Use the Fundamental Theorem of Line Integrals to calculate F · dr C exactly. F = (x + 2) i + (2y + 3) j and C is the line from (3, 0) to (5, 2).
Answers
To use the Fundamental Theorem of Line Integrals, we first need to find a potential function, say φ(x, y), such that F = ∇φ.
Given F = (x + 2)i + (2y + 3)j, let's find the potential function φ(x, y) by integrating F with respect to x and y:
∂φ/∂x = x + 2 => φ(x, y) = (1/2)x^2 + 2x + g(y)
∂φ/∂y = 2y + 3 => φ(x, y) = y^2 + 3y + h(x)
Comparing the two expressions for φ(x, y), we can deduce that g(y) = y^2 + 3y and h(x) = (1/2)x^2 + 2x. Thus, the potential function is:
φ(x, y) = (1/2)x^2 + 2x + y^2 + 3y
Now, we can apply the Fundamental Theorem of Line Integrals:
∫(F · dr) = φ(B) - φ(A)
Where A = (3, 0) and B = (5, 2).
φ(A) = (1/2)(3)^2 + 2(3) + (0)^2 + 3(0) = 13.5
φ(B) = (1/2)(5)^2 + 2(5) + (2)^2 + 3(2) = 37
Now, subtract:
∫(F · dr) = φ(B) - φ(A) = 37 - 13.5 = 23.5
So, the exact value of the line integral is 23.5.
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Related Questions
Pls help me somebody plzzz
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Answer:
the answer is
X + 40 + 155 =180
any help? I'll give brainlest
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Answer:
C
Step-by-step explanation:
Answer:
answer is a
Step-by-step explanation:
Bob uses 16 planks of wood to make 4 beehives. How many planks of wood will he need to make 9 beehives?
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Answer:
36
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Determine the common difference, the fifth term, the nth term, and the 100th term of the arithmetic sequence. 2, 7, 12, 17,...d= ________a5= _______an=_______
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The common difference of the arithmetic sequence 2, 7, 12, 17, ... is 5. The fifth term is 22. The nth term is given by the formula an = 2 + 4(n - 1). The 100th term is 497.
The common difference is calculated by subtracting the first term from the second term. In this case, 7 - 2 = 5. The fifth term is calculated by using the formula an = a1 + (n - 1)d, where a1 is the first term, n is the term number, and d is the common difference. In this case, a5 = 2 + (5 - 1)5 = 22. The nth term is also calculated using the same formula. In this case, an = 2 + 4(n - 1). The 100th term is calculated by plugging in n = 100 into the formula, which gives an = 497.
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for the following grouped frequency distribution table, what is the width of each class interval? x f 20-29 2 30-39 5 40-49 4 50-59 1
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The width of each class interval from the given grouped frequency distribution table is 9.
What is a class interval?In a frequency distribution, the numerical width of a class is referred to as the class interval. Data is organised into classes in a grouped frequency distribution. The class interval is determined by the difference between the upper and lower class limits.
The give class intervals are 20-29, 30-39, 40-49 and 50-59.
The size, or width, of a class interval is the difference between the lower and upper class boundaries and is also referred to as the class width, class size, or class length.
So, class width = Upper class - Lower class
29-20=9
39-30=9
Therefore, the width of each class interval is 9.
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50 decreased by twice craig's score use the variable c to represent craig's score
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Answer:
50 - 2c
Step-by-step explanation:
i think its incomplete
What percentage of young women and teens in the united states is clinically bulimic?
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An eating disorder indicated by binge eating and following purging, usually by induced vomiting or usage of laxatives
1 to 3 percent of female teenagers and youthful adults in the United States exist clinically bulimic.
What are the symptoms of bulimia?Above average body weight.Recurrent episodes of binge eating and fear of not being able to stop eating.Self-induced vomitingExcessive exercise.Excessive fasting.Peculiar eating habits.Inappropriate use of laxatives or diuretics.As numerous as one in ten college women suffer from a clinical or about clinical eating disorder, including 5.1% who sorrow from bulimia nervosa. Around five percent of adolescent and adult women and one percent of men contain anorexia nervosa, bulimia nervosa, or binge eating disorder.
An eating disorder indicated by binge eating and following purging, usually by induced vomiting or usage of laxatives
1 to 3 percent of female teenagers and youthful adults in the United States exist clinically bulimic.
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PLEASE HELP!! THIS IS MY LAST QUESTION PLEASE!!
(Answeer all the problems)
Answers
Answer:
49, 1.5i, 0.25 + 0.5i
Step-by-step explanation:
Answer:
see explanation
Step-by-step explanation:
Given a complex number in standard form a ± bi , then
the conjugate is a ∓ bi
That is the real part is unchanged while the sign of the imaginary part changes.
(22)
The conjugate of \(\sqrt{7}\) is - \(\sqrt{7}\) ( treat the real part as 0 )
(23)
The conjugate of - 6i is + 6i
(24)
The conjugate of - 2 + i is - 2 - i
Leah invests in an account paying a simple interest of 8% per year. If no money will be added or removed from the investment, what should she multiply her current balance by to find her total balance in a year in one step
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Leah should multiply her current balance by 1.08 to find her total balance in one year.
To find Leah's total balance in one year, we can use the simple interest formula:
Simple Interest = Principal x Rate x Time
where:
Principal is the amount of money invested
Rate is the interest rate per year
Time is the time period in years
In this case, we know that Leah's account pays a simple interest rate of 8% per year, and no money will be added or removed from the account during the year. Hence, we may condense the formula to:
Total Balance = Principal + Simple Interest
Total Balance = Principal + (Principal x Rate x Time)
Total Balance = Principal x (1 + Rate x Time)
So to find Leah's total balance in one year, she should multiply her current balance by (1 + 0.08 x 1), which simplifies to 1.08:
Total Balance = Current Balance x 1.08
For example, if Leah currently has 1,000 in her account, her total balance in one year would be:
Total Balance = 1,000 x 1.08
Total Balance = 1,080
So Leah should multiply her current balance by 1.08 to find her total balance in one year.
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Jack Frost was decorating the car windows in the teacher’s parking lot. He puts 2 snowflakes on each of the four windows of the cars. If he makes 90 snowflakes, how many cars can he completely decorate?
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Explanation: since he puts 2 snowflakes on each of the 4 windows on one car, in total Jack uses 8 snowflakes per car. If he makes 90 snowflakes, 90 divided by 8 gives you the amount of cars he can decorate. 90/8 = 11.25, but since you can’t have .25 cars the answer is 11.
If 3% of a number equals 2, find 18% of that number.
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Answer:
3% of 66.7 is 2.001 or 2.00
18% of 66.7 is 12.006 or 12.01
PLS HELP! LATE ASSIGNMENT!!!
It was due last night and it is worth 15% of class mark! pls show all steps!!! I WILL MAKE U BRAINLIST
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The key elements of the graph are completed as below
a b
Direction of Opening Down up
Vertex (x, y) (-3, 4) (2, 1)
Is there a max or min? max min
What is the max or min? 4 1
Axis of symmetry x = -3 x = 2
X-intercepts (-4, 0) and (-1, 0) imaginary
Y-intercept (0, -5) (0, 3)
What is x intercept of a parabola?The x-intercept of a parabola is the point(s) where the parabola intersects the x-axis. Geometrically the x-intercepts are the points where the y coordinate of the parabola is equal to zero, this is also called the roots
In the case of b, the roots are imaginary and hence are not shown on the graph
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if p = 2-5 and q = 8x3 find 3p-q
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Answer:
3p - q = 3(2-5) - (8x3) = -9 - 24 = -33. Therefore, 3p-q=-33.
PLEASE ANSWER THANK YOU!
Match each equation to a value that makes it true.
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1.) 1/2x = -5 goes with x=-10
2.) -2x = -9 goes with x = 4 1/2
3.) x+ (-2) = 1/2 goes with x= -4.5
4.) -2x = 7 goes with x= -3.5
5.) -2 + x = 1/2 goes with x = 2 1/2
6.) -1/2x = 1/4 goes with x= -1/2
2 and 1/2 miles in 3/4 hours
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Answer:
2/(3/4) = 2(4/3) = 8/3 = 2 2/3 miles per hour
Answer:
Step-by-step explanation:
5/2 / 3/4
5/2 x 4/3
20/6= 3 2/6 = 3 1/3 miles per hour
Solve for k.
6 (k + 2) +1 = 13
Answers
Answer:
k=0
Step-by-step explanation:
6(k+2)+1=13 first distribute
6k+12+1=13 then combine like terms
6k+13=13 then subtract 13
-13 -13
6k=0 then divide 6
/6 /6
k=0
For items 7-10, use the figure shown. Find the coordinates of the specified vertex after the given sequence of transformations.
Quadrilateral Q R S T plotted on a coordinate plane with vertices at, Q, (1, 3), R, (3, negative 3), S, (zero, negative 2), and T, (negative 2, 1).
a translation 2 units right, then a reflection across x = 0
Q' = ( , )
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The coordinates of the specified vertex after the given sequence of transformations is given by;
Q' = (3, -3).
What is a translation?In Mathematics, the translation of a geometric figure to the right simply means adding a digit to the value on the x-coordinate (x-axis) of the pre-image of a function while a geometric figure that is translated up simply means adding a digit to the value on the y-coordinate (y-axis) of the pre-image or parent function.
Mathematically, a horizontal translation to the right is modeled by this mathematical expression g(x) = f(x + N) while a vertical translation to the positive y-direction (upward) is modeled by this mathematical expression g(x) = f(x) + N.
Where:
N represents an integer.g(x) and f(x) represent a function.By translating the coordinate Q (1, 3) two (2) units to the right, we have the following:
Coordinate Q (1, 3) → Coordinate Q' (1 + 2, 3) = Q (3, 3)
In Mathematics, a reflection across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate would change from positive to negative. Therefore, a reflection over the x-axis is given by this transformation rule:
(x, y) → (x, -y)
Coordinate Q' (3, 3) → (3, -3).
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Ron made 3 pitchers of lemonade to sell at a lemonade stand. The cost of making the lemonade is 50 cents per pitcher of lemonade.
He sold each cup of lemonade for 15 cents, and he sold a total of 36 cups of lemonade.
How much money did Ron make selling lemonade after paying out the cost of making the lemonade?
Answers
Answer:
3.90 I think.
Step-by-step explanation:
first, 15×36. then subtract 50 three times.
Answer:
$3.90
Step-by-step explanation:
The cost of making them all is 1.50 subtracted from the profit he made
the ages of students at a university are normally distributed with a mean of 20. what percentage of the student body is at least 20 years old?
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The percentage of the student body that is at least 20 years old is 50%.
The ages of students at a university are normally distributed with a mean of 20. To find the percentage of the student body that is at least 20 years old solution to the is as follows: As the Mean of ages of students at a university, μ = 20. Age is normally distributed so it will be divided into two groups Therefore, the mean age is equal to the median age. As the age is normally distributed, 50% of the population will have an age greater than or equal to 20 years old. Thus, the percentage of the student body that is at least 20 years old is 50%. Hence, the required percentage of the student body that is at least 20 years old is 50%.
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if you saved 25 thousand dollars an hour, 40 hours per week for 2020 years, how much money do u have saved up ?!
Answers
Answer:
i dunno
Step-by-step explanation:
A certain forest covers an area of 2300 km. Suppose that each area decreases by 5.75%. What will be the area after 12 years?
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Answer:
The correct answer is 2043 km².
Step-by-step explanation:
Given:
Starting area,
A = 2300 km²
Rate of decreasing,
r = 5.75%
Time,
t = 12 years
As we know,
⇒ \(y = A(1-r)^t\)
By substituting the values, we get
\(=2300(1-0.0575 )^{12}\)
\(=2300(0.9425)^{12}\)
\(=2300\times 0.8883\)
\(=2043 \ km^2\)
PLEASE HELPPPPPPPPPPP
Answers
Answer:
45
Step-by-step explanation:
So a triangle adds up to 180 and so does a line.
180-93=87
And because of vertical angle proof, 48 is the other angle of the triangle.
87+48=135
180-135=45
Answer:
x = 45
Step-by-step explanation:
To get to the interior of the triangle from 48 use vertically opposite angles. That means that the two remote interior angles = 93
Put in simpler language
48 = interior angle
93 = x + 48
93 - 48 = x
x = 45
f m<1 = 2x + 3 and the m<4 = 3x + 2, then find the m<3.
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Answer:
https://www.mrelisara.com/uploads/5/8/7/3/58736993/chapter_2_review_ws_answers.pdfStep-by-step explanation:
for a posttest following anova, there are four different treatment groups. how many pairwise comparisons must be made to gain a complete understanding of which treatment effects differ significantly from others? a. 4 b. 6 c. 12 d. 24
Answers
There are 6 pairwise comparisons that need to be made in order to gain a complete understanding of which treatment effects differ significantly from others.
In order to determine which treatment effects differ significantly from others, we need to make pairwise comparisons between all possible pairs of treatment groups. Let's consider an example with four treatment groups labeled A, B, C, and D.
To determine which treatment effects differ significantly from others, we need to compare the mean scores of each treatment group with the mean scores of every other treatment group. This means we need to make the following pairwise comparisons:
A vs B
A vs C
A vs D
B vs C
B vs D
C vs D
Therefore, there are 6 pairwise comparisons that need to be made in order to gain a complete understanding of which treatment effects differ significantly from others.
It's worth noting that when making multiple pairwise comparisons, there is an increased risk of making a Type I error (i.e., rejecting the null hypothesis when it is actually true) due to the multiple testing problem. To control for this, researchers may choose to adjust the significance level or use methods such as the Bonferroni correction to adjust the p-values of the individual tests.
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consider the following. find an equation of the ellipse. (x−2)2 32 (y−3)2 22=1 find its foci. (x, y) = (smaller x-value) (x, y) = (larger x-value)
Answers
The standard form of the equation of an ellipse with center at (h, k), semi-major axis of length a, and semi-minor axis of length b is:
(x-h)²/a² + (y-k)²/b² = 1
Comparing this to the given equation, we have:
(x-2)²/3² + (y-3)²/2² = 1
So the center is (h, k) = (2, 3), the semi-major axis is a = 3, and the semi-minor axis is b = 2.
To find the foci, we need to use the formula:
c = sqrt(a² - b²)
where c is the distance from the center to each focus. In this case, we have:
c = sqrt(3² - 2²) = sqrt(5)
So the foci are located at a distance of sqrt(5) from the center along the major axis, which is parallel to the x-axis. The x-coordinates of the foci are:
2 + sqrt(5) and 2 - sqrt(5)
So the foci are:
(smaller x-value) = (2 - sqrt(5), 3)
(larger x-value) = (2 + sqrt(5), 3)
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Can You fail 5th grade elementary? i have a 68 in math and im scared
Answers
Answer: I mean technically no. grades don't really matter until high school. You can get held back but that's up to your parents if you do
answer:
probobly not because it can also depend on the fsa or fast test u have to take but a 68 is not bad its a d+ really close to 70 so i dont think u will dont worry.
Simplify 5(−4−8w)−3w.
Answers
Answer:
-20-43w
Step-by-step explanation:
5(-4-8w)-3w
-20-40w-3w
-20-43w
HELP!!!!!!!!!!!!!!!!!!1
Answers
Answer:
The second option "They highlight the unnatural feeling of moving without gravity
Step-by-step explanation:
The probability of a teenager in a locality owning a bike is 0.33, of owning a car is 0.16 and of owning both is 0.11. If a teenager is chosen random from the locality, what is the probability that the teenager owns a bike or a car? a)0.34 b)0.36 c)0.38 d)0.40
Answers
Answer:
C) 0.38
Explanation:
Given the below;
p(owning a bike) = 0.33
p(owning a car) = 0.16
p(owning both) = 0.11
We'll use the below formula to determine the probability of owning a bike or a car;
\(\begin{gathered} p(\text{ owning a bike or a car) = }p(\text{ owning a bike) + p(owning a car) - p(owning both)} \\ \end{gathered}\)Substituting the given values, we'll have;
\(\begin{gathered} p(\text{ owning a bike or a car) = }0.33+0.16-0.11 \\ =0.38 \end{gathered}\)PLS HELP I NEED TO KNOW THIS LIKE RN
WILL GIVE BRAINLIEST TO FIRST GOOD ANSWER!!!!
Answers
Answer:
it's 4x-2y<12
Step-by-step explanation: