The discussion of the ""greatest of poets and philosophers"" in the third paragraph suggests that the author believes the audience of the speech will.

Answers

Answer 1

There a lot of great poets and philosophers in the world today. The author believes the audience of the speech will recognize an implicit reference to Shakespeare and agree about his importance.

There have been a lot of English literature. One of the greatest poet of all time is William Shakespeare.

He was known for his famous quote where He said that All the world, is a stage, and all the men and women merely players.

The author recognize the importance of Shakespeare and his impressive work of literature.

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Related Questions

How did fossils help and teach scientists about the history of the earth brainly.

Answers

Answer:

Fossils helped teach scientist by giving information of the species that lived back then.

Explanation:

The fossils have clues on them help identify what they are.

What amount of heat, in kj, would be involved in condensing 15. 8 g of ch₃oh? (∆hvap = 38. 0 kj/mol).

Answers

Considering the definition of enthalpy of vaporization, the amount of heat that would be involved in condensing 15.8 g of CH₃OH is 18.8 kJ.

When a material changes phase from solid to liquid, or from liquid to gas, a certain amount of energy is involved in this phase change.

In case of phase change from liquid to gas, this amount of energy is known as enthalpy of vaporization (symbolized as ∆Hvap).

So, the enthalpy of vaporization is the amount of energy required to change a unit of mass (for example, moles or kg) of a substance from phase liquid to phase gas at constant temperature and pressure.

In this case, you know that enthalpy of vaporization is ∆Hvap = 38.0 kJ/mol.

On the other side, you want to vaporize 15.8 g of CH₃OH.  Then, being the molar mass of the compound 32 [tex]\frac{g}{mole}[/tex] (that is, the amount of mass contained in one mole), the number of moles to be vaporized is:

[tex]15.8 gramsx\frac{1 mole}{32 grams} =0.49375 moles[/tex]

Then, the amount of heat that would be involved in condensing 15.8 g of CH₃OH is calculated taking into account the enthalpy of vaporization by:

0.49375 moles× 38.0 [tex]\frac{kJ}{mol}[/tex]= 18.76 kJ ≅ 18.8 kJ

Finally, the amount of heat that would be involved in condensing 15.8 g of CH₃OH is 18.8 kJ.

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Which of the following people worked with government in the 1960s to get laws passed to help noncitizens and farm workers? Huey Newton Bobby Seale Stokely Carmichael Dolores Huerta.

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Government often makes laws that aim for the betterment of its people. Cesar Chavez worked with government in the 1960s to get laws passed to help noncitizens and farm workers.

Cesar was known to be a man who use nonviolent means as used by Mahatma Gandhi and Martin Luther King Jr. in all his works.

He was known to be the founder of the National Farm Workers Association (later the United Farm Workers of America)

He won a lot of significant victories so as the pay and improve working conditions for farm workers is improve in the late 1960s and 1970s

See full question below

Which of the following people worked with government in the 1960s to get laws passed to help noncitizens and farm workers?

Huey Newton

Bobby Seale

Cesar Chavez

Dolores Huerta

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Answer:

Explanation:

Dolores huerta

Which of the following statements is true of postmodern cultural spaces.

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Postmodern cultural spaces are the current state of culture and society in the world. it describes events, norms, activities, art, practices etc. Thus, Existences is based upon the current cultural space in the world

A bat emits a sound at a frequency of 3. 00 × 104 hz as it approaches a wall. The bat detects beats such that the frequency of the echo is 900 hz higher than the frequency the bat is emitting. The speed of sound in air is 340 m/s. The speed of the bat is closest to.

Answers

Taking into account the Doppler efect, the speed of the bat is closest to 5.02 [tex]\frac{m}{s}[/tex].

The Doppler effect is defined as the apparent frequency change of a wave produced by the relative movement of the source with respect to its observer. In other words, this effect is the change in the perceived frequency of any wave motion when the sender and receiver, or observer, move relative to each other.

The following expression is considered the general case of the Doppler effect:

[tex]f1=f\frac{v+-vR}{v+-vE}[/tex]

Where:

f1, f: Frequency perceived by the receiver and frequency emitted by the issuer respectively. Its unit of measurement in the International System (S.I.) is the hertz (Hz), which is the inverse unit of the second (1 Hz = 1 s⁻¹) v: Velocity of propagation of the wave in the medium. It is constant and depends on the characteristics of the medium. In this case, the speed of sound in air is considered to be 340 m/s vR, vE: Speed ​​of the receiver and the sender respectively. Its unit of measure in the S.I. is the m/s ±, ∓: We will use the + sign:

In the numerator if the receiver approaches the sender

In the denominator if the sender moves away from the receiver

We will use the sign -:

In the numerator if the receiver moves away from the emitter

In the denominator if the sender approaches the receiver

In this case, a bat emits a sound at a frequency of 3.00×10⁴ Hz= 30,000 Hz as it approaches a wall.

The bat detects beats such that the frequency of the echo is 900 hz higher than the frequency the bat is emitting.

So, you know:

f1= 30,000 Hz + 900 Hz= 30,900 Hzf= 30,000 Hzv= 340 m/svE=vR= vbatReceiver approaches the sender

Replacing in the expression is considered the general case of the Doppler effect:

[tex]30900 Hz=30000 Hz\frac{340 \frac{m}{s} +vbat}{340\frac{m}{s} -vbat}[/tex]

Solving:

[tex]\frac{30900 Hz}{30000 Hz} =\frac{340 \frac{m}{s} +vbat}{340\frac{m}{s} -vbat}[/tex]

[tex]1.03=\frac{340 \frac{m}{s} +vbat}{340\frac{m}{s} -vbat}[/tex]

1.03×( 340[tex]\frac{m}{s}[/tex] -vbat)= 340 [tex]\frac{m}{s}[/tex] + vbat

1.03×340[tex]\frac{m}{s}[/tex] - 1.03×vbat= 340 [tex]\frac{m}{s}[/tex] + vbat

350.2[tex]\frac{m}{s}[/tex] - 1.03×vbat= 340 [tex]\frac{m}{s}[/tex] + vbat

350.2[tex]\frac{m}{s}[/tex] - 340 [tex]\frac{m}{s}[/tex]= vbat + 1.03×vbat

10.2 [tex]\frac{m}{s}[/tex]= 2.03×vbat

10.2 [tex]\frac{m}{s}[/tex]÷ 2.03 = vbat

5.02 [tex]\frac{m}{s}[/tex]= vbat

In summary, the speed of the bat is closest to 5.02 [tex]\frac{m}{s}[/tex].

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An entrepreneur is considering the purchase of a coin-operated laundry. The current owner claims that over the past 5 years, the mean daily revenue was $6750 with a population standard deviation of $265. A sample of randomly selected 30 days reveals a daily mean revenue of $6280. If you were to test the null hypothesis that the daily mean revenue was $6750, which test would you use?.

Answers

Based on the given value of sample size and the population standard deviation, the required test for the null hypothesis is the Z test of population mean.

The Z test of population is used to test the null hypothesis of a Normal distribution.

Since, the sample size is large (>30) and the variance of the population is known, the most appropriate test of hypothesis to use is the Z test of population mean.

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How many grams of iron (density = 7. 87 g/ml ) would occupy the same volume as 96. 4 g of aluminum (density = 2. 70 g/ml)?.

Answers

Explanation:

density = mass / volume

96.4 g of aluminum / volume = 2.7

therefore,

96.4 = 2.7 × volume

volume = 96.4 / 2.7 = 35.7037037... ml

back to iron :

7.87 = mass / 35.7037037...

mass = 7.87 × 35.7037037... = 280.9881481... g

it would be 280.9881481... g of iron in the same volume.

Inspectors for a hospital chain with multiple locations randomly select some of their locations for a cleanliness check of their operating rooms. The inspectors check every operating room in the hospitals that were chosen. What type of sample is this?.

Answers

Considering the situation described in the question, the type of sample Inspector used is called Simple Random Sampling

What is Simple Random Sampling?

Simple Random Sampling is a random sampling technique in which participants are taken randomly from a given population.

How this situation is a Simple Random Sampling.

Here, the given population is the hospital chain with multiple locations; since the inspector only randomly selects some of their locations, this is a simple random sampling.

Type of Random Sampling Techniques

Simple Random SamplingCluster Random SamplingSystematic Random SamplingStratified Random Sampling

Hence, in this case, it is concluded that the correct answer is Simple Random Sampling

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1) What is The Full form Of WWW ?
2)What Is The full form of 3 Cs ?


2nd question is little confused .
plz Don't Spam .​

Answers

WWW is an abbreviation for World Wide Web ...

Corporation, Customer, Competitors (3C's) Model is a business model that focuses on the three key factors needed for success. This model was conceived by the famous business strategist, Kenichi Ohmae.

Answer:

• Full form of WWW :-

World Wide Web

WWW is an abbreviation for World Wide Web and it is commonly called the WEB. It is a system that is used to store documents and is accessed using URL(Uniform Resource Locator) which are accessible through Internet .

• The full form of 3 Cs :-

Corporation, Customer, Competitors

(3C's) Model is a business model that focuses on the three key factors needed for success. This model was conceived by the famous business strategist, Kenichi Ohmae.

Explanation:

Hope this helps you !!

What is the entropy change for the freezing process of 1 mole of liquid methanol at its freezing temperature (–97. 6˚c) and 1 atm? please report your answer two points past the decimal with the unit j/molk. ∆h˚fus = 3. 17 kj/mol.

Answers

Answer:

the value of change in entropy for freezing process of 1 mole of liquid methanol is -18.07 J/mol.k

Explanation:

Entropy is defined as the measure of possible arrangement of the atoms or molecules. Entropy change for the the freezing process mole of 1 liquid methanol at -97.6 degree celcius and 1 atm will be -18.07 J/mol.

The entropy change can be calculated by the given formula:

 

Thus, the value of change in entropy for freezing process of 1 mole of liquid methanol is -18.07 J/mol.k

(Hope this helps can I pls have brainlist (crown)☺️)

 

For the freezing of 1 mole of liquid methanol, the change in entropy value is -18.07 J/mol.

What is entropy value?

The measure of atoms' or molecules' potential arrangements is known as entropy. Entropy change for one mole of liquid methanol frozen at -97.6 degrees Celsius and one atmosphere is -18.07 J/mol.

The range of entropy is 0 to 1. Depending on the number of classes in your dataset, entropy can be greater than 1, but it still indicates a very high level of chaos. For the sake of convenience, the examples in this blog will have entropy values between 0 and 1.

The "information" a variable contains is measured by its entropy. Now, based on the surprise that this value of the bholds and the number of distinct values that are present in the variable, this amount is approximated.

Thus, it is -18.07 J/mol.

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A gardener bought 5 rabbits.

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...

Thats really nice

Dark matter absorb light emit radiation and been observed directly.

Answers

Answer:

does not,

does not,

and has not

Explanation:

Dark matter doesn't absorb light, doesn't emit radiation and has not been observed directly.

What is dark matter?

Dark matter can be defined as often invisible substances that are difficult to spot and observed because they don't absorb, emit or reflect light.

This ultimately implies that, dark matter doesn't affect human view because they do not interact or interfere with electromagnetic radiation.

However, humans can see through the (weakly interacting) dark matter but it has not been observed directly by scientists and astronmers.

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In casabianca the boy's defining character trait is his (points 5).

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The character trait that defines the boy in Casabianca is: his loyalty.

Casabianca is the name of the young boy who is one of the major character sin "Casabianca".

In Casabianca, the boy, out of loyalty and having a strong sense of duty, had to put up bravery to remain at his post on a French ship that caught fire during the attack of the British.

In spite of the obvious dangers, he remained at his post while waiting for orders from the captain (his dad).

Therefore, the character trait that defines the boy in Casabianca is: his loyalty.

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A 20~\mu f20 μf capacitor has previously charged up to contain a total charge of q = 100~\mu cq=100 μc on it. The capacitor is then discharged by connecting it directly across a 100-k\omega100−kω resistor. At what point in time after the resistor is connected will the capacitor have 13. 5~\mu c13. 5 μc of charge remaining on it?.

Answers

Answer:

10385877.377477km

Explanation:

kasi po i plas lang po

A generic x by x grid has c blocks moved to create a (x-c) by (x+c) grid. Which of the following equations represents the difference of squares for this situation?.

Answers

The expression that represents the difference of squares for this situation is [tex]x^2-c^2[/tex]

Given the generic x by x grid has c blocks moved to create an (x-c) by (x+c) grid, to get the difference of two squares, we will take the product of the expressions as shown:

[tex]=(x-c)(x+c)\\=x^2+xc-xc-c^2\\=x^2 -0-c^2\\=x^2-c^2[/tex]

Hence the expression that represents the difference of squares for this situation is [tex]x^2-c^2[/tex]

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A cell with 10 chromosomes undergoes mitosis.

Answers

Answer:

Yes every cell undergoes mitosis

Select the two recommendations that would decrease the margin of error of the interval.

Answers

From the options given, the only option which would result on having a reduced error margin, is to increase the sample size.

Recall :

Margin of Error = Zcritical × σ/√n

Increasing the mean would have no impact on the margin of error as it is not a part of the factors which affects the margin of error value.

Increasing the standard deviation, which is the Numerator will result in an increased margin of error value.

By raising the confidence level, the critical value of Z will increase, hence widening the margin of error.

The sample size, n being the denominator, would reduce the value of error margin.

Hence, only the sample size will cause a decrease in the margin of error of the interval.

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Bills new porch is rectangular with an area of 50 square feet.

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Considering the whole question, the perimeter of the porch is 30 feet.

This is because the whole question is, "Bill's new porch is rectangular, with an area of 50 square feet. if the length of the porch is two times the width, what is the perimeter of the porch?"

Therefore to find the perimeter of the porch, here is the step by step:

Given that the length of the porch is two times the width.

Hence, assume that the width = X

Also, assume that the length = 2X

The general formula here:

50 = X × 2X

X × 2X50 = 2x²

2x²X² = 50 ÷2

50 ÷2X² = 25

25X = √25

√25X = 5

Hence, width = 5 feet.

Given that length of the porch is two times the width = 5 × 2 => 10 feet

Therefore Perimeter of a rectangle = 2 (Length + Breadth)

Put all the values in the formula

Perimeter of the porch = 2 × (5 + 10)

= 2 × (15)

= 30 feet.

Hence, in this case, it is concluded that the correct answer is 30 feet.

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What is the equation for g, which is f(x) = 2x2 + 3x − 1 reflected across the y-axis?.

Answers

The answer to your question is:

How does the first amendment protect religious freedom for american citizens? choose exactly two answers that are correct.

Answers

Answer:

It lets people practice their faith in their own way and

It prevents the government from establishing a national religion.

Explanation:

A cube with volume 8 $\text{cm}^3$ is cut in half by a plane which contains two edges of the cube. What is the area of the rectangle formed by the intersection of the plane and cube? express your answer in simplest radical form.

Answers

The area of the rectangle formed by the intersection of the plane and cube is [tex]4\sqrt{2}cm^2[/tex]

The diagram of the cube and the cutting plane is shown in the figure.

Since the cube has a volume of [tex]8cm^3[/tex], the sides are [tex]2cm[/tex] each.

From the diagram, the red rectangle is the rectangle formed from the intersection of the plane and the cube. It has an area given by

[tex]A_R=\text{side of cube}\times \text{diagonal}=2d[/tex]

and, using Pythagoras' theorem,

[tex]d=\sqrt{2^2+2^2}\\d=\sqrt{8}=2\sqrt{2}\text{ cm}[/tex]

Substituting the value of [tex]d[/tex] into the formula for [tex]A_R[/tex]

[tex]A_R=2\times d\\=2\times 2\sqrt{2}\\=4\sqrt{2}[/tex]

The area of the rectangle formed by the intersection of the plane and cube is [tex]4\sqrt{2}cm^2[/tex]

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Answer:

4 square root of 2cm

Explanation:

cool stuff

A student runs 120 meters in 10 seconds. What is their speed?.

Answers

Answer:

12

Explanation:

speed=distance÷time

Given three consecutive integers, if the sum of the first and third integers is increased by 8, the result is 5 less than triple the second integer. Find the three consecutive integers. *

Answers

Answer:

The three consecutive integers are 12, 13 & 14

Explanation:

Let the three consecutive integers be x, x + 1, x + 2

According to the question,

If sum of the first and third integers is increased by 8, the result is 5 less than triple the second integer.

[tex] \therefore [/tex] x + x + 2 + 8 = (x + 1) × 3 - 5

2x + 10 = 3x + 3 - 5

2x + 10 = 3x - 2

10 + 2 = 3x - 2x

x = 12

So, The three consecutive integers are 12, 13 & 14

Which historical figure does old major most likely represent? tsar nicholas ii an imperial guard vladimir lenin joseph stalin.

Answers

Answer: old major is based on Karl Marx and partially Vladimir Lenin

Explanation:

The historical figure that old major most likely to represent is Vladimir Lenin. The correct option is C.

Who does old Major represent and why?

The pig Snowball represents the intellectual revolutionary Leon Trotsky, while Old Major represents either Karl Marx or Vladimir Lenin.

Old Major represents Karl Marx and Vladimir Lenin, who promoted the Communist Party but died before their vision could be realized.

As a result, those who did not fully comprehend the dream were able to twist it to their advantage.

Old Major describes a dream he had the night before in which animals are free, happy, well fed, and treated with dignity in a world free of human tyranny.

He exhorts the animals to do everything possible to make this dream a reality, including overthrowing the humans who claim to be their owners.

Thus, the correct option is C.

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Evaluate the line integral, where c is the given curve. C (x/y) ds, c: x = t3, y = t4, 1 ≤ t ≤ 4.

Answers

We have

x = t³   ===>   dx/dt = 3t²

y = t⁴   ===>   dy/dt = 4t³

Then with the given parameteriztion, the line integral along C of x/y is

[tex]\displaystyle \int_C \frac xy \, ds = \int_1^4 \frac{t^3}{t^4} \sqrt{(3t^2)^2 + (4t^3)^2} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \int_1^4 \frac1t \sqrt{9t^4 + 16t^6} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \int_1^4 \frac{\sqrt{t^4}}t \sqrt{9 + 16t^2} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \int_1^4 \frac{t^2}t \sqrt{9 + 16t^2} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \int_1^4 t \sqrt{9 + 16t^2} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \frac1{32} \int_1^4 32t \sqrt{9 + 16t^2} \, dt[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \frac1{32} \int_1^4 \sqrt{9 + 16t^2} \, d\left(9+16t^2\right)[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \frac1{32} \cdot \frac23 \left(9+16t^2\right)^{\frac32}\bigg|_1^4[/tex]

[tex]\displaystyle \int_C \frac xy \, ds = \frac1{48} \left(265^{\frac32} - 25^{\frac32}\right) = \boxed{\frac{265\sqrt{265}-125}{48}}[/tex]

The line integral will be "[tex]\frac{265\sqrt{265} - 125 }{48}[/tex]". To understand the calculation, check below.

Line integral

According to the question,

Curves, x = t³ or,

            [tex]\frac{dx}{dt}[/tex] = 3t²

             y = t⁴ or,

            [tex]\frac{dx}{dt}[/tex] = 4t³  

Now, the line integral along C will be:

→ [tex]\int\limits_C {\frac{x}{y} } \, ds[/tex] = [tex]\int\limits^4_1 {\frac{t^3}{t^4} }[/tex] √(3t²)² + (4t³)² dt

             = [tex]\int\limits^4_1 {\frac{1}{t} }[/tex] √9t⁴ + 16t⁶ dt

             = [tex]\int\limits^4_1 {\frac{t^4}{t} }[/tex] √9 + 16t² dt

             = [tex]\int\limits^4_1 {\frac{t^2}{t} }[/tex] √9 + 16t² dt

             = [tex]\int\limits^4_1[/tex] t√9 + 16t² dt

             = [tex]\frac{1}{32}[/tex] [tex]\int\limits^4_1[/tex] √9 + 16t² d (9 + 16t²)

             = [tex]\frac{1}{32}\times \frac{2}{3}[/tex] [tex](9+16t^2)^{\frac{3}{2} } |_1^4[/tex]

             = [tex]\frac{1}{48}[/tex] ([tex]265^{\frac{3}{2}} - 25^{\frac{3}{2}[/tex])

             = [tex]\frac{265 \sqrt{265 } -125 }{48}[/tex]

Thus the above approach is correct.

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If there is more than one speaker, how should they be all noted?.

Answers

Answer:

By their names. If the names are not known you should mark them as speaker 1, speaker 2, speaker 3, etc.

Explanation:

Based on the diagram, can point d be the centroid of triangle acf? explain. Yes, point d is the point of intersection of segments drawn from all three vertices. Yes, de is three-quarters of the length of the full segment. No, de should be longer than ad. No, the ratio between ad and de is 3:1.

Answers

Answer:

A is the correct answer

Explanation:

EDGE

A centroid in a triangle, is the point where all the medians meet.

now a peculiar thing about the centroid is that, it cuts all medians in a 2:1 ratio. So, if that's true, then the median AE, is being cut in two segments AD and DE and AD : DE are on a 2 : 1 ratio.

What is centroid?

In a triangle, the centroid is the point of intersection of the three medians, which are line segments joining each vertex to the midpoint of the opposite side.

The centroid divides each median into two segments, with the distance from the centroid to the vertex being twice the distance from the centroid to the midpoint of the opposite side.

Let M be the midpoint of side AB, N be the midpoint of side AC, and P be the midpoint of side BC. Then, we have:

CD = 2DP

CE = 2DN

We are also given that DE = 15. Since D is the centroid, we have:

AD = BD = CD/3

AE = CE/3

DC + DP = PC

CE + DN = NE

AD + AE = DE

Using these relationships, we can find CD and CE:

CD = 2DP = 2(PC - DC) = 2((AD + DP) - (AD/3)) = 2(2AD/3) = 4AD/3

CE = 2DN = 2(NE - CE) = 2((AE + DN) - (AE/3)) = 2(2AE/3) = 4AE/3

To find AD and AE, we can use the fact that AD + AE = DE and AD = BD:

AD + BD = DE

AD + (2AD/3) = 15

5AD/3 = 15

AD = 9

AE = DE - AD = 15 - 9 = 6

Substituting these values into the equations for CD and CE, we get:

CD = 4AD/3 = 4(9)/3 = 12

CE = 4AE/3 = 4(6)/3 = 8

Therefore, CD = 12 and CE = 8.

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Write your own lyrics in english using the music/rhythm of your favorite song.

Answers

Fall into sea
Another way to go, peacefully
Take your time
The money holder says get in line
Forget the colored stars in your mind

Back, back and fourth doubled as a curse, you’re falling short
Both, both are wrong
Excessive as it is, it’s way to long
Push your way through, you’re too strong

I’ve got something to say
Tied in your macramé tears
Fighting off evil things space is what love can bring, now

Persistent like a dog, locked up in your demagogue, go
Play it off like a joke, locked up in gentle strokes, no

Too and forth i make mistakes
Remember about this place, i know
We never know what to say, i know.

Do you ever wish your dreams could come true?

Dream of things that won't make you boo.

Dreams can come real,

just make sure you make them feel.

If it's a nightmare make sure you hide up the back stairs,

and if it's a good dream make sure you make it seen.

So I really wish dreams were real,

that way I could start to feel.

Explanation: Sorry it isn't that good I did the best I could. It's also rrrreeeealllllllyyyyy short sooo.......

Use stokes' theorem to evaluate s curl f · ds. F(x, y, z) = x2 sin(z)i + y2j + xyk, s is the part of the paraboloid z = 4 − x2 − y2 that lies above the xy-plane, oriented upward.

Answers

Stokes' theorem says that the surface integral of the curl of the vector field F across the surface S is equal to the line integral of F along the boundary of S.

In this case, the boundary is a circle in the x,y-plane with radius 2. (To see this, set z = 0 and rearrange the equation for the paraboloid as x² + y² = 4.)

Then

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_{\partial S} \vec F \cdot d\vec r[/tex]

where

[tex]\vec r = 2\cos(t) \,\vec\imath + 2\sin(t) \,\vec\jmath[/tex]

parameterizes the circular boundary of S with 0 ≤ t ≤ 2π. Then

[tex]d\vec r = \dfrac{d\vec r}{dt}\,\dt = (-2\sin(t)\,\vec\imath + 2\cos(t)\,\vec\jmath)\,dt[/tex]

and the integral becomes

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_{\partial S} \vec F \cdot d\vec r[/tex]

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} \vec F(\vec r(t)) \cdot \frac{d\vec r}{dt}\,dt[/tex]

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s \\= \int_0^{2\pi} ((2\cos(t))^2 \sin(0)\,\vec\imath + (2\sin(t))^2\,\vec\jmath + (2\cos(t))(2\sin(t))\,\vec k) \cdot(-2\sin(t)\,\vec\imath + 2\cos(t)\,\vec\jmath)\,dt[/tex]

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} 8\sin^2(t)\cos(t) \, dt[/tex]

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} 8\sin^2(t) \, d(\sin(t))[/tex]

[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \frac83 (\sin^3(2\pi)-\sin^3(0)) = \boxed{0}[/tex]

Consider the following line integral. C xy dx + x2y3 dy, c is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 4).

Answers

It looks like the integral is

[tex]\displaystyle \int_C xy \, dx + x^2y^3 \, dy[/tex]

Let's close the loop by adding a line integral over the line segment joining (1, 4) to (0, 0). Then the closed loop is the triangular region

T = {(x, y) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 4x}

Since the integrand has no singularities over or on the boundary of T, we have by Green's theorem

[tex]\displaystyle \int_C xy \, dx + x^2y^3 \, dy = \iint_T \left(\frac{\partial(x^2y^3)}{\partial x} - \frac{\partial(xy)}{\partial y}\right) \, dx \, dy = \int_0^1 \int_0^{4x} (2xy^3 - x) \, dy \, dx[/tex]

Compute the double integral:

[tex]\displaystyle \int_0^1 \int_0^{4x} (2xy^3 - x) \, dy \, dx = \int_0^1 \int_0^{4x} \left(128x^5 - 4x^2\right) \, dx = \frac{60}3[/tex]

From this result, we subtract the line integral over the extra line segment we added. Parameterize this path by

C' : {(1 - t, 4 - 4t) : 0 ≤ t ≤ 1}

The line integral over C' is

[tex]\displaystyle \int_{C'} xy \, dx + x^2y^3 \, dy = \int_0^1 (1-t)(4-4t)(-dt) + (1-t)^2(4-4t)^3(-4\,dt)[/tex]

[tex]\displaystyle \int_{C'} xy \, dx + x^2y^3 \, dy = \int_0^1 \left(256 t^5-1280 t^4+2560 t^3-2564 t^2+1288 t-260 \right) \, dt = -44[/tex]

so that the line integral over C alone is 60/3 - (-44) = 64.

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