Solve for X
7(8x + 6) = -1
9(5+ x) = 15 (10+ x)

Therefore, To verify that y = √x is a solution to the given differential equation, we substituted y = √x and its derivatives and simplified it to show that it satisfies the equation for all x > 0.

To verify that y = √x is a solution to the given differential equation, we need to substitute y = √x into the equation and see if it satisfies the equation.
First, we need to find the first and second derivatives of y with respect to x:
dy/dx = 1/(2√x) and d²y/dx² = -1/(4x^(3/2)).
Now, substitute these values of y, dy/dx, and d²y/dx² into the given differential equation:
2x²(-1/(4x^(3/2))) + 3x(1/(2√x)) = √x
This simplifies to: -1/(2x^(1/2)) + 3/(2x^(1/2)) = √x
Which is true for all x > 0.
Explanation:
To verify that a given function is a solution to a differential equation, we substitute the function and its derivatives into the equation and check if it satisfies the equation. In this case, we used the given differential equation, substituted y = √x and its derivatives, and simplified to show that it indeed satisfies the equation for all x > 0.

Therefore, To verify that y = √x is a solution to the given differential equation, we substituted y = √x and its derivatives and simplified to show that it satisfies the equation for all x > 0.

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The combination has a refractive power of 0.167 diopters.

The refractive power of a lens is given by the formula P = 1/f, where f is the focal length of the lens in meters. The focal length of a lens in diopters (d) is given by f = 1/d.

To find the refractive power of the combination of a 10 d lens and a 15 d lens, we need to find the equivalent focal length of the combination. The equivalent focal length of two lenses in contact can be found using the formula:

1/f = 1/f1 + 1/f2

where f1 and f2 are the focal lengths of the individual lenses.

Substituting the values for the focal lengths of the two lenses, we get:

1/f = 1/10 + 1/15

Simplifying, we get:

1/f = 1/6

Multiplying both sides by 6, we get:

f = 6 meters

Therefore, the refractive power of the combination of the 10 d and 15 d lenses is:

P = 1/f = 1/6 = 0.167 d^-1.

Thus, the combination has a refractive power of 0.167 diopters.

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In a right triangle where angle BAC is 90°, and given the lengths AB = 4.4 mm and CA = 4.7 mm, the length of BC, is approximately 6.3 mm which is found using the Pythagorean theorem.

In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AB and CA).

Using the given values, AB = 4.4 mm and CA = 4.7 mm, we can apply the Pythagorean theorem to find BC. The equation is:

[tex]BC^{2}[/tex]= [tex]AB^{2}[/tex] + [tex]CA^{2}[/tex]

Substituting the values, we have:

[tex]BC^{2}[/tex]= [tex]4.4 mm^{2}[/tex] +[tex]4.7 mm^{2}[/tex]

[tex]BC^{2}[/tex] = 19.36 [tex]mm^{2}[/tex] + 21.81 [tex]mm^{2}[/tex]

[tex]BC^{2}[/tex] = 41.17 [tex]mm^{2}[/tex]

Taking the square root of both sides to solve for BC, we get:

BC ≈ √41.17 mm

BC ≈ 6.411 mm (rounded to three decimal places)

Rounding to one decimal place, the length of BC is approximately 6.3 mm.

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The matrix[tex]A=\begin{bmatrix}14&3 &1 \\k&0 &0\end{bmatrix}[/tex]has three distinct real eigenvalues if and only if -16.33... < k < 4.33...,

To find the eigenvalues of a matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the identity matrix and det denotes the determinant. For the matrix A given above, we have

det(A - λI) =[tex]\begin{vmatrix}14 - \lambda&3 &1 \\k&-\lambda &0\end{vmatrix}[/tex]

= (14 - λ)(-λ) - 3k = λ² - 14λ - 3k.

The roots of this quadratic equation are the eigenvalues of A, which are given by the formula

λ = (14 ± √(196 + 12k))/2.

For A to have three distinct real eigenvalues, we need the discriminant Δ = 196 + 12k to be positive and the two roots to be different. This implies that

196 + 12k > 0 and 14 - √(196 + 12k) ≠ 14 + √(196 + 12k).

Simplifying the second inequality, we get

√(196 + 12k) > 0, which is always true.

Therefore, the condition for A to have three distinct real eigenvalues is

-16.33... < k < 4.33...,

where the values -16.33... and 4.33... are obtained by solving the equation 14 - √(196 + 12k) = 14 + √(196 + 12k) and dividing the resulting equation by 2.

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Complete Question:

The matrix A = [tex]\begin{bmatrix} 14&3 &1 \\ k&0 &0 \end{bmatrix}[/tex] has three distinct real eigenvalues if and only if

____ < K < ____

The mean of 3X is 6 and the variance of 3X is 36

Let X and Y be independent random variables with μX = 2, σX = 2, μY = 2, and σY = 3. To find the mean and variance of 3X, we can use the properties of linear transformations for means and variances.

The mean of 3X is found by multiplying the original mean of X (μX) by the scalar value (3):
Mean of 3X = 3 * μX = 3 * 2 = 6

The variance of 3X is found by squaring the scalar value (3) and then multiplying it by the original variance of X (σX²):
Variance of 3X = (3^2) * σX² = 9 * (2^2) = 9 * 4 = 36

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1. Mean of 3X = 3 * μX = 3 * 2 = 6
2. Variance of 3X = (3^2) * σX^2 = 9 * (2^2) = 9 * 4 = 36

To find the mean and variance of 3X, we use the following properties:
Since X and Y are independent random variables with given means (μX and μY) and standard deviations (σX and σY), we can find the mean and variance of 3X.
Mean: E(aX) = aE(X)
Variance: Var(aX) = a^2Var(X)

Using these properties, we can find the mean and variance of 3X as follows:

Mean:
E(3X) = 3E(X) = 3(2) = 6
Therefore, the mean of 3X is 6.

Variance:
Var(3X) = (3^2)Var(X) = 9(2^2) = 36
Therefore, the variance of 3X is 36.

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The dimensions of the rectangle are width = 20 centimeters and length = 1 centimeter.

Let's denote the width of the rectangle as "w" centimeters. According to the problem, the length of the rectangle is 19 centimeters less than its width, so the length can be expressed as "w - 19" centimeters.

The formula for the area of a rectangle is length multiplied by width. In this case, the area is given as 20 square centimeter

Area = Length × Width

20 = (w - 19) × w

To solve this equation, we can expand it:

20 = [tex]w^2[/tex] - 19w

Rearranging the equation to bring everything to one side:

[tex]w^2[/tex] - 19w - 20 = 0

Now, we can factor the quadratic equation:

(w - 20)(w + 1) = 0

Setting each factor equal to zero and solving for "w":

w - 20 = 0 --> w = 20

w + 1 = 0 --> w = -1

Since a negative width doesn't make sense in this context, we discard w = -1.

Therefore, the width of the rectangle is 20 centimeters (w = 20).

To find the length, we substitute this value back into the expression for length:

Length = w - 19

Length = 20 - 19

Length = 1 centimeter

So, the dimensions of the rectangle are width = 20 centimeters and length = 1 centimeter.

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A chi-square test of independence should be performed.

A chi-square test of independence should be performed in this case. A chi-square test of independence, also known as a chi-square test for association, is a statistical hypothesis test used to determine whether two categorical variables are independent of one another or not.The observed and expected frequency counts for two categorical variables are compared using this test. 

The test is appropriate when the variables are categorical, the observed frequencies are frequency counts, and the expected frequencies are also frequency counts based on sample data.Here, the biological anthropologists are interested in determining whether there is any association between two variables, fingerprint patterns, and blood types.

The sample is random and consists of male students from a certain region. Both fingerprint patterns and blood types are categorized as categorical variables. As a result, a chi-square test of independence should be performed.

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Statement b)  p-value is the probability to find the observed or more extreme value for the test statistic given that the null hypothesis is true.  is false.

The correct definition of the p-value is given in statement a), which states that the p-value is the probability of observing the test statistic or a more extreme value, assuming that the null hypothesis is true. In other words, the p-value measures the strength of evidence against the null hypothesis.

Statement c) is true. A high p-value for X2 suggests that there is insufficient evidence to reject the null hypothesis that X2 has no effect on the response variable.

Statement d) is generally true. A low p-value for X1 indicates that there is strong evidence to reject the null hypothesis that the coefficient for X1 is equal to zero, which suggests that X1 has a statistically significant effect on the response variable. However, it's important to note that statistical significance alone does not necessarily imply practical significance or causation.

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The first three nonzero terms in the Taylor polynomial approximation to $y(t)$ are $t + \frac{1}{3}t^2 + O(t^3)$.

Using these initial conditions, we can write the first few terms of the Taylor polynomial approximation as:

\begin{align*}

y(t) &\approx y(0) + y'(0)t + \frac{y''(0)}{2!}t^2 \

&= t + \frac{1}{2}y''(0)t^2 \

&= t + \frac{1}{2}\left(\frac{6\cos(0)}{9\cdot 0 + 9}\right)t^2 \

&= t + \frac{1}{3}t^2

\end{align*}

Therefore, the first three nonzero terms in the Taylor polynomial approximation to $y(t)$ are $t + \frac{1}{3}t^2 + O(t^3)$.

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Answer:

ALL YOU HAVE TO DO IS LOOK AT THE NUMBER IN THE MIDDLE AND THE ONE AT THE LEFT AND PUT THEM TOGETHER FOR INSTENTSET

1 | 3

Is 13

Or

3 | 4

Is 34

And if you see

1 | 3, 4, 5

It stands for 13, 14, and 15

Suppose that in a random sample of size 200, standard deviation of the sampling distribution of the sample mean 0. 8. Researcher wanted to reduce the standard deviation to 0. 4. What sample size would be required?

The formula to calculate the standard error of the mean(SEM) is given by the ratio of the standard deviation and the square root of the sample size. Hence,SEM = SD/√nWhere,SD is the standard deviation of the sampling distribution of the sample mean is the sample sizeTherefore, to reduce the standard deviation to 0.4, the formula can be modified as follows:SEM = 0.4/√nSquaring both sides of the above equation and cross-multiplying, we get:0.16 = 0.8²/nSo, n = (0.8²/0.16) = 4. Hence, the sample size required to reduce the standard deviation to 0.4 is 400.

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Answer:

Step-by-step explanation:

The answer is choice B.

No matter what the equation for each angle,

they still add to 180°.  All interior angles of a triangle

add to 180°.

Step-by-step explanation:

the answer is 2k(2ucos)2usin(vi)

The given argument "K ⊃ Q / Q ⊃ ∼ K // K ≡ Q" is valid.

To determine the validity of the argument "K ⊃ Q / Q ⊃ ∼ K // K ≡ Q," we construct an ordinary truth table. The argument consists of two premises and a conclusion. The symbol "⊃" represents the conditional implication, "∼" represents negation, and "≡" represents equivalence.

We assign truth values (T or F) to the atomic propositions K and Q and evaluate the truth values of the premises and the conclusion based on the given argument. By systematically filling out the truth table, we can examine all possible combinations of truth values for K and Q.

After constructing the truth table, we observe that in every row where the premises K ⊃ Q and Q ⊃ ∼ K are true, the conclusion K ≡ Q is also true. Therefore, the argument is valid.

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The equation represented by the graph is given as follows:

[tex]y = e^x[/tex]

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

For a logarithm with base e, with intercept of y = 1, the equation is given as follows:

[tex]y = e^x[/tex]

Which is the equation for this problem.

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

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Answer:

Original volume = (4/3)π(6^3) = 288π

New volume = (4/3)π(18^3) = 7,776π

7,776π ÷ 28π = 27

When the radius of a sphere is tripled, the volume of the new sphere is 27 times the volume of the old sphere.

The amount of golden delicious apples than red delicious apples that Mr. Myers picked would be 14 1/8.

The extra amount of golden delicious apples that Mr. Myers picked in comparison to the red delicious apples that Mr. Myers picked would be gotten by subtracting the amount of golden delicious apples from red delicious apples as follows:

27 2/8 - 13 1/8

= 14 1/8

So, the amount with which the number of golden delicious apples that Mr. Myers got was greater than the red delicious apples is 14 1/8

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Complete Question:

Mr.Myers picked 13 1/8 bushels of red delicious apples and 27 2/8 bushels of golden delicious apples. How many bushels of golden delicious apples than of red delicious apples did he pick?

Answer:

(x-y)^2 + (x^2+2xy+y^2) = 2x² + 2y²

Step-by-step explanation:

We have : (x-y)² + (x²+2xy+y²)

So :  ( x - y )( x - y ) + ( x² + 2 xy + y² )

So :  x ( x - y ) - y ( x - y ) + ( x² + 2 xy + y² )

So :  x² - xy - y ( x - y ) + ( x² + 2 xy + y² )

So :  2x² + 2y²

Please pick me as brailiest

The both angles are 75 degrees.

When parallel lines are cut by a transversal line angles relationships are formed such as corresponding angles, alternate exterior angles, alternate interior angles, same side interior angles, vertically opposite angles etc.

Therefore, let's use the angle relationships to find the angles in the parallel lines as follows:

Hence,

15x = 12x + 15(alternate interior angles)

15x - 12x = 15

3x = 15

divide both sides by 3

x = 15 / 3

x = 5

Therefore,

15(5) = 75 degrees

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the mass moment of inertia about the x-axis is ixx = (2/7)[tex]mr^{2}[/tex] and about the y-axis is iyy = (1/7)[tex]mr^{2}[/tex] + (2/3)[tex]mh^{2}[/tex]

To find the mass moment of inertia, we consider the solid of revolution as a collection of infinitesimally thin disks or cylinders stacked together along the axis of revolution. Each disk or cylinder has a mass element dm.

For the mass moment of inertia about the x-axis (ixx), we integrate the contribution of each mass element along the axis of revolution:

ixx = ∫ [tex]r^{2}[/tex] dm

Since the solid is homogeneous, dm = ρ dV, where ρ is the density and dV is the volume element. For a solid of revolution, dV = πr^2 dh, where h is the height of the solid.

Substituting the expressions and performing the integration, we get:

ixx = ∫ [tex]r^{2}[/tex] ρπr^2 dh

= ρπ ∫ [tex]r^{4}[/tex] dh

= [tex](1/5)\beta \pi r^{4}[/tex] h

Since the solid is homogeneous, the mass m = [tex]\beta \pi r^{2}[/tex] h. Substituting this in the equation above, we get:

ixx = (1/5)m [tex]r^{2}[/tex]

Similarly, for the mass moment of inertia about the y-axis (iyy), we integrate along the radius r:

iyy = ∫[tex]r^{2}[/tex]  dm

= ∫ [tex]r^{2}[/tex] [tex]\beta \pi r^{2}[/tex] dh

= ρπ ∫ [tex]r^{4}[/tex] dh

= (1/5)[tex]\beta \pi r^{4}[/tex] h

Since the height of the solid is h, substituting [tex]\beta \pi r^{2}[/tex] h = m, we get:

iyy = (1/5)m [tex]r^{2}[/tex] + [tex](2/3)mh^{2}[/tex]

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Answer:

22666.02986

Step-by-step explanation:

In the Heads or Tails game, the player pays 50 cents and chooses either heads or tails. After that, the player tosses a fair coin. If the coin matches the player's call, then the player wins a prize. if 100 players play the game, then we can expect that the students will raise $20.00 as the cost of prizes to be given to the winners.

Answer to part (a):

Probability of winning the game = 1/2

Probability of losing the game = 1/2

Expected number of players who would win = Number of players × Probability of winning

= 100 × 1/2= 50

Expected number of players who would lose = Number of players × Probability of losing

= 100 × 1/2= 50

Therefore, we can expect 50 players to win the game.

Answer to part (b):

The cost of each prize is 40 cents. The expected number of players who would win the game is 50.

Therefore, the total cost of prizes would be:

The total cost of prizes = Cost of each prize × Expected number of players who would win

= 40 × 50

= 2000 cents or $20.00

Therefore, if 100 players play the game, then we can expect that the students will raise $20.00 as the cost of prizes to be given to the winners.

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Let's say we have a graph with four nodes: A, B, C, and D. The edges and their lengths are as follows:
- A to B: 3
- A to C: 1
- B to D: 2
- C to D: -5

Using this we can show that the Dijkstra's algorithm fails if negative edge lengths are allowed

If we use Dijkstra's algorithm to find the shortest path from A to D, we would start at A and initially assign a distance of 0 to it. We would then look at its neighbors, B and C, and update their distances accordingly (3 for B and 1 for C). We would then choose C as the next node to visit since it has the shortest distance so far. However, when we update the distance to D through C, we would get a distance of -4 (since -5 + 1 = -4).

This negative distance causes a problem because Dijkstra's algorithm assumes that all edge weights are non-negative. When we update the distance to D through C, it becomes shorter than the distance we assigned to it when we initially looked at it through B. This means that we would have to revisit D and potentially update its distance again, leading to an infinite loop.

Therefore, Dijkstra's algorithm fails if negative edge lengths are allowed.

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To find the total number of scoops of ice cream served, we need to add the number of scoops of each flavor:

6 ¼ + 5 ¾ + 2 ¾

We can convert the mixed numbers to improper fractions to make the addition easier:

6 ¼ = 25/4

5 ¾ = 23/4

2 ¾ = 11/4

Now we can add:

25/4 + 23/4 + 11/4 = 59/4

So the ice cream parlor served 59/4 scoops of ice cream in total. We can simplify this fraction by dividing the numerator and denominator by their greatest common factor, which is 1:

59/4 = 14 3/4

Therefore, the parlor served 14 3/4 scoops of ice cream in total.

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The height from the top of the Ferris wheel to the ground is 154.06 feet.

The Ferris wheel has a diameter of 64 feet and the bottom of the wheel is 15 feet off the ground.

The Ferris Wheel takes 60 seconds to complete a full rotation.

The radius of the Ferris wheel is = diameter/2

                                                      = 64/2

                                                      = 32 feet.

The bottom of the Ferris wheel is 15 feet off the ground. Therefore, the distance from the center of the wheel to the ground is (radius+15) feet.

So, the height from the top of the Ferris wheel to the ground is :

        height = distance covered by Ferris wheel in 60 seconds - distance from center to ground .

        The distance covered by the Ferris wheel = Circumference of the Ferris wheel= π × diameter

        3.14 × 64= 201.06 feet.∴

 In 60 seconds, distance covered by the Ferris wheel = 201.06 feet.

The distance from the center of the wheel to the ground = radius + 15= 32 + 15= 47 feet.

 Height from the top of the Ferris wheel to the ground = 201.06 - 47 = 154.06 feet.

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To the nearest year, it will take 10 years for the population to reach 24,600.

Now, For this problem, we can use the formula for exponential growth:

P(t) = P₀  (1 + r)ⁿ

Where:

P(t) is the population after t years

P₀ is the initial population

r is the annual growth rate (as a decimal)

n is the number of years

Plugging in the values given:

P₀ = 15,000

r = 0.03

P(t) = 24,600

We can solve for n by dividing both sides by P0 and then taking the logarithm of both sides:

(1 + r)ⁿ = P(t) / P₀ t log(1 + r)

= log(P(t) / P0)

t = log(P(t) / P₀) / log(1 + r)

Plugging in the values given:

t = log(24,600 / 15,000) / log(1 + 0.03) t

t ≈ 10 years

Therefore, to the nearest year, it will take 10 years for the population to reach 24,600.

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The expression 3x² + 5x + 1 can be factored completely as (3x + 1)(x + 1).Explanation:We are given an expression 3x² + 5x + 1.

To factor this expression, we need to look for two factors such that when they are multiplied, we get 3x² + 5x + 1.

For this, we need to find two numbers whose product is 3 and whose sum is 5.

It can be observed that 3 and 1 are two such numbers. Therefore, we can write:3x² + 5x + 1 = (3x + 1)(x + 1)

Hence, the expression 3x² + 5x + 1 can be factored completely as (3x + 1)(x + 1).

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a. The probability that the individual purchased a small cup 24% and probability that the individual purchased a cup of decaf coffee is 20%

b.  If we learn that the selected individual purchased a small cup, the probability that he/she chose decaf coffee is  0.182.

c. If we know the individual purchased decaf, the probability that he/she chose a small cup is 0.5 or 50%.

d. The conditional probability of selecting a small cup given that decaf coffee was chosen is higher than the unconditional probability of selecting a small cup (24%).

(a) The probability that the individual purchased a small cup is 24% or 0.24. The probability that the individual purchased a cup of decaf coffee is 20% or 0.20.

(b) We need to find the conditional probability of choosing decaf given that the individual purchased a small cup. Let D denote the event that decaf coffee is chosen, and S denote the event that a small cup is chosen. Then, using Bayes' theorem, we have:

P(D|S) = P(S|D) * P(D) / P(S)

P(S) = P(S and R) + P(S and D) = 24% + 20% = 44%

P(D) = 20%

P(S|D) = 20% / 50% = 0.4

Therefore, P(D|S) = 0.20 * 0.4 / 0.44 = 0.1818 or approximately 0.182. This means that if we know the individual purchased a small cup, the probability that he/she chose decaf coffee is about 0.182. We can interpret this probability as the proportion of small cup purchases that are decaf.

(c) If we learn that the selected individual purchased decaf, we can find the conditional probability of choosing a small cup as follows:

P(S|D) = P(S and D) / P(D) = 10% / 20% = 0.5

This means that if we know the individual purchased decaf, the probability that he/she chose a small cup is 0.5 or 50%.

(d) The conditional probability of selecting a small cup given that decaf coffee was chosen is higher than the unconditional probability of selecting a small cup (24%). This is because the proportion of small cups among decaf coffee purchases (50%) is higher than the overall proportion of small cups (24%).

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We need to solve the equation 9x - 6y = 12 and determine the values of x and y. Here are the steps to solve this equation:

Step 1: To simplify the equation, first find the greatest common divisor (GCD) of the coefficients. In this case, the GCD of 9, 6, and 12 is 3.

Step 2: Divide the entire equation by the GCD (3). This gives us:

(9x - 6y = 12) ÷ 3

3x - 2y = 4

Step 3: Now, the equation is in its simplest form. However, we cannot find unique values for x and y since we have only one equation with two unknowns. You would need an additional equation involving x and y to determine their specific values. But you can express one variable in terms of the other, like:

y = (3x - 4) / 2

Now, you can substitute any value for x and find the corresponding value for y. The missing entries will depend on the specific values chosen for x and y.

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