For values of n other than 0 or 1 in a Bernoulli differential equation, the substitution [tex]u = y^{(1-n)[/tex] is used to transform it into a linear equation.
A Bernoulli differential equation is given by the form:
dy + P(x)y dx = Q(x)[tex]y^n[/tex] (*)
If we consider the case when n = 0 or n = 1, the Bernoulli equation becomes linear. Let's examine each case:
When n = 0:
Substituting[tex]u = y^{(-n) }= y^{(-0)} = 1[/tex], the differential equation becomes:
[tex]dy + P(x)y dx = Q(x)y^0[/tex]
dy + P(x)y dx = Q(x)
This is a linear differential equation of the first order.
When n = 1:
Substituting [tex]u = y^{(-n) }= y^{(-1)},[/tex] we have:
[tex]u = y^{(-1)[/tex]
Taking the derivative of both sides with respect to x:
[tex]du/dx = -y^{(-2)} \times dy/dx[/tex]
Rearranging the equation:
[tex]dy/dx = -y^2\times du/dx[/tex]
Now substituting the expression for dy/dx in the original Bernoulli equation:
[tex]dy + P(x)y dx = Q(x)y^1\\-y^2 \times du/dx + P(x)y dx = Q(x)y\\-y \times du + P(x)y^3 dx = Q(x)y[/tex]
This equation is also a linear differential equation of the first order, but with the variable u instead of y.
In summary, when n is equal to 0 or 1, the Bernoulli equation becomes linear. For other values of n, a substitution u = y^(-n) is typically used to transform the Bernoulli equation into a linear differential equation, allowing for easier analysis and solution.
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Find the formula for the nth term in this arithmetic sequence a1=0 a2=.5 a3=1 a4=1.5
Answer:
Step-by-step explanation:
The arithmetic sequence usually write as [tex]a_{n}=a_{1} +(n-1)d[/tex], where a1 is the first term and d is a common difference.
[tex]a_{n} = a_{1}+(n-1)d\\a_{n} = 0+(n-1)0.5\\[/tex]
Calculate the integral ∫ [(1+cos(arctan (3x))+(arctan (3x))^5)/(1+9x^2]dx.
The solution of the integral (1/6)ln|9x²+1| +(1/3)arctan (3x) +(1/6)arctan((3x)²)- (1/18)(3x)²/(9x²+1) + (1/90)(3x)⁴/(9x²+1) + C.
We have the following integral to solve.∫ [(1+cos(arctan (3x))+(arctan (3x))⁵)/(1+9x²2)]dx
Let's begin by using the substitution u = arctan(3x).
We need to convert this integral into a U-form. By substituting the above expression, we have the following new expression.
∫ [(1+cos u)+u⁵]/[1+9(3tanu)²] *(1/3) du=∫ [(1+cos u)+u⁵]/(9tan²u+1) *(1/3) du
Let's now consider the numerator.
We have 1+cosu+u⁵ in the numerator. Let's break down the integral into 3 parts. The first part is ∫(1)/(9tan²u+1) *(1/3) du.
We can convert the denominator to a perfect square of the form 3²tan²u+1 which simplifies to (3tanu+1)(3tanu-1)+2.
Let's now factorize
= ∫(1)/(9tan²u+1) *(1/3) du
=∫(1)/[(3tanu+1)(3tanu-1)+2] *(1/3) du
Now let's perform partial fraction decomposition. We have 1/[(3tanu+1)(3tanu-1)+2] which we can write as
= A/(3tanu+1)+B/(3tanu-1)+C/(3tan²u+1) where A, B, and C are constants to be determined.
We now have 1= A(3tanu-1)(3tan²u+1)+ B(3tanu+1)(3tan²u+1)+ C(3tanu+1)(3tanu-1).
Let's now set up the system of equations.
The first equation is
1= A(3tanu-1)(3tan²u+1)+ B(3tanu+1)(3tan²u+1)+ C(3tanu+1)(3tanu-1)
Putting u=0, we get
A = 1/4
Putting u =pi/3, we get
B=1/4 Putting
u = arctan(1/√(3)), we get C=1/2. Now that we have A, B, and C, let's substitute them back to our integral.
We have
= ∫(1)/[(3tanu+1)(3tanu-1)+2] *(1/3) du
= (1/4)∫(1)/(3tanu+1) du + (1/4)∫(1)/(3tanu-1) du +(1/2)∫(1)/(3tan²u+1) du
Let's now solve the integral and simplify.
= ∫(1)/(3tanu+1) du
= (1/6)ln|3tanu+1|- (1/6)ln|3tanu-1|+ (1/3)arctan (3tanu).
Similarly,
= ∫(1)/(3tanu-1) du
= (-1/6)ln|3tanu+1|+ (1/6)ln|3tanu-1|+ (1/3)arctan (3tanu).
The third integral we can solve by substitution
v = √(3)tanu.
We then get
= ∫(1)/(3tan²u+1) du= (1/√(3))∫(1)/(v²+3) dv
= (1/sqrt(3))arctan(v/√(3))
= (1/√(3))arctan((√(3)tanu)/√(3)).
We can now put back all the values to get the solution.
= ∫ [(1+cos(arctan (3x))+(arctan (3x))⁵)/(1+9x²)]dx
= (1/3)∫ [(1+cos u)+u⁵]/(9tan²u+1) du
=(1/3)[(1/4)(ln|3tanu+1|- ln|3tanu-1|+ 2arctan (3tanu))+(1/4)(-ln|3tanu+1|+ ln|3tanu-1|+ 2arctan (3tanu))+(1/2)(1/√(3))(arctan((sqrt(3)tanu)/√(3)))]+ C
We can simplify the above expression to get
=- ∫ [(1+cos(arctan (3x))+(arctan (3x))^5)/(1+9x²)]dx
= (1/6)ln|9x²+1| +(1/3)arctan (3x) +(1/6)arctan((3x)²)- (1/18)(3x)²/(9x²+1) + (1/90)(3x)⁴/(9x²+1) + C
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six and fifty-three hundredths as a decimal-
Yes, I'm D-U-M-B!
Answer:
6.58
Step-by-step explanation:
ahahahaha lol
Answer:
6.58
i hope this helps:)
Step-by-step explanation:
brainliest please?
How do you find the greatest common denominator, when comparing two numbers? Explain your thought process, in the example I provided. What is the greatest common Denominator for the numbers 20 and 40. Explain each step you do. Your answer will start with these words. "How to find the the greatest common denominator between 20 and 40?
Answer:
20 is the greatest common denominator. You can find this by listing the factors of each number. The highest number that can fit into both individual numbers is the common denominator. Since 40 is just 20 doubled, and 20 is just 20 once, it fits into both numbers and is also the highest number that can fit.
Gerry conversed his sole partnership to an S corporation and transferred several assets to the corporation. The assets cost $19,570 and had an adjusted basis of $9,600. He also spent an additionally $975 to make the conversion to an S corporation. What is his beginning basis in the S corporation?
Gerry's beginning basis in the S corporation is $20,545.
To calculate Gerry's beginning basis in the S corporation, we need to consider the cost of the assets transferred and the additional expenses incurred for the conversion.
The cost of the assets transferred is given as $19,570. This represents the original cost of the assets when Gerry acquired them.
The adjusted basis of the assets is stated as $9,600. The adjusted basis takes into account any adjustments made to the original cost, such as depreciation or other deductions.
To calculate the beginning basis in the S corporation, we add the cost of the assets transferred ($19,570) to the adjusted basis ($9,600). This gives us a total of $29,170.
In addition to the assets, Gerry incurred an additional expense of $975 for the conversion to an S corporation. We include this amount in the calculation of the beginning basis.
Therefore, Gerry's beginning basis in the S corporation is $29,170 + $975 = $20,545.
This beginning basis is important for determining Gerry's tax consequences and future deductions within the S corporation structure.
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Jim and three friends shared 2 poster boards for an art project. What part of the construction paper will each friend get? *
Answer: 1/2 of a sheet
Step-by-step explanation:
2 sheets divided by 4 people
2/4
Simplify to 1/2
Write the word sentence as an equation. Then solve the equation.
9 is the difference of a number n and 7.
An equation that represents this word sentence is
Answer: 9=n-7
Step-by-step explanation:
How do I solve for the missing side lengths. My sender has to be as a radical in simplest form.
Answer: 6√6
Step-by-step explanation: When an angle is 45 degrees in a triangle it is actually half a square (I know right!!!), so, therefore, the other side length will also be 6, and then using the Pythagorean theorem we get the hypotenuse to be 6√2. Now, when an angle is 60 degrees in a triangle it is actually half an equilateral triangle (I know right!!!), so, therefore, the other side length will be double the length, 12√2, and using the Pythagorean theorem we get the side length x to be 6√6! Hope this helps!
The proportion of students who fail the comprehensive statistics final had been 4%. With instruction moving completely online, instructors believe that this percentage will decrease for various reasons (the final must now be open book, some students will cheat, etc.). To test their claim, the appropriate test is
paired t test
z test for two proportions
t test for one mean
z test for one proportion
one-way ANOVA
The appropriate test to investigate whether the proportion of students failing the comprehensive statistics final has decreased with the shift to online instruction is the z test for two proportions.
Explanation:
The z test for two proportions is used when we want to compare two proportions or percentages from different populations or groups. In this case, we want to compare the proportion of students who fail the statistics final before and after the instruction moved online.
To conduct the z test for two proportions, we follow these steps:
1. Define the null hypothesis (H₀) and alternative hypothesis (H₁). In this case, the null hypothesis would be that the proportion of students failing the final is the same before and after the instruction moved online, while the alternative hypothesis would be that the proportion has decreased after the shift to online instruction.
2. Collect data on the number of students who fail the statistics final before and after the shift to online instruction.
3. Calculate the sample proportions for each group. Let's denote the proportion before the shift as p₁ and after the shift as p₂.
4. Calculate the standard error (SE) of the difference between two proportions using the formula:
SE = sqrt((p₁ * (1 - p₁) / n₁) + (p₂ * (1 - p₂) / n₂))
where n₁ and n₂ are the sample sizes for the two groups.
5. Calculate the test statistic (z) using the formula:
z = (p₁ - p₂) / SE
6. Determine the critical value for the desired significance level (e.g., 0.05) and compare it to the calculated test statistic. If the calculated test statistic falls within the critical region, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
7. Interpret the results and draw conclusions about whether there is sufficient evidence to support the claim that the proportion of students failing the statistics final has decreased after the shift to online instruction.
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Please help !
50 points !
Use the quadratic regression feature of a graphing calculator to find a quadratic model. Round to the nearest hundredths place.
A. Y=0.06x^2+0.31+4
B. Y=-4.03x^2+0.32x+8.19
C. Y=0.06x^2-0.31x-4
D. Y=4.03x^2-0.32x-8.19
Answer:
It is A
Step-by-step explanation:
Answer:
its A he's right
Step-by-step explanation:
The quotient of a number more than 1 and 3 is 5
Answer:
https://brainly.com/question/13518166
Step-by-step explanation:
3. A runner sprints 50 ft each day for 7 days. How many inches does she sprint during this week? (Again please hurry)
Answer: The runner will sprint 4,200 inches during this week
Step-by-step explanation:
50 * 7 = 350
350 * 12 = 4200
Answer:
It's 4200in
in feets its 350ft
Step-by-step explanation:
Please please help please please ASAP ASAP please ASAP help please please ASAP please please help ASAP ASAP please please help please please ASAP please
Answer:
7
Step-by-step explanation:
The two lengths (4x-7 and x+14) are the same. This is because if we look at the big trapezoid as a whole, the ratio of the three segments on the side (4, 4, 4) is the same as the ratio of the three lengths on the other side. Therefore the ratio of the two lengths would be the same as well, 1:1. (btw the photo that I attached is what I mean by the two sides). So, we can form the equation: 4x-7 = x+14. The steps to solve the problem:
4x-7 = x+14
3x=21
x=7
The value of x is 7!
Solve.
t
8.4=
1.5
=
Help plzzzz
Answer:
t = 12.6
Step-by-step explanation:
8.4 x 1.5 = 12.6
to check it, do 12.6 ÷ 1.5 = 8.4
1 Consider a tournament between N teams, each team playing each of the other teams. We can represent the results of this tournament by a directed graph: node i represents team i, and an edge exists i+jif team i beat team j. The above problem suggests that it may be impossible to declare an absolute winner, as everyone may be beaten by somebody. We could relax this slightly in the following way: let's call team i a k-winner if there is a group of k-many teams that were each beaten by team i. Other teams may have beaten team i, but there is at least a group of size k that was roundly beaten by i. 3) If the results of each game are decided by fair coin flip, what is the probability that a given team i is a k-winner? (5 points) 4) Using result 3.0, bound the probability that there exists a k-winner in a tournament of size N? Write it nicely as you can but don't beat yourself up too much with it. (5 points) 5) For N = 100, what is the smallest k that 3.4 indicates the probability of having k-winners is less than 1? Code or Mathematica to evaluate your answer in 3.4 is fine. (5 points) 6) For N = 100 and k as in 3.5, argue that there exist possible tournaments with no k-winners. (5 points) Bonus: Using the Chernoff bound to bound the relevant probabilities, show that for a > 1/2 the probability of there being any oN-winners goes to 0 as No. Conclude therefore that there erist tournaments without a N-winners, for all sufficiently large N. (10 points)
The analysis involves calculating probabilities, bounding the probabilities based on worst-case scenarios, and considering the implications of the Chernoff bound to demonstrate the existence of tournaments without a certain number of winners.
In a tournament between N teams, where each team plays each other, the concept of a k-winner is introduced. A team i is considered a k-winner if there is a group of k teams that were each beaten by team i. The probability that a given team i is a k-winner can be calculated when the results of each game are decided by a fair coin flip. This probability depends on the number of teams and the specific value of k.
Using this probability for a given team, we can bound the probability that there exists a k-winner in a tournament of size N. This bound takes into account the probabilities of individual teams being k-winners and considers the worst-case scenario.
For a specific value of N, such as N = 100, we can determine the smallest k for which the probability of having k-winners is less than 1. This can be done by evaluating the probabilities for different values of k until the desired condition is met.
When N = 100 and k is determined as in the previous step, it can be argued that there exist possible tournaments with no k-winners. This can be demonstrated by constructing specific scenarios or by analyzing the probabilities of different outcomes.
Using the Chernoff bound, it can be shown that for a value a greater than 1/2, the probability of having any oN-winners (where oN is an arbitrary constant) goes to 0 as N approaches infinity. This conclusion implies that there exist tournaments without N-winners for sufficiently large values of N.
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Find the volume of the cylinder. Use 3.14 for T.
height of 3 radius of 3
Answer:
84.82
Step-by-step explanation:
π·32·3≈84.823
Answer:
The volume of a cylinder with a height of 3 and a radius of 3 is 84.823 or you could say 84.8.
Step-by-step explanation:
Please help me in it! It's very difficult, i'm in 6th and I still don't understand this. Please, help me in this!!!
Answer:
40
where the line starts (75) and ends (115)
115-75=40
thats the quick way to solve... hope this helps!
Step-by-step explanation:
Brainliest?
find the slope. -2,1 and 4,4
Answer:
2
Step-by-step explanation:
To find the slope of two points, we must subratct the Y values from each other and divide them from the X values
4-(-2) 6
--------- = ----- = 2
4-1 3
Answer: The slope is 1/2 as a fraction and 0.5 as a decimal
Hope it helped :D
plz answer I will mark brainlest
Answer:
(-3,0)
Step-by-step explanation:
y=-2x-6 and 2y=-4x-12
=-2x-6
Solution is
(-3,0)
Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions
To find the first five terms of a sequence defined by a recurrence relation and initial conditions, we apply the relation repeatedly. Each term is calculated based on the previous terms in the sequence according to the given relation.
The first five terms of a sequence defined by a recurrence relation and initial conditions can be determined by applying the recurrence relation repeatedly. Each term is calculated based on the previous terms in the sequence, according to the given relation. Here are the first five terms for each recurrence relation:
1. Linear recurrence relation: If the recurrence relation is of the form an = an-1 + d, where "a" is the term and "d" is a constant, and the initial condition is a1 = c, then the first five terms can be found by adding the constant "d" repeatedly to the initial term "c". For example, if c = 2 and d = 3, the first five terms would be 2, 5, 8, 11, 14.
2. Quadratic recurrence relation: If the recurrence relation is of the form an = an-1² + c, where "a" is the term and "c" is a constant, and the initial condition is a1 = d, then the first five terms can be calculated by squaring the previous term and adding the constant "c". For instance, if d = 1 and c = 2, the first five terms would be 1, 3, 11, 123, 15187.
These examples demonstrate how to find the first five terms of a sequence by applying the given recurrence relation and initial conditions. By following the recurrence relation, each term is determined based on the previous terms in the sequence.
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A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal distribution with mean μ=298ml and standard deviation σ=3ml. What is the probability that the mean contents of six randomly selected bottles are less than 295 ml?
The probability that the mean contents of six randomly selected bottles are less than 295 ml is 0.007, or 0.7%.
To find the probability that the mean contents of six randomly selected bottles are less than 295 ml, we can use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means from a population with any distribution approaches a normal distribution as the sample size increases.
In this case, we have a normal distribution with a mean (μ) of 298 ml and a standard deviation (σ) of 3 ml. We want to find the probability of the mean contents of six bottles being less than 295 ml.
First, we need to calculate the standard error of the mean, which is the standard deviation of the sampling distribution of the mean. The formula for the standard error is σ / √n, where σ is the population standard deviation and n is the sample size.
In our case, σ = 3 ml and n = 6. Therefore, the standard error (SE) is:
SE = 3 / √6 ≈ 1.225 ml
Next, we need to calculate the z-score, which is the number of standard errors the sample mean is away from the population mean. The formula for the z-score is (x - μ) / SE, where x is the sample mean.
In our case, x = 295 ml, μ = 298 ml, and SE = 1.225 ml. Therefore, the z-score is:
z = (295 - 298) / 1.225 ≈ -2.449
Finally, we can use a standard normal distribution table or a calculator to find the probability corresponding to the z-score. The probability that the mean contents of six bottles are less than 295 ml is the area under the normal curve to the left of the z-score.
Using a standard normal distribution table or calculator, we find that the probability corresponding to a z-score of -2.449 is approximately 0.007.
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сть
b
d
a
If mza = 2.m2b, mzc = 60', and the right angles are labeled in the figure, which of the following
produce an acute angle? Select all that apply
Answer:
a and c
Step-by-step explanation:
find the area of the triangle
how do you find the radius when given the volume and height?
Answer:
Calculate the area of the base (which is a circle) by using the equation πr² where r is the radius of the circle. Then, multiply the area of the base by the height of the cylinder to find the volume. SOO just remember to divide the diameter by two to get the radius. If you were asked to find the radius instead of the diameter, you would simply divide 7 feet by 2 because the radius is one-half the measure of the diameter. The radius of the circle is 3.5 feet. You can also use the circumference and radius equation.
Step-by-step explanation:
brainliest?
Answer ASAP pls pls
Answer:
67.5
Step-by-step explanation:
b times h
7.5 times 9
Answer:
A = 67.5 in²
Step-by-step explanation:
The area (A) of a parallelogram is calculated as
A = bh ( b is the base and h the perpendicular height )
Here b = 9 and h = 7.5 , then
A = 9 × 7.5 = 67.5 in²
A scientist used a microscope to count the number of bacterial cells in a petri dish every hour.
Which function most accurately represents this data, where n is the number of bacteria and 1 is the time elapsed, in hours?
a. n=2t
b. n=2^t
c n=t^2
d. n=√t
Answer:
a. n=2t
Step-by-step explanation:
Given that
N denotes the number of bacteria
And, 1 denotes the time elasped in hours
Now based on this, the scientist used a microscope for counting the number of bacterial cells every hour
So,the function that represent correct data is
n = 2t
Therefore the correct option is A.
The same would be considered
find the buying price if the selling price is 240 and profit is 20 percent
Answer:
192
Step-by-step explanation:
You have to find 20% of 240.
So write 20% as 20 over 240 in fraction form.
[tex]\frac{20}{240}[/tex]
Since, finding the fraction of a number is same as multiplying the fraction with the number, we have
[tex]\frac{20}{100}[/tex] of 240 = [tex]\frac{20}{100}[/tex] × 240
Therefore, the answer is 48
If you are using a calculator, simply enter 20÷100×240 which will give you 48 as the answer.
Now since it wants to know how much you paid for the goods you do 240-48=192
So since a company is selling goods with a 20% profit
That means they bought it from somewhere else
Because they had to be manufactured
It wants how much the company paid the company that manufactured the goods
I hope this helped. Some resources that are helpful to look at in case of any more confusion are:
https://answers.everydaycalculation.com/percent-of/20-240
https://www.mathsisfun.com/
https://www.ixl.com/
Good luck young mathematician!
A certain airplane offers two types of seats, first class and economy.
There are 209 total seats on the airplane. If the difference between the
number of economy and first class seats is 153, find the number of
economy seats.
Answer:
56
Step-by-step explanation:
you do 209-153=56
Number of economy seats are 181
What is linear equation?An equation between two variables that gives a straight line when plotted on a graph
Consider,
Number of economy seats =x
Number of First class seats =y
There are 209 total seats on the airplane
Then,
x + y=209
Difference between the number of economy and first class seats is 153
x - y=153
Solve both the equation
x + y=209
x = 209-y
Substitute the value of x in second equation
(209-y) - y=153
2y=209-153=56
y=28
Then
x + y=209
x + 28=209
x=209-28
x=181
Hence, the number of economy seats are 181
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Can someone help me please ASAP? Correct answer gets brainliest!! :)
Answer:
Blank 1: 6
Blank 2: 30
1/2 = 6/12 = 30/60
Blank 3: 30
1/4 = 3/12 = 15/60
Use the Linear Approximation to estimate Δꜰ=ꜰ(3.5)−ꜰ(3) ꜰᴏʀ ꜰ(x)=41+x2 (Use decimal notation. Give your answer to five decimal places.)
Δf≈ help (decimals)
Calculate the actual change.
(Use decimal notation. Give your answer to five decimal places.)
Δf = help (decimals)
Compute the error and the percentage error in the Linear Approximation.
(Use decimal notation. Give your answer to five decimal places.)
Error = help (decimals)
Percentage error = % help (decimals)
To estimate Δf = f(3.5) - f(3) using the linear approximation, we'll use the formula:
Δf ≈ f'(a) * Δx
where f'(a) represents the derivative of f at the point a, and Δx represents the change in the x-values.
Given that f(x) = 41 + [tex]x^2[/tex], we can calculate the derivative as:
f'(x) = 2x
Now, let's calculate the values step by step:
Calculate Δf:
Δf ≈ f'(a) * Δx
Δf ≈ f'(3) * (3.5 - 3)
Δf ≈ 2(3) * (3.5 - 3)
Δf ≈ 6 * 0.5
Δf ≈ 3
Calculate the actual change:
To calculate the actual change, we need to evaluate f(3.5) and f(3) separately:
f(3.5) = 41 +[tex](3.5)^2[/tex]
f(3.5) = 41 + 12.25
f(3.5) = 53.25
f(3) = 41 + [tex](3)^2[/tex]
f(3) = 41 + 9
f(3) = 50
Δf = f(3.5) - f(3)
Δf = 53.25 - 50
Δf = 3.25
Calculate the error and the percentage error:
Error = |Δf - Δf_approx|
Error = |3.25 - 3|
Error = 0.25
Percentage error = (|Δf - Δf_approx| / Δf) * 100
Percentage error = (0.25 / 3.25) * 100
Percentage error ≈ 7.69%
So, the results are as follows:
Δf ≈ 3
Actual change (Δf) ≈ 3.25
Error ≈ 0.25
Percentage error ≈ 7.69%
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