Consider the circular annulus (a plane figure consisting of the area between a pair of concentric circles) specified by the range: 1 1 cases. b) Find the potential that satisfies the following boundary conditions 1 u (1,0) = sin? (0) ), u (2,0) = 0. ) = + (1 - cos (20),
Answers
The potential that satisfies the given boundary conditions in part (a) and (b) is: \(\[u(r, \theta) = \sin(\theta)\]\) and \(\[u(r, \theta) = \sin(\theta)\]\) respectively.
Consider the circular annulus (a plane figure consisting of the area between a pair of concentric circles) specified by the range:
\($1 \leq r \leq 2$.\)
a) Find the potential that satisfies the following boundary conditions:
\(\[\begin{aligned}u(1,0) &= \sin(\theta) \\u(2,0) &= 0 \\u(\theta, 1) &= 1 + (1 - \cos(2\theta))\end{aligned}\]\)
b) Find the potential that satisfies the following boundary conditions:
\(\[\begin{aligned}u(1,0) &= \sin(\theta) \\u(2,0) &= 0 \\u(\theta, 1) &= 1 + (1 - \cos(20\theta))\end{aligned}\]\)
To solve this problem, we can use separation of variables and assume a solution of the form:
\(\[u(r, \theta) = R(r)\Theta(\theta)\]\)
Plugging this into Laplace's equation \($\nabla^2u = 0$\) and separating variables, we get:
\(\[\frac{1}{R}\frac{d}{dr}\left(r\frac{dR}{dr}\right) + \frac{1}{\Theta}\frac{d^2\Theta}{d\theta^2} = 0\]\)
Solving the radial equation gives us two solutions:
\(\[R(r) = A\ln(r) + B\quad \text{and} \quadR(r) = C\frac{1}{r}\]\)
For the angular equation, we have:
\(\[\Theta''(\theta) + \lambda\Theta(\theta) = 0\]\)
The general solution to this equation is given by:
\(\[\Theta(\theta) = D\cos(\sqrt{\lambda}\theta) + E\sin(\sqrt{\lambda}\theta)\]\)
To satisfy the boundary conditions, we can impose the following restrictions on \($\lambda$\) and choose appropriate constants:
For part (a)
\(\[\begin{aligned}R(1) &= 0 \implies B = -A\ln(1) = 0 \implies B = 0 \\R(2) &= 0 \implies A\ln(2) + B = 0 \implies A\ln(2) = 0 \implies A = 0 \\\Theta(0) &= \sin(0) \implies D = 0 \\\Theta(0) &= \sin(0) \implies E = 1\end{aligned}\]\)
Therefore, the potential that satisfies the given boundary conditions in part (a) is:
\(\[u(r, \theta) = \sin(\theta)\]\)
For part (b)
\(\[\begin{aligned}R(1) &= 0 \implies B = -A\ln(1) = 0 \implies B = 0 \\R(2) &= 0 \implies A\ln(2) + B = 0 \implies A\ln(2) = 0 \implies A = 0 \\\Theta(0) &= \sin(0) \implies D = 0 \\\Theta(0) &= \sin(0) \implies E = 1\end{aligned}\]\)
Therefore, the potential that satisfies the given boundary conditions in part (b) is:
\(\[u(r, \theta) = \sin(\theta)\]\)
Please note that in both parts (a) and (b), the radial solution does not contribute to the potential due to the boundary conditions at r=1 and r=2. Thus, the solution is purely dependent on the angular part.
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Related Questions
7th grade math I need some help
Answers
Answer:
$33.60
Step-by-step explanation:
First we take 80% of $40.
1. $40 * 0.80 = $32
Next we take 105% of $32.
2. $32 * 1.05 = $33.60
Can someone help me with this question??
Answers
Answer:
rotate it across the Y axis by 90°. it should be in Quad ||
a geology class is studying a sample of rock and a sample of dry sponge the rock sample has a mass of 1 x 10¹ kg. The dry sponge sample has a mass of 2x 10⁻³ kg. Which sample has a greater mass? How many times greater?
Answers
The sample that has a greater mass is given as follows:
The sample of rock.
The sample of rock has a mass that is 5000 times the mass of the sample of dry sponge.
What is scientific notation?A number in scientific notation is given by the notation presented as follows:
\(a \times 10^b\)
With the base being \(a \in [1, 10)\), meaning that it can assume values from 1 to 10, with an open interval at 10 meaning that for 10 the number is written as 10 = 1 x 10¹, meaning that the base is 1, justifying the open interval at 10.
To obtain which amount is greater, we must verify the bases when the exponent is the same, hence:
Rock: 1 x 10¹ kg = 10 kg.Sponge: 2 x 10^-3 kg = 0.2 x 10^-2 kg = 0.02 x 10^-1 kg = 0.002 kg.Hence the rock has the greater mass, and the ratio is given as follows:
10/0.002 = 5000.
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PLEASE DONT USE ANY APPS TO SOLVE THIS QUESTION.
A regression model is desired relating temperature and the proportion of impurities passing through solid helium. Temperature is listed in degrees centigrade. The data are as follows:
Temperature (°C) Proportion of impurities
-260.5 0.425
-255.7 0.224
-264.6 0.453
-265.0 0.475
-270.0 0.705
-272.0 0.860
-272.5 0.935
-272.6 0.961
-272.8 0.979
-272.9 0.990
a) Construct the linear regression model
Answers
The linear regression model for the data is Y ≈ 0.0012X + 0.608
A linear regression model relating temperature and the proportion of impurities passing through solid helium, we'll use the method of least squares to find the equation of a line that best fits the given data.
Let's denote the temperature as X and the proportion of impurities as Y. We have the following data points:
X: -260.5, -255.7, -264.6, -265.0, -270.0, -272.0, -272.5, -272.6, -272.8, -272.9
Y: 0.425, 0.224, 0.453, 0.475, 0.705, 0.860, 0.935, 0.961, 0.979, 0.990
We want to find the equation of a line in the form Y = aX + b, where a is the slope and b is the y-intercept.
To calculate the slope a and y-intercept b, we'll use the following formulas:
a = (nΣ(XY) - ΣXΣY) / (nΣ(X²) - (ΣX)²)
b = (ΣY - aΣX) / n
where n is the number of data points.
Let's calculate the necessary summations:
ΣX = -260.5 + (-255.7) + (-264.6) + (-265.0) + (-270.0) + (-272.0) + (-272.5) + (-272.6) + (-272.8) + (-272.9) = -2704.6
ΣY = 0.425 + 0.224 + 0.453 + 0.475 + 0.705 + 0.860 + 0.935 + 0.961 + 0.979 + 0.990 = 7.017
Σ(XY) = (-260.5)(0.425) + (-255.7)(0.224) + (-264.6)(0.453) + (-265.0)(0.475) + (-270.0)(0.705) + (-272.0)(0.860) + (-272.5)(0.935) + (-272.6)(0.961) + (-272.8)(0.979) + (-272.9)(0.990) = -2517.384
Σ(X²) = (-260.5)² + (-255.7)² + (-264.6)² + (-265.0)² + (-270.0)² + (-272.0)² + (-272.5)² + (-272.6)² + (-272.8)² + (-272.9)² = 729153.05
Now, let's substitute these values into the formulas for a and b:
a = (10(-2517.384) - (-2704.6)(7.017)) / (10(729153.05) - (-2704.6)^2)
b = (7.017 - a(-2704.6)) / 10
Simplifying the calculations, we find:
a ≈ 0.0012
b ≈ 0.608
Therefore, the linear regression model for the given data is:
Y ≈ 0.0012X + 0.608
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What is the solution of the equation (x – 5)^2 + 3(x – 5) + 9 = 0? Use u substitution and the quadratic formula to solve.
A) x=-3+3i√3/2 B) x=7+3i√3/2 C) x=2 D) x=8
Answers
Answer:
\( x = \dfrac{7 \pm 3i\sqrt{3}}{2} \)
Step-by-step explanation:
(x – 5)^2 + 3(x – 5) + 9 = 0
This is a quadratic equation in x - 5.
Let u = x - 5, then the quadratic equation becomes:
u^2 + 3u + 9 = 0
We can use the quadratic formula to solve for u.
\( u = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
\( u = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(9)}}{2(1)} \)
\( u = \dfrac{-3 \pm \sqrt{9 - 36}}{2} \)
\( u = \dfrac{-3 \pm \sqrt{-27}}{2} \)
\( u = \dfrac{-3 \pm 3i\sqrt{3}}{2} \)
Since u = x - 5, now we substitute x - 5 for u and solve for x.
\( x - 5 = \dfrac{-3 \pm 3i\sqrt{3}}{2} \)
\( x = \dfrac{-3 \pm 3i\sqrt{3}}{2} + 5 \)
\( x = \dfrac{-3 \pm 3i\sqrt{3}}{2} + \dfrac{10}{2} \)
\( x = \dfrac{7 \pm 3i\sqrt{3}}{2} \)
Answer:
x = 7+/-3i sqrt3 over 2
Step-by-step explanation:
Which is the smallest fraction? *
7/10
4/5
17/20
3/4
Too tired to work it out myself just do it for me instead person get 15 points
Answers
Solve for x. Round to the nearest tenth
Answers
*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆**☆*――*☆*――*☆*――*☆
Answer: 6) x = 21.04
Explanation:
I hope this helped!
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*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆**☆*――*☆*――*☆*――*☆
what is (2/5)^-4 please answer:)
Answers
Answer:
My answer got deleted because It contained a site, but The answer is
625/16
As a decimal 39.06
Step-by-step explanation:
Find 33⅓% of ⅘ of $4.50.
Answers
I hope this helps you :)
Write the linear equation in slope-intercept form ( y = mx + b )
Answers
Answer:
B. Y= 2x - 4
Step-by-step explanation:
I think, so sorry if wrong
I needes y on its own so I devided the equation by 3
[A→(B→C)]→[B→(A→C)] (A∧B)→(A→B
′
)
′
(A→C)∧(C→B
′
)∧B→A
′
[A→(B∨C)]∧C
′
→(A→B)
Answers
(A→C)∧(C→B′)∧B→A′ is the given proposition. Now, we have to prove that [A→(B→C)]→[B→(A→C)].Proof:We have to prove that [A→(B→C)]→[B→(A→C)] is a tautology.By using the conditional proof method, we have to assume that [A→(B→C)] is true and then show that [B→(A→C)] is also true.
For this, we have to use the rules of inference. Let's begin:1. Assume A → (B → C) is true.2. By Simplification, A is true because we have (A ∧ B) given in the premises.3. By Simplification, B is also true because we have (A ∧ B) given in the premises.4. By Modus Ponens, (B → C) is true.5. By Modus Ponens, (A → C) is true.6. By Simplification, we get C is true.7. By Modus Ponens, (B → (A → C)) is true.8. By Modus Ponens, [(A → (B → C)) → (B → (A → C))] is true. Thus, the proof is completed.Note: Please provide the complete question to receive the correct answer.
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Here’s the other one, i just need two more that i will be posting, ty your and amazing person
Answers
Answer:
See below.
Step-by-step explanation:
X-intercepts: (-7, 0) & (-3, 0)
Axis of Symmetry: x = -5
Y-intercept: (0, -21)
Vertex: (-5, 4)
pls answer this question brainliest will be given
√8/98
Answers
solve this question plz
Answers
Answer:
answer is 69 degree
Step-by-step explanation:
a+33+78=180 degree
a+111=180
a=180-111
a=69
Could someone help me with this due in 2 hours!!!!! Been working on this for 6 hours choosing brainliest!!!!
Answers
Step-by-step explanation:
The Triangle Inequality Theorem states that any two sides of the triangle must add up to be greater than the third side.
Question #1Here we have a triangle that has side lengths of 10 in, 11 in, and ? in.
This last side length must add up to 10 in to be greater than 11 in.
The least possible whole number that would fit this description would be 2 in.
10 in + 2 in > 11 in10 in + 11 in > 2 in 11 in + 2 in > 10 inThe missing side length should be labeled as 2 inches.
Question #2We are given a triangle with side lengths of 6 in and 8 in. We are asked to choose all answers that apply from this list:
2 in3 in4.5 in 6.5 in10 in13.5 in14 in15.5 inWe can tell that 2 in cannot be an answer since 6 in + 2 in is not greater than 8 in.
We also know that 14 in and 15.5 in cannot be part of the answer choices since 8 in + 6 in = 14 in, and this is not greater than 14 in or 15.5 in.
The rest of the answer choices will form a triangle if we follow the Triangle Inequality Theorem. You can test it out yourself to check.
Therefore, the answer choices are:
B. 3 inC. 4.5 inD. 6.5 inE. 10 inF. 13.5 in Question #3We are given these possible side lengths:
2, 4, 5, 6, 10Let's test out which combinations will create a triangle.
2 + 4 > 5 5 + 2 > 44 + 5 > 2The straw lengths 2, 4, and 5 will form a triangle.
2 + 5 > 6 5 + 6 > 22 + 6 > 5The straw lengths 2, 5, and 6 will form a triangle.
2 + 6 > 4 4 + 2 >/ 6The straw lengths 2, 4, and 6 will NOT form a triangle.
4 + 5 > 66 + 5 > 4 4 + 6 > 5The straw lengths 4, 5, and 6 will form a triangle.
5 + 6 > 10 10 + 5 > 66 + 10 > 5The straw lengths 5, 6, and 10 will form a triangle.
We have no more straw lengths that will form a triangle, since adding up each remaining pair of given straw lengths do not output a value greater than another number.
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what is 14/3 divided by 7
Answers
Answer:0.6
Step-by-step explanation:
Answer:
14/3 divided by 7 = 0.66 or 2/3
Step-by-step explanation:
14/3 divided by 7 = 0.66
0.66 as a fraction is: 2/3
Brainliest please! I am so close to getting my next ranking! I would really appreciate it, and it would make my day! Thank you so much, and have a wonderful rest of your day!
What is the final price (before sales tax) for a set of golf clubs that cost $800 if they are on sale for 30% off?
A. $730
B. $770
C. $660
D. $560
Answers
Answer:
D. $560
Step-by-step explanation:
800 x .3 equals 240.
800 minus 240 gives you 560
Mark me as brainliest if this helps!
what is the definition of vertical angles
Answers
Answer:
each of the pairs of opposite angles made by two intersecting lines.Step-by-step explanation:
Answer:
vertical angles
geometry the pair of equal angles between a pair of intersecting lines; opposite angles Also called: vertically opposite angles
Step-by-step explanation:
can you help me ?…………….
Answers
Answer: b
Step-by-step explanation: i think
The area of a rectangle 320in^2. The ratio of the length to the width is 5:4. Find the length and the width
Answers
Answer:
30 in length 24inches width
Step-by-step explanation:
Multiply the multiples if 5 and 4 together until you reach 720.
5*4=20
10*8=80
15*12=180
20*16=320
25*20=500
30*24=720
Your length is 30 in and your width is 24in.
PLEASE I NEED THIS QUICKLY!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Solve the following equation for B. Be sure to take into account whether a letter is capitalized or not F=-m+B/q³
Answers
After solving the given expression → F = - m + B/q³ for [B], we get -
B = q³(F + M).
What is an expression? What is a expression? What is a mathematical equation? A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.We have the following equation -
F = - m + B/q³
We have the following equation -
F = - m + B/q³
On solving for [B], we get -
F + M = B/q³
B = q³(F + M)
Therefore, the given expression after solving for [B], we get -
B = q³(F + M).
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A card is selected to from a standard deck of 52 card what are the odds of selecting a red 9
Answers
The odds of selecting a red 9 is 1/26.
Probability of an event E is represented by P(E) can be defined as (the number of favorable outcomes) / (Total number of outcomes).
Given the total number of cards in a standard deck = 52
there can be only two red9 as one 9 from heart and one red from diamond.
So the number of outcome for red 9 =2
the probability of odds of selecting red 9 is \(\frac{2}{52}\) which can be further simplified into \(\frac{1}{26}\).
Therefore , The odds of selecting a red 9 is 1/26.
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if a student is chosen randomly, what is the probability that he or she is taking exactly one language class?
Answers
The probability of a student being chosen at random and taking exactly one language class is 65%.
Probability is the likelihood or chance of an event happening. In this case, the event is a student taking exactly one language class.
First, we need to find the total number of students taking at least one language class. We have 28 students in the Spanish class, 26 students in the French class, and 16 in the German class. However, some students are taking multiple classes, so we need to subtract these students from the total. The 12 students taking both Spanish and French, 4 taking both Spanish and German, and 6 taking both French and German should only be counted once. In addition, the 2 students taking all three classes should also only be counted once.
So, the total number of students taking at least one language class is
=> 28 + 26 + 16 - 12 - 4 - 6 + 2 = 40 students.
To find the number of students taking Spanish only, we need to subtract the number of students taking both Spanish and French (12) and the number of students taking both Spanish and German (4) from the number of students taking Spanish (28).
So, the number of students taking Spanish only is
=> 28 - 12 - 4 = 12 students.
Similarly, the number of students taking French only is
=> 26 - 12 - 6 = 8 students
and the number of students taking German only is
=> 16 - 4 - 6 = 6 students.
Finally, to find the probability of a student being chosen at random and taking exactly one language class, we divide the number of students taking exactly one language class by the total number of students taking at least one language class.
So, the probability of a student taking exactly one language class is
=> (12 + 8 + 6) / 40 = 26 / 40 = 0.65 or 65%.
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Complete Question:
An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 students in the French class,, and 16 in the German class. There are 12 students that are in both Spanish and French, 4 that are in both Spanish and German, and 6 that are in both French and German. In addition, there are 2 students taking all 3 classes.
If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?
in the expression 3x+7 what is the coefficient
Answers
Answer:
3
Step-by-step explanation:
Coefficient in the number behind the variable.
In 3x + 7,
x is the variable.
3 is the coefficient.
What does "when x is an integer" mean?
Answers
Answer:
x = whole-valued numbers without fractions or decimal.
Step-by-step explanation:
An integer in mathematics refers to a whole number that can either be a positive or negative number but without any fractional part. Integers are numbers such as -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 etc.
Hence, when we say "x" is an integer, it simply means that the value of x should be a whole number that can either be a positive or negative number but without any fractional or decimal value or part.
In Mathematics, when solving for an unspecified or unknown integer we can use a lowercase alphabet "x" to denote the integer.
For instance;
2x + 5 = 11
2x = 11 - 5
2x = 6
Dividing both sides by 2, we have;
x = 3.
How many packs of dvds can u but with 165 if one costs 15 dollars
Answers
Answer: You can buy 11 packs of dvds
Answer: You can buy 11 DVD’s with $165
Step-by-step explanation:
165 divided by 15= 11
Write an equation in point-slope form of the line that passes through the given point and with the given slope m. A. (1, -8) m= -9
Answers
The point-slope form of a line is:
y = mx + b
where m is the slope and b is the y-intercept
Replacing with point (1, -8) and m= -9, we get:
-8 = (-9)(1) + b
-8 = -9 + b
-8 + 9 = b
1 = b
Then, the equation is:
y = -9x + 1
as. The proportional of an iceberg above the water is describe by the ratio 1. 9. If 60 ct bic meters of the iceberg are visible above the water: a) How much of the iceberg s below the water? (mks) b) What the total volume of the iceberg? ( 2mks)
Answers
a. 540 cubic meters of the iceberg is below the water .
b. 480 cubic meters is volume of the iceberg are below the waterline.
What is proportions?Proportion is a mathematical concept that relates two or more quantities in a fixed relationship or ratio. In other words, a proportion is a statement that two ratios are equal. For example, the statement "2/5 equals 4/10" is a proportion, because both ratios simplify to 2:5.
Proportions are often used in problem-solving and real-world applications, such as in cooking recipes, financial calculations, and engineering designs. Proportions can be used to find missing values or to scale up or down quantities while maintaining the same ratio. They can also be used to compare two different sets of data by calculating their relative sizes or growth rates.
a) If 60 cubic meters of the iceberg are visible above the water and the ratio of visible iceberg to total iceberg is 1:9, then we can set up the proportion:
visible iceberg/total iceberg = 1/9
We can solve for the total iceberg by cross-multiplying and simplifying:
visible iceberg/1 = total iceberg/9
total iceberg = 9 * visible iceberg
total iceberg = 9 * 60 cubic meters
total iceberg = 540 cubic meters
b) The total volume of the iceberg is 540 cubic meters, and we know that the visible portion of the iceberg is 60 cubic meters. Therefore, the volume of the iceberg below the waterline is:
total volume - visible volume = 540 - 60 = 480 cubic meters
So, 480 cubic meters of the iceberg are volume.
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find the surface area for a sphere with a radius of 10 feet. round to the nearest whole number. a. 1,256 ft2b. 4,189 ft2c. 1,089 ft2d. 1,568 ft2
Answers
The surface area of a sphere with a radius of 10 feet is approximately 1,256 ft².
The surface area of a sphere refers to the total area that covers the surface of the sphere. It is the sum of all the areas of the small flat faces that make up the sphere.
To find the surface area of a sphere with a radius of 10 feet, we need to use the formula:
Surface Area = 4πr²
Where r is the radius of the sphere.
Plugging in the value of r=10 into the formula, we get:
Surface Area = 4π(10) = 400π
Since π is an irrational number, we cannot calculate its exact value. However, we can approximate it to 3.14. Therefore,
=> Surface Area = 400 * 3.14 = 1256
Rounding to the nearest whole number, we get the surface area of the sphere with a radius of 10 feet as 1,256 ft², which is option (a).
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Each marble bag sold by Ahmad's Marble Company contains 5 green marbles for every 8 orange marbles. If a bag has 25 green marbles, how many orange
marbles does it contain?
Answers
Answer:
40 orange marbles
Step-by-step explanation:
Set up and solve this proportion:
\( \frac{5}{8} = \frac{25}{x} \)
\(5x = 200\)
\(x = 40\)