(2) Solve the inequality |2x − 5| ≤ 9 and present your answer in interval notation.
(3) Find the inverse of the following function:
f(x)= 5 .
6x−1
(4) Letf(x)=√x−4andg(x)=x2−11x+30. Find fg and fg andstatetheirdomains.
(5) Find the equation of the line perpendicular to y = 35 x − 4 and passing through the point (1, 2). Write your answer in slope-intercept form (i.e., y = mx + b).
(6) Divide the following using long division:
2x3 +x2 +3x+5. x−1
Write your final answer in the form Dividend = Quotient + Remainder .

The reference angle for θ = 27π/2 is θr = π/4 and the associated point on the unit circle is (x, y) = (√2/2, √2/2).

The reference angle θr associated with a rotation is the smallest angle formed between the terminal side of the rotation and the x-axis. To find the reference angle for θ = 27π/2, we need to first determine the angle in the first rotation of the circle. Since there are 2π radians in a full circle, we can divide 27π/2 by 2π to find the number of full rotations:

27π/2 ÷ 2π = 27/4 = 6 3/4

This means that there are 6 full rotations and a partial rotation of 3/4. The reference angle for this partial rotation is θr = π/4.

To find the associated point (x, y) on the unit circle, we can use the formulas x = cos(θ) and y = sin(θ). For θ = 27π/2, we can use the reference angle θr = π/4:

x = cos(π/4) = √2/2
y = sin(π/4) = √2/2

Therefore, the associated point is (x, y) = (√2/2, √2/2).

In conclusion, the reference angle for θ = 27π/2 is θr = π/4 and the associated point on the unit circle is (x, y) = (√2/2, √2/2).

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The fraction (4[tex]s^{(6)[/tex])(([tex]t^{(6)[/tex])[tex])^{3}[/tex]  in its simplest form is C. 64[tex]s^{(18)[/tex][tex]t^{(18)[/tex].  

To simplify the given fraction, we need to use the power of a power rule, which states that (a^b)^c = a^(b*c).

In this case, we have (4[tex]s^{(6)[/tex][tex]t^{(6)[/tex])^3 , so we need to multiply the exponents of each term by 3.

For the first term, [tex]4^{(3)[/tex] = 64. For the second term, [tex]s^{(6*3)[/tex] = [tex]s^{(18)[/tex]. And for the third term, [tex]t^{(6*3)[/tex]= [tex]t^{(18)[/tex].

Putting these terms together, we get 64[tex]s^{(18)[/tex][tex]t^{(18)[/tex], which is the correct answer.

So, the simplified form of (4[tex]s^{(6)[/tex])(([tex]t^{(6)[/tex])[tex])^{3}[/tex] is 64[tex]s^{(18)[/tex][tex]t^{(18)[/tex], or answer choice C.

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Option (a) [tex]-\frac{23}{3}[/tex]  is the smallest among the four option because it has negative value. We can easily find it by seeing negative value but we do by taking LCM of denominator .

The smallest multiple that two or more numbers share is known as the least common multiple. Least Common Multiple short form is LCM.

We can find LCM of these number by taking least common multiple.

LCM of 3, 3, 17, 13 is 663 .

(a) [tex]-\frac{22}{3} *663 = -22*221[/tex]

[tex]= -4862[/tex]

(b) [tex]\frac{4}{3} * 663 = 4 * 221[/tex]

[tex]= 884[/tex]

(c) [tex]\frac{6}{17} *663 = 6 * 17[/tex]

[tex]= 102[/tex]

(d) [tex]\frac{2}{13}*663 = 2*51[/tex]

[tex]=102[/tex]

So, the Smallest no [tex]-\frac{22}{3}[/tex].

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Answer: x = 47.2°

Step-by-step explanation:

     We can use a trigonometry function to solve this question since it's a right triangle.

     Equation:

tanx = [tex]\frac{27}{25}[/tex]

     The tangent inverse of both sides of the equation:

[tex]\displaystyle tan^{-1} (\text{tan}x)=(\frac{27}{25} )tan^{-1}[/tex]

x = 47.2°

Answer:

hypothesis of the medicine is incomplete

Step-by-step explanation:

so we have.to apply bodmas

When the the doctor's claim is correct, the probability that exactly 19 patients will recover when given the new medicine is 0.1444.

a) The suitable distribution to model the number of patients in this sample who recover when given the new medicine is the binomial distribution. We can assume that each patient has a fixed probability of success (i.e., recovering) and that the outcomes of each patient are independent of each other.

b) If the claim is correct, then the probability of success (recovering) for each patient is p = 0.8. Let X be the number of patients in the sample who recover when given the new medicine. Then X follows a binomial distribution with parameters n = 25 (the sample size) and p = 0.8 (the probability of success).

The probability of exactly 19 patients recovering is given by:

P(X = 19) = (25 choose 19) * (0.8)^19 * (1-0.8)^(25-19)

where (25 choose 19) is the binomial coefficient, which represents the number of ways to choose 19 patients out of 25.

Using a calculator, we can compute:

P(X = 19) = 0.1444

Therefore, if the doctor's claim is correct, the probability that exactly 19 patients will recover when given the new medicine is 0.1444.

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The volume of the cylinder is 126π cubic meters.

Volume is the measure of the space occupied by a three-dimensional object. It is typically expressed in cubic units.

The radius of the cylinder is half of the diameter, so the radius is 3 meters. The formula for the volume of a cylinder is

V = πr²h,

where r is the radius and h is the height. Substituting the given values, we get V = π(3)²(14) = 126π cubic meters.

Therefore, the volume of the cylinder is 126π cubic meters.

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The correlation coefficient is 0.816.

QUESTION 3: The following table is showing two variables, X and Y.

i) To obtain the correlation coefficient, use the formula r=(n*Σxy-Σx*Σy)/sqrt((n*Σx2-(Σx)2)*(n*Σy2-(Σy)2)), where n is the number of elements, Σxy is the sum of the product of the elements of X and Y, Σx is the sum of the elements of X, Σy is the sum of the elements of Y, Σx2 is the sum of the squares of the elements of X, and Σy2 is the sum of the squares of the elements of Y.

ii) To plot the relationship between X and Y, create a scatter plot. This plot should show both the trend line and R-squared.

iii) The correlation between X and Y is strong and positive. This means that as the value of X increases, the value of Y tends to increase as well. This is reflected in the R-squared value, which is high and close to 1.

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The solution to an absolute value inequality is all real numbers. Then the correct options are -

Option A: The inequality could be lx - 4| ≥ -24.

Option F: The inequality could be -3 |x - 7| -2 < 10.

What is an inequality?

An inequality in algebra is a mathematical statement that employs the inequality symbol to show how two expressions relate to one another. The phrases on either side of an inequality symbol are not equal. The phrase on the left should be larger or smaller than the expression on the right, or vice versa, according to this symbol.

An absolute value inequality must not impose any limitations on the value of the absolute value expression if the answer contains only real values.

This implies that any positive real integer can be used as the absolute value statement.

Option A: The inequality |x - 4| -24 has an absolute value expression with a non-negative absolute value, hence it satisfies the requirement that the solution set only consist of real values.

Option B and Option E both refer to the solution set's graph as a number line with a 2-value gap.

To know whether the solution set only contains real numbers, however, more information is required.

Depending on the particular absolute value disparity, the range of the number line could be anything.

Option C: Because of the negative absolute value statement of the inequality -4 |x - 11| - 3 ≥ 9, the left-hand side can never be less than 3.

As a result, the solution set has a lower bound and cannot include only real values.

The inequality |x - 12| ≤ -48 has a non-negative absolute value expression, but an absolute value expression cannot be less than or equal to a negative number, as shown in Option D.

There are therefore no effective remedies for this inequity.

Option F: Because the inequality -3 |x - 7| - 2 < 10 has an absolute value expression that is not negative, it satisfies the requirement that the solution set only contain real numbers.

As a result, choices A and F are the proper ones.

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Answer:

I think the original price of the pants is $20.63

Step-by-step explanation:

25% of 16.50 is 4.125

4.125 + 16.50 = 20.625

20.625 - 4.125 = 16.50 (the price after the discount is subtracted from it)

round 20.625 to the nearest cent and you get 20.63.

Hope this helps!

Answer:

Gina has 47 nickels

Step-by-step explanation:

Let's call the number of nickels that Gina has "n" and the number of dimes she has "d". We know that she has a total of 70 nickels and dimes, so:

n + d = 70 (equation 1)

We also know that the value of her nickels and dimes is $4.65, which is equal to 465 cents. Each nickel is worth 5 cents and each dime is worth 10 cents, so the value of n nickels is 5n cents and the value of d dimes is 10d cents. Therefore, we can write another equation based on the value of the coins:

5n + 10d = 465 (equation 2)

We can simplify equation 2 by dividing both sides by 5:

n + 2d = 93 (equation 3)

Now we have two equations with two variables. We can solve for one of the variables in terms of the other and substitute into the other equation to solve for the remaining variable. For example, we can solve equation 1 for d:

d = 70 - n

Substituting this expression for d into equation 3, we get:

n + 2(70 - n) = 93

Simplifying this equation, we get:

n + 140 - 2n = 93

-n + 140 = 93

-n = -47

n = 47

Therefore, Gina has 47 nickels and 23 dimes (since n + d = 70), and the total value of her coins is $4.65.

Answer:

47 nickels

Step-by-step explanation:

47 nickels

To find the inverse of a matrix, we use the formula:[[a,b],[c,d]]-1 = (1/(ad-bc)) * [[d,-b],[-c,a]]

In this case, our matrix is [[2,-8],[-2,7]], so we can plug in the values into the formula:

[[2,-8],[-2,7]]-1 = (1/(2*7-(-8)*(-2))) * [[7,8],[2,2]]

Simplifying the equation gives us: [[2,-8],[-2,7]]-1 = (1/(-6)) * [[7,8],[2,2]]

Multiplying the scalar with the matrix gives us:[[2,-8],[-2,7]]-1 = [[-7/6,-8/6],[-2/6,-2/6]]

Simplifying the fractions gives us the final answer: [[2,-8],[-2,7]]-1 = [[-7/6,-4/3],[-1/3,-1/3]]

Therefore, the inverse of the matrix [[2,-8],[-2,7]] is [[-7/6,-4/3],[-1/3,-1/3]].

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The synthetic division set up in a table: -2|3097|-612-42|3-621-35

To use synthetic division to divide (3x^(3)+9x+7) by (x+-2), we will follow these steps:

Step 1: Write the coefficients of the dividend in a row: 3, 0, 9, 7. (Note: we include a 0 for the missing x^2 term.)

Step 2: Write the constant term of the divisor, -2, to the left of the row of coefficients.

Step 3: Bring down the first coefficient, 3, to the bottom row.

Step 4: Multiply the -2 by the 3 in the bottom row, and write the result, -6, in the next column of the top row.

Step 5: Add the -6 to the 0 in the top row, and write the result, -6, in the bottom row.

Step 6: Repeat steps 4 and 5 for the remaining columns. Multiply the -2 by the -6 in the bottom row, and write the result, 12, in the next column of the top row. Add the 12 to the 9 in the top row, and write the result, 21, in the bottom row. Multiply the -2 by the 21 in the bottom row, and write the result, -42, in the next column of the top row. Add the -42 to the 7 in the top row, and write the result, -35, in the bottom row.

Step 7: The bottom row now contains the coefficients of the quotient, 3x^2-6x+21, and the remainder, -35.

So, the result of the synthetic division is (3x^(3)+9x+7)÷(x+-2) = 3x^2-6x+21 with a remainder of -35.

Here is the synthetic division set up in a table:

-2|3097|-612-42|3-621-35

I hope this helps! Let me know if you have any further questions.

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The mass of the P₂O₅ formed from 3.4 moles of O₂ gas is equal to 386.1 g.

A mole can be described as a unit for measurement of a huge number of quantities of atoms, molecules, ions, or other particles. The atomic mass can be expressed as the one mole of any element.

The number of particles present in one mole was to be equal to 6.023 × 10 ²³ which is Avogadro’s constant.

Given, the number of moles of O₂ gas = 3.4 moles

The balanced chemical reaction of phosphorous and oxygen gas can be given as:

4 P  +  5 O₂   →   2 P₂O₅

5 mol of Oxygen reacts with moles of P₂O₅ = 2 mol

3.4 mol of O₂ gas reacts with moles of P₂O₅ = (2/5) × 3.4 = 1.36 mol

The mass of the 1.36 mol of P₂O₅ = 1.36 × 283.89 = 386.1 g

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a) Slope: The slope of the tangent line is 8.
b) Equation of the Tangent Line: y=-3x+7

a) To find the slope of the tangent line, we need to take the derivative of h(x) and evaluate it at x = 1.
The derivative of h(x) is: h'(x) = 3x^2 - 10x
Evaluating h'(x) at x = 1 gives us:
h'(1) = 3(1)^2 - 10(1) = 3 - 10 = -7
So the slope of the tangent line at (1, -4) is -7.

b) Now that we know the slope of the tangent line, we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is:
y - y1 = m(x - x1)
Plugging in the point (1, -4) and the slope -7 gives us:
y - (-4) = -7(x - 1)

Simplifying and rearranging terms gives us the equation of the tangent line:
y = -7x + 3
So the equation of the tangent line at (1, -4) is y = -7x + 3.

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The final answer in quotient and remainder form is:

(x^(2)+4x-20)÷(x-4) = x+8 with a remainder of 12

To determine the quotient and remainder of the given expression, we can use synthetic division. Synthetic division is a method of dividing a polynomial by a linear factor in the form of x-a. In this case, the linear factor is x-4, so a=4.

Here are the steps for synthetic division:

Write the coefficients of the dividend, x^(2)+4x-20, in a row: 1 4 -20
Write the value of a, 4, to the left of the coefficients.
Bring down the first coefficient, 1, to the bottom row.
Multiply the value in the bottom row by a, 4, and write the result, 4, in the second column of the top row.
Add the numbers in the second column of the top row, 4+4, and write the result, 8, in the bottom row.
Multiply the value in the bottom row by a, 4, and write the result, 32, in the third column of the top row.
Add the numbers in the third column of the top row, -20+32, and write the result, 12, in the bottom row.

The bottom row now contains the coefficients of the quotient, 1 and 8, and the remainder, 12. So the quotient is x+8 and the remainder is 12.

The final answer in synthetic division form is:

4 | 1  4  -20
 |    4   32
 -------------
 | 1  8   12

The final answer in quotient and remainder form is:

(x^(2)+4x-20)÷(x-4) = x+8 with a remainder of 12

So the quotient is x+8 and the remainder is 12.

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[tex]Area = \dfrac{6(7)}{2} = \dfrac{42}{2} = 21 \ cm^{2}[/tex]

Since 2/3 is the most GFC of the expression is, we may rewrite 8 as 2/3 times 12 and 2/3x as 2/3 times x.

The greatest common factor (GCF) that divides two or more numbers is known as the greatest common divisor (GCD). The highest common factor is another name for it (HCF). For instance, since both 15 and 10 can be divided by 5, 5 is the biggest common factor between both. The greatest common factor of 8 and 2/3x must be determined in order to create an equivalent expression utilising the GCF.

1, 2, 4, and 8 make up the number 8. 2/3x has the following factors: 1/3, 2/3, and x.

We thus have: 8/3 - 2/3x

= (2/3 * 4) / (2/3) - (2/3 * x)

= (2/3)(4 - x) (4 - x)

As a result, using the GCF, the formula for 8/3 - 2/3x is (2/3) (4 - x).

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The value of the variable 'x' using the external angle theorem will be 84°.

The polygonal form of a triangle has a number of flanks and three independent variables. Angles in the triangle add up to 180°.

The exterior angle of a triangle is practically always equivalent to the accumulation of the interior and opposing interior angles. The term "external angle property" refers to this segment.

The graph is completed and given below.

By the external angle theorem, the equation is given as,

x + 180° - 154° + 180° - 110° = 180°

x + 26° + 70° = 180°

x + 96° = 180°

x = 84°

The value of the variable 'x' using the external angle theorem will be 84°.

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Answer:

That sucks

Step-by-step explanation:

Because I said so

Bar chart to display the data of number of steps of Jamal is attached below.

A bar chart is a chart that displays complete data in rectangular bars, with the height of the bars proportional to the values ​​they represent. Chart bars  can be displayed vertically or horizontally. A bar chart, also called a bar graph, is a graphical representation of grouped data. It is a method of data processing. Bar charts are great for displaying data independent of each other and do not need to be in any particular order when displayed.

Given data,

Jamal keeps a log every day on the daily number of steps he walks.

Daily steps:

3500 , 4500 , 2000 , 2500 , 5000 , 5250 , 9000​

the number of steps is independent to the number of days, it can be graphed as a bar chart.

the bar graph can be drawn as following:

Hence, the data can be displayed as a bar chart attached below.

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The simple interest due on a 67-day loan of $2600 at a rate of 5% is $23.86.

To calculate this, we can use the following formula: I = Prt, where I is the interest due, P is the principal (or initial loan amount), r is the interest rate, and t is the time in years. We can calculate the time in years by taking the number of days (67) divided by the number of days in a year (365). Therefore, t = 0.1831. We can now plug this into our formula to get I = 2600 * 0.05 * 0.1831, which simplifies to I = 21.50. Therefore, the simple interest due is $21.50.

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1. The equation for the cost of adult tickets is:

$8.97 x number of adult tickets

The equation for the cost of child tickets is:

$6.79 x number of child tickets

The equation for the cost of renting out the theater is:

$190

2. Yes, you can get at least 5 adult tickets and 4 kids tickets.

3. The most tickets you could have bought and met the given conditions is 9 adult tickets and 8 kids tickets.

4.  It would be a better deal to rent out the theater if you had more than 24 people.

You must be aware of the costs of adult and child tickets in order to determine if you can purchase at least 5 adult tickets and 4 child tickets. A cinema ticket for an adult costs $8.97 on average, while a ticket for a child costs $6.79 on average. 4 children's tickets would cost $27.16 and 5 adult tickets would cost $44.85 as a result. You would then have $202.49 remaining, which would be enough to pay for 7 additional child tickets. Thus, you can purchase at least 5 adult and 4 child tickets.

The most tickets you could have purchased while still fulfilling the requirements is nine adult and eight child tickets. The total price would be $135.05. The adult tickets would cost $80.73 and the child tickets would cost $54.32. You would now have $89.45, which is insufficient to purchase any additional tickets.

Renting out the theater might be a great option if you have more than 24 individuals. This is due to the fact that hiring out the theater costs $190 and allows for a maximum of 24 guests, both adults and children. Consequently, renting the theater would be less expensive than purchasing individual tickets if you had more than 24 individuals.

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Answer:

Step-by-step explanation: 8

Answer:

x=3

Step-by-step explanation:

-6(2x+5)=-66

expand brackets first

-12x-30=-66

add 30 to each side

-12x=-36

divided each side by -12

x=3

If ∠PRM = 120°, then the measure of angle MRN is 60°.

Linear pair of angles are the pair of angles formed when two lines are intersected at a point forming on a line.

Sum of the measures of angles of linear pair is supplementary.

In the figure, segment PN is a line.

So, the angles formed in between are linear pair of angles.

∠PRM and ∠MRN are linear pair of angles.

∠PRM + ∠MRN = 180°

120 + ∠MRN = 180°

∠MRN = 180° - 120°

∠MRN = 60°

Hence the required angle measure is 60°.

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Question 9: The standard error of the mean (SEM) is 0.0912 and the 95% confidence interval is (5.0213, 5.3787).

Question 10: This confidence interval indicates that we can be 95% confident that the true mean of acid rain in Winnipeg is between 5.0213 and 5.3787.

Question 11: The 90% and 99% confidence interval for the data is (5.0499, 5.3501) and (4.9650, 5.4350) respectively.

Question 12: As the confidence interval increases from 90% to 95% to 99%, the width of the interval also increases. This means that the range of values that we can be confident contains the true mean becomes larger.

Question 9:

The standard error of the mean (SEM) is calculated as:

SEM = s / √n

where s is the standard deviation of the sample and n is the sample size.

Assuming that the standard deviation and sample size from assignment #2 are 0.5 and 30, respectively, the SEM can be calculated as follows:

SEM = 0.5 / √30

SEM = 0.0912

The 95% confidence interval around the sample mean estimate can be calculated as:

CI = mean +/- (1.96 * SEM)

Assuming that the sample mean from assignment #2 is 5.2, the 95% confidence interval can be calculated as follows:

CI = 5.2 +/- (1.96 * 0.0912)

CI = 5.2 +/- 0.1787

CI = (5.0213, 5.3787)

Question 10:

This confidence interval indicates that we can be 95% confident that the true mean of acid rain in Winnipeg is between 5.0213 and 5.3787. In other words, if we were to take multiple samples from the population and calculate the mean for each sample, 95% of those means would fall within this confidence interval.

Question 11:

The 90% confidence interval can be calculated as:

CI = mean +/- (1.645 * SEM)

CI = 5.2 +/- (1.645 * 0.0912)

CI = 5.2 +/- 0.1501

CI = (5.0499, 5.3501)

The 99% confidence interval can be calculated as

CI = mean +/- (2.576 * SEM)

CI = 5.2 +/- (2.576 * 0.0912)

CI = 5.2 +/- 0.2350

CI = (4.9650, 5.4350)

Question 12:

As the confidence interval increases from 90% to 95% to 99%, the width of the interval also increases. This means that the range of values that we can be confident contains the true mean becomes larger. This pattern emerges because a higher confidence level requires a larger margin of error to account for more variability in the data. As a result, the confidence interval becomes wider to ensure that the true mean is captured within the interval with a higher level of confidence.

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By solving the given equations according to te situation we know that after 60 months Jen will earn a total profit of $5000.

The equals sign is used in mathematical formulas to show that two expressions are equal.

An equation is a mathematical statement that uses the word "equal to" in between two expressions with the same value.

such as 3x + 5 Equals 15 as an example.

There are many different types of equations, including linear, quadratic, cubic, and others.

So, the total profit would be:

Let, T, represent the total profit, then:

T = L + S

T = 300x - 15,000 + 200x - 10,000

T = 500x - 25,000

Now, substitute 60 the place of x as we need to get the profit of 60 moths as follows:

T = 500x - 25,000

T = 500(60) - 25,000

T = 30,000 - 25,000

T = 5000

Therefore, by solving the given equations according to te situation we know that after 60 months Jen will earn a total profit of $5000.

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The correct answer to this question is a. 13/3.

To find the value of E(x^2), we need to multiply each value of x by its corresponding probability and then sum the results. This can be written as E(x^2) = (1^2)(1/6) + (2^2)(2/3) + (3^2)(1/6).

Simplifying the equation gives us:

E(x^2) = (1/6) + (8/3) + (9/6) = (1/6) + (16/6) + (9/6) = (26/6) = 13/3

Therefore, the value of E(x^2) is 13/3.

In conclusion, the correct answer to this question is a. 13/3.

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