computer is shown below
We know that
Z-score = (x - mean) ÷ standard deviation
-0.8 = (x - 949) ÷ 75
x = (-0.8 × 75) + 949
= -60 + 949
= $889
Thus, the price of the computer is $889
The above formula should be used in order to get the accurate value
As noted in lecture, it is possible to have a game where a Nash Equilibrium does not exist. In such a game, are there dominated strategies? What about dominant strategies?
In game theory, "dominant strategy" refers to the choice that is best for a player regardless of how their opponents may play, and "inferior strategy" refers to the other choice.
Explain about the dominant strategies?Game theory holds that a player should always choose the dominant strategy, regardless of what other players choose to do. The game is said to be in a Nash equilibrium when both players make their best plays while simultaneously considering those of their rivals.
It is referred described as being dominant when a method exceeds all others. In a Nash equilibrium, reactions between individuals are optimized. A set of strategies is in Nash equilibrium if nobody wants to change from how they are going about things.
Sometimes a combination of two or more pure strategies can defeat a pure strategy that cannot be defeated by another pure approach. Such methods are described by strictly dominant mixed strategies.
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Respond to all parts of the question. In your response, use substantive examples where
appropriate.
Use the pie charts to answer the following questions.
A. Identify the percentage of the budget that was mandatory spending in 1968.
B. Describe a change over time in the pie charts.
A. 28%
B. The mandatory spending has been increasing annually while the discretionary spending has been decreasing over time, resulting in a fluctuation in the net interest.
What is a budget?A budget is a financial plan that outlines estimated revenues and expenses for a particular period, usually a month, quarter, or year. It helps individuals and organizations to manage their finances effectively by providing a framework for allocating resources and tracking progress towards financial goals.
A typical budget consists of two main components: income and expenses. Income includes all sources of revenue, such as salaries, investments, and grants. Expenses include all expenditures, such as rent, utilities, salaries, and supplies.
The goal of a budget is to ensure that expenses do not exceed income and that financial resources are allocated effectively. By creating a budget, individuals and organizations can identify areas where they can reduce costs, prioritize spending, and plan for future needs. Additionally, a budget can help individuals and organizations to track their progress towards financial goals and make adjustments as needed.
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1. How would you describe the economic transactions described in the document? To what extent were they conducted between equal parties? Who, if anyone, held the upper hand in these dealings?
2.How might an African merchant have described the same transaction? How might Equiano describe it?
3.Notice the outcome of Phillips voyage to Barbados in the last two paragraphs. What does this tell you about European preferences for slaves, about the middle passage, and about the profitability of the enterprise?
Source: The Interesting Narrative of the Life of Olaudah Equiano (1789)
The agreement was made between the white men and the local king. They were paid by the king in cowries (money), and the king also provided slaves. Although the King and the Blacks took note of this, it mostly harmed the Whites.
What is White slavery?White slavery, often known as the "white slave trade" or "white slave trafficking," is the practise of enslaving Europeans, whether by non-Europeans (such as people from West Asia and North Africa) or by other Europeans (such as the Vikings' thralls or slaves on naval galleys). Both the Ottoman Empire and ancient Rome had slaves who were of European descent.There were many different kinds of white slaves. Under feudalism, those who were indentured or made to work for free had a variety of statuses that applied to them, including serf, bordar, villein, vagrant, and slave.Under Muslim administration, invasions into European countries or the capture of Caucasian children as part of a blood tax were frequent sources of funding for the Arab slave trades, which also included Caucasian prisoners.To learn more about White slavery refer to:
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The writer wants to combine sentences 11 and 12 (reproduced below) into a single sentence.
But when the stakes are high, the risks of making biased decisions are simply too great. An example of a high-stakes situation would be when jurors are deliberating a defendant's fate.
Which of the following revisions to the underlined portion of sentences 11 and 12 most effectively accomplishes this goal?
A
high, then the risks
making biased decisions-one example of which would be when jurors are deliberating a defendant's fate-are simply too great
B
high, one good example of which is a defendant having his or her fate being deliberated by jurors, the risks of making biased decisions are simply too great
c) high (for example, when jurors are deliberating a defendant's fate, the stakes would be high), the risks of making biased decisions are simply too great
D
high, such as when jurors are deliberating a defendant's fate, the risks of making biased decisions are simply too great
E high-like a defendant whose fate is being deliberated by jurors-the risks of making biased decisions are simply too great
Answer:
D. high, such as when jurors are deliberating a defendant's fate, the risks of making biased decisions are simply too great
Explanation:
This version combines the two sentences in a concise and clear manner. Of the options presented, it is the most effective way to express the ideas in one sentence.
The revision that correctly fulfills the underlined part of sentence 11, as well as, 12 would be:
D). high, such as when jurors are deliberating a defendant's fate, the risks of making biased decisions are simply too great.
What is a sentence?The sentence is described as the formation of words together to convey a complete meaning.
The above sentence correctly revises the highlighted part in sentences 11 and 12 as it gives it clarity and precision.
This joins the ideas together in a comprehensive and whole manner which makes it convenient for understanding them effectively.
Thus, option D is the correct answer.
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Write a summary of the following excerpt from “Why I Wrote 'The Yellow Wallpaper" by Charlotte Perkins
Gilman.
For many years I suffered from a severe and continuous nervous breakdown tending to melancholia--and beyond
During about the third year of this trouble I went, in devout faith and some faint stir of hope, to a noted specialist in
nervous diseases, the best known in the country. This wise man put me to bed and applied the rest cure, to which
a still-good physique responded so promptly that he concluded there was nothing much the matter with me, and
sent me home with solemn advice to "live as domestic a life as far as possible," to "have but two hours' intellectual
life a day," and "never to touch pen, brush, or pencil again" as long as I lived. This was in 1887
Answer:
Gilman felt very melancholy and was having a severe nervous breakdown. This mental state was maintained for three years, which made her look for a specialist in nervous diseases, who would examine her and help her to heal. The specialist stated that she did not have any serious illness, but that she needed to rest as long as she lived to be cured.
Explanation:
To make a summary of a text, you must rewrite the text in a reduced form, pointing out only the most important elements of the story. Therefore, anyone who reads the summary is able to understand the most important and relevant parts of the original text, even if they do not have many details of how it happened, since the details are presented only in the original text.
Where are the most mountains found on Africa
Answer:
North-western, central and eastern regions of Africa have large number of mountains.
Explanation:
North-western, central and eastern regions of Africa have large number of mountains. However, 50% of the Africa's countries have high mountains (higher than 2000 m)
Some of the prominent mountains are -
Mount Kilimanjaro (5,895 m) in Tanzania, Mount Kenya (5,195 m) in Kenya, Mount Stanley (5,119 m), Mount Speke (4,890 m) and Mount Baker (4,844 m) in Congo-Uganda etc.
The seventeenth century saw the heyday of the [European] East India Companies. They became an inevitable part of the politics and economy of South India, finding their way into the society and even the vocabulary of the local people. The ‘kumbini’,* as the local populace called it, was not to be ignored. As with the rest [of the population], the spinners and weavers, the washers, the dyers and the once powerful textile-merchant guilds had their links with the company [warehouses]. Of course there were also those merchants and weavers who operated independently of the company, but their proportion was small. With the establishment of the rival European companies in the seventeenth century, the lives of the weavers [of South India] no longer revolved around the temple but around the European [warehouses] and the towns [on the outskirts of European trading posts]. The weavers gradually began losing their bargaining power and independence. . . . Their creativity also suffered since they were compelled to copy mechanically the [designs] provided to them [by the companies’ officials]. Economically they were much worse off.” *a Tamil pronunciation of the word “company”
Vijaya Ramaswamy, Indian historian, book published in 198
In your response, be sure to address all parts of the question. Use complete sentences; an outline or bulleted list alone is not acceptable.
Use the passage to answer all parts of the question that follows.
a) Identify ONE claim made in the passage.
b) Describe ONE reason why European trading companies “became an inevitable part of the politics and economy of South India” and other regions of Asia in the period 1450–1750.
c) Explain ONE piece of evidence from the period 1450–1750 that would complicate the portrayal of the economic relationship between Asian and European producers and merchants provided in the second paragraph.
1. One of the claim made in the passage was that during the 17th century, South India's politics and economy were significantly influenced by the European East India Companies.
2. The Europeans believed that the key to wealth was selling goods to other countries to stockpile as much gold and silver as possible.
3. One piece of evidence that could complicate the portrayal of the economic relationship between Asian and European producers and merchants was the passage states that weavers were worse economically
What was the passage about?Selling goods to other nations in order to accumulate as much gold and silver as possible was, in the eyes of the Europeans, the secret to wealth. To help achieve this, European trading firms established joint stock companies that let investors purchase stock in organizations like the Dutch East India Company, which supported funding for Asian trade exploration. Dutch ships had a competitive advantage because they were lighter and faster.
The passage that claims that weavers had worse economic conditions is one piece of evidence that could make it more difficult to portray the trade relations between Asian and European producers and merchants. It is difficult to believe that the weavers who actually produced the goods were worse off economically as a result of the growth given the fact that organizations like the Dutch East India Company made significant profits from trading in Asia.
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The writer is considering replacing the underlined portion of sentence 11 (reproduced below).
In one study, test subjects did a bunch of things with a humanlike robot called Nao.
Which of the following versions of the underlined text most closely fits the style of the passage?
A. (as it is now)
B. performed a series of tasks
C. finished a ton of work
D. completed loads of experiments
E. executed an array of assigned enterprises
Answer:
B. performed a series of tasks
Explanation:
Answer B
Correct. This version of the underlined text most closely fits the style of the passage in terms of diction, syntax, and conventions. It relays information directly and uses formal yet accessible language.
Answer:
B) performed a series of tasks
Explanation:
I just took the test, it is correct :)
The end of the civil war between Arab Mulim and Chritian Black African in Sudan coincided with the eruption of an ethnic war in the Sudanee region of , along the Chadian border. A. N’Djamena
B. Ouagadougou
C. Abyei
D. Darfur
E. Juba
Answer:
Based on the information provided, the correct answer is D. Darfur.
The civil war between Arab Muslim and Christian Black African in Sudan ended around the same time as the eruption of an ethnic war in the region of Darfur, along the border with Chad. This conflict, which took place in the early 21st century, involved fighting between the Sudanese government and rebel groups in Darfur, and resulted in widespread human rights abuses and the displacement of millions of people. The conflict was eventually resolved through a peace agreement in 2010, but tensions and violence continue to persist in the region.
Explanation:
Which of the following best describes why the author includes the phrase “unless you believe in objective, scientific polling” (paragraph 1, sentence 4)?
(A) It reveals the author’s doubts about the reliability of scientific data.
(B) It reinforces the author’s argument that our opinions are frequently
based on science.
(C) It helps to establish the author’s ironic, humorous tone.
(D) It supports the author’s feelings about the increasing importance of
technology.
(E) It introduces a comparison between the author’s view of himself
and the way others view him.
The "elastic clause" expands the power of what branch of government to do what is "necessary and proper" to carry out its enumerated powers?
A. The Executive Branch
B. The judiciary
C. Congress
D. None of the above
Answer:
C. Congress
The "elastic clause," also known as the "necessary and proper clause," is a provision of the United States Constitution that grants Congress the power to pass any laws it deems necessary and proper to carry out its enumerated powers. This clause allows Congress to expand its powers and adapt to changing circumstances, enabling it to effectively fulfill its role as the legislative branch of government.
Explanation:
Examine Item A in your test documents. This diagram best illustrates the concept of:
A. division of power (federalism)
B. separation of powers (Checks and balances)
C. power of the people (popular sovereignty)
D. enumerated powers (constitutional powers)
Test Documents Are Attached
The diagram best illustrates the concept of division of power that is federalism.
What do you mean by federalism?The ability to govern is divided between the federal government and provincial/state governments under modern federalism, a political system founded on democratic principles and institutions. Depending on the context, the term "federalist" can refer to a variety of political ideologies.
Federalism is a form of political structure that enables the union of several states or other polities within a larger political system while still allowing each to preserve its own integrity.
The political ideas that drive federal systems place a premium on negotiations and negotiated cooperation among various power centers; they highlight the benefits of dispersed power centers as a way to protect personal and local liberty.
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Examine Item A in your test documents. Where in the diagram does the supremacy clause come most into play?
State laws that affect federal laws are prohibited since the Constitution is the highest law of the land.
What are supremacy clause?The Supremacy Clause's main point is straightforward state laws that contradict with the Constitution or federal legislation (of the kinds specified in the first portion of the Clause) are superseded.The provision gave the Supreme Court the power to contribute to building a robust federal government. The U.S. Constitution's Supremacy Clause was added because the Articles of Confederation did not contain one. According to the Articles, federal laws did not and could not take precedence over state laws.The Supremacy Clause creates domains of power where the federal government cannot meddle with certain state affairs and vice versa, so long as the Constitution is obeyed. This enables the U.S. federal system of government function.To learn more about supremacy clause refer to:
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What is the money multiplier when RRR = 9% and the ERR =3%?
The money multiplier when RRR = 9% and the ERR =3% is 8.333.
What is money multiplier?A fractional-reserve banking system's money multiplier is one of many closely related ratios of commercial bank money to central bank money. Given a certain amount of central bank money, it has to do with the maximum amount of commercial bank money that can be produced.
Loans are how banks make money. To increase interest earnings, a bank lends or invests its excess reserves. The money supply grows by more than one dollar for every dollar increase in the monetary base. The money multiplier is the expansion of the money supply.
Money multiplier = 1 ÷ reserve ratio
Here reserve ratio = 9%+3%
= 12%.
= 0.12
Money multiplier = 1 / 0.12
= 8.333
So money multiplier is 8.333
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Evaluate the extent to which American migration fostered social change in the western territories from 1862 to 1900.
Mining possibilities and the gold rush (silver in Nevada) the chance to work as a "cowboy," to be involved in the cattle business Railroad-related advantages include quicker travel to the West and increased supply options. under the Homestead Act, a low-cost option for land ownership.
What is Reasons for Westward Expansion?An introduction to a lesson about westward expansion could utilise this activity. You could also use this assignment as a review at the conclusion of a unit, or you could have the whole class work through it progressively over the course of a unit.Start off by having the class come up with a theory as to why Americans would have wished to emigrate to the West in the middle of the 19th century. Compare and contrast the broad motivations behind people's moves from one location to another with the specific political and cultural environment that existed in the United States at this time. For future use, ideas could be written down.The class should start the sequencing exercise. Analyze a few of the documents in greater detail. Examine the documents with the class and see if anyone can name and describe any of them without further inspection. Inquire about proposals for "earlier" and "later" materials. The ability to defend assumptions is a must for students. (Remind students that at this time, since they haven't yet conducted any detailed examination, it is perfectly fine to make a guess with limited evidence to boost their confidence and readiness to make educated assumptions.)To Learn more About the gold rush refer To:
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HELP ASAP
what famous painter is known to have used a camera obscura make his paintings more accurate?
A.) Pablo Picasso
B.) Michelangelo Buonarroti
C.) Leonardo de Vinci
D.) Rembrandt van Rijn
Need help for #2!! Person w correct answer will be marked brainliest!!
survival of the fittest, a term made famous in the fifth edition (published in 1869) of On the Origin of Species by British naturalist Charles Darwin, suggested that organisms best adjusted to their environment are the most successful in surviving and reproducing.
What is the environment?The environment can be defined as a sum absolute of all the realistic and non-living atmospheric condition and their effects that influence human life. While all living or biotic elements are animals, plants, forests, fisheries, and birds, non-living or abiotic weather conditions include water, land, sunshine, rocks, and air.
"Fittest" simply means "most successful over the long run in passing on your genes." If you are physically fit, powerful, and healthy yet don't pass your genes on. In terms of evolution, you are not fit.
Natural selection is a mechanism that propels evolutionary change, and it is often referred to as "survival of the fittest" in popular culture. Natural selection operates by favoring individuals who have a higher level of environmental adaptation over those who have a lower level of adaptation.
Therefore, The environment is the most successful in surviving and reproducing.
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What is ultimately responsible for the size of the money supply?
A. The United States treasury
B. The money multiplier
C. Government
D. The Fed
E. Banks
The organization that is ultimately responsible for the size of the money supply is the bank. Thus, the correct option for this question is E.
Who determines the size of the money supply?The Federal Reserve i.e. known as America's central bank is responsible for controlling the supply of U.S. dollars. The Fed creates money by purchasing securities on the open market and adding the corresponding funds to the bank reserves of commercial banks.
When the Fed limits the money supply via contractionary or hawkish monetary policy, interest rates rise and the cost of borrowing increases. This can dampen inflationary pressures, but also risk slowing down economic growth. This federal reserve is the bank that regulates the utilization of money and services across the country USA.
Therefore, the organization that is ultimately responsible for the size of the money supply is the bank. Thus, the correct option for this question is E.
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7, 5, 10, 4, 4
What is the median of the numbers above?
Responses
Compared to distilled water, is water isotonic?
Answer:
Isotonic solutions have the same concentration of solutes as a reference solution, usually expressed as milliosmoles per liter (mOsm/L). Distilled water is a pure substance that does not contain any solutes, so it has an osmolarity of 0 mOsm/L. Therefore, compared to distilled water, most solutions are hypertonic, meaning that they have a higher concentration of solutes, or hypotonic, meaning that they have a lower concentration of solutes.
However, it is important to note that the term "isotonic" is often used to describe solutions that are similar in concentration to the fluids inside the human body. For example, saline solutions are often used in medical settings because they have an osmolarity similar to the fluids in the human body, making them isotonic in this context. In comparison to these solutions, distilled water is hypotonic.
So, to answer your question, it depends on the reference solution that is being used. Compared to distilled water, which has an osmolarity of 0 mOsm/L, most solutions are either hypertonic or hypotonic. However, compared to solutions that are similar in concentration to the fluids in the human body, distilled water is hypotonic.
Explanation:
What happens when two copies of information for a
trait are not the same?
Genes come in different varieties, called alleles. Somatic cells contain two alleles for every gene, with one allele provided by each parent of an organism. Often, it is impossible to determine which two alleles of a gene are present within an organism's chromosomes based solely on the outward appearance of that organism. However, an allele that is hidden, or not expressed by an organism, can still be passed on to that organism's offspring and expressed in a later generation.
When you are about to make a meal or bake something, after finding a recipe what is the first thing you do?
When you are preparing foods that you've made before, you only have to prepare ingredients first that you know you won't have time to ready while cooking.
What is nutrient and calories content?Proteins, lipids (fats), and carbohydrates are the nutrients that "produce energy." For every gram of carbohydrates, lipids, or protein that we eat, we receive 4 calories, 9 calories from fats, and 4 calories from proteins. When it comes to macronutrients (protein, carbohydrates, and fat) and micronutrients (vitamins, minerals, and phytonutrients like antioxidants), the amount of calories in a food tells us very little about its nutritional makeup. The food we eat provides us with energy. More specifically, alcohol, protein, fat, and carbohydrates are the four types of energy sources. These substances each provide a variable quantity of energy, expressed in calories, when they are burned (metabolized) by the body.
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over a whole lifetime about how much someone with a professional degree expects to earn compared to with a high school diploma who didn't attend collage
In terms of median lifetime earnings, men with graduate degrees surpass high school graduates by $1.5 million. The salary increase for women with graduate degrees is $1.1 million.
What is Education and Lifetime Earnings?The typical lifetime income of men with bachelor's degrees is $900,000 higher than that of men with only high school diplomas. Bachelor's degree-holding women make $630,000 more than males do.Males with bachelor's degrees would have a median lifetime income that is $655,000 higher than men with only a high school diploma, according to regression projections. The median lifetime income for women with bachelor's degrees is $450,000 higher than that of high school grads.Additionally, we use the discounted present dollar value to calculate the lifetime returns of educational achievement.In comparison to a high school certificate, a bachelor's degree at age 20 has a net present lifetime worth of $260,000 for males and $180,000 for women, using a real discount rate of 4% annually. Men pay $400,000 and women $310,000 for those with graduate degrees.All things considered, these data support the substantial long-term economic advantages of a college education.To Learn more About median lifetime earnings refer To:
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What is News Media Bias?
Rather than the viewpoint of a single journalist or article, the phrase "media bias" suggests a systemic or widespread bias that violates the rules of journalism.
What does media bias mean?Rather than the viewpoint of a single journalist or article, the phrase "media bias" suggests a systemic or widespread bias that violates the rules of journalism. There is considerable debate over the type and severity of media bias in different nations. The three types of biases mentioned above—confirmation bias, sampling bias, and brilliance prejudice—can all have an impact on our capacity to evaluate information critically. To help us better grasp how cognitive biases affect our daily life, Jono Hey of Sketchplanations talks us through some examples.Bias is a disproportionate preference for or opposition to an idea or object, typically in an unreflective, prejudiced, or unfair manner. Biases may be ingrained or acquired. Biases for or against a person, a group, or a belief can arise in people. A bias in engineering and science is a type of systematic inaccuracy.To learn more about media bias refer to:
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Explain how supranationalist organization membership can limit sovereignty of member states.
The way that supernationalist organization membership can limit sovereignty of member states is that : Members of supranational organizations, such as the EU and World Trade Organization, consent to giving up some of their sovereignty.
What steps does a state need to take in order to join a supranational organization?Member states do have to give up some sovereignty because the organization is supranational.
Military alliances aid states in fending off common foes, and economic alliances promote free trade on a greater scale. Supranational organizations are another name for these coalitions, and they always include states ceding some of their sovereign authority in the name of the greater good.
Therefore, In order to establish worldwide laws governing nations' affairs and avoid wars, a supranational body is a viable option. Countries that belong to supranational organizations like the EU and World Trade Organization consent to give up some of their sovereignty to the group on certain matters.
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How has the introduction of the automobile and, subsequently, the highway begun to fundamentally change the way cities are designed and function?
Answer:
The introduction of the automobile and, subsequently, the highway begun to fundamentally change the way cities are designed and function because these technological advances have allowed citizens to travel greater distances in less time: before the automobile, the horse was used to get around, which travels distances at about 30 kilometers per hour on average, and needs to rest after long stretches. Today, the car travels at average speeds of about 60 kilometers per hour and does not need breaks, other than the logical routine mechanical conditioning.
Thus, cities have been able to begin to expand horizontally, enlarging their contours because people no longer necessarily need to live near their centers, thanks to the mobility facilities that exist today.
What is the rate of change of the radius to the thousandths?
Area of a Circle Formula:
[tex]\displaystyle A = \pi r^2[/tex]
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]:
[tex]\displaystyle [u(v)]' = u'(v)v'[/tex]
Let's organize what the problem gives us.
We are told that the area of a circle is decreasing at 122 in²/min.
⇒ [tex]\displaystyle \frac{dA}{dt} = -122 \frac{\text{in}^2}{\text{min}}[/tex]
We are also told an instantaneous point: the area of the circle is 16π in².
We are asked to find the rate of change of the radius.
Step 2: WorkLet's first differentiate our area of a circle formula with respect to time. Apply basic differentiation rules listed above under "Calculus":
[tex]\displaystyle\begin{aligned}\frac{d}{dt}[A] & = \frac{d}{dt}[\pi r^2] \\\frac{dA}{dt} & = \pi \frac{d}{dt}[r^2] \\\frac{dA}{dt} & = 2 \pi r \frac{dr}{dt} \\\end{aligned}[/tex]
∴ we have found the relationship between the change in area and the change in radius w/ respect to time.
Now we need to determine what the radius is when the area of the circle is equal to 16π in²:
[tex]\displaystyle\begin{aligned}16 \pi \ \text{in}^2 & = \pi r^2 \\r^2 & = 16 \ \text{in}^2 \\r & = \boxed{ \pm 4 \ \text{in} } \\\end{aligned}[/tex]
∴ the radius of the circle when the area is equal to 4 in, since the radius of the circle can't be negative.
With all of our variables, we can now substitute them into our derivative and solve for the rate of change of the radius:
[tex]\displaystyle \frac{dA}{dt} = 2 \pi r \frac{dr}{dt} \Rightarrow -122 = 2 \pi (4) \frac{dr}{dt}[/tex]
[tex]\displaystyle\begin{aligned}\frac{-122 \frac{\text{in}^2}{\text{min}}}{2 \pi (4 \text{in})} & = \frac{dr}{dt} \\\frac{dr}{dt} & = \frac{-122 \frac{\text{in}^2}{\text{min}}}{8 \pi \ \text{in}} \\& = -4.8542257643 \frac{\text{in}}{\text{min}} \\& \approx \boxed{ -4.854 \frac{\text{in}}{\text{min}} } \\\end{aligned}[/tex]
∴ the rate of change of the radius is equal to -4.854 inches per minute rounded to the thousandths place. Reworded, the rate of change of the radius decreases by 4.854 inches per minute.
Answer∴ at the instant when the area of the circle is 16π square inches, the change of the radius is equal to:
[tex]\displaystyle \boxed{ \frac{dr}{dt} \approx -4.854 \frac{\text{in}}{\text{min}} }[/tex]
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Topic: Calculus I
Unit: Differentiation (Related Rates)