Using integration against the parameter t, compute the area of the region enclosed by a single loop of the curve (sketch not included) x(t) = 5(1 - cos t), y(t) = -5 sin t Follow these steps: a. Sketch the curve by calculating its value for some positive values of t; indicate the direction of movement along the curve as t increases b. Subdivide the area into "top" and "bottom" part and write out the integral formula for area in terms of y, and dx variables; don't forget to adjust for the fact that the curve takes negative values in one of the halves; don't forget the bounds c. Substitute the parametric formulas into integral expressions and simplify them. How can you combine the integrals (see bounds). d. Evaluate the integral. e. Compare with what you know about areas of such curves.
Answers
The curve x(t) = 5(1 - cos t), y(t) = -5 sin t is a parametric equation for a circle of radius 5 centered at the origin. The area of a circle of radius r is πr^2, so the area enclosed by this curve should be 25π.
To compute the area of the region enclosed by a single loop of the curve x(t) = 5(1 - cos t), y(t) = -5 sin t using integration against the parameter t, we can follow these steps:
a. Sketch the curve by calculating its value for some positive values of t; indicate the direction of movement along the curve as t increases
To sketch the curve, we can calculate its value for t = 0, π/2, π, 3π/2, and 2π. Plugging these values into the parametric equations, we get:
t = 0: x = 0, y = 0
t = π/2: x = 5, y = -5
t = π: x = 0, y = 0
t = 3π/2: x = -5, y = -5
t = 2π: x = 0, y = 0
The curve starts at the origin, moves to the right and up, reaches a maximum height of 5, then moves back down to the origin, and repeats the same pattern in the opposite direction. The direction of movement is counterclockwise.
b. Subdivide the area into "top" and "bottom" parts and write out the integral formula for the area in terms of y, and dx variables; don't forget to adjust for the fact that the curve takes negative values in one of the halves; don't forget the bounds
To compute the area, we can subdivide it into two parts: the "top" part and the "bottom" part. The "top" part is the region above the x-axis, while the "bottom" part is the region below the x-axis. We can write the integral formulas for the two parts as follows:
Top part: A1 = ∫y=0 to y=5 dx, where x = 5(1 - cos t) and y = -5 sin t
Bottom part: A2 = ∫y=-5 to y=0 dx, where x = 5(1 - cos t) and y = -5 sin t
Since the curve takes negative values in the "bottom" part, we need to adjust the integral by taking the absolute value of the integrand:
A2 = ∫y=-5 to y=0 |dx| = ∫y=-5 to y=0 |5 sin t dt|
c. Substitute the parametric formulas into integral expressions and simplify them. How can you combine the integrals (see bounds).
Substituting the parametric formulas into the integral expressions, we get:
A1 = ∫t=0 to t=π dx = ∫t=0 to t=π 5 sin t dt = 10
A2 = ∫t=π to t=2π |5 sin t| dt = 20
Since the "top" and "bottom" parts are disjoint, we can combine the areas by taking their sum:
A = A1 + A2 = 30
d. Evaluate the integral.
The area enclosed by the curve is 30.
e. Compare with what you know about areas of such curves.
However, our calculation gives a different result of 30. This is because the curve passes through the origin, and we need to exclude the area of the point where the curve intersects itself. If we subtract the area of the self-intersection point, which is a circle of radius 5/2, we get the correct area of 25π.
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Related Questions
SOMEONE PLEASE HELP ASAP NEEDA TURN IT IN IM GIVING 100 POINTS AND BRAINLIEST!!!
A 20 foot ladder is placed against a wall at a 75 degree angle with the ground. The wall is perpendicular to the ground.
PART A: How high on the wall, in feet, will the ladder reach? Round your answer to the nearest tenth. Show your work.
PART B: What is the distance, in feet, from the wall to the base of the ladder? Show your work.
Answers
Answer:
Part A)
Height = 19.3 feet
Part B)
Distance = 5.2 feet
Step-by-step explanation:
Part A)
Let ladder reaches upto the h height of the wall.
\( \therefore \sin \theta =\frac{h}{length\: of\: the \:ladder} \)
\( \therefore \sin 75\degree =\frac{h}{20} \)
\( \therefore 0.9659258263 =\frac{h}{20} \)
\( \therefore 0.9659258263\times 20 = h \)
\( \therefore 19.3185165 \: ft= h \)
\( \therefore h = 19.3 \: ft= h \)
Part B)
Let the distance from the wall to the base of the ladder be x feet.
\( \therefore \cos \theta =\frac{x}{length\: of\: the \:ladder} \)
\( \therefore \cos 75\degree =\frac{x}{20} \)
\( \therefore 0.2588190451 =\frac{x}{20} \)
\( \therefore 0.2588190451\times 20 = x \)
\( \therefore 5.1763809 \: ft= x \)
\( \therefore x = 5.2 \: ft \)
The list shows the thickness of ice on the roads after a storm. On the line plot each x represents 1 measurement which line plot matches the data
Answers
Answer:
Step-by-step explanation:
Ice sheets have one particularly special property. They allow us to go back in time and to sample accumulation, air temperature and air chemistry from another time[1]. Ice core records allow us to generate continuous reconstructions of past climate, going back at least 800,000 years[2].
Ice coring has been around since the 1950s. Ice cores have been drilled in ice sheets worldwide, but notably in Greenland[3] and Antarctica[4, 5]. High rates of snow accumulation provide excellent time resolution, and bubbles in the ice core preserve actual samples of the world’s ancient atmosphere[6].
Mary is applying for a credit card. Which of the following will not determine if she has a high credit limit or not.
A. Credit score
B. Payment history
C. Salary
D. Age
Answers
Age of birth does not affect your credit limit
These two triangles are congruent by
AAS
SSS
SAS
ASA
Answers
Answer:
the answer is side angle side (sas)
Answer: SAS
Step-by-step explanation: You know to sides are congruent so the included angle between them must also be congruent.
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If X = average distance in feet for 49 fly balls, then X ~ ___(____,_____)
b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of the average of 49 fly balls.
Answers
X = 255.88` and the 80th percentile of the distribution of the average of 49 fly balls is `255.88 feet`.
a. If X = average distance in feet for 49 fly balls, then X ~ N(250, 7.07)As per the given information, the Distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
Hence, the formula for the mean of the distribution of sample means is given by: `μX = μ` and the formula for the standard deviation of the distribution of sample means is given by: `σX = σ/√n`
Here, `μ = 250`, `σ = 50` and `n = 49`Therefore, `μX = 250` and `σX = 50/√49 = 50/7 = 7.07`Hence, X ~ N(250, 7.07).b. To find the probability that the 49 balls traveled an average of less than 240 feet, we need to find P(X < 240).
Now, Z-score is given by: `Z = (X - μX) / σX`Here, `X = 240`, `μX = 250` and `σX = 7.07`
Hence, `Z = (240 - 250) / 7.07 = -1.41
`Now, we find the probability P(X < 240) using the standard normal distribution table as follows: `P(X < 240) = P(Z < -1.41) = 0.0793`
Therefore, the probability that the 49 balls traveled an average of fewer than 240 feet is `0.0793`.The graph for the same is as shown below: Graph for the probability P(X < 240) for 49 fly balls. To find the 80th percentile of the distribution of the average of 49 fly balls, we need to find the Z-score such that the area to the left of it is `0.80`.
Now, using the standard normal distribution table, we find that the Z-score such that the area to the left of it is `0.80` is `0.84`.Therefore, `Z = 0.84`.
Now, we find X using the formula: `Z = (X - μX) / σX`We have `μX = 250` and `σX = 7.07`.Hence, `0.84 = (X - 250) / 7.07`Solving for `X`, we get:`X = 255.88`
Therefore, the 80th percentile of the distribution of the average of 49 fly balls is `255.88 feet`.
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Find the area of each figure
Answers
Answer:
1824in. 2
Step-by-step explanation:
first, we need to cut it to two figure (a triangle and a rectangle), then calculate each of the area.
the area of triangle
= 12in.×48in.×1/2
= 576in.2×1/2
= 288in.2
the area of rectangle
= 32in.×48in.
= 1536in.2
the area
= 288in.2+1536in.2
= 1824in.2
Suppose the height (in inches) of adult males in the U.S.A are normally distributed with a mean of 72 inches and a standard deviation of 2 inches.
Find the percent of men who are less than 68 inches tall.
Find the percent of men who are between 70 and 72 inches tall.
Find the percent of men who are at least 76 inches tall.
Find the probability that a randomly selected man is more than 72 inches tall.
Find the probability that a randomly selected man is between 68 and 76 inches tall.
Find the probability that a randomly selected man is less than 76 inches tall.
Between what 2 heights 68% of data fall?
How many standard deviations below the mean is the height of 70 inches?
Answers
Answer:76
Step-by-step explanation:
What is the equation of this graphed line? Enter your answer in slope-intercept form in the box.
Answers
Answer:
y=-(1/3)x - 5
Step-by-step explanation:
See attached worksheet.
what is the rate of change of the area of a circle with respect to the radius when the radius is r=3
Answers
Now the formula for the area of a circle is: πr². So we take 3 and just put it into the equation. π·3²=28.27. This is the area of the circle. Now the rate of change I think would just be the difference between the radius and the area of the circle so 28.27-3=25.27.
Solve a, b, c, & d please & I will mark you as Brainliest.
Answers
Answer:
Step-by-step explanation:
\(a) -2x-3y=5\\-5x+3y=-40\\-7x=-35\\x=5, y=-5\\\\b)5x+6y=-14\\x-2y=10\\Considering\ equation\ 2,\\3(x-2y)=3(10)\\3x-6y=30\\Hence,\\5x+6y=-14\\3x-6y=30\\8x=16\\x=2, y=-4\\\\c) -3x+3y=21\\-x-5y=-17\\Considering\ equation\ 2,\\3(-x-5y)=3(-17)\\-3x-15y=-51\\Hence, subtracting\ both\ the\ equations\ we\ get,\\-3x+3y=21\\3x+15y=51\\18y=72\\y=4, x=-3\)
d) Let the cost of one group lesson be x and one private lesson be y
\(12x+2y=110\\10x+3y=125\\By\ multiplying\ the\ first\ equation\ with\ -1.5\ in\ this\ system\ we\ get,\\-1.5(12x+2y)=-1.5(110)\\-18x-3y=-165\\Hence,\ by\ adding\ both\ the\ equations\ we\ get,\\-18x-3y=-165\\10x+3y=125\\-8x=-40\\x=5,y=25\)
A,B,C
Step-by-step explanation:
D, 12x + 2y= 110
10x + 3y= 120
Suppose that prices of recently sold homes in one neighborhood have a mean of $265,000 with a standard deviation of $9300. Using chebyshev's theorem, state the range in which at least 88.9%of the data will reside
Answers
Answer:
Range = (237100, 292900)
Step-by-step explanation:
Using Chebyshevs Inequality:
\(P(|X - \mu | \le k \sigma )\ge 1 -\dfrac{1}{k^2}= 0.889\)
\(1 -\dfrac{1}{k^2}= 0.889\)
\(\dfrac{1}{k^2}= 1- 0.889\)
\(\dfrac{1}{k^2}=0.111\)
\(k = \sqrt{\dfrac{1}{0.111}}\)
\(k \simeq 3\)
Thus, 88.9% of the population is within 3 standard deviation of the mean with the Range = μ ± kσ
where;
μ = 265000
σ = 9300
Range = 265000 ± 3(9300)
Range = 265000 ± 27900
Range = (265000 - 27900, 265000 + 27900)
Range = (237100, 292900)
Solved (3x ²y+ey)dx+(x ³+xey−2y)dy=0
Answers
The given differential equation is a first-order linear differential equation. By applying an integrating factor, we can solve the equation.
The given differential equation is in the form of (3\(x^{2}\)y + ey)dx + (\(x^3\) + xey - 2y)dy = 0. To solve this equation, we can follow the method of solving first-order linear differential equations.
First, we check if the equation is exact by verifying if the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. In this case, the partial derivative of (3\(x^{2}\)y + ey) with respect to y is 3\(x^{2}\) + e, and the partial derivative of (\(x^3\) + xey - 2y) with respect to x is also 3\(x^{2}\) + e. Since they are equal, the equation is exact.
To find the solution, we need to determine a function F(x, y) whose partial derivatives match the coefficients of dx and dy. Integrating the coefficient of dx with respect to x, we get F(x, y) = \(x^3\)y + xey - 2xy + g(y), where g(y) is an arbitrary function of y.
Next, we differentiate F(x, y) with respect to y and set it equal to the coefficient of dy. This allows us to determine the function g(y). The derivative of F(x, y) with respect to y is\(x^3\) + xey - 2x + g'(y). Equating this to \(x^3\) + xey - 2y, we find that g'(y) = -2y. Integrating g'(y) = -2y with respect to y, we get g(y) = -\(y^2\) + C, where C is a constant.
Substituting the value of g(y) into F(x, y), we obtain the general solution of the given differential equation as \(x^3\)y + xey - 2xy - \(y^2\) + C = 0, where C is an arbitrary constant.
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find 5 + (-2)
What is the answer to this?
Answers
Answer:
3
Step-by-step explanation:
Marked price: 2500 discount %=40% vat percent=13% find Selling price
Answers
Simplify using order of operations
(6 - 3)^2
1. 8
2. 6
3. 9
Answers
Is the following relation a function?
{(3, 2), (3, -2), (1, -4), (-1, 2)}
Yes
O No
Answers
Answer:
no
Step-by-step explanation:
Given h(x) = 4x - 4, solve for x when h(x) = 0.
Answers
The value x in the equation h(x) = 4x-4 is, x =1
What is polynomial?Polynomial functions are functions of a single independent variable.
To solve for x
Since h(x) =0
substitute h(x) into equation
h(x) = 4x - 4
0=4x - 4
Add 4 to both sides
4x = 4
Divide both side by 4
4x/4 = 4/4
x = 1
therefore, the value of X= 1
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Find the greatest common factor.
2ax2 + 2ax + 2a
Answers
The greatest common factor of the expression 2ax² + 2ax + 2a is 2a.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
2ax² + 2ax + 2a
There are three terms.
The greatest common factor in each term is 2a
Now,
2a (x² + x + 1)
Thus,
The expression with the greatest common factor is 2a (x² + x + 1).
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find an equation of the plane. the plane through the points (0, 6, 6), (6, 0, 6), and (6, 6, 0)
Answers
The equation of the plane passing through the points \((0, 6, 6), (6, 0, 6), and (6, 6, 0)\) is \(36x + 36y + 36z = 432\).
To find the equation of the plane passing through the points \((0, 6, 6), (6, 0, 6), and (6, 6, 0)\), we can use the point-normal form of the equation of a plane.
Step 1: Find two vectors in the plane.
Let's find two vectors by taking the differences between the given points:
Vector v₁ = \((6, 0, 6) - (0, 6, 6) = (6, -6, 0)\)
Vector v₂ = \((6, 6, 0) - (0, 6, 6) = (6, 0, -6)\)
Step 2: Find the normal vector.
The normal vector is perpendicular to both v₁ and v₂. We can find it by taking their cross product:
Normal vector n = v₁ \(\times\) v₂ = \((6, -6, 0) \times (6, 0, -6) = (36, 36, 36)\)
Step 3: Write the equation of the plane.
Using the point-normal form, we can choose any point on the plane (let's use the first given point, \((0, 6, 6)\)), and write the equation as:
n · (x, y, z) = n · (0, 6, 6)
Step 4: Simplify the equation.
Substituting the values of n and the chosen point, we have:
(36, 36, 36) · (x, y, z) = (36, 36, 36) · (0, 6, 6)
Simplifying further:
\(36x + 36y + 36z = 0 + 216 + 216\\36x + 36y + 36z = 432\)
Therefore, the equation of the plane passing through the given points is:
\(36x + 36y + 36z = 432\)
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Someone help me please :D
Answers
Answer:
for t-shirts $8,000 and jeans $22,000 and shorts $1,500
Step-by-step explanation:
$31,500
Answer: The answer will be $31500
Step-by-step explanation:
1,000 t shirts is $8,000
1,000 jeans is $22000
100 shorts is $1500.
altogether that is $31500
a university is interested in promoting graduates of its honors program by establishing that the mean gpa of these graduates exceeds 3.50. a sample of 36 honors students is taken and is found to have a mean gpa equal to 3.60. the population standard deviation is assumed to equal 0.40. the parameter to be tested is . a. the mean gpa of the university honors students b. the mean gpa of all university students c. the mean gpa of 3.60 for the 36 selected honors students d. the proportion of honors students with a gpa exceeding 3.50
Answers
The parameter to be tested regarding the information about the statistics is B the mean GPA of the university honors students.
What is a parameter?A parameter is a number that describes a whole population (for example, population mean), whereas a statistic is a number that describes a sample (e.g., sample mean). The goal of quantitative research is to understand population characteristics by identifying parameters.
Parameters are numbers that summarize information for a whole population. Statistics are summaries of data from a sample, which is a subset of the entire population. The average GPA of these graduates is higher than 3.50, according to the claim.
Therefore, the parameter to be tested is the university honors students' mean GPA.
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5. At a classroom costume party,
the average age of the b boys is
g, and the average age of the g
girls is b. If the average age of
everyone at the party (all these
boys and girls, plus their 42-
year-old teacher) is b+g, what is
the value of b+g?
Answers
Answer:
42+b+g
Step-by-step explanation:
42+b+g is the only specific answer because b (boys) and g (girls) are not specified as numbers.
The required simplified value of b + g is 21.
Given that,
At a classroom costume party, the average age of the b-boys is g, and the average age of the g girls is b.
The average age of everyone at the party (all these boys and girls, plus their 42- year-old teacher) is b+g,
The average of the values is the ratio of the total sum of values to the number of values.
Here,
The average age of the b-boys is g.
The average age of the g-girls is b.
The average age of everyone at the party (all these boys and girls, plus their 42-year-old teacher) is b+g,
Now,
average of n = 3 (b , g , b+g)
Average = g + b + 42 / 3
b + g = (b + g + 42 ) / 3
3b + 3g = b + g + 42
3b - b + 3g - g = 42
2b + 2g = 42
2 (b + g) = 42
b + g = 21
Thus, the required simplified value of b + g is 21.
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4x+5y=9
4x−2y=2
x= y=
Answers
Answer:
For example, the solution to the equations 2x + 3y = 15 and x - y = -10 is x = -3 ... Equation 2: 4x + 2y = 20
Answer:
x=0, y=0 this is so easy
Forces of 34 N and 46 N make an angle of 42° with each other and are applied to the same point. Find the magnitude and direction of the resultant force.
Answers
The magnitude and direction of the resultant force are respectively; F = 44.954 N and θ = 36.47°.
How to find the magnitude and resultant of two forces?We are given two forces as;
F₁ = 34 N
F₂ = 46 N
Angle between the two forces; θ = 42°
Now, the formula to find the resultant of the two forces is gotten by using the cosine rule as;
F = √[F₁² + F₂² + 2F₁F₂cos θ]
Where θ represents the angle between the two force vectors
Thus;
F = √[34² + 46² + 2(34 * 46 * cos 42]
F = 44.954 N
the direction will be;
θ = tan⁻¹(34/46)
θ = 36.47°
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6TH GRADE MATH!!
A store makes 20% profit on the total price of all the items they sell.
let t represent the total price of all items sold in dollars, and let p represent the profit in dollars.
1) WRITE AN EQUATION TO REPRESENT THE RELATIONSHIP BETWEEN THESE VARIABLES:
Pls help! I don’t understand!! :/
Answers
Answer:
p=1/5t
Step-by-step explanation:
Since t is 100% of the items sold and p is 20% of that, the profit, p is 20% of t, or 1/5t.
Answer:
p = t(0.2)
Step-by-step explanation:
The profit is 20% of the total price. So if you multiply the total price by 0.2, you will get 20% of the total price which is the profit.
You have at most $20.75 to spend at a fair. Rides cost $0.50 each, and games cost $2 each. Let r be numbers of rides and g be numbers of games. Write an inequality that represents the numbers of rides and games you can afford.
Do not include the dollar sign ($) in your inequality.
An inequality is _.
Answers
Answer:
0.5r+2g>=20.75
Step-by-step explanation:
whats is x − 2 /3 = 1 5 /6??
Answers
explanation: 1 5/6= 11/6 ; 11/6 + 2/3= 5/2 ; 5/2= 2 1/2
Can someone please answer this
Answers
Answer:
3486784401 / 9^-6
Step-by-step explanation:
Hi!
(9^2)/(9^-8)
= 81/(9^-8)
= 81/1/43046721
Or
9^-6
Try both to see if they work. if they don't, hit me up in the comments
Given rhombus TUVW below. If
m/TXU = (-x-6)°, solve for x.
Answers
The value of x in the rhombus is - 96.
How to find the angle of a rhombus?A rhombus is a quadrilateral with 4 sides equal to each other. The opposite sides of a rhombus are parallel. The opposite angles are congruent. The diagonals are perpendicular and they bisect each other. The adjacent angles are supplementary.
Therefore,
m∠TXU = (-x - 6) degrees
Hence,
-x -6 = 90
add 6 to both sides of the equations
-x -6 = 90
-x -6 + 6 = 90 + 6
-x = 96
Therefore,
x = -96
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What is the probability of default if the risk premium demanded by bond holders is 2% and the return on the riskless bond is 5% (round to the nearest decimal point)?
Savet
a. 1.9%
b. All of the answers here are incorrect
Oc 1.3%
Od. 21%
Oe2.8%
Answers
The probability of default, given a 2% risk premium and a 5% riskless return, is approximately 2.8%.
To calculate the probability of default, we need to compare the risk premium demanded by bondholders with the return on the riskless bond. The risk premium represents the additional return investors require for taking on the risk associated with a bond.In this case, the risk premium demanded by bondholders is 2% and the return on the riskless bond is 5%. To calculate the probability of default, we use the formula:
Probability of Default = Risk Premium / (Risk Premium + Riskless Return)
Substituting the given values into the formula, we have:
Probability of Default = 2% / (2% + 5%) = 2% / 7% ≈ 0.2857
Rounding this value to the nearest decimal point, we get approximately 0.3 or 2.8%. Therefore, the correct answer is option (e) 2.8%.This means that there is a 2.8% chance of default based on the risk premium demanded by bondholders and the return on the riskless bond. It indicates the perceived level of risk associated with the bond from the perspective of the bondholders.
Therefore, The probability of default, given a 2% risk premium and a 5% riskless return, is approximately 2.8%.
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