Evaluate the expression.

The value of f(2π) is:π + 2√(2π).

The given differentiable function is: f′(x) = sin²(x) + x^(-1/2)

Given that: f(0) = 5.420

To find:f(2π)

The function is differentiable.

Therefore, f(x) must be continuous.

Let's first integrate the derivative of the function.

∫f′(x) dx = ∫sin²(x) + x^(-1/2) dx

∫sin²(x) dx = x/2 - (sin x cos x)/2 = (x - sin x cos x)/2

∫x^(-1/2) dx = 2x^(1/2) = 2√x

The integral is equal to: f(x) = (x - sin x cos x)/2 + 2√x

Now we need to substitute x with 2π:

f(2π) = [(2π - sin(2π) cos(2π))/2] + 2√(2π)

f(2π) = [(2π - 0 x (-1))/2] + 2√(2π)

f(2π) = [π + 2√(2π)]

Therefore, the value of f(2π) is:π + 2√(2π).

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The function f(x,y) = 6 + x ln(xy-7) is differentiable at the point (4,2).

We can find the partial derivative fx(x,y) by applying the chain rule of differentiation to the function f(x,y) = 6 + x ln(xy-7), as follows:

fx(x,y) = ln(xy-7) + x(1/(xy-7))(ydx/dx)

= ln(xy-7) + 1/(y-7)*x

where dx/dx = 1 is the derivative of x with respect to itself. Similarly, the partial derivative fy(x,y) can be obtained as:

fy(x,y) = x(1/(xy-7))(xdy/dy)

= x/(xy-7)

where dy/dy = 1 is the derivative of y with respect to itself.

To show that fx and fy are continuous at the point (4,2), we need to evaluate them at that point and show that the resulting values are finite. Substituting x = 4 and y = 2 into the equations for fx and fy, we get:

fx(4,2) = ln(1) + 1/(2-7)4 = -4/5

fy(4,2) = 4/(42-7) = -4/3

Since both fx(4,2) and fy(4,2) are finite, we can conclude that the partial derivatives of f exist and are continuous at (4,2).

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Complete Question:

Explain why the function is differentiable at the given point.

f(x, y) = 6 + x ln(xy − 7), (4, 2)

The partial derivatives are fx(x, y) =

and fy(x, y) =

Sonali purchased some pants and skirts the numbers of skirts is 7 less than eight times the number of pants purchase also number of skirt is four less than five times the number of pants purchased is 1 pant and 1 skirt.

Let's denote the number of pants Sonali purchased as P and the number of skirts as S. We're given two pieces of information:

1. The number of skirts (S) is 7 less than eight times the number of pants (8P). This can be represented as S = 8P - 7.

2. The number of skirts (S) is also 4 less than five times the number of pants (5P). This can be represented as S = 5P - 4.

Now we have a system of two linear equations with two variables, P and S:

S = 8P - 7
S = 5P - 4

To solve the system, we can set the two expressions for S equal to each other:

8P - 7 = 5P - 4

Solving for P, we get:

3P = 3
P = 1

Now that we know P = 1, we can substitute it back into either equation to find S. Let's use the first equation:

S = 8(1) - 7
S = 8 - 7
S = 1

So, Sonali purchased 1 pant and 1 skirt.

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any finite set is countable yes  (a) True

A finite set is countable because it has a specific number of elements that can be put into one-to-one correspondence with a subset of natural numbers. In other words, you can assign a unique natural number to each element in the finite set without running out of natural numbers to assign.


1. A finite set has a limited number of elements, meaning there is an exact count for the elements in the set.
2. A countable set is a set whose elements can be put into one-to-one correspondence with a subset of natural numbers (0, 1, 2, 3, ...).
3. Since a finite set has a specific number of elements, we can assign a unique natural number to each element without running out of numbers.
4. This one-to-one correspondence between the elements in the finite set and the natural numbers demonstrates that the set is countable.

Finite sets are countable because they contain a specific, limited number of elements that can be matched up with a subset of natural numbers. By establishing a one-to-one correspondence between the elements in the finite set and natural numbers, we can effectively count the elements in the set, making it countable.

Any finite set is countable because its elements can be placed in one-to-one correspondence with a subset of natural numbers, allowing for the set to be effectively counted.

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We are asked to compute the matrix U, formed by concatenating u1 and u2 as columns, and to compute U'U and UUT. Additionally, we are asked to compute the projection of y onto the subspace spanned by u1 and u2, as well as (uuT)y and (UU)y.

We can compute the matrix U by concatenating u1 and u2 as columns. Thus, we have:

U = | 2/3 -2/3 |

| 2/3 1/3 |

| 1/3 2/3 |

Next, we can compute U'U and UUT as follows:

U'U = | 2 0 |

| 0 2 |

UUT = | 8/9 4/9 2/9 |

| 4/9 4/9 4/9 |

| 2/9 4/9 8/9 |

For the second part of the problem, we can compute the projection of y onto the subspace spanned by u1 and u2 using the formula,

[tex]projwy[/tex]= (y'u1/u1'u1)u1 + (y'u2/u2'u2)u2. Plugging in the given values, we get:

[tex]projwy[/tex]= | 22/9 |

| 20/9 |

| 4/9 |

We can also compute [tex](uuT)y[/tex]and (UU)y as follows:

[tex](uuT)y[/tex]= [tex]uuT y[/tex]= | 10 |

| 0 |

| 0 |

(UU)y = UU (4 5 1)' = | 14 |

| 14 |

| 7 |

We also computed the projection of y onto the subspace spanned by u1 and u2, as well as [tex](uuT)y[/tex] and (UU)y.

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Answer:

In ascending order: 1/2, 3/5, 6/11, 8/9

The train is travelling at an initial velocity of 15 m/s. After 8 seconds, the train is travelling at a final velocity of 30 m/s. We need to calculate the acceleration of the train during this time period.

The formula for acceleration is given by the equation a = (v_f - v_i) / tWhere a is the acceleration, v_f is the final velocity, v_i is the initial velocity and t is the time taken .So, substituting the values we have: [tex]a = (30 - 15) / 8a = 1.875 m/s^2[/tex]Therefore, the acceleration of the train is [tex]1.875 m/s^2[/tex].The train is accelerating at a rate of 1.875 m/s^2. This means that every second, the train is increasing its velocity by 1.875 m/s. If the train continues to accelerate at this rate, it will reach a velocity of 60 m/s in 24 seconds.

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Answer:

13x

Step-by-step explanation:

+3y and -3y cancel out

Therefore, we have 4x+9x

13x

Answer:

C

Step-by-step explanation:

the sum of the interior angles of a polygon is

sum = 180° (n - 2) ← n is the number of sides

a pentagon has 5 sides , that is n = 5

sum = 180° × (5 - 2) = 180° × 3 = 540°

since the 5 angles are congruent then divide the sum by 5 , that is

∠ F = 540° ÷ 5 = 108°

Step-by-step explanation:

Formula of calculating total angles with n side (Polygon) : (n-2) . 180°

total pentagon angles :

= (5 - 2) . 180

= 3 . 180

= 540°

all of the angle is congruent, then :

m<F = 540/5

m<F = 108° (C)

Subject : Mathematics

Level : JHS

Chapter : Geometry

The maximum possible error is 2/3.

The Maclaurin series for cosine function is given by:

[tex]cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...[/tex]

Using the first three nonzero terms, we get:

[tex]cos(x) ≈ 1 - x^2/2! + x^4/4![/tex]

To estimate the error, we can use the Taylor remainder formula:

[tex]Rn(x) = f(n+1)(c) * (x-a)^(n+1) / (n+1)![/tex]

where f(n+1)(c) is the (n+1)th derivative of f evaluated at some value c between a and x.

In this case, we have:

f(x) = cos(x)

a = 0

n = 2

x = 2

To find an upper bound for the error, we need to find the maximum value of the absolute value of the third derivative of cosine function over the interval [0,2]. Since the third derivative of cosine is -cos(x), the maximum value of its absolute value is 1.

Therefore, we have:

[tex]|R2(2)| ≤ 1 * (2-0)^(2+1) / (2+1)![/tex]

≤ 4/3!

≤ 2/3

So the maximum possible error is 2/3.

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If Joe’s marginal cost was $40 per additional hour, it would make sense for him to keep the restaurant open longer for a maximum of 2 hours because after this point the marginal cost exceeds the marginal benefit.

Explanation:Marginal cost is the additional cost of producing an extra unit of output while marginal benefit is the additional benefit gained from producing an extra unit of output.

To maximize profits, businesses should continue producing units of output until the marginal cost equals the marginal benefit.The question states that Joe’s marginal cost is $40 per additional hour. This implies that for every additional hour the restaurant is kept open, it would cost Joe $40. In order to decide if it is economically beneficial to keep the restaurant open longer, Joe would need to compare the marginal cost with the marginal benefit.

If Joe’s marginal benefit is higher than his marginal cost, then it would make sense for him to keep the restaurant open longer. However, if his marginal cost is higher than his marginal benefit, then it would not be economical to keep the restaurant open longer.

The opportunity cost of an economic decision is the next best alternative foregone. In this case, Joe would need to consider what he would have gained or lost if he did not keep the restaurant open for an additional hour.

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The maximum number of barrels per day imported in september 2003 was 1.72 million

To find the year when oil imports were greatest, we need to find the maximum value of the function I(t) = -0.015t^2 + 0.1t + 1.4, where t is in years since the start of 2000.

The maximum value of a quadratic function occurs at the vertex, which has x-coordinate equal to -b/2a for a function in the form [tex]ax^2 + bx + c.[/tex]For this function, a = -0.015 and b = 0.1, so the x-coordinate of the vertex is:

x = -b/2a = -0.1 / (2*(-0.015)) = 3.33

Since t is in years since the start of 2000, the year when oil imports were greatest is 2003.33 (or approximately September 2003).

To find the number of barrels per day imported that year, we can simply plug in t = 3.33 into the function I(t):

[tex]I(3.33) = -0.015(3.33)^2 + 0.1(3.33) + 1.4[/tex]= 1.72 million barrels per day

Therefore, the maximum number of barrels per day imported was approximately 1.72 million, and this occurred in September 2003.

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Answer: I know why its because

you see your acount on the right corner and you see how many crowns and points you have welll you need this all the way filled up the thing around your name like mine its almost full

Step-by-step explanation:

Answer:

A'(-8,6)

B'(6,4)

C'(-8,0)

Step-by-step explanation:

 Since this transformation is a violation we can we can use the skill factor to multiply both the x-coordinate and the Y And the Y coordinate of the Of the point to make sure that everything is consistent and that the figures stay similar.

The value of vector A is approximately 17.08.

To find the magnitude of a vector with components = 4.43 and = -16.5, you can use the Pythagorean theorem.

The formula for the magnitude of a vector (|A|) is:

|A| = √(x² + y²)

In this case, x = 4.43 and y = -16.5.

Plugging these values into the formula, you get:

|A| = √((4.43)² + (-16.5)²)

|A| = √(19.5849 + 272.25)

|A| = √(291.835)

Calculating the square root, you find that the magnitude of vector A is approximately 17.08.

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To find the area between two z-scores on a calculator, we use the command "normalcdf" on most scientific calculators.

This command calculates the area under the normal distribution curve between two specified z-scores. We need to input the two z-scores and the mean and standard deviation of the normal distribution, which can be obtained from the problem statement or by calculating them from the given data.

Another command that is used in conjunction with "normalcdf" is "invNorm". This command can be used to find the z-score corresponding to a given area under the normal distribution curve. It is used when we are given the area and we need to find the corresponding z-score.

Together, these two commands are useful for solving problems that involve normal distributions, such as finding probabilities, finding critical values, or constructing confidence intervals. It is important to understand how to use these commands properly in order to perform accurate and efficient calculations.

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Triggering the secretion of vasopressin is not a mechanism by which angiotensin II elevates ECF volume.

Angiotensin II is a hormone that plays a crucial role in regulating blood pressure and fluid balance in the body. It is produced by the renin-angiotensin-aldosterone system (RAAS) in response to low blood pressure or decreased blood flow to the kidneys.

When released, angiotensin II acts on various targets to increase blood pressure and restore fluid balance. One of its effects is to stimulate the secretion of aldosterone from the adrenal glands, which promotes salt and water retention in the kidneys.

This, in turn, increases extracellular fluid (ECF) volume. Additionally, angiotensin II can also stimulate thirst, which encourages the intake of fluids, further increasing ECF volume. However, angiotensin II does not directly cause the release of vasopressin, which also promotes water retention.

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The period of the function in this problem is given as follows:

20 units.

A periodic function is a function that has the behavior repeating over intervals in the domain of the function.

Then the period of the function has the concept defined as the difference between two points in which the function has the same behavior.

Looking at the peaks of the function, they are given as follows:

Hence the period of the function is given as follows:

20 - 0 = 20 units.

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The interval on which f(x) is increasing is (-3, 3).

To determine on what interval the function f(x) = x 0 (9 − t2) et2 dt is increasing, we need to find the derivative of f(x) and then examine its sign.

We can use the Leibniz rule to find the derivative of f(x):

f'(x) = (d/dx) x 0 (9 − t2) et2 dt = (9 − x2) ex2

Now we need to determine the sign of f'(x) on different intervals. Notice that the factor (9 - x^2) is always positive for x in the interval [-3, 3], and ex^2 is always positive for any x. Therefore, the sign of f'(x) is determined by the sign of (9 - x^2)ex^2.

If x < -3 or x > 3, then (9 - x^2) is negative, and so is f'(x). Therefore, f(x) is decreasing on (-∞, -3) and (3, ∞).

If -3 < x < 3, then (9 - x^2) is positive, and so is f'(x). Therefore, f(x) is increasing on (-3, 3).

Therefore, the interval on which f(x) is increasing is (-3, 3).

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The total number of ways to distribute a total inventory of twenty batteries among the six different types is 10 x 120,332,228 = 1,203,322,280.

To determine how many ways a total inventory of twenty batteries can be distributed among the six different types, we need to use the concept of combinations. We know that there are seven different types of batteries, but we are given that there are only ten a7b batteries and at least twenty of each of the other types. This means that the maximum number of batteries that can be used from the other six types is 20 x 6 = 120.

So, to distribute a total of twenty batteries among the six types, we need to consider the number of a7b batteries and the number of batteries from the other six types. Since we are given that there are only ten a7b batteries, we can distribute them in 10 different ways among the six types.

For the other six types, we have a maximum of 120 batteries to use. To distribute 20 batteries among these six types, we can use the formula for combinations, which is nCr = n! / r!(n-r)!. In this case, we have 120 batteries to choose from, and we want to choose 20 batteries, so the formula becomes 120C20 = 120! / 20!(120-20)! = 120,332,228 ways.

Therefore, the total number of ways to distribute a total inventory of twenty batteries among the six different types is 10 x 120,332,228 = 1,203,322,280.

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A good way to make sense of qualitative data is through quantitative analysis.

Qualitative data refers to non-numerical data that is descriptive, such as textual responses, interviews, observations, and open-ended survey questions. To make sense of qualitative data, quantitative analysis techniques are not applicable. Instead, qualitative data analysis methods are used.

Quantitative analysis is focused on numerical data and involves statistical techniques, such as linear regression, hypothesis testing, and data modeling. While these techniques are valuable for analyzing quantitative data, they are not suitable for analyzing qualitative data.

To make sense of qualitative data, researchers typically employ methods such as thematic analysis, content analysis, coding, categorization, and pattern recognition. These techniques involve organizing, coding, and interpreting the qualitative data to identify themes, patterns, and relationships.

Therefore, among the options provided, the most appropriate way to make sense of qualitative data is through qualitative analysis techniques rather than mental blocks, thinking units, or linear regression.

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The average value of the function f(x) = 6x over the interval [0, 9] is 27.

To find the average value of a function over a given interval, you need to take the definite integral of the function over that interval, and divide by the length of the interval. In this case, the function is f(x) = 6x, and the interval is [0, 9].

So first, we need to find the definite integral of 6x over [0, 9]:

∫[0,9] 6x dx = 3x^2 |[0,9] = 243

Next, we need to find the length of the interval, which is simply 9 - 0 = 9.

Finally, we divide the definite integral by the length of the interval:

Average value of f(x) = (1/9) * 243 = 27

So the average value of the function f(x) = 6x over the interval [0, 9] is 27.

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The issue with the claim that a heat engine takes in 21 kJ of heat, discharges 16 kJ of heat to the environment, and performs 3 kJ of work is that the work performed does not equal the difference between the heat input and the heat output. Therefore, the correct option  is A.

1. According to the first law of thermodynamics, the work performed by a heat engine is equal to the difference between the heat input (Qin) and the heat output (Qout).
2. In this case, Qin is 21 kJ and Qout is 16 kJ.
3. The difference between the heat input and heat output is 21 kJ - 16 kJ = 5 kJ.
4. However, the claim states that the work performed is 3 kJ, which is not equal to the difference between the heat input and the heat output (5 kJ).

Hence, the claim is incorrect because the work performed does not equal the difference between the heat input and the heat output. The correct answer is option A.

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The percentage of data less than 61 on the box and whisker plot is given as follows:

100%.

A box and whisker plots shows these five metrics from a data-set, listed and explained as follows:

The metrics for this problem are given as follows:

The problem is given by the image presented at the end of the answer.

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Pool A has a diameter of approximately 88 feet, and Pool B has a diameter of approximately 27.02 meters. The circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.

In summary, Pool A has a diameter of approximately 88 feet, while Pool B has a diameter of approximately 27.02 meters. The circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.
The diameter of a circle is twice the radius. Since the radius of Pool A is given as 44 feet, the diameter of Pool A would be (2 * 44) = 88 feet.
To compare Pool A and Pool B in the same unit, we need to convert the diameter of Pool B from meters to feet. Given that 1 meter is equal to 3.281 feet, the diameter of Pool B in feet would be (27.02 * 3.281) = 88.63 feet (rounded to the nearest hundredth).
The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius. For Pool A, the circumference would be (2 * 3.14159 * 44) = 276.46 feet (rounded to the nearest hundredth).
For Pool B, the circumference would be (2 * 3.14159 * 88.63) = 556.80 feet (rounded to the nearest hundredth).
Comparing the circumferences, we find that the circumference of Pool A is greater than the circumference of Pool B by approximately (556.80 - 276.46) = 280.34 feet (rounded to the nearest hundredth), which is equivalent to approximately 85.34 meters.
Therefore, the circumference of Pool A is greater than the circumference of Pool B by approximately 77.22 meters.

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The value of k that divides the solid into two solids of equal volume is (A) 4.

To find the value of k that divides the solid into two solids of equal volume, we need to determine the intersection points of the curves y = 3√x and x = 8.

Setting the equations equal to each other, we have:

3√x = 8

Squaring both sides, we get:

9x = 64

Solving for x, we find:

x = 64/9

This intersection point determines the value of k, as x = k. Therefore, k = 64/9, which is approximately 7.111.

Comparing the given answer choices, the closest option to 7.111 is (A) 4. Thus, the correct value of k is 4.

The line x = 4 divides the solid into two equal-volume parts.

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The singular value decomposition of the given matrices.

Singular value decomposition (SVD) is a factorization method used to decompose a matrix into three separate matrices: U, Σ, and V^T. The U matrix represents the left singular vectors, Σ is a diagonal matrix containing the singular values, and V^T represents the right singular vectors.

To find the singular value decomposition, we can apply the SVD algorithm to each of the given matrices. By performing SVD, we can analyze the structure and properties of the matrices, such as their rank, null space, and condition number. The decomposition can also be used for various applications, including dimensionality reduction, image compression, and solving linear equations.

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The mirror should be inclined at an angle of 35°

To determine the angle at which the mirror should be inclined, we need to use trigonometry. Let's first draw a diagram:

In the below diagram, A represents the customer's eyes, and B represents the customer's feet. The angle we need to find is θ.

We know that A = 140cm (the height of the customer's eyes off the floor) and B = 50cm (the distance from the customer to the mirror). We want to find θ, the angle at which the mirror should be inclined.

We can use the tangent function to find θ:

tan2θ = A/B
θ = 1/2 [tex]tan^{-1}[/tex] A/B
θ = 1/2 [tex]tan^{-1}[/tex] 140cm/50cm
θ = 35°

Therefore, the mirror should be inclined at an angle of approximately 35° so that a person standing 50cm from the mirror with eyes 140cm off the floor can see her feet.

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To find the sum of the telescoping series 1 - 1/2, we need to calculate the first few terms of the series. The series is formed by subtracting consecutive terms, leading to cancellation of most terms, resulting in a simplified expression for the sum.

The given telescoping series is 1 - 1/2. To find the sum, let's calculate the first few terms.

When we plug in n = 3 into the series, we get: 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6. Notice that many terms in the series cancel each other out. For example, the positive 1/3 cancels out with the negative 1/3, and the positive 1/5 cancels out with the negative 1/5. This cancellation continues for all terms except the first and last terms.

Therefore, after canceling out terms, the simplified expression for the sum of the telescoping series becomes: 1 - 1/2 + 1/5 - 1/6.

To find the actual sum, we can evaluate this expression. Adding the terms together, we get: 1 - 1/2 + 1/5 - 1/6 = 3/10.

Hence, the sum of the telescoping series 1 - 1/2 is 3/10.

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Both conditions of the alternating series test are satisfied, so the series ∑ (-1)^n a_n converges.

To apply the alternating series test, we need to verify the following two conditions:

The sequence {a_n} = 1/(n^2 - 5) is positive, decreasing, and approaches 0 as n approaches infinity.

The series ∑ (-1)^n a_n = ∑ (-1)^n/(n^2 - 5) converges.

To check the first condition, we can take the derivative of a_n:

a'_n = -2n/(n^2 - 5)^2

Since n ≥ 3, we have n^2 - 5 ≥ 4, so (n^2 - 5)^2 ≥ 16. This implies that a'_n ≤ 0 for n ≥ 3. Therefore, the sequence {a_n} is decreasing.

To check that the sequence approaches 0, we can use the limit comparison test with the convergent p-series ∑ 1/n^2:

lim n→∞ a_n/(1/n^2) = lim n→∞ n^2/(n^2 - 5) = 1

Since the limit is finite and positive, we conclude that {a_n} approaches 0 as n approaches infinity.

Thus, both conditions of the alternating series test are satisfied, so the series ∑ (-1)^n a_n converges.

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