Stokes' theorem relates the surface integral of the curl of [tex]\vec F[/tex] across [tex]S[/tex] to the line integral of [tex]\vec F[/tex] along the boundary of [tex]S[/tex].
The boundary of [tex]S[/tex] is a circle with radius 7 centered at the origin in the [tex]x,y[/tex]-plane. Parameterize this path by
[tex]\vec r(t) = 7\cos(t)\,\vec\imath + 7\sin(t)\,\vec\jmath[/tex]
with [tex]0\le t\le2\pi[/tex]. Observe that [tex]z=0[/tex], so [tex]\cos(z) = 1[/tex] and the [tex]\vec\jmath[/tex]-component of [tex]\vec F[/tex] contributes nothing. The double integral then reduces to
[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 \\\\ ~~~~~~~~ = \int_0^{2\pi} \left(e^{49\cos(t)\sin(t)}\,\vec\imath + 49\cos(t)\sin(t)\,\vec\jmath\right) \cdot \left(-7\sin(t)\,\vec\imath + 7\cos(t)\,\vec\jmath\right) \, dt \\\\ ~~~~~~~~ = -7 \int_0^{2\pi} e^{49\cos(t)\sin(t)} \sin(t) \, dt[/tex]
Observe that by substituting [tex]t=u+\pi[/tex], we have
[tex]\sin(t) = \sin(u+\pi) = \sin(u)\cos(\pi) + \cos(u)\sin(\pi) = -\sin(u)[/tex]
so that the integral over [tex][\pi,2\pi][/tex] can be expressed in terms of the integral over [tex][0,\pi][/tex] as
[tex]\displaystyle \int_\pi^{2\pi} e^{49\cos(t)\sin(t)} \sin(t) \, dt = \int_0^\pi -e^{49\cos(t)\sin(t)} \sin(t) \, dt[/tex]
Then the integrals over [tex][0,\pi][/tex] and [tex][\pi,2\pi][/tex] cancel each other and integral of the curl of [tex]\vec F[/tex] is
[tex]\displaystyle -7 \int_0^{2\pi} e^{49\cos(t)\sin(t)} \sin(t) \, dt = -7 \int_0^\pi 0 \, dt = \boxed{0}[/tex]
Mark decided to make Christmas presents for his classmates. he bought 10lb of two types of candies: caramel candies for $2.80 per pound and chocolate candies for $3.50 a pound.
Write a function f(x) that represents the total cost of candy, where x is the weight of the less expensive candy
Brainliest for correct answer
The function for the given situation is f(x)=35.0-0.7x.
Given that, Mark bought 10 pound of two types of candies, caramel candies for $2.80 per pound and chocolate candies for $3.50 a pound.
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
Now, the function for the given situation is as follows:
If x is the weight of the less expensive candy, the 10-x is the weight of the high expensive candy.
So, f(x)=2.80x+3.50(10-x)
⇒ f(x)=2.80x+35.0-3.50x
⇒ f(x)=35.0-0.7x
Therefore, the function for the given situation is f(x)=35.0-0.7x.
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If Art has a 7/1 ARM, how long will the fixed interest rate be applied to his loan?
Answer:
step-by-step explanation:
A 7/1 ARM is a mortgage that has a fixed interest rate in the beginning, then switches to an adjustable or variable one. The 7 in 7/1 indicates the initial fixed period of seven years. After that, the interest rate adjusts once yearly based on the index stated in the loan agreement, plus a margin set by the lender.
3 squares are positioned to form a triangle. the small square is labeled 6 units, medium square is 8 units, and large square is not labeled. answer these questions about the right triangle. what is the area of the square of the leg 6? what is the area of the square of the leg 8? what is the area of the hypotenuse square? what is the length of the hypotenuse?
The area of the square of the leg 6 is 36 square units.
The area of the square of the leg 8 is 64 square units.
The area of the square of the hypotenuse square is 100 square units.
The length of the hypotenuse is 10 units.
Formula Used
Area of square, A = [tex](a)^{2}[/tex]
Here, [tex]a[/tex] is the length of the side of the square.
Pythagoras Theorem is given as,
[tex]h^{2} =p^{2} +b^{2}[/tex]
Here, [tex]h, p, b[/tex] are the hypotenuse, perpendicular, and the base of a right angle triangle respectively.
Area of the Square of the Leg 6
Length of the edge of the square = 6 units
Area of square = 6*6
= 36 square units
Area of the Square of the Leg 8
Length of the edge of the square = 8 units
Area of square = 8*8
= 64 square units
Area of the Hypotenuse Square and the Length of the Hypotenuse
Applying Pythagoras theorem,
[tex]h^{2} = (8)^{2} +(6)^{2}[/tex]
[tex]h^{2} = 64+36[/tex]
[tex]h = \sqrt{100}[/tex]
[tex]h = 10 units[/tex]
Therefore, the area of the hypotenuse square is 100 square units and the length of the hypotenuse comes out to be10 units.
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Answer: in the picture
Step-by-step explanation:
2. A point A (x, y) is translated 10 units to the right and 3 units down. What is the mapping to the new point A'?
Answer:
(x+10,y-3)
Step-by-step explanation:
MATH: GRAPHS AND FUNCTIONS...HELP!
Solution: Check the picture I uploaded.
Content attribution
QUESTION 5
.
1 POINT
Find the mode(s) of
1, 8, 6, 2, 5, 5, 3, 6, 6, 2, 7,2
Answer:
6
Step-by-step explanation:
The mode of a data set, is the value that occurs the most often. There can be multiple modes, if two values have the same amount of frequency in a data set as well as having the highest frequency. So in this case case it appears there is only 1 mode with that being 6, since it appears 3 times. with no other value appearing this much in the data set.
Find the H.C.F. of the following expressions:36(x³+x²y-12xy²), 54(x³y+4x³y²-21xy)
Answer:
18x
Step-by-step explanation:
The HCF is the highest factor common to both expressions. We can find it by factoring both expressions and identifying common factors.
Factorization
36(x³+x²y-12xy²) = 18x·2·(x +4y)(x -3y)
54(x³y+4x³y²-21xy) = 18x·3y·(x +7)(x -3)
The highest common factor is 18x.
Order the following numbers from least to greatest.
4.12310562562 4.5 4.23
Answer:
4.12310562562, 4.23, 4.5
I need help with my homework
The required points of x where the slope of tangent and secant is equal at boundary points 4 and 9.
Slope is defined as the rate of change or angle at which curve is having inclination or declination.
Since, the curve of 1/2√x defined for interval [4,9]
The curve is of increasing order so the tangent and secant having equal slope only possible at the boundaries of the curve.
Boundary conditions = (4,1) and (9, 1.5)
Equation can be given as,
y = 0.1x + 0.5
Thus, the required points of x where the slope of tangent and secant is equal at boundary points 4 and 9.
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there 56 people invited to marks family reunion. each table at the park where the reunion is held can hold 7 people
Write the equation of the line that passes through the points (1,1) and (2,4). Write your answer in slope intercept form. Step by step please
Answer: y = 3x - 2
Step-by-step explanation:
gradient = (4 - 1)/(2 - 1) = 3/1 = 3
y = 3x + c needs to work for both points
Substitute (1,1) into equation
1 = 3 x 1 + c
c = 1 - 3 = -2
y = 3x - 2
-7 + p = 3 for p
help before my dad comes home
Answer:
p=10
Step-by-step explanation:
-7+p=3
+7-7+p=3+7
p=10
Answer:
Hello! The answer to this question is 10.
Step-by-step explanation:
You have to get p by itself so we will add 7 to both sides:
-7 + p = 3
+7 +7
-------------
p = 10
When doing so, we get p = 10.
You can check your work:
-7 + 10 ⇒ 10 - 7 = 3
Therefore, p = 10 is the correct answer.
Refrigerators-’N-More sells appliances with a 20% markup from their purchase cost. Alice makes a 5% commission rate on her daily sales. If the store buys a washer and dryer set for $600, what would Alice’s commission be if she sold those appliances?
The price of the washer and dryer set after the markup is
.
The amount of Alice’s commission would be
The price of the washer and the dryer set is $720.
The commission earned by Alice is $36.
What is the price of the washer and dryer set ?Percentage can be described as a fraction an amount expressed as a number out of hundred. Percentage is a measure of frequency.
Price after the mark up = ( + percentage mark up) x cost price
(1 + 0.20) x $600 = $720
Commission earned by Alice = 5% x 720 = $36
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Anne has n pairs of shoes, and Fiona has 3 times as
many pairs of shoes. If Fiona gives Anne 6 pairs of
shoes, the girls will have an equal number of shoes.
How many pairs of shoes did Anne have originally?
Answer:
Anne has n
Fiona has 3n
3n=6
so n=2
so Anne have 2
Longhorn Corporation provides low-cost food delivery services to senior citizens. At the end of the year on December 31, 2021, the company reports the following amounts:
Cash $ 1,300 Service revenue $ 65,700
Equipment 27,000 Cost of goods sold (food expense) 53,000
Accounts payable 4,000 Buildings 36,000
Delivery expense 2,200 Supplies 3,000
Salaries expense 5,100 Salaries payable 900
In addition, the company had common stock of $36,000 at the beginning of the year and issued an additional $3,600 during the year. The company also had retained earnings of $17,400 at the beginning of the year.
Prepare the statement of stockholders’ equity for Longhorn Corporation.
The statement of stockholders’ equity for Longhorn Corporation is: $62,800
Statement of stockholders’ equity
Common stock Retained earning Total stakeholder equity
Beginning balance $36,000 $17,400 $53,400
Issuance of common stock $4,000 - $4,000
Add net income - $5,400 $5,400
[65,700-(53,000+2,200+5,100)]
Ending balance $40,000 $22,800 $62,800
Therefore the statement of stockholders’ equity for Longhorn Corporation is: $62,800.
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please can you hurry
What is the truth value for the following conditional statement?
p: true
q: false
p → q
ANSWERS
T F → T
F F → T
T F → F
F T → T
In the case above, the truth value for the following conditional statement is Option B: F F → T.
What is the conditional statement about?The truth value that was given in the statements are:
p : trueq : trueSo one can write that the truth values of :
p : false
q : false
Then, one need to find the truth value of (p → q)
Note that when as it is written, it means or implies that the truth value of any two statements is false.
If P is true and Q is false then one can say that in every condition, the truth value holds true. So we can then say that p → q: True
In the case above, the truth value for the following conditional statement is Option B: F F → T
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Evaluate.
—4 1/2+3(6.4+3/4)
Answer:
16.95
Step-by-step explanation:
use a calculator
65 people took a clarinet exam. Before the exam, 23 people predicted that they would pass, and the rest predicted they would fail. Of the people who predicted they would pass, 18 actually did. In total, 31 people passed the clarinet exam. What fraction of those who thought that they would fail actually did fail?
Answer:
The answer is 29/42
Step-by-step explanation:
First, let's calculate how many people predicted they would fail. The question states that 65 people took the exam and 23 predicted they would pass, so we can find the number of people that predicted they would fail by the following calculation:
Let FP be the people who predicted they'd fail
65 = 23 + FP
65 - 23 = FP
42 = FP
Now, let's move on to the next part. The question states that a total of 31 people passed the test, from those 18 being the people who predicted they would pass and the rest are people who had predicted they would fail but ended up passing.
Let's set x as the number of people who predicted they would fail but have passed.
31 = 18 + x
31 - 18 = x
13 = x
Since 13 of the 42 FP have passed, we can calculate how many of them failed. Let y be the number of people that predicted to fail and ended up failing:
13 + y = 42
y = 42 - 13
y = 29
Finally, now we have the fraction of those who predicted that they would fail actually did fail and that's 29/42
Help me with this question please!! :)
Answer:
60 ways
Step-by-step explanation:
combination C 4 and 6= (6*5*4*3)/1*2*3=360/6=60 ways
Express - 6 + √25 as a complex number in the form a + bi.
- 6 + √-25 =
Answer:
[tex]-6+5i[/tex]
Step-by-step explanation:
[tex]-6+\sqrt{-25}\\\\=-6+5i[/tex]
Can you help me I need to send this to my teacher like TODAY
A: Three collinear points are A, T, and B
B: Line BA is another name for line AB
C: Line L is in plane Z
D: line AB is the intersection of planes Z and W
Hi, can someone please help me? trying to graduate :(
Write an equation in slope-intercept form of the line that passes through the point (3,-2) with slope -2.
Answer:
D
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here m = - 2 , then
y = - 2x + c ← is the partial equation
to find c substitute (3, - 2 ) into the partial equation
- 2 = - 6 + c ⇒ c = - 2 + 6 = 4
y = - 2x + 4 ← equation of line in slope- intercept form
^
The graph below shows the graphs of the linear function = 7x and the exponential function y =
Which statement is true?
HURRY TIMED
Answer:
its D
Step-by-step explanation:
I was able to get the first two parts and now I am stumped.
The correct response that would be expected based on the probability information is 164
How to depict the values?From the information, there were 352 trials that were made and we're correct 164 times.
Therefore, the correct response that would be expected is 164.
The success rate will be:
= 164/352
= 0.466
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Write the equation of a sine or cosine function to describe the graph.
Step-by-step explanation:
The equation of a trigonometric function is
[tex]y = a \sin(bx + c) + d[/tex]
or
[tex]y = a \: cos(bx + c) + d[/tex]
Let define some variables,
D is the midline, this refers to the midpoint of the highest y value and lowest y value. Some textbooks call it the vertical translations but it is the same thing.
A is the amplitude. The amplitude is the distance from the midline to the highest y value. Some distance is non negative, the formula for the amplitude is
[tex] |a| [/tex]
The period is how often the wave repeats itself on a interval.
Period can't be negative so the formula for period is
[tex] \frac{2 \pi}{ |b| } [/tex]
To find the period, look at the extreme points.
The phase shift tells us if the sinusoid have been shifted to the right or left. The formula for the phase shift
[tex]bx + c = 0[/tex]
Solving for x gives us
[tex]x = \frac{ - c}{b} [/tex]
If our x is negative, we have a phase shift to the right
If our x is Positve, we have a phase shift to the left.
Let solve this equation, Let use Sine since sin(0)=0,
The smallest y value here is 0, and the highest is 2, so the amplitude is 1.
[tex]d = 1[/tex]
Next, the distance from the max to the midline is 1, as well the min to the midline is also 1.
So
[tex]a = 1[/tex]
We have minimum at 0, and 8, so our period is 8.
[tex] \frac{2\pi}{b} = 8[/tex]
Solve for b,
[tex] \frac{\pi}{4} = b[/tex]
Plug this in the equation.
[tex]1 \sin( \frac{\pi}{4} x) + [/tex]
[tex]y = \sin( \frac{\pi}{4} x) + 1[/tex]
The graph passes through (4,2) so let see if that holds true for our equation
[tex]2 = \sin( \frac{\pi}{4} (4)) + 1[/tex]
[tex]2 = \sin(\pi) + 1[/tex]
[tex]2 = 0 + 1[/tex]
[tex]2 = 1[/tex]
This doesn't hold true, so we must have a phase shift
Notice that
[tex] \sin( \frac{\pi}{2} ) = 1[/tex]
So if we shift this pi/2 to the right, we can get our equation to be true.
[tex]y = \sin( \frac{\pi}{4} x - \frac{\pi}{2} ) + 1[/tex]
This equation works so our equation is
[tex]y = \sin( \frac{\pi}{4} x - \frac{\pi}{2} ) + 1[/tex]
Find any stationary points of the graph of [tex]y = 2x^2 + e^-x^4[/tex]
[tex]y = 2x {}^{2} + e {}^{ - x {}^{4} } [/tex]
[tex] \frac{dy}{dx} = 4x +( e {}^{ - x {}^{4} } )( - 4x {}^{3} )[/tex]
[tex] \frac{dy}{dx} = 4x - 4x {}^{3} e { }^{ - x {}^{4} } [/tex]
[tex] \frac{dy}{dx} = 4x(1 - x {}^{2} e {}^{ - x {}^{4} } )[/tex]
[tex] \frac{dy}{dx} = 0 \\ 4x = 0 \: \: \: \: \: 1 - x {}^{2} e {}^{ - x {}^{4} } = 0[/tex]
[tex]4x = 0 \\ x = 0 \: \: \: \: \: \: y(0) = 0 + e {}^{0} = 1[/tex]
[tex] \frac{x {}^{2} }{e {}^{x {}^{4} } } = 1 \\ x {}^{2} = e {}^{x {}^{4} } \\ 2ln(x) = x {}^{4} \\ these \: 2 \: functions \: do \: not \: intersect[/tex]Stationary point ( 0 , 1 )Find the monthly interest payment in the situation described below. Assume that the monthly interest rate is 1 divided by 12 of the annual interest rate.
You maintain an average balance of $675 on your credit card, which carries a 24%
annual interest rate.
The monthly interest payment is
The monthly interest payment is 162
What is Interest?Interest, in its most simple form, is calculated as a percent of the principal.
First,
monthly interest rate,
= 0.24/12
= 0.02.
Now,
Average monthly interest,
=0.02×$675
= $13.50.
Thus, the annual interest payments,
= 12×$13.50
= $162
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For the polynomial function ƒ(x)= −2x2 − 4x + 16, find all local and global extrema.
For the given quadratic equation we only have a maximum at y = 18.
How to find the extrema of the given function?
Here we have:
[tex]f(x) = -2x^2 - 4x + 16[/tex]
Notice that this is a quadratic equation of negative leading coefficient.
Then we have a maximum at the vertex, and both arms tend to negative infinity as x tends to infinity or negative infinity.
The vertex is at:
x = -(-4)/(2*(-2)) = -1
The maximum is:
[tex]f(-1) = -2*(-1)^2 - 4*(-1) + 16 = -2 + 4 + 16 = 18[/tex]
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a rectangle is a parallelogram with 4 equal sides true or false
Answer:
False
Step-by-step explanation:
A rectangle does not have 4 equal sides, it has 2 sets of equal sides.
Cedric applied the rule to figure WXYZ.
Figure W X Y Z is reflected across the line of reflection m to form W prime X prime Y prime Z prime. Figure W prime X prime Y prime Z prime is rotated 180 degrees about point P to form W double-prime X double-prime Y double-prime Z double-prime.
What mistakes did he make? Select two options.
He applied the reflection to the pre-image first.
He applied the rotation to the pre-image first.
He changed the size of the figure instead of just applying a rotation.
He used point P as the center of rotation.
He used an incorrect angle of rotation around point P
The mistakes that were made regarding the transformation are:
He applied the reflection to the pre-image first.He used an incorrect angle of rotation around point PHow to illustrate the information?It should be noted that a transformation simply means the changing of the position of an object
In this case, the transformation is incorrect as he applied the reflection to the pre-image first and used an incorrect angle of rotation around point P.
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