Mathematics College

G verify that the divergence theorem is true for the vector field f on the region
e. f(x, y, z) = z, y, x , e is the solid ball x2 + y2 + z2 ≤ 36

Answers

Answer 1
[tex]\mathbf f(x,y,z)=\langle z,y,x\rangle\implies\nabla\cdot\mathbf f=\dfrac{\partial z}{\partial x}+\dfrac{\partial y}{\partial y}+\dfrac{\partial x}{\partial z}=0+1+0=1[/tex]

Converting to spherical coordinates, we have

[tex]\displaystyle\iiint_E\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=6}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=288\pi[/tex]

On the other hand, we can parameterize the boundary of [tex]E[/tex] by

[tex]\mathbf s(u,v)=\langle6\cos u\sin v,6\sin u\sin v,6\cos v\rangle[/tex]

with [tex]0\le u\le2\pi[/tex] and [tex]0\le v\le\pi[/tex]. Now, consider the surface element

[tex]\mathrm d\mathbf S=\mathbf n\,\mathrm dS=\dfrac{\mathbf s_v\times\mathbf s_u}{\|\mathbf s_v\times\mathbf s_u\|}\|\mathbf s_v\times\mathbf s_u\|\,\mathrm du\,\mathrm dv[/tex]
[tex]\mathrm d\mathbf S=\mathbf s_v\times\mathbf s_u\,\mathrm du\,\mathrm dv[/tex]
[tex]\mathrm d\mathbf S=36\langle\cos u\sin^2v,\sin u\sin^2v,\sin v\cos v\rangle\,\mathrm du\,\mathrm dv[/tex]

So we have the surface integral - which the divergence theorem says the above triple integral is equal to -

[tex]\displaystyle\iint_{\partial E}\mathbf f\cdot\mathrm d\mathbf S=36\int_{v=0}^{v=\pi}\int_{u=0}^{u=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(\mathbf s_v\times\mathbf s_u)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle36\int_{v=0}^{v=\pi}\int_{u=0}^{u=2\pi}(12\cos u\cos v\sin^2v+6\sin^2u\sin^3v)\,\mathrm du\,\mathrm dv=288\pi[/tex]

as required.

Related Questions

Jemma has 24 balls. Out of the 24 balls,
12 are yellow, 4 are pink, and the rest
are red. What is the ratio of the number
of red balls to the number of balls that
are either yellow or pink?

Answers

The ratio of red balls to either yellow or pink balls is 8:24, or simplified to the max it would be 1:3

Mark earns $135 for selling a $1500 refrigerator. Which sale listed has the same commission rate? $36 commission on a $400 dryer or B) $40 commission on a $500 dryer sale C) $210 commission on $1400 oven sale d) $30 commission on a $600 freezer sale?
salelisted has the same commssion rate. ?

Answers

The answer is A

36/400 * 3.8 = 136.6/1520
40/500 * 3 = 120/1500
210/1400 (not even close)
30/600 * 3 = 90/1800 (not even close)

so the two answer choices that are close are A and B, A has the closest numbers to the original commission rate.

Answer:

Option A is the correct choice.

Step-by-step explanation:

We have been given that Mark earns $135 for selling a $1500 refrigerator.

Since proportions states that two fractions are equal , so we will use proportions to find the same commission rate with the listed price from our given choices.

We can set proportion for commission and listed price as:

[tex]\frac{\text{Commission}}{\text{Listed price}}=\frac{135}{1500}[/tex]

Let us compare our given choices one by one.

A. $36 commission on a $400 dryer.

[tex]\frac{36}{400}=\frac{135}{1500}[/tex]

Upon simplifying our proportion we will get,

[tex]\frac{9}{100}=\frac{9}{100}[/tex]

[tex]0.09=0.09[/tex]

We can see that commission rates for both items are equal, therefore, option A is the correct choice.

B. $40 commission on a $500 dryer sale.

[tex]\frac{40}{500}=\frac{135}{1500}[/tex]

Upon simplifying our proportion we will get,

[tex]\frac{2}{25}=\frac{9}{100}[/tex]

[tex]0.08\neq 0.09[/tex]

We can see that both commission rates are different, therefore, option B is not a correct choice.

C. $210 commission on $1400 oven sale.

[tex]\frac{210}{1400}=\frac{135}{1500}[/tex]

Upon simplifying our proportion we will get,

[tex]\frac{3}{20}=\frac{9}{100}[/tex]

[tex]0.15\neq 0.09[/tex]  

Since the commission rates are different, therefore, option C is not a correct choice.

D.  $30 commission on a $600 freezer sale.

[tex]\frac{30}{1400}=\frac{135}{1500}[/tex]

Upon simplifying our proportion we will get,

[tex]\frac{3}{140}=\frac{9}{100}[/tex]

[tex]0.0214\neq 0.09[/tex]  

Since the commission rates are different, therefore, option D is not a correct choice.

Find the integer a such that
a.a ≡ 43 (mod 23) and −22 ≤ a ≤ 0.

Answers

The integer a such that a ≡ 43 (mod 23) and −22 ≤ a ≤ 0 is -3. A zero, a positive natural number, or a negative integer with a minus sign is an integer.

What do we mean when we say that integers are congruent to modulo?When two positive integers a and b are divided by a positive integer n, they are said to be congruent modulo n (or an is congruent to b modulo n), or if a b is divisible by n. It can be expressed as a b mod n, where n is the modulus.If n is a positive integer, then the integers a and b are said to be congruent modulo n, as denoted by the expression a b (modn).Division algorithm Allow dd and aa to be both positive integers. Following that, there are two distinct integers, qq and rr, where rd0 and rd are equal, resulting in a = dq + ra = dq + r

Therefore,

-3 is in between -22 and 0 so a =-3

a ≡ 43 (mod 23) and −22 ≤ a ≤ 0.

a ≡ 43 (mod 23)

≡ 43-22 (mod 23)

Simplifying,

≡ 20 (mod 23)

≡ 20-23 (mod 23)

Then we get,

≡ -3 (mod 23)

-3 is in between -22 and 0 so a =-3

To learn more about congruent modulo refer to:

brainly.com/question/13427492

#SPJ2

Select all numbers that are irrational. 1049−−√ 3.14 π 0.5

Answers

The answer would be 10/49 and π which are irrational numbers from the definition of the irrational number.

What is an irrational number?

It is defined as the numbers in all real numbers which cannot be represented as rational numbers, in other words, the irrational number cannot be expressed in the form of p/q form.

A rational number is a sort of real number that has the form p/q and does not equal zero.

It is given that:

The numbers are:

10/49,  3.14,  π,  0.5

As we know, from the definition of the irrational, the numbers which cannot be represented as rational numbers, in other words, the irrational number cannot be expressed in the form of p/q form.

The numbers 10/49 and π are irrational numbers.

Thus, the answer would be 10/49 and π which are irrational numbers from the definition of the irrational number.

Learn more about the irrational number here:

brainly.com/question/17450097

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What is the value of x?

Answers

The value of x is 35°

Evaluate 3(y-5) for y=7

Please help

Answers

So if y= 7 you plug in the variable into the equation.

3(7-5)
3(2)
6
The answer is 6. I hope this helps love!:)
3(7-5)= exchange y for 7
3(2)= subtract 5 from 7
6 multiply 3 times 2

A)- -17
B)- -8
C)- -2
D)- 2
E)- 5
F)- 8
G)- 15
SELECT ALL THAT APPLY

Answers

B -8
D -2
G 15

That can represent an function that sentences

how many fossil kits with 10 samples each have the same number of fossils as 30 fossil kits with 8 samples each?

Answers

((( answer 24 )))as 30×8= 240 and 10×24 = 240

Answer:

24 fossil kits.

Step-by-step explanation:

Given,

The number of fossil kits = 30,

the number of samples each has  = 8,

So, the total samples = number of fossil kits × the number of samples contained by each kit

= 30 × 8

= 240,

Let x be the number fossil kits with 10 samples each,

Total samples = 10x

According to the question,

10x = 240

[tex]\implies x =\frac{240}{10}=24[/tex]

Hence, there would be 24 fossil kits with 10 samples each.

Chris answered 61/66 questions correctly. What is his test average to the nearest thousandth?

Answers

61/66=x/100
x=61*100/66=92.424

A seafood company sold 9,125 pounds of fish last month. If 6 seafood companies sold the same amount of fish, how much fish did the 6 companies sell last month in all ?

Answers

The 6 seafood companies sold 54,750 pounds of fish

solve the following system of equations 3x+4y=17 -4x-3y=-18

Answers

3x + 4y = 17.....multiply by 4
-4x - 3y = -18 ...multiply by 3
------------------
12x + 16y = 68 (result of multiplying by 4)
-12x - 9y = - 54 (result of multiplying by 3)
------------------add
7y = 14
y = 14/7
y = 2

3x + 4y = 17
3x + 4(2) = 17
3x + 8 = 17
3x = 17 - 8
3x = 9
x = 9/3
x = 3

ur solution is (3,2)

At an automobile factory, 1849 parts are made in 4 hours. What is the average rate at which parts are made per hour?

Answers

Just divide "1849 parts" by "4 hours:"  the average rate is 1849/4 parts/hour,

or 1849 parts/ 4 hours, or 462.25 parts/hour, or (rounded off) 462 parts/hour.

Answer:

462.25

Step-by-step explanation:

At an automobile factory, the number of parts are made in 4 hours = 1849.

We have to calculate the average rate of production per hour.

We will divide the production of parts in 4 hours by 4 to get the number of parts made per hour.

The average rate of production in one hour = [tex]\frac{1849}{4}[/tex]

                                                                         = 462.25/hour

462.25 parts are made per hour.

differentiate please..

y= √x +6 ÷ √x-6

Answers

[tex]\bf y=\cfrac{\sqrt{x}+6}{\sqrt{x}-6}\implies \cfrac{dy}{dx}=\stackrel{quotient~rule}{\cfrac{\frac{1}{2}x^{-\frac{1}{2}}(\sqrt{x-6})~~-~~(\sqrt{x}+6)\frac{1}{2}x^{-\frac{1}{2}}}{(\sqrt{x}-6)^2}}[/tex]

[tex]\bf \cfrac{dy}{dx}=\cfrac{\frac{1}{2}x^{-\frac{1}{2}}~(\underline{\sqrt{x}}-6~-~\underline{\sqrt{x}}-6)}{(\sqrt{x}-6)^2}\implies \cfrac{dy}{dx}=\cfrac{\frac{1}{2}x^{-\frac{1}{2}}(-12)}{(\sqrt{x}-6)^2} \\\\\\ \cfrac{dy}{dx}=\cfrac{\frac{-12}{2\sqrt{x}}}{(\sqrt{x}-6)^2}\implies \cfrac{dy}{dx}=\cfrac{\frac{-6}{\sqrt{x}}}{(\sqrt{x}-6)^2}\implies \cfrac{dy}{dx}=\cfrac{-6}{\sqrt{x}(\sqrt{x}-6)^2}[/tex]

I thought of a number, then divided it by another number, and got a result of 7. Think of the possible values for my number and the number I divided it by. What number is impossible to divide by? What number can my original number not be?

My number is a multiple of ?
The number I divided my number by can be any number but ?
My number cannot be a number which we cannot divide by ?

Answers

Let the first unknown number be [tex]x[/tex]
And the second unknown number be [tex]y[/tex]

We know that [tex]x[/tex] ÷ [tex]y[/tex] = 7

My number is a multiple of 7 and also multiple of [tex]y[/tex]
This is because of 7 × [tex]y[/tex] = [tex]x[/tex]
-----------------------------------------------------------------------------------------------------------

The number I divided my number by can be any number but ZERO
Because [tex]x[/tex] ÷ 0 is undefined
--------------------------------------------------------------------------------------------------------------

My number cannot be a number which we cannot divide by SEVEN
Because [tex]x[/tex] ÷ [tex]y[/tex] = 7, so [tex]x [/tex] must be divisible by 7 to obtain [tex]y[/tex]

[tex] \frac{x}{y} =7[/tex]
[tex] \frac{x}{7} =y[/tex]
[tex]7y=x[/tex]


what is the sum of 2 3/8 and 3 3/7

Answers

first off, we'd convert the mixed fractions to "improper", then get the LCD to sum them up.

[tex]\bf \stackrel{mixed}{2\frac{3}{8}}\implies \cfrac{2\cdot 8+3}{8}\implies \stackrel{improper}{\cfrac{19}{8}} \\\\\\ \stackrel{mixed}{3\frac{3}{7}}\implies \cfrac{3\cdot 7+3}{7}\implies \stackrel{improper}{\cfrac{24}{7}}\\\\ -------------------------------\\\\ \cfrac{19}{8}+\cfrac{24}{7}\impliedby \textit{so our LCD is just 56}\implies \cfrac{(7\cdot 19)~+~(8\cdot 24)}{56} \\\\\\ \cfrac{133+192}{56}\implies \cfrac{325}{56}\implies 5\frac{45}{56}[/tex]

A scatterplot is produced to compare the size of a lake to the number of fish that are in it. There are 15 data points, each representing a different lake. The points are widely dispersed on the scatterplot with no pattern of grouping. Interpret what the results of the scatterplot tell you about the relationship between the two variables.

Answers

A: Since there is no cluster formed in the scatterplot, the two variables are not related. Therefore, based on the data shown in the scatterplot, the number of fish in a lake is not dependent on the size of the lake.

Based on the dispersion of the scatterplot, we can infer that there is no relationship between the variables.

Relationships in scatterplots Data points are used to predict relationships. If a line can be drawn based on these data points, then a relationship exists.

The fact that these data points are dispersed however and offer no form of grouping, means that the variables are not related and show no correlational trend.

Find out more on scatterplots at https://brainly.com/question/2749543.

write a word problem that can be solved by ordering three decimals to thousandths.

Answers

Angelo wants to grow the smallest pea which will be 2.6 times smaller than worlds smallest pea currently is 5 millimeters , what is 5 divided by 2.6? Given: smallest pea = 2.6 millimeters, worlds smallest pea = 5 millimeters

[Solution: 5 / 2.6 = 1.923076923076923 millimeters or approximately 1.923 millimeters]

kalen's truck has a 60 gallon fuel tank The fuel gauge points to 1/4 how much fuel does he need to buy to fill the tank show Your work

Answers

He should need 45 more gallons. 60/4=15 15gallons for every 1/4 of the tank. 60-15=45

Alice leaves her house and walks to school. She walks 45 meters south and 336 meters east. How far is Alice from her house?

Answers

she is 381 meters away from her house
The way to figure this out is to first set it up on an x/y graph. it also looks like a compass, which tells you which direction east and south is. So first draw a point where 45 would be (also south). Now draw a point around 336 east. You are now going to use the Pythagorean Theorem (Asquared +Bsquared =Csquared) to figure it out. 336 squared, + 45 squared. When you get that answer, square it, giving you your distance. Answer = 339

What is the total rounded to the nearest tenth of 94.23

Answers

The total is 94.2.

Hope this helps!
See 94.23 is the number you are rounding.

94.23 the 2 is in the tenth place so you look at the next number 3 since it’s less then 5 the total rounded to the nearest tenth would be 94.2 because you take off the 3

Use the cosine double-angle identity cos(2θ) = 1 − 2sin2θ to derive an identity for 2sin2θ

Answers

then is just a matter of linear simplification

[tex]\bf cos(2\theta )=1-2sin^2(\theta )\implies 2sin^2(\theta )=1-cos(2\theta )[/tex]
Final answer:

To derive an identity for 2sin2θ using the cosine double-angle identity cos(2θ) = 1 − 2sin²θ, we can rearrange the equation and substitute the value of sin²θ to obtain the identity 2sin²θ = 1 - cos(2θ).

Explanation:

To derive an identity for 2sin2θ using the cosine double-angle identity cos(2θ) = 1 − 2sin²θ, we can start by rearranging the equation to solve for sin²θ. This gives us sin²θ = (1 - cos(2θ))/2.

Next, we can substitute this value into the expression for 2sin²θ. This gives us 2sin²θ = 2(1 - cos(2θ))/2 = 1 - cos(2θ).

Therefore, the identity for 2sin²θ is 1 - cos(2θ).

Line IJ has an equation of a line y = −3x − 8. Which of the following could be an equation for a line that is parallel to line IJ?

A) y=3x+7
B) y=1/3x+7
C) y= -3x+7
D) y= -1/3x+7

Answers

Answer:

C) y = -3x +7

Step-by-step explanation:

It's all about recognizing patterns.

The form of the equation is ...

... y = mx + b

In this form, the coefficient of x, which we called "m" in the above, is the slope of the line. When two lines have the same slope, they are parallel. So, we're looking for an equation that has an x-coefficient of -3, the same as in the given equation.

The coefficient of x in selection C is -3, so that is our answer choice.

... C) y = -3x +7

Answer:

y = -3x +7 which is option C

got it right on my test.

What produces a graph in a straight line

Answers

Linear equations in one variables shows always graph in straight line,

For ex: x = 3, y = 4, x = -1.5, y = -2/3

Also two variables linear equations with degree of 0 or 1 (Each term) shows always graph in straight line,

For ex: x + y = 5, 2x - 4y = 7 etc.

write the equation of the line MN, which is formed by the points M(-3,5) and N(2,0)

Answers

The equation is y = -x + 2

Step 1

Find the slope of the given line

Let

[tex]M(-3,5)\ N(2,0)[/tex]

we know that

the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

substitute the values

[tex]m=\frac{0-5}{2+3}[/tex]

[tex]m=\frac{-5}{5}[/tex]

[tex]m=-1[/tex]

Step 2

Find the equation of the line

we know that

the equation of the line into point-slope form is equal to

[tex]y-y1=m(x-x1)[/tex]

we have  

[tex]m=-1[/tex]

point N [tex](2,0)[/tex]

substitute the values

[tex]y-0=-1(x-2)[/tex]

[tex]y=-x+2[/tex]

therefore

the answer is

[tex]y=-x+2[/tex]

Ari’s teacher says he may have his report grade based on either the mean or the median of his last six test scores.

88%, 73%, 97%, 76%, 90%, 80%

Answers

To find the mean you: add up all the numbers, then divide by how many numbers there are.  So the mean for these numbers would be: 84%

To find the median: You put the numbers in order (From least to greatest), and then find the number exactly in the middle. Here it is an even amount of numbers. No problem, just add the two numbers in the middle and then divide by 2. The answer here is: 84%

Answer: The Mean

The mean is when you add up all of those grades and divide by the amount of exams.

Car A began a journey from a point at 9 am, traveling at 30 mph. At 10 am car B started traveling from the same point at 40 mph in the same direction as car A. At what time will car B pass car A

Answers

Car a 30(t+1)

Car b 40t

 30(t+1) =40t

30t+30 =40t

30=10t

T = 30/10 = 3hours

9 +3 = 12 noon


Find the point of intersection of the tangent lines to the curve r(t) = 5 sin(πt), 6 sin(πt), 3 cos(πt) at the points where t = 0 and t = 0.5.

Answers

the answer your looking for is c hope i helped you

put these in order from the smallest
6² 3 to the power of 4 2 to the power of 5 5³

Answers

Hello!

A = 6 to the power of 2, which is 36
B = 3 to the power of 4, which is 81
C = 2 to the power of 5, which is 32
D = 5 to the power of 3, which is 125

Therefore, the order is C, A, B, D

Hope that helped!
Helloo.

From least to greatest.
1. 6^2= 36
2. 3^4 = 81
3. 5^3 = 125
4. 2^5 = 128

208,000,478 in expanded form

Answers

200,000,000+8,000,000+400+70+8

How is adding 4.56 + 2.31 similar to adding $2.31 + $4.56?

Answers

They're all whole/natural numbers. The only difference is that one problem is in the form of currency. Anyhoo, you'll end up with the same answer. Currency, and an equation. The similarities are the factors, and the operations are the same.
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