Let D be a region bounded by a simple closed path C in the xy-plane. The coordinates of the centroid (xˉ,yˉ) of D are xˉ=2A1∮Cx2dyyˉ=−2A1∮Cy2dx where A is the area of D. Find the centroid of a quarter-circular region of radius a. (xˉ,yˉ)=___
Answers
The centroid of a quarter-circular region of radius $a$ is $\left(\frac{a^2}{2\pi}, \frac{a^2}{4}\right)$.
The centroid of a region is the point that is the average of all the points in the region. It can be found using the following formulas: xˉ=2A1∮Cx2dyyˉ=−2A1∮Cy2dx
where $A$ is the area of the region, $C$ is the boundary of the region, and $x$ and $y$ are the coordinates of a point in the region.
For a quarter-circular region of radius $a$, the area is $\frac{a^2\pi}{4}$. The integrals in the formulas for the centroid can be evaluated using the following substitutions:
x = a \cos θ
y = a \sin θ
where $θ$ is the angle between the positive $x$-axis and the line segment from the origin to the point $(x,y)$.
After the integrals are evaluated, we get the following expressions for the centroid:
xˉ=a22π
yˉ=a24
Therefore, the centroid of a quarter-circular region of radius $a$ is $\left(\frac{a^2}{2\pi}, \frac{a^2}{4}\right)$.
The first step is to evaluate the integrals in the formulas for the centroid. We can do this using the substitutions $x = a \cos θ$ and $y = a \sin θ$.
The integral for $xˉ$ is:
xˉ=2A1∮Cx2dy=2A1∮Ca2cos2θdy
We can evaluate this integral by using the double angle formula for cosine: cos2θ=12(1+cos2θ)
This gives us: xˉ=2A1∮Ca2(1+cos2θ)dy=2A1∮Ca2+a2cos2θdy
The integral for $yˉ$ is:
yˉ=−2A1∮Cy2dx=−2A1∮Ca2sin2θdx
We can evaluate this integral by using the double angle formula for sine:
sin2θ=2sinθcosθ
This gives us:
yˉ=−2A1∮Ca2(2sinθcosθ)dx=−2A1∮Ca2sin2θdx
The integrals for $xˉ$ and $yˉ$ can be evaluated using the trigonometric identities and the fact that the area of the quarter-circle is $\frac{a^2\pi}{4}$.
After the integrals are evaluated, we get the following expressions for the centroid:
xˉ=a22π
yˉ=a24
Therefore, the centroid of a quarter-circular region of radius $a$ is $\left(\frac{a^2}{2\pi}, \frac{a^2}{4}\right)$.
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Related Questions
solve the following equations
63= 12x -21
Answers
Answer:
X=7
take the -21 to the other side of the equation, when you do this it goes from a negative to a positive so we now have 63+21=12x
if we add 63 and 12 we get 84 decide by 12 because we have 12x and we get 7
x=9 and y=4 what does xy/2
Answers
Answer:
18
Step-by-step explanation:
Equation 9 x 4 / 2 = 18
First do 9 x 4 which is 36
Next do 36 / 2 which is 18
Also the "/" is another way to say to divide
Explain the HOW and WHY of each step when solving the equation.
Use algebra to determine: x-axis symmetry, y-axis symmetry, and origin symmetry.
y = x9
Answers
To determine the x-axis symmetry, y-axis symmetry, and origin symmetry of the equation y = x^9, we need to analyze the properties of the equation and understand the concepts of symmetry.
The x-axis symmetry occurs when replacing y with -y in the equation leaves the equation unchanged. The y-axis symmetry happens when replacing x with -x in the equation keeps the equation the same. X-axis symmetry: To determine if the equation has x-axis symmetry, we replace y with -y in the equation. In this case, (-y) = (-x^9). Simplifying further, we get y = -x^9. Since the equation has changed, it does not exhibit x-axis symmetry.
Y-axis symmetry: To check for y-axis symmetry, we replace x with -x in the equation. (-x)^9 = x^9. Since the equation remains the same, the equation has y-axis symmetry.
Origin symmetry: To determine origin symmetry, we replace x with -x and y with -y in the equation. The resulting equation is (-y) = (-x)^9. This equation is equivalent to the original equation y = x^9. Hence, the equation has origin symmetry.
In summary, the equation y = x^9 does not have x-axis symmetry but possesses y-axis symmetry and origin symmetry.
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How do you write 35.4 in scientific notation?
Answers
Answer:
3.54x10^1
Step-by-step explanation:
hi amariharrisdagoat :D
Answers
Answer:
heyyy even though you are not talking to me
BAHBAHA
Step-by-step explanation:
For all values of x, f(x)=x+5, g(x)=x²-2 Solve fg(x) = gf(x)
Answers
Answer:
x=1
Step-by-step explanation:
given- f(x)=x+5 g(x)=x²-2to solve,
fg(x)=gf(x)so,
first of all let's put the values,
f(x²-2)=g(x+5)
x(x²-2)+5=x²(x+5)-2
x³-2x+5=x³+5x-2 (opened the bracket)
x³-x³-2x-5x+5+2=0(arranged the like terms)
-7x+7=0
-7x=-7{brought (+7) to RHS which is (-7)}
x=(-7)/(-7) [ (-7) was in multiplayer state by putting it in another side it changes into division state]
x = 1 { (-7)/(-7) divided by 1 and minus cancelled}
any confusion ask in CMT!!!A bag contains 20 coloured marbles. Copy and
complete the table below to show the probability of
picking each colour at random and the number of
each colour marble in the bag. What is the
probability, as a percentage (%), of picking a purple
marble at random? How many purple marbles are in
the bag?
Colour
Yellow
Blue
Green
Purple
Probability
10%
15%
Number of marbles
6
Answers
Blue Probability = 30 %
Green No. of balls = 3
Purple Probability = 45 %
Purple No. of balls = 9
The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z =4
Answers
volume= 16/3, A coordinate plane is a graphing and description system for points and lines. A vertical (y) axis and a horizontal (x) axis make up the coordinate plane. There are four quadrants in the coordinate plane. The point where these lines connect is called the origin (0, 0).
limits
z= 0 to z = 4-y-2x
y= 0 to y = 4- 2x
x= 0 to x= 2
volume
v= \(\int\limits^2_0 \int\limits^4_0 \int\limits^4_0 dzdydx\)
v= \(\int\limits^2_0 \int\limits^4_0 (4-y-2x) dydx\)
v= \(\int\limits^2_0 ( 4y- y^{2} / 2 - 2xy) ^4^-^2^x _0\)
dx= \(\int\limits^2_0 [ 16-8x - 16+ 4x^2 - 16x / 2 - 8x+ 4x^2 ] dx\)
v= \(\int\limits^2_0 [ 8+ 2x^2- 8x] dx\\\)
= [ 8x + 2x^3 / 3 - 8x^2 / 2 ] ^2_0
= [16+ 16/3- 16]
v= 16/3
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Complete Question
Question: Sketch The Tetrahedron Enclosed By The Coordinate Planes And The Plane 2x+Y+Z=4. Use A Triple Integral To Find The Volume Of The tetrahedron
solid fats are more likely to raise blood cholesterol levels than liquid fats. suppose a nutritionist analyzed the percentage of saturated fat for a sample of 6 brands of stick margarine (solid fat) and for a sample of 6 brands of liquid margarine and obtained the following results: we want to determine if there a significant difference in the average amount of saturated fat in solid and liquid fats. what is the test statistic? (assume the population data is normally distributed)
Answers
The test statistic would be a two-sample t-test, which compares the means of two independent samples by calculating the difference in means between the two samples and dividing the difference by the standard error of the difference.
How does math define a percent?In essence, percentages are fractions with a denominator of 100. The percent symbol (%) is placed next to a number to indicate that it is a percentage. For instance, you would have received a 75% on the examination if you correctly answered 75 out of 100 questions (75/100).
Describe a percentage example.
The hundredth component, one percent (symbolized as 1%), is expressed as 100 percent, and 200 percent denotes double the quantity specified. For example, 1% of 1,000 chickens equals 1/100 of 1,000, or 10 chickens, and 20% of the total is equal to 20% of 1,000, or 200.
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2460 concert tickets were sold for a total of $29,545 if students paid $9 and non students paid $14 how many tickets were sold to each group
Answers
The tickets sold to the students is 979 and the tickts sold to non students is 1481.
According to the statement
we have to find the equation which is linear with the help of the given data.
So, For this purpose, we know that the
A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.
From the given statement
Let the students buy a ticket is x
And let the non students buy a ticket is y
Then from the information the equation become
x+y = 2460
And the
9x+14y = 29,545
put y = 2460 -x then
9x + 14(2460-x) = 29,545
9x -14x+ 34440 = 29,545
-5x = 29,545 -34440
-5x = -4895
x = 979
And the y become
y = 2460 -x
y = 2460 -979
y = 1481.
So, The tickets sold to the students is 979 and the tickts sold to non students is 1481.
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A mountaineer climbed 1,000 feet at a rate of x feet per hour. He climbed an additional 5,000 feet at a different rate. This rate was 10 feet per hour less than twice the first rate.
Which expression represents the number of hours the mountaineer climbed?
Answers
who's good in math? please help me
find the value of p
\(4p = \frac{64}{4 {}^{p - 3} }\)
Answers
Answer:
p would equal 4
p=4
Step-by-step explanation:
I don't know if it was squared by p.
![who's good in math? please help mefind the value of p[tex]4p = \frac{64}{4 {}^{p - 3} }[/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/oWzzxxNrndeEsdTMZ7hQXFjeL1KbF16J.jpeg)
Answer:
p=4
is it?
this is the answer I guess.
The linear function f(x) = 0.4x+81 represents the average test score in your math class, where x is the number of the tests taken. The linear function g(x) represents the average test score in your science class, where x is the number of the test taken. x g(x) 1 86 2 84 3 82 Part A: Determine the test average for your math class after completing Test number 3. (3 points) Part B: Determine the test average for your science class after completing Test number 2. (3 points) Part C: Which class had a higher average after completing Test 3? Show work to support your answer. (4 points
Answers
Part A: The test average for your math class after completing Test number 3 is 83.6., Part B: The test average for your science class after completing Test number 2 is 84 and Part C: The math class had a higher average after completing Test 3.
What is test average?Test Average is a measure of a student's performance in a given set of tests or assessments. It is calculated by adding the scores of all the tests together and dividing the total by the number of tests taken. Test Average can be used to assess how well a student is doing in a particular course, or to measure overall academic performance. Test Average is an important indicator for teachers to assess the progress of their students over time.
Part A: The test average for your math class after completing Test number 3 is 83.6, calculated as f(3) = 0.4 × 3 + 81 = 83.6.
Part B: The test average for your science class after completing Test number 2 is 84, calculated as g(2) = 84.
Part C: The math class had a higher average after completing Test 3, since the test average for the math class after completing Test number 3 is 83.6, while the test average for the science class after completing Test number 2 is 84.
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The base area of a square-based prism is 25 ft squared. One of the lateral faces of the prism is 30 ft squared. What is the surface area of the prism?
Answers
Step-by-step explanation:
Given data
lateral surface area= 30ft
base area of squared prism = 25 ft means height= 25
(therefore, square object length breadth and height same )
total surface area = lateral surface * height
= 30ft*25ft
=750 ft square.
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thanks.
The table below gives beverage preferences for random samples of teens and adults. Teens Adults Total Coffee 50 200 250 Tea 100 150 250 Soft Drink 200 200 400 Other 50 50 100 400 600 1000 We are asked to test for independence between age (i.e., adult and teen) and drink preferences. The expected number of adults who prefer coffee is a. 150. b. 200. c. .25. d. .33.
Answers
The expected number of adults who prefer coffee is 150.
We must determine the anticipated number of adults who prefer coffee in order to test for independence between age and drink preferences.
First, let's look at the totals of the rows and columns to figure out the expected values for each category. The formula is available to us:
Anticipated esteem = (Line absolute * Section complete)/Stupendous aggregate
The stupendous complete for this situation is 1000.
Number of adults expected to prefer coffee:
= (Adults total minus Coffee total) / Grand total = (600 minus 250) / 1000 = 150; consequently, 150 adults are anticipated to prefer coffee.
In this way, the right response is (a) 150.
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failure to include some units, or entire sections, of the defined survey population in the actual operational sampling frame represents:
Answers
Failure to include some units, or entire portions, of the defined survey population in the actual operational sample frame represents: a noncoverage error.
One of the components of coverage error that results from a flawed sample frame that excludes a certain percentage of the population is noncoverage.
Sampling errors need to be controlled and reduced to a level at which their presence does not defeat or obliterate the usefulness of the final sample results.
The difference between the observed values of a variable and the long-run average of the observed values in repetitions of the measurement is the sampling error.
Noncoverage, another word for these frames, inadequate coverage is also known as undercoverage.
Sampling errors can be decreased by increasing the sample size.
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A point P lies in a plane and is a distance of r = 37 units from the origin of a Cartesian coordinate system. If the line joining the point and the origin makes an angle of = 350 degrees with respect to the x-axis, what are the (x, y) coordinates of the point P?
Answers
The (x, y) coordinates of point P are approximately (31.19, 20.67).
It is stated that the point P lies at a distance of r = 37 units from the origin and forms an angle of θ = 35° with respect to the x-axis, we can use trigonometry to find the x and y coordinates.
Using the trigonometric definitions, we have,
x = r * cos(θ) = 37 * cos(35°) ≈ 31.19
y = r * sin(θ) = 37 * sin(35°) ≈ 20.67
Therefore, the approximate (x, y) coordinates of point P are (31.19, 20.67). The coordinates (31.19, 20.67) represent the position of point P in the Cartesian coordinate system based on the given distance and angle measurements.
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Complete question - A point P lies in a plane and is a distance of r = 37 units from the origin of a Cartesian coordinate system. If the line joining the point and the origin makes an angle of = 35° degrees with respect to the x-axis, what are the (x, y) coordinates of the point P?
y=2x+5;x+2y=10
What is the answer using substitution PLEASE
Answers
Answer:
x = 0 and y = 5
Step-by-step explanation:
y = 2x + 5
x + 2y = 10
Substitute y = 2x + 5 in x + 2y = 10
x + 2(2x + 5) = 10
x + 4x + 10 = 10
5x = 10 - 10
5x = 0
x = 0/5
x = 0
Substitute x = 0 in x + 2y = 10
0 + 2y = 10
2y = 10
y = 10/2
y = 5
Alexa can follow the sidewalk from the library to the school or she can travel across the grass
directly. Which route is longer?"
Sidewalk-78m
13m
Answers
The longer distance that Alexa can follow to the school is the sidewalk since 70m is longer than 13m.
How to calculate the distance?From the information given, Alexa can follow the sidewalk from the library to the school or she can travel across the grass.
In this case, the distance of the sidewalk is 78m and the distance of the grass is 13m. Therefore, the longer distance is the sidewalk.
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Show your work pleaseeILL MARK YOU BRAINLIST
Answers
Answer: The correct answer is the first number = 9, the second number = 3
Step-by-step explanation:
x = the first number
y = the second number
What we know:
The first number is 3 more than twice the second number.The sum of twice the first number and three times the second number is 27.What we need to determine:
What are the first and second numbers?Solve using a system of equations:
The first equation (the first number is 3 more than twice the second number):
x = 2y + 3
The second equation (sum of twice the first number and three times the second number is 27):
2x +3y = 27
Substitute the first equation in the second:
2(2y + 3) +3y = 27
Solve for y:
y = 3
Substitute for y and solve for x:
x = 2y + 3
x = 2(3) + 3
x = 6 + 3
x = 9
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A study of the effect on the human digestive system of potato chips made
with a fat substitute
Answers
Answer:
Perform an experiment
Step-by-step explanation:
The experiments state that it is the test which is to be carried to confirm or refute the hypotheses they had about a specific subject. Therefore, the study findings will be based on the outcome of these experiments.
Now, according to the given situation, an analysis of the impact of potato chips on the human digestive system with fat replacement is refer to perform an experiment.
So, the right answer is to perform an experiment.
A.359.5
B.2,178
C.18.0
Answers
Answer:
C, or 18.0
Step-by-step explanation:
The SAS theorem provides that the area is 17.97, which rounds up to 18.0, giving you C as the answer. The other two are far too high for the given values.
There were 70 enrolled students in STAT 3355 during the year 2020 . The population of adults, 18 years or older, in the United States was 258.3 million in 2020 . A student surveyed 30 of her classmates in 2020 and found that 22 students liked to play video games. If this student computed a 95% confidence interval, would it have contained the value of 65%, which was known to be the proportion of adults that liked to play video games in the United States in 2020. (Hint: Calculate the confidence interval by hand at first, and then try to use R ).
Answers
The confidence interval for proportion of adults who liked video game is (0.575091,0.8915756 ) from sample. It has a parameter 65%.
Number of enrolled students in STAT during year 2020 = 70
The population of adults that is 18 or above in 2020 = 258.3 million
Number of students are classmates= 30
Out of 30, number of students who like video game = 22
level of significance = 0.95
Let p denote the proportion of students in sample who liked video games. it is p = 22/30 = 0.733.
Using the distribution table, value of z for 95% is equals to the 1.96. From the formula of confidence interval, \(CI = p ± z\sqrt{ \frac{ p( 1 - p)}{n}}\)
\(= 0.733 ± \sqrt{ \frac{0.733( 1 - 0.733)}{30}}\)
= (0.575091, 0.8915756), the lower limit and upper limit of interval. This interval contains 65% which is population parameter. Hence, required value is (0.575091, 0.8915756).
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In a weighted grading system, students are graded on quizzes, tests, and a project, each with a different weight. Matrix W represents the weights for each kind of work, and matrix G represents the grades for two students, Felipe and Helena.
Q T P
W = [0.40 0.50 0.10] Felipe Helena
G= Q {80 70}
T {60 80}
p { 90 60}
Final grades are represented in a matrix F. If F = WG, what is F?
A. [7174]
B. [7174]
C. [7471]
D. [7471]
Answers
For Felipe and Helena's final grades, the solution is option C, [74 71].
How to calculate final grades?Using the given values for Q, T, and P weights and Felipe and Helena's grades, calculate their final grades as follows:
Felipe's final grade:
0.40 x 80 + 0.50 x 60 + 0.10 x 90 = 32 + 30 + 9 = 71
Helena's final grade:
0.40 x 70 + 0.50 x 80 + 0.10 x 60 = 28 + 40 + 6 = 74
To represent the final grades for Felipe and Helena in a matrix F, given formula F = WG, where W = matrix of weights and G = matrix of grades:
[0.40 0.50 0.10] [80 70]
F = WG = [0.40 0.50 0.10] x [60 80]
[0.40 0.50 0.10] [90 60]
Performing matrix multiplication:
[32 + 30 + 9 28 + 40 + 6]
F = WG = [32 + 40 + 6 28 + 40 + 3]
[36 + 25 + 6 36 + 20 + 3]
Simplifying:
[71 74]
F = WG = [78 71]
[67 59]
Therefore, [74 71] for Felipe and Helena's final grades, respectively.
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find the missing number of the unit rate
4/2 = ?/1
Answers
Answer:
2
Step-by-step explanation:
4/2 = 2
2/1 = 2
4/2=2/1
EZ
What is the factored form of 27a6+8g12?
Answers
Answer:
\(\left(2g^4+3a^2\right)\left(4g^8-6g^4a^2+9a^4\right)\)
Step-by-step explanation:
\(27a^6+8g^{12}\)
\(\mathrm{Rewrite\:}27a^6+8g^{12}\mathrm{\:as\:}\left(3a^2\right)^3+\left(2g^4\right)^3\)
\(\mathrm{Apply\:Sum\:of\:Cubes\:Formula:\:}x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(\left(3a^2\right)^3+\left(2g^4\right)^3=\left(3a^2+2g^4\right)\left(3^2a^4-3\cdot \:2g^4a^2+2^2g^8\right)\)
\(=\left(2g^4+3a^2\right)\left(2^2g^8-3\cdot \:2g^4a^2+3^2a^4\right)\)
\(=\left(2g^4+3a^2\right)\left(4g^8-6g^4a^2+9a^4\right)\)
Answer:
b
Step-by-step explanation:
correct on edge 2021
the other one isnt even an option
Homer's Home Supply Store ships a specific model of a bathroom vanity in cube-shaped boxes. Each box has a volume of one cubic meter. The boxes are loaded onto a truck that is shaped like a rectangular prism. The floor of the truck can be completely covered with a layer of 8 boxes. If 2 layers will fill the entire truck, what is the volume of the truck?
Answers
If 2 layers will fill the entire truck then the volume of the truck is 16 cubic meters.
Since the floor of the truck can be completely covered with a layer of 8 boxes, we know that the area of the truck's floor is equal to the total area of 8 boxes.
The volume of each box is one cubic meter, which means each box has a height, length, and width of 1 meter. Therefore, the area of the truck's floor is 8 square meters (8 boxes x 1 square meter per box).
If two layers of boxes will fill the entire truck, we need to multiply the area of the truck's floor by the height of two layers. Since each box has a height of 1 meter, the height of two layers will be 2 meters.
The volume of the truck = Area of the floor x Height
Volume of the truck = 8 square meters x 2 meters
Volume of the truck = 16 cubic meters
Therefore, the volume of the truck is 16 cubic meters.
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Pleaseee helppp ASAP
Extra 100 points
And brainliest
Answers
Answer:
what help I can do for you
Answer:
where is the questions
Step-by-step explanation:
you should write correct question
6 ≥ -6(a +2)
pls help i don't understand
Answers
Answer:
a ≥ -3
Step-by-step explanation:
Answer:
a ≥ -3
Step 1: Simplify both sides of the inequality.
6 ≥ − 6a − 12
Step 2: Flip the equation.
−6a − 12 ≤ 6
Step 3: Add 12 to both sides.
−6a − 12 + 12 ≤ 6 + 12
−6a ≤ 18
Step 4: Divide both sides by -6.
−6a/−6 ≤ 18/−6
a ≥ −3
I used a calculator, hope this helps
sketch the graph of the following functions: y=5x+2
Answers
Answer:
This is the graph sketch
Step-by-step explanation:
Answer:
the app is called Desmos Graphing Calculator
Step-by-step explanation:
