Let C be the boundary of the region in the first quadrant bounded by the x-axis, a quarter-circle with radius 9, and the y-axis, oriented counterclockwise starting from the origin. Label the edges of the boundary as C1,C2,C3 starting from the bottom edge going counterclockwise. Give each edge a constant speed parametrization with domain 0≤t≤1.
Answers
Solution :
Along the edge [tex]$C_1$[/tex]
The parametric equation for [tex]$C_1$[/tex] is given :
[tex]$x_1(t) = 9t , y_2(t) = 0 \ \ for \ \ 0 \leq t \leq 1$[/tex]
Along edge [tex]$C_2$[/tex]
The curve here is a quarter circle with the radius 9. Therefore, the parametric equation with the domain [tex]$0 \leq t \leq 1 $[/tex] is then given by :
[tex]$x_2(t) = 9 \cos \left(\frac{\pi }{2}t\right)$[/tex]
[tex]$y_2(t) = 9 \sin \left(\frac{\pi }{2}t\right)$[/tex]
Along edge [tex]$C_3$[/tex]
The parametric equation for [tex]$C_3$[/tex] is :
[tex]$x_1(t) = 0, \ \ \ y_2(t) = 9t \ \ \ for \ 0 \leq t \leq 1$[/tex]
Now,
x = 9t, ⇒ dx = 9 dt
y = 0, ⇒ dy = 0
[tex]$\int_{C_{1}}y^2 x dx + x^2 y dy = \int_0^1 (0)(0)+(0)(0) = 0$[/tex]
And
[tex]$x(t) = 9 \cos \left(\frac{\pi}{2}t\right) \Rightarrow dx = -\frac{7 \pi}{2} \sin \left(\frac{\pi}{2}t\right)$[/tex]
[tex]$y(t) = 9 \sin \left(\frac{\pi}{2}t\right) \Rightarrow dy = -\frac{7 \pi}{2} \cos \left(\frac{\pi}{2}t\right)$[/tex]
Then :
[tex]$\int_{C_1} y^2 x dx + x^2 y dy$[/tex]
[tex]$=\int_0^1 \left[\left( 9 \sin \frac{\pi}{2}t\right)^2\left(9 \cos \frac{\pi}{2}t\right)\left(-\frac{7 \pi}{2} \sin \frac{\pi}{2}t dt\right) + \left( 9 \cos \frac{\pi}{2}t\right)^2\left(9 \sin \frac{\pi}{2}t\right)\left(\frac{7 \pi}{2} \cos \frac{\pi}{2}t dt\right) \right]$[/tex]
[tex]$=\left[-9^4\ \frac{\cos^4\left(\frac{\pi}{2}t\right)}{\frac{\pi}{2}} -9^4\ \frac{\sin^4\left(\frac{\pi}{2}t\right)}{\frac{\pi}{2}} \right]_0^1$[/tex]
= 0
And
x = 0, ⇒ dx = 0
y = 9 t, ⇒ dy = 9 dt
[tex]$\int_{C_3} y^2 x dx + x^2 y dy = \int_0^1 (0)(0)+(0)(0) = 0$[/tex]
Therefore,
[tex]$ \oint y^2xdx +x^2ydy = \int_{C_1} y^2 x dx + x^2 x dx+ \int_{C_2} y^2 x dx + x^2 x dx+ \int_{C_3} y^2 x dx + x^2 x dx $[/tex]
= 0 + 0 + 0
Applying the Green's theorem
[tex]$x^2 +y^2 = 81 \Rightarrow x \pm \sqrt{81-y^2}$[/tex]
[tex]$\int_C P dx + Q dy = \int \int_R\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx dy $[/tex]
Here,
[tex]$P(x,y) = y^2x \Rightarrow \frac{\partial P}{\partial y} = 2xy$[/tex]
[tex]$Q(x,y) = x^2y \Rightarrow \frac{\partial Q}{\partial x} = 2xy$[/tex]
[tex]$\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) = 2xy - 2xy = 0$[/tex]
Therefore,
[tex]$\oint_Cy^2xdx+x^2ydy = \int_0^9 \int_0^{\sqrt{81-y^2}}0 \ dx dy$[/tex]
[tex]$= \int_0^9 0\ dy = 0$[/tex]
The vector field F is = [tex]$y^2 x \hat i+x^2 y \hat j$[/tex] is conservative.
Related Questions
What would y be? If x is 9 m
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I need to evaluate the limit and I also need to show the work for it. Please help me out if you can.
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Answer:
-6
Step-by-step explanation:
You can solve all math problems via Photo math application, you can download it from the store for free, it is easy to write the question via the camera and quickly display the final result with the solution steps.
Idil owns a small business selling clothing. She knows that in the last week 3
customers paid cash, 25 customers used a debit card, and 53 customers used a credit
card.
with
Based on these results, express the probability that the next customer will pay
cash or a credit card as a percent to the nearest whole number.
Answers
The probability that the next customer will pay cash or a credit card is 56%, since 3 customers paid cash and 53 customers used a credit card out of a total of 81 customers.
What is customers?Customers are individuals or organizations who purchase goods and services from a business. They play an important role in a company's success as they provide the revenue needed to sustain and grow the business. Customers become loyal to a business when they are provided with quality products, excellent customer service, and competitive prices. Companies that focus on customer satisfaction often develop long-term relationships with their customers, as they understand the importance of customer loyalty. Companies must strive to continually meet the changing needs of their customers in order to remain successful in the long term.
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What is the area of a circle with a diameter of 14 meters? Leave the answer in terms of π.
49π square meters
7π square meters
196π square meters
14π square meters
Answers
Therefore , the solution of the given problem of circle comes out to be Area = 154 meter square or 49π meter square option 1 is correct.
Circle – what is it?A moving point on a plane is followed so that its distance from a specific point remains constant to produce the circular form known as a circle. The English term circle derives from the Greek word kirkos, which meaning hoop or ring. Area of circle= πr²
Here,
Given :
Diameter of circle : 14
Radius =14/2 = 7 meters
and
Using formula
Area= πr²
=> Area = π7²
=>Area = 22/7 * 7²
=> Area = 22*7
=> Area = 154 meter square or 49π meter square
Therefore , the solution of the given problem of circle comes out to be Area = 154 meter square or 49π meter square option 1 is correct.
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Answer: A the first one 49π square meters
The density of a certain material is such that it weighs 6900 grams for every 4.5 quarts of volume. Express this density in pounds per cubic foot. Round your answer to the nearest whole number.
Answers
The density in terms of pounds per cubic foot is 101.2 pounds/ft³
What is the unitary method?The unitary method is a method in which you find the value of a unit and then the value of a required number of units.
Given here; 6900 g for every 4.05 quarts thus expressing in terms of pounds per cubic foot we get = 15.22/0.15039
= 101.2
Hence, The density in terms of pounds per cubic foot is 101.2 pounds/ft³
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Is a function that can be graphed
as a solid line discrete or continuous?
Answers
Linear function Is a function that can be graphed as a solid line discrete or continuous
What is a linear function?Linear functions are simply known as functions whose graph is a straight line.
They have one dependent variable and one independent variable.
These functions have either one or two variables without exponents.
Continuous graphs are used to represent functions that are continuous along their entire or whole domain.
Whereas, discrete graphs are used to represent values at specific points along the number line from different directions.
Types of the linear function;
Direct variationslope-intercept formstandard formpoint-slope formHence, it is a linear function
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For every boy taking classes at the music school, there are 3 girls who are taking classes at the school. If there are 128 students taking classes, write and solve a proportion to predict the number of girls taking classes at the school.
Answers
Answer:
96
Step-by-step explanation:
Pls answer will mark brainliest! Ashlyn is having new hardwood flooring installed at the living room kitchen and dining room if the flooring cost $4.25 per square foot, how much will she spend?
Answers
Answer:
$580.13Step-by-step explanation:
The area of the flooring:
A = 1/2*8*(13.5 - 7.5) + 1/2(8 + 4 + 18)(7.5) = 136.5 ft²Cost of flooring:
136.5*$4.25 = $580.13The graph of the function y = f(x + 26) can be obtained from the graph of
y = = f(x) by one of the following actions:
Answers
Answer:
left 26 units
Step-by-step explanation:
inside the parentheses means left or right, and it also means opposite direction.
Answer:
this question answer Iis left 26 units..
Find the area of the parallelogram below.
Answers
Answer: 58 i think
Step-by-step explanation: i guess you have to add sorry if it wrong
In ΔHIJ, the measure of ∠J=90°, the measure of ∠H=15°, and JH = 54 feet. Find the length of HI to the nearest tenth of a foot
Answers
Answer:
46
Step-by-step explanation:
A couch and coffee table cost a total of $1172. The cost of the couch is three times the cost of the coffee table. Find the cost of each item.
Answers
Answer:
x + 3x = 1172, where x is the cost of the coffee table and 3x equals the cost of the couch.
4x = 1172
x = 293
thus, 3x = 293
293 x 3= 879
The couch is $879 and the coffee table is $293
Find the exact value of x.
Answers
Answer:
[tex]x = 4 \sqrt{3} [/tex]
Step-by-step explanation:
[tex] \frac{8}{2} = 4 \\ 4 \times \sqrt{3} = 4 \sqrt{3} [/tex]
Mr Paul has 4 bars of chocolate. He distributes 1/3
of each bar to 4 children.
a) Write the fraction of chocolate distributed.
b) Write the fraction of cholates left with Mr. Paul
Answers
a) The fraction of chocolate distributed is given as follows: 4/3.
b) The fraction of chocolates left with Paul is given as follows: 8/3.
How to obtain the fractions?The fractions are obtained applying the proportions used in the context of this problem.
He distributes 1/3 of each bar to 4 children, hence the total number of bars distributed is given as follows:
4 x 1/3 = 4/3.
(multiplication of fractions, multiply the numerators and the denominators, keeping the quotient).
The total number of bars that Paul had initially is given as follows:
4 bars.
Then the fraction of chocolate remaining with Paul is given by the subtraction of the total amount of bars by the amount given to the children, as follows:
4 - 4/3 = 12/3 - 4/3 = 8/3 bars.
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please help me this is. big
Answers
Answer:
C. = 69.5
Step-by-step explanation:
The median or the average between 80 and 59 is 69.5
Hope this helps! :)
pls help i’ve been stuck on this question ‼️
Answers
Find a formula for RN for the function f(x) = (5x)² on [-1, 5] in terms of N.
Could you please show the work? I was able to get the area but can't exactly figure out what they want for the formula.
Compute the area under the graph as a limit.
lim RN=1050
Answers
Therefore , R(N) = 100 + 400 (N+1)(2N+1) + 800 x N2 N3 and area are the formulas for R(N) for the function f(x) = (5x)2 on [1, 5] and the function f(x) = (5x)2 in terms of N. R(N) = 767(approx) (approx).
What is function?Each x value in the domain has one but only one y value range, according to the rule of a function. Definition. If there is only one and only one value of x variable such that f(x) = y, then the function is one-to-one.
Here,
In the given question, we have to find a formula for R(N) for the function f(x) =(5x)^2 on [1, 5] in terms of N.
The given function is f(x)=(5x)^2 on [1, 5].
Ax = (5-1)/N
Ax = 4/N
X = 1+iAx
X = 1+i4/N
f(x) = 25X^2
f(x) = 25(1+i4/N)^2
f(x) = 25(1+i8/N+16/N^2 i^{2})
R(N) = 1ƒ(X) Ax
Now putting the value of f(x) and Ax
R(N)==1 [25 (1+ &i + 16i2)]
R(N) = 1001 (1 +Ni + 16i2)
N100 (1+i+12)N
100 (N += i +122)+16 N(N+1)(2N+1)(+1)(2N+1)
R(N)=100 (N + √2/ N(N+1) x 100
R(N) =2/6× N + 100 × 5 × N(N+1) + 100 x 2
[tex]\lim_{n \to \infty}[/tex] R(N) = 100 + 400 + 800/3
[tex]\lim_{n \to \infty}[/tex] R(N) = 766.67
[tex]\lim_{n \to \infty}[/tex] R(N) = 767(approx)
Therefore , the solution of the given problem of function comes out to be [tex]\lim_{n \to \infty}[/tex] R(N) = 767(approx).
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A satellite dish is in the shape of a parabolic surface. Signals coming from a satellite strike the surface for the dish and are reflected to the focus, where the receiver is located. The satellite dish has a diameter of 14 feet and a depth of 3 feet. How far from the base of the dish should the receiver be placed? Round your answer to the nearest tenth.
Answers
Answer:
Given that we know the base of the parabola is at the origin, and that it is symetrical about the y-axis, we can simplify the equation to
y=ax2=14px2" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">y=ax2=14px2y=ax2=14px2
which corresponds to the given equation a=14p" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">a=14pa=14p given above.
Since y=6" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">y=6y=6 when x=2" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">x=2x=2,
6=a(2)2=4a" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">6=a(2)2=4a6=a(2)2=4a
and
a=32" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">a=32a=32
Thus we have
32=14p12p=2p=16" role="presentation" style="display: table-cell !important; line-height: 0; text-indent: 0px; text-align: center; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 3.337em; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; width: 10000em; position: relative;">32=14p12p=2p=1632=14p12p=2p=16
So, the receiver should be placed 16" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 16.66px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;">1616 of a foot, or 2 inches, above the base.
Johnny bought 8 toy cars. The cars came in p packages. Write an expression that shows how many toy cars were in each package.
Answers
Answer:
p=8+x
Step-by-step explanation:
Edgar accumulated $9,000 in credit card debt. If the interest rate is 30% per year and he does not make any
payments for 2 years, how much will he owe on this debt in 2 years by compounding continuously?
Answers
Answer:
$16399.06Step-by-step explanation:
Given
Debt amount P = $9000Interest rate r = 30% = 0.3Time t = 2 yearsCompound number = continuousUse formula
A = Pe^(rt)A = 9000*e^(0.3*2) = $16399.06What is the value of the expression below when x=2x=2?
8x^2 +6x-8
8x
2
+6x−8
Answers
The value of the expression 8x² + 6x - 8 below when x=2 is 36.
The value of the expression when x = 2The expression can be evaluated by following the steps below:
8x² + 6x - 8 in x = 2 means that the variable x will be replaced by the given number which is 2.
8*2² + 6*2 - 8
8*4 + 12 - 8
32 + 16 - 8 = 36
This therefore means that when x= 2 in 8x² + 6x - 8 , the value gotten will be 36.
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Which model can be used to represent
the number below?
Answers
Answer:
0.7
Step-by-step explanation:
The ratio of the measure of an exterior angle of a triangle to the measure of the adjacent interior angle is 1:4. Is the
triangle acute or obtuse?
Justify your answer.
A.) Acute; the measure of the interior angle must be 144º.
B.)Acute; the measure of the interior angle must be 36°.
C.)Obtuse; the measure of the interior angle must be 144º
D.)The triangle could be acute or obtuse, depending on the measures of the other two angles.
Answers
The ratio of the measure of an exterior angle of a triangle to the measure of the adjacent interior angle is 1:4, which means that for every 1 degree of the exterior angle, there are 4 degrees of the interior angle. However, this does not provide enough information to determine the overall shape of the triangle.
The triangle could be acute (all angles less than 90 degrees), right (one angle equal to 90 degrees), or obtuse (one angle greater than 90 degrees) depending on the measures of the other two angles. For example, if the interior angle measure is 36 degrees, the exterior angle would be 9 degrees, and the triangle could be acute. On the other hand, if the interior angle measure is 144 degrees, the exterior angle would be 36 degrees, and the triangle could be obtuse.
A truck driver drives at an average rate of 43 miles per hour.
At this rate, how many miles will he drive in 11 hours?
A: 473 miles
B: 54 miles
C: 483 miles
D: 55 miles
Answers
the sum of three numbers is 124. the third number is 4 times the first. the second number is 10 more than the first what are the numbers
Answers
Answer:
19,29,76
Step-by-step explanation:
x=first number
x+10=second number
4x=third number
add three numbers to equal 124.
x+(x+10)+4x=124
6x=114
x=19 first number
x+10 0r
19+10=29 second number
4x or
4(19)=76 third number
19+29+76=124
124=124
Mark's school is 5 kilometers away from his home. There are about 0.62 miles in 1 kilometer. About how many miles is Mark’s school from his home?
Answers
Answer:
3.1 miles
Step-by-step explanation:
(5 km)(0.62 mi/km) = 3.1 miles
Alisa needs 4.65 feet of fencing to
complete the fence she is building.
She has a piece of fencing that is 4 ft
long. Does she have enough fencing?
Explain.
Answers
No Alisa do not have enough fence.
What is perimeter?The perimeter of an object is given by adding the side lengths of the object.
Given that Alisa needs 4.65 feet of fencing to complete the fence she is building. She has a piece of fencing that is 4 ft long.
We need to determine whether she have enough fencing or not,
Since we know that the fencing can be considered as perimeter of the building,
So, if she only has 4 ft long fence, she cannot cover a 4.65 ft long fence.
Hence Alisa do not have enough fence.
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Select all the relations which represent a function.
- (1,3), (2,5), (2,7), (4,9) (1,3), (1,5), (1,7), (4,9) 0 (1,2), (2,5), (2,1), (4,5)
(1,3), (2,5), (3,7), (4,9) (1,3), (2,3), (3,3), (4,3)
Answers
Therefore , (1, 3), (2, 5), (3, 7) (4, 9) and (1, 3), (2, 3), (3, 3) (4, 3) are the relations that represent a function.
Function is what?According to a function's rule, every x value in the domain has one and only one range of y values. Definition. A one-to-one function exists when there is just one and only one value of x such that f(x) = y.
Here,
The relationship is (1, 3), (2, 5), and (2, 7) (4, 9)
Given that 2 is repeated, the connection is not a function.
The relationship is (1, 3), (2, 5), and (3, 7) (4, 9)
Since there are no repeating numbers, the relation is a function.
The relationship stated is (1, 3), (1, 5), and (1, 7). (4, 9)
The relationship is not a 1 repeated function.
The relationships are (1, 3), (2, 3), and (3, 3). (4, 3)
Since there are no repeating numbers, the relation is a function.
The relationship stated is (1, 2), (2, 5), and (2, 1) (4, 5)
As 2 repeated, the relationship is not a function.
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Pls help me, if you get it right I will give you Brainlist!
Answers
Answer:
Step-by-step explanation:
3.14x (3)^2
28.3ft
Answer:
The area of the circle is 28.3 [tex]ft^2[/tex].
Step-by-step explanation:
The formula to find the area of a circle is [tex]\pi r^2[/tex], where [tex]r[/tex] represents the radius of the circle. In this case, the area would be [tex]3.14 * 3^2 = 3.14 * 9 = 28.26[/tex] ≈ 28.3 [tex]ft^2[/tex].
Maya is solving the equation 2(1+7x)=72. She lists an explanation for each step as shown. Which is the missing explanation for Maya's first step after writing the given equation?
Answers
Answer:
apply the distrbutive property
Step-by-step explanation:
...........................