Solution: Given boundary value problem is Au = 0, 0 < x < R, 0 < a < 27, u(R, 6) = 4+3 sin 0, 0 << 27. = Using separation of variables let the solution be: u(x,θ) = X(x)Θ(θ)
Now, we need to solve the equation Au = 0 by using the method of separation of variables. Let us first start with Θ(θ) part. Let Θ(θ) = A sin(mθ) + B cos(mθ), Where A, B are constants and m is a constant to be determined, and let the boundary condition at θ = 6 be u(R, 6) = 4 + 3sin(0)∴ 4 + 3sin(0) = X(R)Θ(6)= X(R) (A sin(6m) + B cos(6m))…
(1)Next we need to determine the value of m. For this we will use the boundary condition that u(0,θ) = 0, which gives usΘ(θ) = A sin(mθ) + B cos(mθ)= 0, θ ≠ 6⇒ B cot(m6) = -A …
(2)Hence we obtainΘ(θ) = A sin(m(θ - 6)) + B cos(m(θ - 6))Now let us move to the X(x) part which satisfies: X''(x)/X(x) = - λLet λ = m² + k² …
(3)⇒ X(x) = C₁ cos(mx) + C₂ sin(mx) ...
(4)Hence the general solution to the equation Au = 0 is u(x,θ) = (C₁ cos(mx) + C₂ sin(mx))(A sin(m(θ - 6)) + B cos(m(θ - 6))) ...
(5). Now let us apply the boundary condition u(R, 6) = 4 + 3 sin(0) to get C₁ = 0, C₂ = 3/Θ(R) = A sin(6m) + B cos(6m)= 4 + 3sin(0)⇒ A = 3cos(6m) and B = 4/sin(6m). Now we have the expression for Θ(θ), hence substituting the values of A and B in the expression of Θ(θ), we getΘ(θ) = 3cos(m(θ - 6)) + 4sin(m(θ - 6))/sin(6m). Thus the solution to the boundary value problem is given by: u(x,θ) = C sin(mθ) (3cos(m(θ - 6)) + 4sin(m(θ - 6))), where C = 4/3π(1 - cos(6m)) and m is given by (3). Therefore, u(x,θ) = 4/3π(1 - cos(6m)) sin(mθ) (3cos(m(θ - 6)) + 4sin(m(θ - 6))).
Thus the solution to the boundary value problem is given by u(x,θ) = 4/3π(1 - cos(6m)) sin(mθ) (3cos(m(θ - 6)) + 4sin(m(θ - 6))) and m is given by (3).
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If you roll a six sided die, what is the probability that you do not roll a one or two?
Answer:
66.6%
Step-by-step explanation:
You have a 1/6 chance to roll any number. That is a 16.6% chance per number.
So, that means you have a 4/6 (2/3 simplified) probability of not rolling a one or two. That is a 66.6% chance.
[NOTE: The '6' at the end of '16.6%' and '66.6%' is a repeating decimal]
PLSSS HELP IMMEDIATELY!!!! i’ll give brainiest, if u don’t provide just a link! answer choices: (4,6) , (0,1) , (0,0) , (5,4)
Answer:
try (4,5)
Step-by-step explanation:
that is where the points meet I think it is that I am not sure sorry if it is wrong
One kilogram is approximately 2.2 pounds. Write a direct variation equation that relates x kilograms to y pounds.
Answer:
2.2y=1x or just x
Step-by-step explanation:
Answer: y=2.2x
Step-by-step explanation:
A continuous random variable is said to have a Laplace(μ, b) distribution if its probability density function is given by
fX(x)= 1 exp(−|x−μ|), 2b b
where μ is a real number and b>0.
(i). If X ∼ Laplace(0,1), find E(X) and Var(X).
(ii). If X ∼ Laplace(0,1) and Y = bX + μ, show Y ∼ Laplace(μ, b). (iii). If W ∼ Laplace(2,8), find E(W) and Var(W).
(i) For X ~ Laplace(0,1):
E(X) = 0, Var(X) = 2.
(ii) If X ~ Laplace(0,1) and Y = bX + μ:
Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8):
E(W) can be approximated numerically.
Var(W) = 128.
(i) If X ~ Laplace(0,1), we need to find the expected value (E(X)) and variance (Var(X)).
The Laplace(0,1) distribution has μ = 0 and b = 1. Substituting these values into the PDF, we have:
fX(x) = (1/2) * exp(-|x|)
To find E(X), we integrate x * fX(x) over the entire range of X:
E(X) = ∫x * fX(x) dx = ∫x * [(1/2) * exp(-|x|)] dx
Since the Laplace distribution is symmetric about the mean (μ = 0), the integral of an odd function over a symmetric range is zero. Therefore, E(X) = 0 for X ~ Laplace(0,1).
To find Var(X), we use the formula:
Var(X) = E(X^2) - [E(X)]^2
First, let's find E(X^2):
E(X^2) = ∫x^2 * fX(x) dx = ∫x^2 * [(1/2) * exp(-|x|)] dx
Using the symmetry of the Laplace distribution, we can simplify the integral:
E(X^2) = 2 * ∫x^2 * [(1/2) * exp(-x)] dx (integral from 0 to ∞)
Solving this integral, we get:
E(X^2) = 2
Now, substitute the values into the variance formula:
Var(X) = E(X^2) - [E(X)]^2 = 2 - 0 = 2
Therefore, for X ~ Laplace(0,1), E(X) = 0 and Var(X) = 2.
(ii) To show that Y = bX + μ follows a Laplace(μ, b) distribution, we need to find the probability density function (PDF) of Y.
Using the transformation method, let's express X in terms of Y:
X = (Y - μ)/b
Now, calculate the derivative of X with respect to Y:
dX/dY = 1/b
The absolute value of the derivative is |dX/dY| = 1/b.
To find the PDF of Y, substitute the expression for X and the derivative into the Laplace(0,1) PDF:
fY(y) = fX((y-μ)/b) * |dX/dY| = (1/2) * exp(-|(y-μ)/b|) * (1/b)
Simplifying this expression, we get:
fY(y) = 1/(2b) * exp(-|y-μ|/b)
This is the PDF of a Laplace(μ, b) distribution, thus showing that Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8), we need to find E(W) and Var(W).
The PDF of W is given by:
fW(w) = (1/16) * exp(-|w-2|/8)
To find E(W), we integrate w * fW(w) over the entire range of W:
E(W) = ∫w * fW(w) dw = ∫w * [(1/16) * exp(-|w-2|/8)] dw
This integral can be challenging to solve analytically. However, we can approximate the expected value using numerical methods or software.
To find Var(W), we can use the property that the variance of the Laplace distribution is given by 2b^2, where b is the scale parameter.
Var(W) = 2 * b^2
= 2 * (8^2)
= 2 * 64
= 128
Therefore, Var(W) = 128 for W ~ Laplace(2,8).
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How many solutions does this equation have? 9z = –8 + 7z
-no solution
-one solution
-infinitely many solutions
Answer:
one solution.
The cost of renting a bicycle, y, for
x hours can be modeled by a linear
function. Renters pay a fixed insurance
fee of $12 plus an additional cost of $10
per hour, for a maximum of 6 hours.
What is the range of the function for this
situation?
F {22, 32, 42, 52, 62, 72}
G {1, 2, 3, 4, 5, 6}
H {12, 24, 36, 48, 60, 72}
J {22, 34, 46, 58, 70, 82}
Answer:
F
Step-by-step explanation:
1(10) + 12= 22
2(10) + 12= 32
etc.....
A soccer ball has an interior
diameter of 22cm. How
many cubic centimeters of air
does a soccer ball contain,
rounded to the nearest
hundredth?
Answer:
5575 cm³
Step-by-step explanation:
The volume of a sphere is V = (4/3)πr³. If we substitute r = (d/2 where r is the radius and d the diameter), this formula becomes:
πd³
V = (4/3)π·d³/8, or V = -------------
6
and so, if d = 22 cm, the volume is:
π(22 cm)³
V = ------------------- = 5575 cm³
6
Find the product
(10^2)^3
Answer:
Well first you must do what's in the parenthesis.
10^2 = 100
100^3 = 1000000
Shania is making lasagna. The recipe she uses calls for 2 1/3 cups of spaghetti sauce. If she doubles the recipe, how much spaghetti sauce will she need?
Answer:
4 2/3 cups
Step-by-step explanation:
what is the price of a $600 bike 15% off
Answer: You will pay $510 for a item with original price of $600 when discounted 15%.
AH PLEASE SOMEONE HELP
Answer:
Variant b
Step-by-step explanation:
If you multiply it should be a+b but if you divided
A+b-c
25
What is the solution to the equation 12(x+5) = 4x?
Answer:
x = -7.5
Step-by-step explanation:
12(x+5) = 4x
12x+ 60 = 4x
60 = -8x
-7.5 = x
Let R be the binary relation defined on a set of all integers Z as follows: for all integers m and n, mRn m’ – n’ is divisible by 6. a) Is R an equivalence relation? Check the conditions. b) What is the equivalence class of -17?
Previous question
The required solutions are:
a) Yes, the relation R is an equivalence relation.
b)The equivalence class of -17 is {-17, -23, -29, -35, ...}.
a) In order to determine whether R is an equivalence relation or not, we need to check if it satisfies the following three conditions:
Reflexibility: For all integers m, mRm should hold. In the given case, if we take m=n, we have m-n=n-m=0, which is divisible by 6. So, we can see that the reflexibility is satisfied.Transitivity: For all integers m, n, and p, if mRn and nRp hold, then mRp should also hold. Assume mRn and nRp, which means m-n, and n-p are both divisible by 6. To check transitivity, we need to check if m - p is divisible by 6. By adding the two previous conditions, we have (m-n) + (n-p) = m-p, which is also divisible by 6. Therefore, transitivity is satisfied.Symmetry: For all integers m and n, if mRn holds, then nRm should also hold. If mRn, it means m-n is divisible by 6. In order to check the symmetry, we need to check if n - m is divisible by 6. We can use the fact that a-b = -(b-a), we can rewrite n - m as -(m - n), which is divisible by 6. So, we can say that symmetry is satisfied.We can see that the relation 'R' satisfies all the conditions ( reflexibility, symmetry, and transitivity), so R is an equivalence relation.
b) In order to find the equivalence class of -17, we need to find all integers that are related to -17 under the relation R.
We can rewrite the relation as mRn if and only if m' - n' = 6k for some integer k.
In this case, -17Rn if and only if (-17)' - n' = -17 - n = 6k for some integer k.
To find all integers n that satisfy this equation, we can rearrange it as n = -17 - 6k.
By putting in different values of k, we can find all the integers n that are in the equivalence class of -17.
For example, when k = 0, n = -17 - 6(0) = -17. So, -17 is in the equivalence class of -17.
We can also see that when k = 1, n = -17 - 6(1) = -23. So, -23 is also in the equivalence class of -17.
The equivalence class of -17 consists of all integers that can be obtained by subtracting multiples of 6 from -17. So, the equivalence class of -17 is {-17, -23, -29, -35, ...}.
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The Σx = 68, y = 79, xy = 1200 x² = 600 and N = 12 determine the equation of the least square line. Leave answer in 2 decimal places.
To determine the equation of the least squares line, we need to find the slope (m) and the y-intercept (b) using the given data. The formula for the least squares line is:
y = mx + b
where:
m is the slope of the line
b is the y-intercept of the line
We can use the following formulas to calculate the slope and y-intercept:
m = (NΣxy - ΣxΣy) / (NΣx² - (Σx)²)
b = (Σy - mΣx) / N
Given the values:
Σx = 68
Σy = 79
Σxy = 1200
Σx² = 600
N = 12
Let's substitute these values into the formulas:
m = (12 * 1200 - 68 * 79) / (12 * 600 - (68)²)
b = (79 - m * 68) / 12
Calculating the values:
m = (14400 - 5372) / (7200 - 4624)
m = 9028 / 2576
m ≈ 3.5039
b = (79 - 3.5039 * 68) / 12
b = (79 - 238.2752) / 12
b ≈ -13.2738
Therefore, the equation of the least squares line is:
y ≈ 3.5039x - 13.2738
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The equation of the least squares line is y = 1.25x + 10.75
To determine the equation of the least squares line, we need to find the slope (b) and y-intercept (a) using the following formulas:
b = (NΣxy - ΣxΣy) / (NΣx² - (Σx)²)
a = (Σy - bΣx) / N
Given the following values:
Σx = 68
Σy = 79
Σxy = 1200
Σx² = 600
N = 12
Let's calculate the slope (b) first:
b = (12 × 1200 - 68×79) / (12×600 - 68²)
b ≈ 1.25
Now we can calculate the y-intercept (a):
a = (79 - 1.25×68) / 12
a = 10.75
Therefore, the equation of the least squares line is: y = 1.25x + 10.75.
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The highest temperature in Las Vegas is 125 degrees Fahrenheit and the lower recorded temperature in Las Vegas is 50 degrees Fahrenheit below zero what is the difference between these two temperatures
Answer:
175 degrees Fahrenheit
Step-by-step explanation:
We are to find the difference between the two temperatures
125 - (-50)
two minuses gives a plus
125 = 50 = 175
help me, please. I'm not very good at math
Answer:
1st, 2nd, 3rd
Step-by-step 1explanation:
30+40+5=75
30+40=70
70+5=75
20+1+50+4
=20+50=70
1+4=5
70+5=75
50+30-5
50+30=80
80-5=75
I hope this helps :)
Find the values of the sine, cosine, and tangent for angle a
[tex]sina=\frac{2\sqrt{13} }{13} \\cosa=\frac{3\sqrt{13} }{13}\\tana=\frac{2}{3}[/tex]
Answer:
A.
Step-by-step explanation:
correct on edge2021
What is the measure of angle C?
Answer:
angle C = 36°
Step-by-step explanation:
Fun fact that I found out:
all interior angles of a triangle added together = 180°
5x + 3x + 2x = 180°
combine like terms:
10x = 180°
divide both sides of the equation by 10:
x = 18°
angle C = 2(18°) = 36°
Help is much needed. You will get lots of point too!
When describing the results of an observational study, the mathematical relationship between the relative risk (RR) and the odds ratio (OR) is referred to as: a. The rare disease assumption b. The built-in bias c. The relative odds approach d. Effect modification e. All of the above
Previous question
The mathematical relationship between the relative risk (RR) and the odds ratio (OR) is (c) The relative odds approach.
How to find the mathematical relationship between the relative risk (RR) and the odds ratio (OR)?The mathematical relationship between the relative risk (RR) and the odds ratio (OR) is often described as the relative odds approach.
This approach quantifies the relationship between RR and OR and provides insights into the association between exposure and outcome in observational studies.
The rare disease assumption (a) refers to the assumption made in case-control studies that the odds ratio approximates the relative risk when the outcome is rare.
It is not specifically related to the mathematical relationship between RR and OR.
The built-in bias (b) does not directly relate to the mathematical relationship between RR and OR.
It refers to biases inherent in the study design or data collection process.
Effect modification (d) refers to a situation where the association between an exposure and outcome differs depending on the levels of another variable.
While effect modification can influence the relationship between RR and OR, it is not the specific term used to describe their mathematical relationship.
In summary, the mathematical relationship between RR and OR is best described as the relative odds approach (c), while the other options (a, b, d) are not directly related to this specific relationship.
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y= 2x-3
y= x+4
Graph each system and determine the number of the solutions that it has. If it has one solution, name it.
Do males or females feel more tense or stressed out at work? A survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table shown to the right. Complete parts (a) through (d) below. Felt Tense or Stressed Out at Work Yes No Total Gender Male 100 200 300 Female 145 125 270 Total 245 325 570 a. What is the probability that a randomly selected person's gender is female?
b. What is the probability that a randomly selected person feels tense or stressed out at work and is female?
c. What is the probability that a randomly selected person feels tense or stressed out at work or is female?
d. Explain the difference in the results in (b) and (c).
A survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table is as follows:
a) The probability that a randomly selected person's gender is female is 270/570 or 0.474, which is approximately 47.4%.Formula used: P (Female) = Number of Females/Total Number of Individuals
b) The probability that a randomly selected person feels tense or stressed out at work and is female is 145/570 or 0.254, which is approximately 25.4%. Formula used: P (Female and Tense) = Number of Females who are Tense/Total Number of Individuals
c) The probability that a randomly selected person feels tense or stressed out at work or is female is: P (Female or Tense) = P(Female) + P(Tense) - P(Female and Tense)P(Tense) = (245/570) or 0.43, which is approximately 43%P(Female or Tense) = 0.47 + 0.43 - 0.254 = 0.646, which is approximately 64.6%.
d) The distinction between the outcomes in (b) and (c) is that the former shows the likelihood of being female and tense at work, whereas the latter shows the likelihood of being female or tense at work.
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Use pie= 22/7 to approximate the circumference of a circle whose diameter is 28 inches.
Answer:
88 inches
Step-by-step explanation:
formula for circumference: 2pir or pi*diameter
22/7*28=88
88 inches
3.1. Two friends. Ben and Mike, take part in a 15km fun
run. Ben took 1 h 23 min 12 sec and Mike took 1 h 39
min 4 sec. How long did Ben wait at the finish line for
Mike?
Answer: 15.867 min
Step-by-step explanation:
Given
Ben took [tex]1\ h\ \text{and}\ 23\ min\ 12\ s[/tex] to complete 15 km
While Mike take [tex]1\ hr\ 39\ min\ 4\ s[/tex] to complete the same
converting time into miniutes
Ben time
[tex]\Rightarrow 60+23+\frac{12}{60}=83.2\ min[/tex]
Mike time
[tex]\Rightarrow 60+39+\frac{4}{60}=99.0667\ min[/tex]
So, mike waited for
[tex]\Rightarrow 99.067-83.2=15.867\ min[/tex]
A 12ft basketball hoop casts an 8 ft shadow. Find the length of the shadow of a 4 ft tall fence.
Set up a ratio of height over shadow for each :
12/8 = 4/x
Cross multiply:
12x = 32
Divide both sides by 12:
X = 2 2/3 feet
The shadow is 2 2/3 feet.
Help pls it is my homework
Can y'all help me?
Answer:
A
Step-by-step explanation:
the mean is what occurs most often
FILL in the blank:AB E M nxn (R) (i) det (A.B) = ____________ . (ii) A is invertible if and only if _____________ .
Answer:
For square matrices A and B of equal size, the determinant of a matrix product equals the product of their determinants: det (A.B) = det (A) det (B) 1. A is invertible if and only if its determinant is nonzero 1.
Step-by-step explanation:
f(x) = 2x+10. If f(x)= -2, find x.
Answer:
x=-5
Step-by-step explanation:
Question is in picture
Answer:
27.5ft
Step-by-step explanation:
a² + b² = c²
Plug in what you have:
a² + 12² = 30²
Get a by itself:
a² = 30² - 12²
Simplify:
a² = 900 - 144
Simplify:
a² = 756
Simplify:
√a = √756
Then you get 27.49 and when you round you get 27.5
The following contingency table gives the results of a sample survey of South African male and female respondents with regard to their preferred fastfood outlet: PREFERRED FAST FOOD OUTLET Burger King McDonalds TOTAL 50 No. of Males No. of Females 20 130 100 120 TOTAL 270 150 110 140 400 0.1.1.1 What is the probablyf randomly selecting a respondent who is male and prefer Burger 01.12 What is the probably selecting a female respondent, even that the preferred fastfood out? 0.1.1.3 What is the probability of selecting a respondent who is female or who prefers McDonalds? 12) (2) Events X and Yare such that PC) = 0.20 and PCXUY) = 0.55. Given that Xand Yare independent and non-mutually taclusive, determine P(Y). Give your final answer as a percentage to two decimal places (5) 13 (2) 2.1.3.1 Helen is the manager of a Finance Department. She has fifteen (15) members of stuff working for her. She has to choose five (5) members of her staff for a research team. How many different teams can she select from the fifteen members of staff 2.1.22 There are twelve (12) teams in a basketball league. What is the probability of correctly predicting the top three teams at the end of the 3) season in the correct order?
Q1.1.1 The probability of randomly selecting a male respondent from the sample is 0.4. Q.1.1.2 The probability of randomly selecting a respondent who is female and prefers HP is 0.275. Q.1.1.3 The probability of selecting a male respondent, given that the preferred brand is Lenovo is 0.4545. Q.1.1.4 The probability of selecting a respondent who is male or who prefers HP is 0.575. Q.1.1.5 The probability of selecting a respondent who does not prefer Lenovo is 0.725.
Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?
The probability of randomly selecting a male respondent is given by the number of male respondents divided by the total number of respondents:
Probability = No. of Males / Total = 160 / 400 = 0.4
Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?
The probability of randomly selecting a respondent who is female and prefers HP is given by the number of females who prefer HP divided by the total number of respondents:
Probability = No. of Females who prefer HP / Total = 110 / 400 = 0.275
Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?
The probability of selecting a male respondent, given that the preferred brand is Lenovo, is given by the number of males who prefer Lenovo divided by the total number of respondents who prefer Lenovo:
Probability = No. of Males who prefer Lenovo / Total No. of respondents who prefer Lenovo = 50 / 110 = 0.4545
Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?
The probability of selecting a respondent who is male or who prefers HP is given by the sum of the probabilities of selecting a male respondent and selecting a respondent who prefers HP, minus the probability of selecting both (to avoid double counting):
Probability = (No. of Males / Total) + (No. of Females who prefer HP / Total) - (No. of Males who prefer HP / Total)
Probability = (160 / 400) + (110 / 400) - (40 / 400) = 0.4 + 0.275 - 0.1 = 0.575
Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?
The probability of selecting a respondent who does not prefer Lenovo is given by the number of respondents who do not prefer Lenovo divided by the total number of respondents:
Probability = (Total - No. of respondents who prefer Lenovo) / Total
Probability = (400 - 110) / 400 = 290 / 400 = 0.725
The complete question is:
The following contingency table gives the results of a sample survey of South African male and female respondents with regard to their preferred brand of notebook:
HP Lenovo Dell Total
No. of Females 110 60 70 240
No. of Males 40 50 70 160
Total 150 110 140 400
Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?
Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?
Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?
Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?
Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?
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