Use separation of variables to solve the following Laplace's equation on the quarter disk ∇u= 0, 0 u(r, 0) = u (r, π/2) = 0, 0
Answers
Given that the Laplace's equation on the quarter disk is ∇u= 0, with the boundary conditions
=u(r, 0)
= u (r, \(\pi/2\))
= 0
The Laplace's equation in polar coordinates is given by:
\(\nabla u = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0\)
where r is the radial distance and θ is the angular coordinate of a point.
To solve the Laplace's equation on the quarter disk, we use separation of variables and assume that u(r, θ) can be written as a product of functions of r and θ as follows:
u(r, θ) = R(r)Θ(θ)
Substituting the above expression in the Laplace's equation, we get:
\(\frac{r}{R} \frac{d}{dr} \left( r \frac{dR}{dr} \right) + \frac{1}{\Theta} \frac{d^2 \Theta}{d\theta^2} = 0\)
Separating the variables and equating the terms to a constant λ, we get:
\(r \frac{d^2R}{dr^2} + \frac{dR}{dr} - \lambda R = 0\)
\(\frac{d^2\Theta}{d\theta^2} + \lambda \Theta = 0\)
The general solution to the second equation is given by:
\(\begin{equation}\Theta(\theta) = A \sin(\sqrt{\lambda \theta}) + B \cos(\sqrt{\lambda \theta})\) where A and B are constants.
The boundary conditions u(r, 0) = u (r, π/2) = 0 give us
\(\Theta(0) = A \sin(0) + B \cos(0) = 0\) and
\(\Theta\left(\frac{\pi}{2}\right) = A \sin\left(\sqrt{\lambda}\frac{\pi}{2}\right) + B \cos\left(\sqrt{\lambda}\frac{\pi}{2}\right)\)
= 0
Solving the above equations, we get:
A = 0 and \(\sqrt{\frac{\lambda\pi}{2}} = \frac{(2n - 1)\pi}{2}\) where n = 1, 2, 3, ...Therefore,
\(\lambda_n = (2n - 1)^2 \pi^{-2}\), n = 1, 2, 3, ...The general solution to the first equation is given by:
\(R(r) = C_1J_0(\sqrt{\lambda_n r}) + C_2Y_0(\sqrt{\lambda_n r})\) where J₀ and Y₀ are Bessel functions of the first and second kind of order zero, respectively, and C₁ and C₂ are constants.
The general solution to the Laplace's equation on the quarter disk is given by:
\(u(r, \theta) = \sum_{n=1}^{\infty} C_n \sin((2n - 1)\pi \theta/2\pi) J_0((2n - 1)r/2)\)
where Cₙ are constants and J₀ is the Bessel function of the first kind of order zero.
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Related Questions
Suppose n balls are distributed randomly into n compartments.
What is the probability that m balls will fall into the first compartment? Assume all the n arrangements are equally like.
Answers
The probability that m balls is P(m balls in first compartment) = (nCm) * ((n-1)^(n-m)) / (n^n).
The total number of possible arrangements of distributing n balls into n compartments is given by n^n. Each ball has n choices for which compartment to fall into, and since there are n balls, we multiply n by itself n times.
Now, let's consider the number of ways m balls can fall into the first compartment. We can choose m balls out of the n balls to go into the first compartment in nCm ways, which is the binomial coefficient.
The remaining (n-m) balls can go into any of the remaining (n-1) compartments in (n-1)^(n-m) ways. Therefore, the probability that m balls will fall into the first compartment is given by:
P(m balls in first compartment) = (nCm) * ((n-1)^(n-m)) / (n^n)
This formula calculates the ratio of favorable outcomes (m balls in the first compartment) to the total number of possible outcomes (all possible arrangements of distributing n balls into n compartments).
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5.98 + 9.2 = ___________ work the answer out
Answers
And 9.2 is 920 and not 92 bc 9.2 equals 9.20.
So 598+920= 1518
Than you Have to put comma like in the previous numbers.
9.20
5.98
15.18
hmm need help please
Answers
Answer:
Hello! answer: b = 5
Step-by-step explanation:
13 × 13 = 169 12 × 12 = 144 169 - 144 = 25 5 × 5 = 25 therefore b = 5 hope that helps!
13x13=169
169-144=25
5^2= 25
B=5 feet
(!!!!!VERY URGENT!!!!!!) find the area of the composite figure
Answers
A man invests $ 16800 in savings plan that pays simple interest at a rate of 5% per annum. Find the Tim’s taken for his investment to grow to $18900
Answers
Answer:
2.5 years
Step-by-step explanation:
The given amount invested, which is the principal, P = $16,800
The simple interest rate, R = 5% per annum
The intended total value of the investment, A = $18,900
The simple interest on the principal, I = A - P
∴ I = $18,900 - $16,800 = $2,100
The formula for the simple interest, I, is given as follows;
\(I = \dfrac{P \times R \times T}{100}\)
Therefore, we have;
\(T = \dfrac{I \times 100}{P \times R}\)
Plugging in the values, gives;
\(T = \dfrac{2,100 \times 100}{16,800 \times 5} =2.5\)
The time it will take the investment to grow to $18,900 is T = 2.5 years
A spinner has 5 equally sized sections, 1 of which is gray and 4 of which are blue. The spinner is spun twice. What is the probability that the first spin lands on blue and the second spin lands on gray ? Write your answer as a fraction in simplest form
Answers
The probability that the first spin lands on blue and the second spin lands on gray is 4/25.
Since the spinner has 5 equally sized sections, the probability of landing on blue on the first spin is 4/5, as there are 4 blue sections out of a total of 5 sections.
After the first spin, there are 4 sections left on the spinner, with 1 gray section and 3 blue sections.
The probability of landing on gray on the second spin is 1/4.
To find the probability of both events occurring, we multiply the probabilities together:
(4/5) * (1/4) = 4/20 = 1/5.
However, the probability is usually expressed in simplest form. The fraction 1/5 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 1.
Therefore, the final answer is 4/25.
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-(4-4b)+6=16+2b solution
Answers
after adding everything up, you get 3 as b, plug that in and check just in case
If the product of the extremes is 8,then the geometric mean is
~2
~4
~2(square root sign)2
Answers
The geometric mean of two numbers is the square root of their product:
\(GM(a,b)=\sqrt{ab}\)
So, if the product is 8, the geometric mean is \(\sqrt{8}=2\sqrt{2}\)
The tenth term of 85
Answers
The 1st term is 10,
2nd term is 10+15=25,
3rd term is 10+2\(\times\)15=40 and
4th term is 10+3\(\times\)15=55.
Given the first term as 10 and the 6th term as 85.
From the arithmetic sequence
The series where the common difference between any two next words stays constant is known as an arithmetic sequence.
As an illustration, the numbers 1, 6, 11, 16,... are an arithmetic sequence because they follow a pattern in which the preceding term is multiplied by 5 to produce each subsequent number.
Formula is \(a_{n} =a+(n-1)d\) where d=\(a_{n} -a_{n-1}\).
So, we can write a=10
6th term is 85
a+5d=85
10+5d=85
5d=85-10
5d=75
d=15
Therefore, the 1st term is 10,
2nd term is 10+15=25,
3rd term is 10+2\(\times\)15=40 and
4th term is 10+3\(\times\)15=55.
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Un conductor circula a 12 m/s. Acelera y pasa a circular a 20 m/s al cabo de 10 segundos. Calcula la aceleración del coche.
Answers
Answer:
Aceleracion = 0.8 m/s²
Step-by-step explanation:
Dados los siguientes datos;
Velocidad inicial = 12 m/s
Velocidad final = 20 m/s
Tiempo, t = 10 segundos
Para encontrar la aceleración;
Aceleración se puede definir como la tasa de cambio de la velocidad de un objeto con respecto al tiempo.
Esto simplemente significa que la aceleración viene dada por la resta de la velocidad inicial de la velocidad final a lo largo del tiempo.
Por lo tanto, si restamos la velocidad inicial de la velocidad final y la dividimos por el tiempo, podemos calcular la aceleración de un objeto. Matemáticamente, la aceleración viene dada por la fórmula;
\( Aceleracion = \frac{final \; velocidad - inicial \; velocidad}{tiempo}\)
Sustituyendo en la fórmula, tenemos;
\( Aceleracion = \frac{20 - 12}{10}\)
\( Aceleracion = \frac{8}{10}\)
Aceleracion = 0.8 m/s²
Which one of the following statements is false? A. The sampling distribution of any statistic becomes approximately normal for large sample sizes. B. The standard deviation of the sample mean decreases as the sample size increases. C. The sampling distribution of the sample mean is exactly normal if the observations are normally distributed D. The sampling distribution of the sample mean becomes approximately normal for large sample sizes.
Answers
The false statement is: C. The sampling distribution of the sample mean is exactly normal if the observations are normally distributed.
Statement A is true. According to the central limit theorem, the sampling distribution of any statistic, including the sample mean, tends to become approximately normal as the sample size increases, regardless of the underlying distribution of the population.
Statement B is true. The standard deviation of the sample mean, also known as the standard error, decreases as the sample size increases. This decrease in standard deviation indicates that the sample mean becomes a more precise estimator of the population mean.
Statement C is false. The sampling distribution of the sample mean follows an approximately normal distribution for large sample sizes, even if the individual observations are not normally distributed. The central limit theorem ensures the approximate normality of the sample mean's distribution through the law of large numbers and the convergence of the sampling distribution to a normal distribution shape.
Therefore, the false statement is C, as the sampling distribution of the sample mean is not exactly normal even if the observations are normally distributed.
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Solve x² = 25.
x= 12.5
x = ±5
x = 5
x = ±12.5
Answers
Answer:
x = ±5
Step-by-step explanation:
x = ±5
Solve x² = 25.
x= 12.5
x = ±5
x = 5
x = ±12.5
x² = 25
x = √25
x = ±5
--------------------
check with 5
5² = 25
check with -5
-5² = 25
solve the quadratic linear below :
6x²-x-2=0
Answers
Answer:
x = 3/2; x = -1/2
Step-by-step explanation:
6x² - x - 2 = 0
⇔ 6x² - 4x + 3x - 2 = 0
⇔ (6x² - 4x) + (3x - 2) = 0
⇔ 2x(3x - 2) + (3x - 2) = 0
⇔ (3x - 2)(2x + 1) = 0
⇔\(\left \{ {{3x - 2=0} \atop {2x + 1=0}} \right.\)
⇔\(\left \{ {{3x=2} \atop {2x=-1}} \right.\)
⇔\(\left \{ {{x=2/3} \atop {x=-1/2}} \right.\)
I hate school I really do!
Answers
Answer:
SAMEEE
Step-by-step explanation:
QUIT SCHOOOOLLL
A student applies 0.3 N of force to a computer mouse. The mouse slides .025 m. How much work did the student do on the mouse?
Answers
Answer:
0.0075 N·m
Step-by-step explanation:
Work is the product of force and distance:
W = fd
W = (0.3 N)(0.025 m) = 0.0075 N·m
__
Additional comment
1 N·m = 1 J
Use the units you like. You could also write this in terms of millijoules: 7.5 mJ.
What is the quotion? -4 ½ ÷ ( -2 ⅔)
Answers
Answer in Exact Form: 27/16
Answer in Decimal Form: 1.6875
Answer in Mixed Number Form: 1 11/16
Hope this helps¡!
A right triangle has a hypotenuse with a length of 10cm. One angle of the three in this triangle is 53.1 degrees. What is the length of the side opposite this angle?Group of answer choices8.0 cm6.0 cm1.33 cm4.0 cm
Answers
The length of the side opposite the angle of 53.1 degrees in a right triangle with a hypotenuse of 10 cm is 8 cm. (A)
This can be found using trigonometry and the Pythagorean theorem. The trigonometric function sine can be used to find the ratio of the length of a side to the length of the hypotenuse given an angle in a right triangle.
In this case, the sine of the angle 53.1 degrees is the ratio of the length of the side opposite the angle to the length of the hypotenuse, which is 10 cm.
Therefore, the length of the side opposite the angle 53.1 degrees can be found by multiplying the sine of the angle by the length of the hypotenuse, which is 10 cm * sin(53.1) = 8 cm.
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2. MGSE9-12.A.REI. 1 Which step in solving this equation is justified by the Distributive Property? Consider the sequence of steps to solve the equation: 6v + 3(v – 5) = 4v - (2v + 1) Given = 6v + 3(v — 5) = 4v - (2v + 1) Step 1 = 6 + 3x – 15 = 4y - (2v + 1) Step 2 => 6v + 3v – 15 = 4v – 2y = 1 Step 3 9v – 15 = 4y – 2y - 1 Step 4 => 9v - 15 -- 2v + 4y = 1 Step 5 =9v = 15 = 2y + 1 Step 6 3 7v - 15 =-1 Step 7 3 Tv = 14 Step 8 = v= 2 a. Step 1 b. Step 7 C. Step 5 d. Step 2
Answers
6v + 3(v – 5) = 4v - (2v + 1)
Given = 6v + 3(v — 5) = 4v - (2v + 1)
step 1
when you open the first parenthsesis in step 1
we have 6v + 3v - 15 = 4v - (2v + 1)
Then in step 2
we open the second parenthesis,
This gives;
6v + 3v - 15 = 4v - 2v - 1
Therefore; step 1 and step 2 justified the distributive property
HELPPPPPPPPPP ASAPPPPPP!
Answers
Answer:
Number 3 is ASA
Number 4 is SAS
Number 5 is SSS
Number 8 is 3
Number 9 is 2
Step-by-step explanation:
Which criteria can be used to prove triangles are congruent select all that apply?
Answers
The four criteria that can be used to prove triangles are congruent and can be used to prove the triangles are congruent.
Triangle congruence: Two triangles are said to be congruent if all three of their corresponding sides and all three of their corresponding angles have the same measurements. These triangles can be moved around, rotated, flipped, and turned to have a same appearance. They match up with one another when moved.
The four criteria that can be used to prove triangles are congruent are SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SSS (Side-Side-Side). Additionally, CPCTC (Corresponding Parts of Congruent Triangles are Congruent) can be used to prove the triangles are congruent.
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Which similarity postulate or theorem can be used to verify that the two
triangles shown below are similar?
A. SAS theorem
B. SSS theorem
C. AA postulate
D. Similarity cannot be determined.
Answers
Answer:
C
Step-by-step explanation:
AA postulate because we are given 2 of the 3 angles of each triangle. They are both the same in each triangle. Also, we can find the 3rd angle in each triangle. Since both triangles have the exact same angles, they are similar.
The given triangles ΔABC and ΔLMN are similar by AA postulate.
What is a triangle?A triangle is a three sided polygon that consist of three edges and three vertices .
Here two given triangles are ΔABC and ΔLMN. In these two triangles ∠A=∠L=53° and ∠B=∠M= 72°.
Therefore, the two corresponding angles of triangles ΔABC and ΔLMN are equal . So the third angle of ΔABC is also equal to the corresponding angle of ΔLMN.
Hence two triangles are similar by AA postulate.
Thus two triangles are similar by AA postulate. So option (C) is correct.
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what does it mean to say a system has a ∆g equal to zero?
a. the reaction is spontaneous at standard conditions
b. the reaction is nonspontaneous at standard conditions
c. the system is at equilibrium at standard conditions
d, the reaction is both non spontaneous and at equilibrium
e. the reaction is both spontaneous and at equilibrium
Answers
A system having ΔG equal to zero means that the system is at equilibrium at standard conditions. This is option c. In a chemical reaction, ΔG represents the change in Gibbs free energy, which is a measure of the energy available to do work.
When ΔG is equal to zero, it indicates that the system is balanced, with the forward and reverse reactions occurring at the same rate. At equilibrium, the concentrations of reactants and products remain constant over time. For example, consider the reaction A ⇌ B. Initially, if we have more A than B, the forward reaction will be favored until equilibrium is reached. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of A and B remain constant. In this case, ΔG is zero.
It is important to note that while a ΔG of zero indicates equilibrium, it does not provide information about whether the reaction is spontaneous or nonspontaneous. The spontaneity of a reaction is determined by the sign of ΔG. If ΔG is negative, the reaction is spontaneous, while a positive ΔG indicates a nonspontaneous reaction.
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8^6/8
please could you help
Answers
Answer:
32768
Do 8 to the power of 6 first since exponents come before dividng and multiplying. Then divide by 6.
Huilan's age is three times Thomas's age. The sum of their ages is 64. What is Thomas's age?
Answers
Answer:
16
Step-by-step explanation:
Answer: 16 years old
Step-by-step explanation:
PLEAESE HELP ME I WILL GIVE BRAINLYEST!!!
(01.04 MC)
What is the sign of the product (3)(−25)(7)(−24)? (10 points)
Group of answer choices
Positive, because the products (3)(−25) and (7)(−24) are positive, and the product of two positive numbers is positive
Negative, because the products (3)(−25) and (7)(−24) are positive, and the product of two positive numbers is negative
Negative, because the products (3)(−25) and (7)(−24) are negative, and the product of two negative numbers is negative
Positive, because the products (3)(−25) and (7)(−24) are negative, and the product of two negative numbers is positive
Answers
Answer:
Positive, because the products (3)(−25) and (7)(−24) are negative, and the product of two negative numbers is positive
HELPPP ASAPPP geometry honors
Answers
Answer:
-2
Step-by-step explanation: has to be -2 postive it has to be but /////////////////////////////////////
Find the circumference of a circle rounded to the nearest tenth with a
diameter of 4 inches
Answers
Answer:
12.6
Step-by-step explanation:
Circumference is πd. So...
(3.14*4)
12.56
12.6
Hope this helps and have a nice day ~(^_^)~
2+2 +2x2x7x2+13 —————————-
Answers
Answer:
73
Step-by-step explanation:
2×2×7×2=56
2+2=4
56+4+13=73
a 12 oz cup costs $4.25, a 16 oz cup costs $4.95 and a 20 oz cup costs $5.25. Which is the better deal?
Answers
Answer:
The 12 ounce cup
Step-by-step explanation:
Just divide the cup size with its price and the lowest number is the best deal
Two different cross sections are taken parallel to the base of a three-dimensional figure. The two cross sections are the same shape, but are not congruent. Which could be the three-dimensional figure? select three options.
Answers
One possible three-dimensional figure that fits this description is a rectangular prism. Another possible option is a cylinder. A third option is a pyramid with a square base.
Based on your question, the three-dimensional figure could be one of the following three options:
1. Cone: If two cross sections are taken parallel to the base, they will both be circles but with different radii, making them similar but not congruent.
2. Pyramid: If two cross sections are taken parallel to the base, they will both be the same shape as the base (e.g., triangles, squares) but with different side lengths, making them similar but not congruent.
3. Frustum: A frustum is a section of a cone or pyramid with the top cut off parallel to the base. The two cross sections taken parallel to the base would be the same shape but with different dimensions, making them similar but not congruent.
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