C F · dr= -6 π by Stokes' Theorem
Stokes' Theorem states that the circulation of the curl of a vector field F around a closed curve C is equal to the flux of the curl of F through any surface bounded by C.
Using Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = 6yi + xzj + (x + y)k, C is the curve of intersection of the plane z = y + 8 and the cylinder x2 + y2 = 1.
Stokes' Theorem:
∫C F · dr = ∬s (curl F) · dS
Here, the given vector field F is: F(x, y, z) = 6yi + xzj + (x + y)k
C is the intersection of the plane z = y + 8 and the cylinder x2 + y2 = 1. The equation of the plane is given as z = y + 8.
The equation of the cylinder is given as x2 + y2 = 1. This can be rearranged as y = sqrt(1 - x2). Now, substitute this value of y in the equation of the plane to get:
z = sqrt(1 - x2) + 8
Therefore, the curve C is given by the intersection of the above two equations. The parameterization of this curve can be given by:
r(t) = xi + yj + zk, where y = sqrt(1 - x2), and z = sqrt(1 - x2) + 8Substitute the values of y and z to get:
r(t) = xi + sqrt(1 - x2)j + (sqrt(1 - x2) + 8)k
Now, we can use the Stokes' Theorem to find the circulation of the vector field F around the curve C. We need to find the curl of the vector field F first.
curl F = ( ∂Q/∂y - ∂P/∂z ) i + ( ∂P/∂z - ∂R/∂x ) j + ( ∂R/∂x - ∂Q/∂y ) k,
where P = 0, Q = 6y, and R = x + y.
Substitute these values to get,
curl F = -6j
Therefore,
∫C F · dr = ∬s (curl F) · dS= ∬s -6j · dS
As viewed from above, the projection of the surface S on the xy plane is the unit circle centered at the origin. Therefore, the surface integral can be calculated using polar coordinates as follows:
S = {(r, θ) : 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π}j = sin(π/2)j (since the unit vector in the j direction is j itself)
Therefore, the surface integral is given by,
∬s -6j · dS= -6 ∬s j · dS= -6 ∬s sin(π/2)j · r dr dθ= -6 ∫0^{2π} ∫0^1 r dr dθ= -6 π
Therefore,
∫C F · dr = ∬s (curl F) · dS= -6 π
Answer is -6π
learn more about Stokes' Theorem here:
https://brainly.com/question/10773892
#SPJ11
Find all the missing elements
Round to the nearest tenth.
Answer:
B = 48.7°
C = 61.3°
b = 12
Step-by-step explanation:
Given:
A = 70°
a = 15
c = 14
Required:
B, C, and b
Solution:
✔️Using the law of sines, let's find C:
Sin C/c = Sin A/a
Plug in the values
Sin C/14 = Sin 70/15
Cross multiply
Sin C × 15 = Sin 70 × 14
Divide both sides by 15
Sin C = (Sin 70 × 14)/15
Sin C = 0.8770
C = Sin^{-1}(0.8770)
C = 61.282566° = 61.3° (nearest tenth)
✔️Find B:
B = 180 - (70 + 61.3) (sum of triangle)
B = 48.7°
✔️Find b using the law of sines:
b/sinB = a/sinA
Plug in the values
b/sin 48.7 = 15/sin 70
Cross multiply
b*sin 70 = 15*sin 48.7
Divide both sides by sin 48.7
b = (15*sin 48.7)/sin 70
b = 11.9921789
b = 12.0 (nearest tenth)
If you flip two coins, what is the probability that both will be heads?
Answer:
1/2 x 1/2 = 1/4
Step-by-step explanation:
P(H,H) = 1/2 x 1/2 = 1/4
P(T,T) = 1/2 x 1/2 = 1/4
P(H,T) = 1/2 x 1/2 = 1/4
P(T,H) = 1/2 x 1/2 = 1/4
1. The angle of depression from the top of the
school to the base of the flag pole in front of
the school is 50°. If the flag pole is 35 feet from
the base of the school, find the height of the
school.
Answer:
41.7feet
Step-by-step explanation:
From the question we are given the following
angle of depression = 50°
Distance of the pole from the base of the feet = 35feet (Adjacent)
Required
height of the school (opposite)
Using the SOH CAH TOA identity
Tan theta = opp/adj
Tan 50 = H/35
H = 35tan 50
H = 35(1.1918)
H = 41.7feet
Hence the height of the school is 41.7feet
What is a volume of this composite solid?
Shape a = 6 x 4 x 3 = 72
Shape b = 4 x 4 x 4 = 64
72 + 64 = 136
A
Answer:
A
Step-by-step explanation:
use mathmatecial induction to prove that for each non negative odd integer n: 24 / (2²³1 +1+1) (n²₁)
Mathematical induction can be used to prove that for every non-negative odd integer n, the expression [tex]24 / (2^{(3n+1)}+1+1) * (n^2+1)[/tex] holds true.
To prove the statement using mathematical induction, we need to follow two steps: the base case and the induction step.
First, we verify if the statement holds true for the base case, which is typically the smallest value of n. In this case, let's consider n = 0. Plugging in n = 0 into the expression, we get [tex]24 / (2^{(3*0+1)}+1+1) * (0^2+1)[/tex]. Simplifying, we have 24 / (2+1+1) * 1, which equals 24 / 4 * 1, resulting in 6. Therefore, the statement holds true for n = 0.
Next, we assume that the statement is true for some arbitrary odd integer k, and we will prove that it holds true for k+2. Assume that [tex]24 / (2^{(3k+1)}+1+1) * (k^2+1)[/tex] holds true.
Now, we substitute k+2 into the expression and aim to show that it holds true for k+2 as well. We have [tex]24 / (2^{(3(k+2)+1)}+1+1) * ((k+2)^2+1)[/tex]. Simplifying the expression, we get [tex]24 / (2^{(3k+7)}+1+1) * (k^2 + 4k + 5)[/tex].
We can manipulate the equation further to demonstrate that it is equal to the assumed expression for k. By performing algebraic manipulations and simplifications, we can equate the expressions and conclude that the statement holds true for k+2.
Since we have verified the base case and shown that the statement holds true for k+2 when it holds true for k, we can conclude that the statement is true for every non-negative odd integer n.
Learn more about integer here: https://brainly.com/question/490943
#SPJ11
1) Which of the following can cause OLS estimators to be biased? Which of the following do not cause the usual OLS t statistics to be invalid (that is, to have t distributions under H0)? (6 points)
Omitting an important independent variable
Multicollinearity
Heteroskedasticity
Including irrelevant variable
The error term non-normally distributed
The following can cause OLS estimators to be biased: Omitting an important independent variable.
Multicollinearity. Heteroskedasticity, Including an irrelevant variable.
The following does not cause the usual OLS t statistics to be invalid (that is, to have t distributions under H0): The error term non-normally distributed
OLS (ordinary least squares) estimates are typically unbiased when calculated.
However, the following problems may cause OLS estimates to be biased:
Omitting an important independent variable: When an important independent variable is omitted from the regression equation, the OLS estimate of the effect of one variable on the dependent variable is biased.
In particular, the estimate of the effect of the variable that is omitted is influenced by the remaining variables' presence in the equation.
Multicollinearity: When the independent variables in a multiple regression model are strongly related, multicollinearity exists.
When there is multicollinearity in a model, the estimated slope coefficients are frequently biased, making them difficult to interpret.
In this scenario, small changes in the data may cause substantial changes in the estimated coefficients.
As a result, the usual tests of hypothesis may fail to produce reliable inferences.
Heteroskedasticity: In the population, heteroskedasticity exists when the variance of the error term is not constant across observations.
Heteroskedasticity can induce OLS estimates' variance to be biased, even if the estimates are unbiased themselves.
When there is heteroskedasticity, the OLS estimates are no longer BLUE (best linear unbiased estimator).
Including irrelevant variable: When an irrelevant variable is included in a regression equation, the OLS estimates of the other variables' effects are biased, and the estimates' standard errors are larger than necessary.
The error term non-normally distributed: When the error term in a regression equation is non-normally distributed, the distribution of the OLS estimates is also non-normal.
However, this does not affect the distribution of the t statistics under H0.
The reason for this is that, even if the error term is non-normally distributed, the sample mean converges to the population mean, according to the central limit theorem.
Furthermore, the standard error of the mean is unaffected by the distribution of the error term, as long as the sample size is large enough.
As a result, the t statistics can be trusted to be asymptotically normally distributed under H0.
To know more about Multicollinearity, visit:
https://brainly.com/question/30691253
#SPJ11
Find the cost of cat food for a 29-day supply, a 30-day supply, and a 31-day supply
Answer:
not a complete question..
Step-by-step explanation:
How many meters are equal to 3,736 centimeters? Use how the placement of the decimal point changes when dividing by a power of 10 to help you.
Answer:
37.36 meters are equal to 3,736 centimeters.
Step-by-step explanation:
We have that 1 meter is equal to 100 centimeters, so if we want to convert 3,736 cm to m we need to divide by 100:
[tex] x = 3,736 cm*\frac{1 m}{100 cm} = 3,736 cm*\frac{1 m}{10^{2} cm} [/tex]
When we divide a number by a power of 10, we move the decimal point to the left as many places as the power indicates.
Since we are dividing by 10², we need to move the decimal point two places to the left, as follows:
[tex] x = 3,736 cm*\frac{1 m}{10^{2} cm} = 37.36 m [/tex]
Hence, 37.36 meters are equal to 3,736 centimeters.
I hope it helps you!
Which of the following factors does not affect the mortgage payment? No A. Interest rates
B. The down payment
C. The borrower's credit score
D. The neighborhood the home is located in
Answer:
The neighborhood the home is located in
The factors which does not affect the mortgage payment is the neighborhood the home is located in.
The correct option is D.
We are aware of this;
A mortgage is a loan when the borrower's property is used as security. The mortgage payment is determined by the cost of the property, the interest rate, the down payment, the length of the loan, taxes, and various insurances like homeowners insurance, among other factors.
Hence, It's not depend on ''The neighborhood the home is located in.''
Learn more about Mortgage payment here:
brainly.com/question/1859113
#SPJ7
(x_6)(x_5)+(x_5)(x_6)
Answer:
(X+6)(x-5) + (X-5)(X+6)
Answer:
[tex]2x ^{2} - 22x + 60[/tex]
What is the perimeter of abcd ?
Answer:
38
Step-by-step explanation:
(10*2)+(9*2)
what is the volume of the right triangle prism shown
Answer:
42cm3
Step-by-step explanation:
v = 0.5× b × h × l
= 0.5 × 4 × 3 × 7
= 42cm3
In macroeconomics, the economy can best be understood through the use of
money.
models.
pricing.
debates.
Answer:
models
Step-by-step explanation:
Answer: B.) Models
Edge
A hockey tournament consists of 16 teams. In the first round, every team is randomly assigned to one of 8 games (2 teams per game). Suppose exactly 3 of the teams are from Alberta. What is the probability all 3 Alberta teams are randomly assigned to different games (call this event A)?
Given that a hockey tournament consists of 16 teams. In the first round, every team is randomly assigned to one of 8 games (2 teams per game). We are supposed to find the probability that all three Alberta teams are randomly assigned to different games. Let A be the event of assigning all three Alberta teams to different games.
Then the number of ways to select 3 teams from 16 teams is $\ dbinom {16}{3}$, the number of ways to assign 3 teams to different games is $8\times7\times6$, and the number of ways to assign the remaining 13 teams to games is $(13!) / (2^6\times6!)$.The probability of event A is given by;$$
P(A) = \frac{\text{number of ways to assign 3 teams from Alberta to different games}}{\text{number of ways to assign all teams to games}} = \frac{8\times7\times6 \times (13!) / (2^6\times6!)}{\dbinom{16}{3} \times (14!) / (2^7\times7!)}
$$Simplifying the above expression,$$
P(A) = \frac{8\times7\times6 \times 13! \times 2}{\dbinom{16}{3} \times 14!} = \frac{8\times7\times6 \times 2}{\dbinom {16}{3}} = \frac{336}{560} = \frac{3}{5}
Therefore, the probability that all three Alberta teams are randomly assigned to different games is $\frac{3}{5}$.
Know more about probability:
https://brainly.com/question/31828911
#SPJ11
Find the slope: numbers are: (1,-3) and (-5,-4)
[tex]\frac{-3 - (-4)}{1 - (-5)}[/tex]
= [tex]\frac{-3 + 4}{1 + 5}[/tex]
= 1/6
Answer: the slope is 1/6How many yards are equivalent to 38 feet? Show your work.
Answer:
12.6
Step-by-step explanation:
divide the length value by 3
Which expression is equivalent to 10(−45x+3)−2x?
−8x+3
−10x+3
−10x+30
−30x+30
Answer:
-10x+30
Step-by-step explanation:
just got it right on edg 2021 :)
None of these are correct
What is the fraction that is equal to 0.534
Answer:
267/500
Step-by-step explanation:
0.534 = 534 / 1000
Simplify to 267/500
Step-by-step explanation:
537/1000
this is the correct answer
a cookie factory uses 1/6 pf a barrel pf oatmeal in each batch of cookies, the factory used 1 1/3 barrels of oatmeal yesterday. how many batches of cookies did the factory make?
Answer:
5 batches
Step-by-step explanation:
1/6 oatmeal can make 1 batch, so 5/6 makes 5 batches
Two numbers sum to 312 and have a difference of 210. What are the two numbers?
Answer:
51 and 261
Step-by-step explanation:
Let the numbers be x and y
x + y =312
x - y =210
2y=102
y=51
x=312-51=261
function project: a day of fun
Answer:
Yay fun day :DDDDDD
Step-by-step explanation:
Consider an election with 129 votes.
(a) If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have? Explain your answer.
(b) If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have? Explain your answer
(a) If there are 4 candidate, the smallest number of votes that a plurality candidate could have is 33.
(b) If there are 8 candidate, the smallest number of votes that a plurality candidate could have is 17.
What is the smallest number of votes obtained?The smallest number of votes that a plurality candidate could have is calculated as follows;
(a) If there are 4 candidate, the number of votes for each candidate;
= 129 / 4
= 32.25
The least number of votes for the plurality candidate = 33
(b) If there are 8 candidate, the number of votes for each candidate;
= 129 / 8
= 16.125
The least number of votes for the plurality candidate = 17
Learn more about smallest number in division here: https://brainly.com/question/29898343
#SPJ4
Last translation I need help with I promise-
The types of transformation in this problem is given as follows:
Vertical and horizontal translation.
What are the translation rules?The four translation rules are defined as follows:
Left a units: x -> x - a. -> horizontal translation.Right a units: x -> x + a. -> horizontal translation.Up a units: y -> y + a. -> vertical translation.Down a units: y -> y - a. -> vertical translation.The translations for this problem are given as follows:
3 units left -> horizontal translation.3 units up -> vertical translation.More can be learned about translation at brainly.com/question/29209050
#SPJ1
A 12-sided solid has faces numbered 1 to 12. The table shows the results of rolling the solid 200 times. Find the experimental probability of rolling a number less than 3 .
Answer:
The experimental probability of rolling anumber less than 3 is 3/20
Step-by-step explanation:
16+14=30
30/200
=3/20
Answer:
7/50
Step-by-step explanation:
Please help me on this question if you would want brianleist!! Tank yah!! ^^
Answer:
x axis
Step-by-step explanation:
(Fermat's Theorem, 5pt) Calculate 2^2873686243768478237864767208 mod 101 using Fermat's little theorem (that is, without computer, and without repeated squaring). Explain how you did it. Hint: 101 is prime.
To calculate[tex]2^2873686243768478237864767208[/tex] mod 101 using Fermat's little theorem, we can simplify the exponent by taking it modulo 100, since 100 is the Euler's totient function value of 101. Therefore,[tex]2^2873686243768478237864767208[/tex] mod 101 is equal to 57.
Fermat's little theorem states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) ≡ 1 (mod p). In this case, p = 101, and we need to find[tex]2^2873686243768478237864767208[/tex]mod 101.
First, we simplify the exponent by taking it modulo 100, since 100 is the Euler's totient function value of 101. The exponent 2873686243768478237864767208 is congruent to 8 modulo 100. So, we need to calculate 2^8 mod 101. Applying Fermat's little theorem, we know that 2^(101-1) ≡ 1 (mod 101), since 101 is prime. Therefore, 2^100 ≡ 1 (mod 101).
We can express [tex]2^8[/tex] in terms of 2^100 as [tex](2^100)^0.08[/tex]. Simplifying this, we get [tex](2^100)^0.08 ≡ 1^0.08[/tex]≡ 1 (mod 101).
Thus, we conclude that[tex]2^8[/tex] ≡ 1 (mod 101), and therefore 2^2873686243768478237864767208 ≡ [tex]2^8[/tex] (mod 101).
Finally, evaluating [tex]2^8[/tex] mod 101, we find that [tex]2^8[/tex] ≡ 57 (mod 101).
Therefore,[tex]2^2873686243768478237864767208[/tex] mod 101 is equal to 57.
Learn more about congruent here:
https://brainly.com/question/30596171
#SPJ11
Please please please help me with this pleasseee!!!
[tex]\Large\boxed{\tt Answer:~A:~y=-3x+5}[/tex]
All we have to do in order to find the equation that is equivalent to 6x + 2y = 10 is to solve for y.
Step 1: Keep y on one side of the equation and move the rest to the other.
[tex]\tt 6x + 2y = 10\\2y = -6x + 10[/tex]
Step 2: Divide all terms by 2 to isolate y.
[tex]\tt 2y \div 2=y\\-6x \div 2 = -3x\\10 \div 2 =5\\\\y=-3x+5[/tex]
Let M = {x +1, x2 – 2,3x}. Which of the following statements is true about M? M spans P3 O the above is true M spans P2 O the above is true O None of the mentioned
The correct statement is : M spans P2.
The set M = {[tex]x+1, x^2-2, 3x[/tex]} consists of three polynomials in the variable x.
To determine whether M spans P3 or P2, we need to consider the highest degree of the polynomials in M.
The highest degree of the polynomials in M is 2 (from [tex]x^2-2[/tex]), which means that M can span at most the space of polynomials of degree 2 or less, i.e., P2.
To check whether M spans P2 or not, we need to see if any polynomial of degree 2 or less can be expressed as a linear combination of the polynomials in M.
We can write any polynomial of degree 2 or less as [tex]ax^2 + bx + c[/tex], where a, b, and c are constants.
To express this polynomial as a linear combination of the polynomials in M, we need to solve the system of equations:
[tex]a(x^2-2) + b(x+1) + c(3x) = ax^2 + bx + c[/tex]
This can be written as:
[tex]ax^2 + (-2a+b+3c)x + (b+c) = ax^2 + bx + c[/tex]
Equating the coefficients of [tex]x^2, x,[/tex] and the constant term, we get:
[tex]a = a,\\-2a+b+3c = b,\\b+c = c.[/tex]
The first equation is always true, and the other two equations simplify to:
[tex]-2a+3c = 0,\\b = 0.[/tex]
Solving for a, b, and c, we get:
[tex]a = 3c/2,\\b = 0,\\c = c.[/tex]
Therefore, any polynomial of degree 2 or less can be expressed as a linear combination of the polynomials in M. This means that M spans P2.
However, M cannot span P3, because P3 includes polynomials of degree 3, which cannot be expressed as a linear combination of the polynomials in M (since the highest degree polynomial in M is [tex]x^2[/tex]).
Therefore, the correct statement is: M spans P2.
Learn more about Polynomial span : https://brainly.com/question/31857690
#SPJ11
Find the surface area of a cylinder with a height of 4 yd and a base radius of 3 yd.
Answer:
131.95yd squared
Step-by-step explanation:
A=2πrh+ 2 r(2)=2*π*3*4+2*π*3(2)~131.94689yd(2)
A person borrows a certain amount of money. He has to pay the debt in equal installments once every month, for 10 years. The first installment was paid on 2016-01-01. Find the date on which he has to pay the final installment.
Answer: January 1, 2025
Step-by-step explanation:
The debt is due to be paid back in 10 years.
If the first payment was in 2016, the last payment should therefore be:
= 2016 + 9 years
= 2025
We used 9 years because 2016 was the first year of payment so the remaining years would be 9 years.
As the first payment was on January 1, 2016, the last payment would have to be on the same date in 2025 which is:
= January 1, 2025