Suppose F⃗ (x,y)=(x+3)i⃗ +(6y+3)j⃗ . Use the fundamental theorem of line integrals to calculate the following.
(a) The line integral of F⃗ along the line segment C from the point P=(1,0) to the point Q=(3,3).
∫CF⃗ ⋅dr⃗ =?
(b) The line integral of F⃗ along the triangle C from the origin to the point P=(1,0) to the point Q=(3,3) and back to the origin.
∫CF⃗ ⋅dr⃗ =?
Answers
In order to use the fundamental theorem of line integrals, you need to find a scalar potential function - that is, a scalar function f(x, y) for which
grad f(x, y) = F(x, y)
This amounts to solving for f such that
∂f/dx = x + 3
∂f/∂y = 6y + 3
Integrating both sides of the first equation with respect to x gives
f = 1/2 x ^2 + 3x + g(y)
Differentiating with respect to y gives
∂f/∂y = dg/dy = 6y + 3
Solving for g gives
g = ∫ (6y + 3) dy = 3y ^2 + 3y + C
and hence
f(x, y) = 1/2 x ^2 + 3x + 3y ^2 + 3y + C
(a) By the fundamental theorem, the integral of F along any path starting at the point P (1, 0) and ending at Q (3, 3) is
∫ F(x, y) • dr = f (3, 3) - f (1, 0) = 99/2 - 7/2 = 46
(b) Now we're talking about a closed path, so the integral is simply 0. We can verify this by checking the integral over the origin-containing paths:
• From the origin to P :
∫ F(x, y) • dr = f (1, 0) - f (0, 0) = 7/2 - 0 = 7/2
• From Q back to the origin:
∫ F(x, y) • dr = f (0, 0) - f (3, 3) = 0 - 99/2 = -99/2
Then the total integral is 7/2 + 46 - 99/2 = 0, as expected.
Related Questions
Please help . Stuck
Answers
Answer:
[tex] - \frac{9}{2} [/tex]
Step-by-step explanation:
Rate of change of y with respect to x
[tex] = \frac{60 - 96}{ - 12 - ( - 20)} \\ \\ = \frac{ - 36}{ - 12 + 20} \\ \\ = \frac{ - 36}{ 8} \\ \\ = - \frac{9}{2} [/tex]
LAST ONE, ON A TIMER. Find the zeros of the function and write in a + bi form.
-(x+1)^2-4=0
Answers
Answer:
Option DStep-by-step explanation:
-(x+1)²- 4 = 0(x+1)² = - 4 x + 1 = ± √-4x = -1 ± 2iCorrect choice is D
Answer:
D
Step-by-step explanation:
Given
- (x + 1)² - 4 = 0 ( add 4 to both sides )
- (x + 1)² = 4 ( multiply both sides by - 1 )
(x + 1)² = - 4 ( take the square root of both sides )
x + 1 = ± [tex]\sqrt{-4}[/tex] = ± 2i ( subtract 1 from both sides )
x = - 1 ± 2i , then
x = - 1 + 2i, x = - 1 - 2i → D
What value of x makes m/n(pls show work)
Answers
Answer:
x = 20
Step-by-step explanation:
For the lines to be parallel, the angles have to be equal
2x+15 = 55
Subtract 15 from each side
2x+15-15 = 55-15
2x = 40
2x/2 = 40/2
x = 20
Rent me $250 washing cars with his mobile car wash company he charges $70 per car and earn $50 in tips write an equation to represent the situation
Answers
Answer:
Try 250 = 70x + 50, or y = 70x + 50.
Step-by-step explanation:
He made 250 total. The variable is how many cars he washes. The y-intercept is 50. Therefore, either 250 = 70x + 50, or y = 70x + 50.
Answer:
250 = 70x + 50
Step-by-step explanation:
this is the equation if you are looking for the profit
Ex.
You washed 2 cars
250 = 70 (2) + 50
250 = 190
-190 -190
60
you are still $60 away from making a profit
What would be the correct answer for this?
Answers
First let's understand the discriminant. Discriminant is an equation of D = b²-4ac. Discriminant is to find how many solutions do the equation have. In case for Quadratic Equation, there are 3 types of solutions which are:
No solutions or D < 0 — When D < 0, the equation does not have real roots.One solution or D = 0 — When D = 0, the equation has one real root. Two solutions or D > 0 — When D > 0, the equation has two real roots.To find discriminant graphically, we need to know that the x-axis plane represents the solutions to the equations. All equations can be drawn as a graph. Some graphs do not intersect x-axis while some graph do.
If a graph intersects x-axis, the equation for that graph has solutions. If a graph doesn't intersect x-axis, the equation doesn't have any real solutions.You may also notice that some graphs intersect x-axis more than one point. Like I said that the x-axis represents the roots or the solutions. Intersecting x-axis "n" times means that the equation has "n" roots.
For example, if a graph intersects x-axis 8 points - the equation has 8 roots or solutions.
From the graph, the parabola intersects on x-axis just only one point. That means there is only one solution. The question asks to find the discriminant of the function. Recall that if D = 0, there's only one real root. Therefore the discriminant for the function is zero.
Answer
The discriminant for function is zero.Let me know if you have any doubts!
Which function of x has a y-intercept of -5?
Answers
Answer:
[tex]y = - 3x - 5 \\ c = - 5[/tex]
Mike works h hours in a week. To find his weekly pay he multiplies his hours by his pay of $24 per
hour and adds $6 for his sales bonus. He's decided that he wanted to take one-sixth of his weekly
pay and place it in his savings. Which simplified expression represents the amount that will go into
his savings each week?
Answers
Step-by-step explanation:
let n be the number of working hour
so total income is 24n + 6
now 1/6 of the income will be into saving
so the saving is (24n + 6) /6
The expression representing the amount that will go into his savings each week is $4h + $1.
What is Expression?An expression is combination of variables, numbers and operators.
Mike's weekly pay is given by:
Pay = (hourly rate) x (hours worked) + sales bonus
Pay = $24h + $6
To find one-sixth of his weekly pay, we can multiply his pay by 1/6:
Savings = (1/6) x Pay
Savings = (1/6) x ($24h + $6)
Savings = $4h + $1
Therefore, the simplified expression representing the amount that will go into his savings each week is $4h + $1.
To learn more on Expressions click:
https://brainly.com/question/14083225
#SPJ2
Determine the radius of a cone that has a volume of 155.521 cubic inches and a height of 9 inches.
Answers
Answer:The answer is 4.06
Step-by-step explanation:
Okay I am 98% sure my math could be wrong since I don’t know what the possible answers to the question are but this is what I got.
Write the equation of a straight line that passes through the point (3,1) and is parallel to the line y=2x-1
Answers
Answer:
Step-by-step explanation:The new line will be parallel to the line represented by y =2x +2
The equation y = 2x + 2 is written in slope- intercept form (i.e. y=mx + b) where m represents the slope. So the slope of this line is 2. A line parallel to y = 2x + 2 will have the same slope. This means the slope of the new line will also be 2.
To determine the equation of the new line, we can use the given point to find the y intercept of the new line:
y = 2x + b
Use x = -1 and y =3 to find b (or the y intercept ) of the new line:
3 = 2 * (-1) + b
3= -2 + b
b=5
Therefore, the equation of the line passing thru (-1,3) with the same slope as that of y = 2x + 2 will be:
y = 2x + 5
Upvote
•
1
Zoey bought a new skateboard that cost 36.79. How much change should she get if she paid for the skateboard with two 20 bills
Answers
Answer:
4 dollars and something back
Step-by-step explanation:
Hi there!
»»————- ★ ————-««
I believe your answer is:
$3.21
»»————- ★ ————-««
Here’s why:
Zoey bought a $36.79 skateboard and paid for it with two $20 bills. This means that Zoey paid $40 in total.⸻⸻⸻⸻
[tex]40.00-36.79=\boxed{3.21}[/tex]
⸻⸻⸻⸻
She should get $3.21 as her change.
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
What is the value of x?
Answers
Answer:
80°
Step-by-step explanation:
Central Angles and the Arc they create are equal
The following stem-and-leaf plot represents the times in minutes required for 26 co-workers to commute to work. Use the data provided to find the quartiles. Commute Times in Minutes Stem Leaves 2 1 3 5 5 5 8 3 1 5 6 7 8 9 4 2 3 4 6 6 6 8 5 2 2 4 5 7 7 8 Key: 2|1
Answers
Answer:
Q1 = 30.25 minutes
Q2 = 42.5 minutes
Q3 = 40 minutes
Step-by-step explanation:
Given, In Question
The stem and leaf diagram (key 2 I 1)Stem | Leaves
2 | 1 3 5 5 5 8
3 | 1 5 6 7 8 9
4 | 2 3 4 6 6 6 8
5 | 2 2 4 5 7 7 8
We need to find the quartiles Q1, Q2 and Q3. We know that,Q1 = (1/4)*(n+1)th value
Q2 = (1/2)*(n+1)th value
Q3 = (3/4)*(n+1)th value
where n is the total number of co-workers 26.
So, Q1 = (1/4)*(26+1)th valueQ1 = 6.75th value
we need to count the leaves in the plot starting from the first one until we reach the 6.75th value. So, by counting, we conclude that the 6.75th value lies between the 6th and 7th value i.e. 28 and 31.
Q1 = 28 + (31-28)*0.75
= 28 + 2.25
Q1 = 30.25 minutes
Now, Q2 = (1/2) * (26+1)th value
= 13.5th value.
From the plot, we find that the 13.5th value lies in the middle of the 13th and 14th values i.e. 42 and 43. So,
Q2 = (42+43)/2
= 85/2
Q2 = 42.5 minutes
And, Q3 = (3/4)*(26+1)th value
= 20.25th value
From the plot. we find that the 20.25th value lies somewhere between the 20th and 21st value i.e. 52 and 52. So,
Q3 = 40 + (52-52)*0.25
= 40 + 0
Q3 = 40 minutes
How many small squares in this rectangle should be shaded to represent
5 of
I
Answers
Answer:
Where's the picture
Step-by-step explanation:
The picture...
Each observation indicates the primary position played by the Hall of Famers: pitcher (P), catcher (H), 1st base (1), 2nd base (2), 3rd base (3), shortstop (S), left field (L), center field (C), and right field (R). a. Construct frequency and percent frequency distributions to summarize the data. Position Frequency Percent Frequency (to one decimal) Pitcher % Catcher % 1st base % 2nd base % 3rd base % Shortstop % Left field % Center field % Right field % b. What position provides the most Hall of Famers
Answers
Answer:
a. See below for the Frequency and Relative frequency Table.
b. Pitcher (P) is the position provides the most Hall of Famers.
c. 3rd base (3) is the position that provides the fewest Hall of Famers.
d. R is the outfield position that provides the most Hall of Famers.
e. Th number of Hall of Famers of Infielders which is 16 is less than the 18 Hall of Famers of those of outfielders.
Step-by-step explanation:
Note: This question not complete. The complete question is therefore provided before answering the question as follows:
Data for a sample of 55 members of the Baseball Hall of Fame in Cooperstown, New York, are shown here. Each observation indicates the primary position played by the Hall of Famers: pitcher (P), catcher (H), 1st base (1), 2nd base (2), 3rd base (3), shortstop (S), left field (L), center field (C), and right field (R).
L P C H 2 P R 1 S S 1 L P R P
P P P R C S L R P C C P P R P
2 3 P H L P 1 C P P P S 1 L R
R 1 2 H S 3 H 2 L P
a. Use frequency and relative frequency distributions to summarize the data.
b. What position provides the most Hall of Famers?
c. What position provides the fewest Hall of Famers?
d. What outfield position (L, C, or R) provides the most Hall of Famers?
e. Compare infielders (1, 2, 3, and S) to outfielders (L, C, and R).
The explanation of the answers is now provided as follows:
a. Use frequency and relative frequency distributions to summarize the data.
The frequency is the number of times a position occurs in the sample, while the relative frequency is calculated as the frequency of each position divided by the sample size multiplied by 100.
Therefore, we have:
Frequency and Relative frequency Table
Position Frequency Relative frequency (%)
P 17 30.91%
H 4 7.27%
1 5 9.09%
2 4 7.27%
3 2 3.64%
S 5 9.09%
L 6 10.91%
C 5 9.09%
R 7 12.73%
Total 55 100%
b. What position provides the most Hall of Famers?
As it can be seen from the frequency table in part a, Pitcher (P) has the highest frequency which is 17. Therefore, Pitcher (P) is the position provides the most Hall of Famers.
c. What position provides the fewest Hall of Famers?
As it can be seen from the frequency table in part a, 3rd base (3) has the lowest frequency which is 2. Therefore, 3rd base (3) is the position that provides the fewest Hall of Famers.
d. What outfield position (L, C, or R) provides the most Hall of Famers?
As it can be seen from the frequency table in part a, we have:
Frequency of L = 6
Frequency of C = 5
Frequency of R = 7
Since R has the highest frequency which is 7 among the outfield position (L, C, or R), it implies that R is the outfield position that provides the most Hall of Famers.
e. Compare infielders (1, 2, 3, and S) to outfielders (L, C, and R).
Total frequency of infielders = Frequency of 1 + Frequency of 2 + Frequency of 3 + Frequency of S = 5 + 4 + 2 + 5 = 16
Total frequency of outfielders = Frequency of L + Frequency of C + Frequency of R = 6 + 5 + 7 = 18
The calculated total frequencies above imply that number of Hall of Famers of Infielders which is 16 is less than the 18 Hall of Famers of those of outfielders.
7,35,175, 875,... find the 10th term
Answers
Explanation: each number is multiplied by 5 to get the next number.
write the number sentence as an inequality: a number n is greater than 1
ill give brainlyiest
Answers
Answer:
n > 1
Step-by-step explanation:
Find the area of the composite/irregular figure. Use 3.14 for pi.
PLEASE SHOW WORK/EXPLANATION FOR BRAINLIEST
Answers
explanation: separate into rectangle and semi circle. you can find out the diameter and radius of the semi circle by subtracting the 6 from the top from the 16 of the bottom because the diameter of the circle plus the 6 take up the same length.
16 - 6 = 10
for radius divide by 2
10 / 2 = 5
area of rectangle: l x w
16 x 7 = 112
area of semi circle: 1/2(pi)r^2
1/2(3.14)5^25 = 39.25
add up the two areas to find the area of the entire figure
112 + 39.25 = 151.25
Answer:
151.25cm
Step-by-step explanation:
First split the the shape into two, a rectangle and a semicircle.
Let's begin with the rectangle. The formula for the area of a rectangle is
A = l × w
A = 16 × 7
A = 112
Now we move onto the semicircle. We know that the length of the long side of the rectangle is 16. Because of the markings, we know that the missing portion of the 6cm line is 16 as well. That means that we would do 16 - 6 to find that small missing length. It gives us 10cm which will act as the diameter for the semicircle.
The formula for the area of a semicircle is
A = [tex]\frac{\pi r^{2} }{2}[/tex] (fill in what you know)
A = [tex]\frac{3.14 (5^{2} )}{2}[/tex] (The radius is half the diameter. So 10 ÷ 2 = 5)
A = [tex]\frac{3.14(25)}{2}[/tex] (Solve what's in the parentheses first)
A = [tex]\frac{78.5}{2}[/tex] (multiply pi and 25)
A = 39.25
Now all we have to do is add the two areas to get the combined area of the whole.
A = 112 + 39.25
A = 151.25
Solve for the value of X pleassee
Answers
Answer:
188
Step-by-step explanation:
SR and ST are equal in length so the arc SR and arc ST also have same measurement
since the perimeter of the circle is equal to 360 and arc RT is given as 64
x = (360 - 64) / 2
x = 188
—1, 0, 2, 5, 9,
一一一
Es geometric, neither, arithmetic
Answers
Answer: Neither since you are adding n, n+1, n+2 etc
Step-by-step explanation:
Multiply.
3,672 × 4
Enter your answer
Answers
That’s is the answer
the ratio equivalent to 3:4.
Answers
Answer:
33:44
Step-by-step explanation:
4*11=44
3*11=33
In which function is x = 2 mapped to 32 ?
Answers
Answer:
Option 2 is the correct answer
[tex]g(x) = 4 {(x + 3)}^{2} - 68 [/tex]
Step-by-step explanation:
[tex]g(x) = 4 {(x + 3)}^{2} - 68 \\ \\ \because \: g(2) = 4 {(2 + 3)}^{2} - 68 \\ = 4 {(5)}^{2} - 68 \\ = 4(25) - 68 \\ = 100 - 68 \\ = 32[/tex]
Answer:
option b
Step-by-step explanation:
See image below:)
FYI you can use the app photo math, you just take a pic of the problem and it gives you the answer and explains the steps and it is free.
solving equation using the completing the square (a).3p(p+4)=p(p+2)
Answers
Answer:
P = 0, -5
Step-by-step explanation:
Answer:
p = 0
p = - 5
Step-by-step explanation:
3p(p + 4) = p(p + 2)
3p² + 12p - (p² + 2p) = 0
3p² + 12p - p² - 2p = 0
3p² + 10p - p² = 0
2p² + 10p = 0
2p(p + 5) = 0
p = 0
p + 5 = 0
p + 5 - 5 = 0 - 5
p = - 5
Use law of sines or law of cosines to find the length of side AB Please show work !!
Answers
Answer:
[tex]AB = 11.3[/tex]
Step-by-step explanation:
Given
The attached triangle
Required
Length AB
To do this, we make use of sine law which is represented as:
[tex]\frac{a}{\sin(a)} = \frac{b}{\sin(b)} = \frac{c}{\sin(c)}[/tex]
So, we have:
[tex]\frac{15}{\sin(70)} = \frac{AB}{\sin(45)}[/tex]
Make AB the subject
[tex]AB = \frac{15}{\sin(70)} * \sin(45)[/tex]
[tex]AB = \frac{15}{0.9397} *0.7071[/tex]
[tex]AB = 11.3[/tex]
Please help me in this question.
Answers
Answer:
a) 24
b) 63
c) 2
d) -2
Step-by-step explanation:
you are supposed to try different numbers, but...
X-9 = 15 add 9 to both sides
X = 24
7x/9 = 49 multiply both sides by 9
7x = 441 divide by 7
x = 63
3(x + 6) = 24 divide both sides by 3
(x + 6) = 8 subtract 6 from both sides
X = 2
11x + 2 = -20 subtract 2 from both sides
11x = -22 divide both sides by 11
X = -2
Jim records the number of cars his team sells
each month for a year. What percent of the
months does his team sell more than 6 cars?
Quick pld
Answers
Answer:
50%
Step-by-step explanation:
There are 12 months in a year.In the problem there are six months the number of cars his team sells more than 40%.=6/12x100%=0.5x100%=50%.Therefore his team sold 50%.
Hope this helped!
The manager of the dairy section of a large supermarket chose a random sample of 250 egg cartons and found that 30 cartons had at least one broken egg. let p denote the proportion of all cartons which have at least one broken egg. Find a point estimate for p and also construct a 90% confidence interval for p.
Answers
Answer:
The point estimate for p is of 0.12.
The 90% confidence interval for p is 0.0862 < p < 0.1538.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
The manager of the dairy section of a large supermarket chose a random sample of 250 egg cartons and found that 30 cartons had at least one broken egg.
This means that [tex]n = 250, \pi = \frac{30}{250} = 0.12[/tex]
The point estimate for p is of 0.12.
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.12 - 1.645\sqrt{\frac{0.12*0.88}{250}} = 0.0862[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.12 + 1.645\sqrt{\frac{0.12*0.88}{250}} = 0.1538[/tex]
The 90% confidence interval for p is 0.0862 < p < 0.1538.
Which lists all the factors of 78?
A. 1, 2, 3, 6, 13, 26, 39, 78
B. 1, 2, 4, 19, 39, 78
C. 1, 2, 6, 13, 39, 78
D. 2, 3, 6, 13, 26, 39
Answers
Answer:
1, 2, 3, 6, 13, 26, 39, 78
Answer:
Factors of 78: 1, 2, 3, 6, 13, 26, 39, and 78.
Step-by-step explanation:
I hope this helps!!!!!
PLEASE HELP!!!
In 1994, the city of Amuel had a population of 1,256 people. That same year a factory opened near the town, and many people moved into the city limits. The population grew to 1,381 people in 1995, and in 1996 the population of Amuel reached 1,519 people. Assume this rate of growth continued until the factory closed in 2007. How many people were living in Amuel when the factory closed? Explain. Round to the nearest whole number, if needed.
Answers
Answer:
Step-by-step explanation:
so if in 1994 it was = 1,256
1995 = 1256 + x = 1,381 {x is 125}
1996 = 1381 + x = 1,519 {x is 138}
2007 = it is eleven years from 1996 to 2007 so take the average of the both digits which we will find through adding both the x quantities and dividing it with 2 the answer is 131.5. back to 2007, so multiply 11 with 131.5 which is 1,446.5. now add 1,446.5 to 1519
and the answer is 2,965.5.
hope it helps
TEST I - Reading and Writing Numbers. (15 points) (15 minutes)
A. Read these numbers. (1 point per item=5 points) (5 minutes)
1.) 3/4
2.) P 12 131 267. 25
3.) 75%
4.) 1256
5.) 12:26
Answers
Answer:
Step-by-step explanation:
1)
Three-Fourth
2)
Twelve Million One Hundred Thirty One Thousand Two Hundred Sixty Seven and Twenty Five Hundredths
3)
Seventy Five Percentage
4)
One Thousand Two Hundred and Fifty Six
5)
Twelve is to Twenty Six
