The solution of Neumann problem, ∇²u= 0 if r < R , Uₙ (R,θ) = f(θ) is u(r,θ) = a'₀+ rⁿ(a'ₙ cosnθ + b'ₙ sinnθ) with boundary conditions uᵣ (r,θ) = ∑n R⁽ⁿ⁻¹⁾(Aₙ cosnθ + Bₙ sinnθ) = f(θ) and
Aₙ=∫(f(θ)cosnθ /π nR⁽ⁿ⁻¹⁾)dθ, where θ∈[-π, π]
Bₙ=∫(f(θ)sinnθ /π nR⁽ⁿ⁻¹⁾)dθ, where θ∈[-π, π]
Given that
The solution of Numann problem
∇²u= 0 if r < R , Uₙ (R,θ) = f(θ)
Use polar co-ordinates (r,θ)
uᵣᵣ + 1/r uᵣ+ 1/uᵣ (uθθ) = 0 ,0 < r< R,
0 <θ <2π and ∂u/∂r(R,θ) = f(θ) is directional derivative
r²d²u/dr² + rdu/dr + d²u/dθ² = 0
Let , r = ε⁻ᵗ , u(r(t),θ)
uₜ = uᵣ(rₜ) = - e⁻ᵗ uᵣ
uₜₜ =( - e⁻ᵗ uᵣ )ₜ = ε⁻ᵗuᵣ + e⁻²ᵗ uᵣᵣ
= r uᵣ+ r²uᵣᵣ
Thus we have, uₜₜ + uθθ = 0
Let u(t,θ) = X(t)Y(θ)
Which gives X''(t)Y(θ) + X(t)Y"(θ) = 0
X"(t)/X(t) = - Y"(θ)/Y(θ) = λ
From Y"(θ) + λ Y(θ) = 0
We get, Yₙ(θ) = aₙ cosnθ + bₙ sinnθ
λ= n² , n =0, 1, ...
With these values of λn we solve,
X"(t) - n² X(t) = 0
If n = 0 , X₀(t) = c₀t + d₀
X₀(r) = -c₀log (r) + d₀
If n not equal to 0 then
Xₙ (t) = cₙeⁿᵗ + dₙ e⁻ⁿᵗ
Xₙ(r) = cₙ(r)⁻ⁿ + dₙ (r)ⁿ
We have u₀(r, θ) = X₀(r)Y₀(θ)
= a₀ ( - c₀(log r) + d₀)
uₙ(r,θ) = Xₙ(r) Yₙ(θ)
= (cₙ r⁻ⁿ+ dₙrⁿ)(aₙ cosnθ + bₙ sinnθ)
But u must be positive at t =0
So, cₙ = 0 ; n = 0,1,2....
u₀ (r,θ) = a₀ d₀
uₙ(r,θ) = dₙ rⁿ( aₙ connθ + bₙ sinnθ)
By superposition , we can write as
u(r,θ) = a'₀+ rⁿ(a'ₙ cosnθ + b'ₙ sinnθ)
Boundary conditions gives
uᵣ (r,θ)=∑n R⁽ⁿ⁻¹⁾(Aₙ cosnθ + Bₙ sinnθ) = f(θ)
the coefficients aₙ , bₙ for n ≥ 1 are determined are Fourier series for f(θ)
but a₀ is not determined from f(θ) therefore , it may take arbitrary value. By using Fourier series,
Aₙ= Integration of(f(θ)cosnθ dθ/π n R⁽ⁿ⁻¹⁾) where θ∈[-π, π]
Bₙ= Integration of (f(θ) sinnθ dθ/π nR⁽ⁿ⁻¹⁾) where θ∈[-π, π]
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simplify the following expression a^2b^-2c^-1 / e^7
Answer:
a²/(b²ce^7)
Step-by-step explanation:
a^2b^-2c^-1 / e^7 =
= a²/(b²ce^7)
If x³ = 1/125 then 625x³ =
Answer: 5 i think
because if x^3 = 1/125 then 625 times 1/125 is 5
A 39-inch board is cut into two pieces. One piece is two times the length of the other. Find the length of the shorter piece.
The shorter piece is
inches long.
Answer:
The shorter piece is 13 inches long.
Step-by-step explanation:
39=2x+x
39=3x
/3 /3
13=x
A certain association provided data on the cost of the most popular home remodeling projects. Sample data on the cost in thousands of dollars for two types of remodeling projects are as follows.
Kitchen Master Bedroom
25.2 18.0 16.4 22.9 21.8 26.4 20.9 24.8 18.7 26.9 22.0 17.8 19.7 24.6 15.9 21.0 21.8 22.6
(a) Develop a point estimate of the difference in thousands of dollars) between the population mean remodeling costs for the two types of projects. (Use Kitchen - Master Bedroom.) thousand (b) Develop a 90% confidence interval for the difference in thousands of dollars) between the two population means. (Use Kitchen - Master Bedroom. Round your answers to one decimal place.)
As per the given confidence interval, the population mean remodeling costs for the two types of projects is -1.6
Confidence interval
The range of values that's likely to include a population value with a certain degree of confidence is known as confidence interval.
Given,
A certain association provided data on the cost of the most popular home remodeling projects.
Here we need to find the the population mean remodeling costs for the two types of projects.
Here we will find the point estimate of the difference between the population mean remodeling costs for the two types of projects using an equation,
=> X1 − X2
In order to calculate the sample mean we will use equation,
=> X = ∑x/n
Now, we have to substitute values we get,
=> X1 = 21.2
=> X2 = 22.8
Here X1 refers the sample mean remodeling costs for the kitchen and X2refers the sample mean remodeling cost for the master bedroom.
Therefore, the point estimate of the difference between the population mean remodeling costs for the two types of projects are,
=> X1 − X2 = −1.6
Therefore, we estimate that the remodeling master bedroom has the mean costs of 1.6 greater than mean remodeling costs for the kitchen.
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What does the statement below mean?
Range: {-7 < y < 12}
The statement -7 < y < 12 meant y is greater than -7 and less than 12.
How to interpret inequality?{-7 < y < 12}
This is a compound inequality.
A compound inequality is a type of inequality which combones two or more simple inequality.
-7 < y < 12
This can be broken down into two parts;
-7 < y (minus 7 less than y)y < 12 (y is less than 12This can also be written together as y is greater than -7 and less than 12.
This means y is between the values that are greater than -7 but less than 12
So,
y could be -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Therefore, y can be said to be within the range of values between -6 and 11
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please help!!!!!!!!!!!!!!
Answer:
-6/7p + 1/7
Step-by-step explanation:
in picking the smoothing constant for an exponential smoothing model, we should look for a value that
In picking the smoothing constant for an exponential smoothing model, we should look for a value that produces values which compare well with actual values based on a standard measure of error.
What is exponential smoothing model?
Using the exponential window function, exponential smoothing is a general method for blending time series data. Exponential functions are used to assign exponentially decreasing weights over time, as opposed to the simple moving average, which weights previous observations equally.
The correct answer is, produces values which compare well with actual values based on a standard measure of error.
Quantitative forecasting utilizes a compilation of past performance data of the organization to arrive at an accurate forecast for any given process.
The smoothing constant selected requires to optimize accuracy the unknown parameters and the initial values need to be considered with the implementation of the sum of squared errors providing the required accuracy.
Hence, in picking the smoothing constant for an exponential smoothing model, we should look for a value that produces values which compare well with actual values based on a standard measure of error.
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Complete question:
Complete question is attached below.
Burton Bush wants to retire in Arizona when he is 80 years of age. Burton, who is now 55, believes he will need $400,000 to retire comfortably. To date, he has set aside no retirement money. If he gets an interest rate of 6% compounded annually, what will he have to invest today (to the nearest dollar) to reach his goal?
Multiple Choice
$160,000
$91,243
$96,500
$93,199
Burton Bush has to invest $93,199 to get $400,000 at the end. Hence, option D is correct.
What is compound interest?Interest that is added to a loan and deposit sum is known as compound interest. In our everyday lives, it is the notion that is employed the most frequently. Compound interest is calculated as a sum using the interest and principal accrued over time. Compound interest versus simple interest differ primarily in this way.
As per the given information in the question,
Rate, r = 6%
Number of years, n = 25
Present value, PV = ?
Final value, FV = $400,000
Use the equation of compound interest:
[tex]PV = FV (1 + r)^-^n[/tex]
Put values from above in the given equation,
PV = 400,000 (1 + 0.06)⁻²⁵
PV = 400,000 × 0.23
PV = $93,199.
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Perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of the answer: cos(x) 1 + sin(x) cos(x) 1 - sin(x) 2tan
After simplifying the given equation the answer will be -2tanx.
What is identity?
An equation is referred to as an identity if it has one or more variables and is true for all alternate values of the variables for which both sides of the equation are defined.
Sine, cosine, tangent, cosecant, secant, and cotangent are some of their names. The adjacent side, opposite side, and hypotenuse side of the right triangle are used to define each of these trigonometric ratios. The six trigonometric ratios are the source of all fundamental trigonometric identities.
Given: [tex]\frac{cosx}{1+sinx}-\frac{cosx}{1-sinx}[/tex]
We have to solve this.
Take cross multiplication.
[tex]= \frac{cosx - cosxsinx-cosx-cosxsinx}{(1-sin^2x)}\\= \frac{-2cosxsinx}{(1-sin^2x)}[/tex]
Since,
[tex]1-sin^2x=cos^2x[/tex]
[tex]= \frac{-2sinx}{cosx}[/tex]
Since, [tex]\frac{sinx}{cosx} = tanx[/tex]
[tex]= -2tanx[/tex]
Hence, after simplifying the given equation the answer will be -2tanx.
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Suppose a power series converges if |6x 12/<78 and diverges if |6x 12| 78 Determine the radius and interval of convergence. The radius of convergence is R =
The radius of convergence is a-R a+R = 6+12 =18 and 6-12 = -6 for the |6x 12/<78 and diverges if |6x 12| 78.
The set of actual numbers x wherein the collection converges is the c program language period of convergence. If there exists a actual wide variety R such that the collection converges for |x−a|R, then R is the radius of convergence.
The radius of convergence is 1/2 of of the duration of the c program language period of convergence. If the radius of convergence is R then the c program languageperiod of convergence will consist of the open c program languageperiod: (a − R, a + R). To discover the radius of convergence, R, you operate the Ratio Test.
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If on a scaled drawing 1/8 of an inch represents 15 feet, how long should a drawing of a 120 ft long pole be?
Here, on solving the provide question to us, we got to know that - 210 feet represents = 825/4
What is equation?Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.
Here,
15 feet represents = 1/8 of an inch
1 feet represents = 1/8 X 15
210 feet represents = 1/8 X 15 X 210 = 825/4
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A dog weighs 80 pounds more than a cat. Their combined weight equals 120 pounds. What is the dog’s weight?
Answer:
100 pounds
Step-by-Step Explanation:
20+100=120
cat weighs 20
dog weighs 100
dog weighs 80 pounds more than the cat (20) = 100
Gina runs every 8 days, cycles every 16 days, and jogs every 24 days. If she did all three activities on Tuesday, in how many days and on what day of the week will she do all three activities again on the same day?
When she did all three activities on Tuesday, the number of days that Gina will do all three activities again on the same day is in 48 days.
How to illustrate the LCM?The least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, denoted by lcm in arithmetic and number theory, is the smallest positive integer that is divisible by both a and b.
LCM is an abbreviation for "Least Common Multiple." The smallest multiple that two or more numbers have in common is defined as the least common multiple.
Since Gina runs every 8 days, cycles every 16 days, and jogs every 24 days. This will be:
Multiples of 8 = 8, 16, 24, 32, 40, 48
Multiples of 16 = 16, 32, 48
Multiples of 24 = 24, 48
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Ten families were randomly selected and their daily water usage (in gallons) before and after viewing a conservation video was recorded. The sample mean difference was found to be 4.8 and the corresponding standard deviation 5.25, where differences were found (Before - After). Assume the data is normally distributed. Find a 90% confidence interval for this data. State the lower bound [x] and the upper bound [y].
lower bound = 1.76
Upper bounds = 7.84
The 90% confidence interval for the given data is 4.8 ± 2.73.
What is Confidence interval?A confidence interval is a type of interval computation used in statistics that determines the true value of an unknown parameter based on the observed data. The interval estimates the deterministic parameter with a level of confidence that is quantified by the confidence level.
Given, sample mean (x) = 4.8
z at 90% confidence level = 1.645
sample standard deviation (s) = 5.25
sample size (n) = 10
We know that, Confidence interval = x ± (z × s) ÷ √n
= 4.8 ± (1.645×5.25)/√10
= 4.8 ± 8.63625/3.1622
= 4.8 ± 2.73
∴ Lower bound = 4.8 - 2.73 = 2.07
and, Upper bound = 4.8 + 2.73 = 7.53
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When used inside a function with a parameter, an asterisk groups a variable number of positional arguments into a single A of parameter values
When used inside a function with a parameter, an asterisk groups a variable number of positional arguments into a single tuple of parameter values
Tuples are utilized to store different things in a solitary variable. A tuple is one of 4 implicit information types in Python used to store assortments of information, the other 3 are Rundown, Set, and Word reference, all with various characteristics and use. A tuple is an assortment that is requested and unchangeable. At the point when we say that tuples are requested, it implies that the things have a characterized request, and that request won't change. Python likewise incorporates the * administrator, which permits you to make another rundown with the components rehashed the predetermined number of times. *args permits us to pass a variable number of non-catchphrase contentions to a Python capability. In the capability, we ought to utilize a bullet ( * ) before the boundary name to pass a variable number of contentions.
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Subtract expression 2 from expression 1
Expression 1: -9x + 8y
Expression 2: -8x - 2y
Match these values of r with the accompanying scatterplots:
0.406,
−1,
0.748,
−0.748,
and
0.994.
Match the values of r to the scatterplots.
Scatterplot 1,
r=
▼
Scatterplot 2,
r=
▼
Scatterplot 3,
r=
▼
Scatterplot 4,
r=
▼
Scatterplot 5,
r=
▼
r = 1 in scatter plot 1, the response. Scatter plot 2, with a r of 0.364 Scatter plot 3, r=-0.995 Scatter plot 4, r = 0.995 Scatter plot 5, with a r of 0.735.
what is scatterplots ?Charts that depict the relationship between two variables are known as scatter charts, sometimes known as scatter plots. They are a very effective type of chart because they enable readers to see relationships or trends right away that are difficult to see in practically any other form.
You can figure out R using the formula R = s t if you've worked in sections. Between 1 and 1 will be the response you receive. Any response that is affirmative demonstrates a positive correlation, with anything over 0.7 often being regarded as a significant link.
r = 1 in scatter plot 1, the response. Scatter plot 2, with a r of 0.364 Scatter plot 3, r=-0.995 Spreadsheet plot 4, r = 0.995 Scatter plot 5, with a r of 0.735.
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Yanilen needs to buy 5/8 pounds of peanuts at the store. The scale only measures in 16ths. What fraction will be equivalent to 5/8 with a denominator of 16
The fraction that will be equivalent to 5/8 with a denominator of 16 is 10/16
How to determine the fraction that will be equivalent to 5/8?From the question, we have the following parameters that can be used in our computation:
Fraction = 5/8
This fraction is to be converted to a fraction with a denominator of 16
This is so because the scale only measures in 16ths
So, we have the following representation
Fraction = 5/8
When 8 is multiplied by 2, the result is 16
So, we multiply 5 and 8 by 16
This gives
Fraction = 10/16
Hence. the solution is 10/16
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Determine the polynomial y(x) that has roots at x=-0.7, x=0.5, x=3.4 and x=5.8
Make a plot of the polynomial in the domain -1<=x<=6
The polynomial with given roots is,
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902.
What is factor of polynomial?
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.
Given:
The polynomial y(x) that has roots at x = -0.7, x = 0.5, x = 3.4 and x = 5.8.
Since if 'a' is a root of the polynomial then (x - a) is the factor of the polynomial.
So, the factors of the polynomial are, (x + 0.7), (x - 0.5), (x - 3.4), (x - 5.8).
⇒ y(x) = (x + 0.7)(x - 0.5)(x - 3.4)(x - 5.8)
= (x^2 - 0.5x + 0.7x - 0.35)(x^2 - 5.8x - 3.4x + 19.72)
= (x^2 + 0.2x - 0.35)(x^2 - 9.2x + 19.72)
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902
The graph of the polynomial on the domain -1 ≤ x ≤ 6 is attached below.
Hence, the polynomial with given roots is,
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902.
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A family of statisticians is trying to decide if they can afford for their child to play youth baseball. The cost of joining a team is normally distributed with a mean of $750 and a standard deviation of $185. If a sample of 40 teams is selected at random from the population, select the expected mean and standard deviation of the sampling distribution below.oz= $185 oz= $29.25 oz = $4.63 nu2 = $118,59 0 nu2 = $18.75 nu2= $750
If a sample of 40 teams is selected at random from the population, the expected mean is $750 and the standard deviation is $29.25
The mean of the cost of joining a team = $750
Standard deviation = $185
A sample of 40 teams is selected at random from the population
The expected standard deviation = [tex]\sqrt{\frac{185^2}{40} }[/tex]
= $29.25
Here the cost of joining a team is normally distributed with a mean of $750
Here the distribution of the cost of joining is normal the mean of the sample distribution, so it will be equal to the mean of the population
The mean = $750
Therefore, the standard deviation is $29.25 and the mean is $750
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Silicone implant augmentation rhinoplasty is used to correct congenital nose
deformities. The success of the procedure depends on various biomechanical
properties of the human nasal periosteum and fascia. The article "Biomechnics in
Augmentation Rhinoplasty" reported that for a sample of 15 (newly deceased)
adults, the mean failure strain (%) was 25.0, and the standard deviation was 3.5.
(p. 277 #35)
a. Assuming a normal distribution for failure strain, estimate true average strain in
a way that converys information about precision and reliability.
b. Predict the strain for a single adult in a way that conveys information about
precision and reliability. How does the prediction compare to the estimate
calculated in part (a)?
A true average strain in a way that converys information about precision and reliability (23.0616, 26.9384)
In this question we have been given:
sample size n = 15
the mean failure strain x(bar) = 25.0
the standard deviation s = 3.5
Assuming a normal distribution for failure strain, we need to estimate true average strain in a way that converys information about precision and reliability.
A 95% confidence interval is necessary; other confidence intervals can be calculated in a similar way.
c = 0.95 or 95%
In the table with the critical values for t distributions in the appendix, locate the row beginning with degrees of freedom.
Here, the degree of freedom :
df = n - 1
df = 15 - 1
df = 14
and the column with α =(1-c)/2
=0.025
To find the t-value:
t_(α/2) = 2.145
Therefore, the error margin is:
E = t_(α/2) × s/√n
E = 2.145× 3.5/√15
E ≈1.9384
The confidence interval's outer limits are thus:
x{bar} - E = 25.0 - 1.9384
= 23.0616
x{bar} + E = 25.0 + 1.9384
= 26.9384
Therefore, the confidence interval is: (23.0616, 26.9384)
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Find the values of x and y.
I need help with this please
Answer:
The two sides of the triangle are the same as marked.
118+Y=180
Y=180-118=62
Y=62
X+Y+Y=180
X+2Y=180
X=180-2Y
X=180-2x62
X=56
Step-by-step explanation:
The function has a maximum point at (-pi/2, 5) and a minimum point at (pi/4, -4)
The sinusoidal function is y = -4.5cos(8(x + π/2)) + 1/2
How to determine the cosine function for the functionFrom the question, we have the following parameters that can be used in our computation:
Maximum = (-π/2, 5)
Minimum = (π/4, -4)
A cosine function is represented as
y = A cos(B(x+C)) + D
Where
A = amplitude, 2π/B = Period, C = phase shift leftD = vertical shift up.The cosine function has a max and min of 1 and -1.
So, we have
Difference = |-4 - 5|
Difference = 9
This means that
Amplitude, A = 9/2
A = 4.5
Also, we have
Vertically shift = (5 - 4)/2
Vertically shift = 1/2 up
So, the function becomes
y = 4.5cos(θ) + 1/2
Next, we calculate the value of θ
The period is calculated as
Period = π/4 + π/2
Period = -π/4
This means that
2π/B = -π/4
So, we have
B = -8
This gives
y = 4.5cos(-8(x + c)) + 1/2
cos(0) = 1,
So we want -8(-π/2) - 8C = 0
Solving for c, we have
8C = 8(π/2)
Divide
C = π/2
Substitute C = π/2 in y = 4.5cos(-8(x + c)) + 1/2
y = 4.5cos(-8(x + π/2)) + 1/2
y = -4.5cos(8(x + π/2)) + 1/2
Hence, the function is y = -4.5cos(8(x + π/2)) + 1/2
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Complete question
The function has a maximum point at (-pi/2, 5) and a minimum point at (pi/4, -4)
Determine a cosine equation for this function.
4. Complete: For the function f(x)=√x-1, the average rate of change to the nearest hundredth over the interval 3 ≤ x ≤ 6 is?
Answer:
the interval is positive
Can somebody help me I’m stuck on this one
Answer:
p=11
Step-by-step explanation:
59+8p+3p=180
59+11p=180
-59. -59
11p=121
/11. /11
p=11
hopes this helps
All of the following expressions have the same result, except
What operation is being done to the variable in the equation -5m = -40?
It's being multiplied by -5.
It's being multiplied by -40.
It's being divided by -5.
It's being divided by -40.
- ÷
The variable is m so it is being divided by 5
What is the arithmetic operations?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
We have,
-5m = -40
It's being multiplied by -5.
-5m = -40
-5m * (-5) = -40(-5)
25m = 200
m = 8
It's being multiplied by -40.
-5m * (-40)= -40 * (-40)
200m = 1600
m = 8
it's being divided by -5
-5m= -40
m = 8
It's being divided by -40.
-5m / (-40)= -40 / (-40)
m = 8
Hence, The variable is m so it is being divided by 5
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the statistical method used to predict values of one numerical variable based on the values of another is___
The statistical method used to predict the values of one numerical variable based on the values of another is regression analysis.
Regression analysis:
Regression analysis is a statistical method that shows the relationship between two or more variables. Usually expressed in a graph, the method tests the relationship between a dependent variable against independent variables.
Regression analysis is used to estimate the relationship between a dependent variable and one or more independent variables.
The common form of regression analysis is linear regression. Regression analysis allows us to infer that there is a relationship between two or more variables.
Therefore regression method is used to predict values of one numerical on another.
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Find the measure of the arc or central angle indicated.
The central angle of the given arc LKJ is; 211°
How to find the length of an arc of a circle?
We want to find the length of the arc LKJ. The central angle for this arc length is; 115° + 96° = 211°
The formula for length of arc is;
L = θ/360 * 2πr
where;
θ is central angle
r is radius
We are not given the radius and as such, we can only find the central angle which is 211°
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find the slope of the question below
Answer:6/3 or 1/2
Step-by-step explanation:
what is the value of the function at x=−2?
Answer: -2
Step-by-step explanation:
Draw that point of intersection. Similarly construct a horizontal line which goes through the point of intersection. The value of the function here at x = 2 is -2.
