let f be a differentiable function such that f(1)=2 and f′(x)=x2 2cosx 3−−−−−−−−−−−−−√. what is the value of f(4) ?

Answer:

Amount originally invested = $700

Step-by-step explanation:

[This answer assumes that the interest is simple interest.]

We are told that upon investing a certain amount of money for 4 years at a rate of 6%, we get $168 in interest. We are then asked to calculate the amount originally invested.

To solve this problem, we have to use the formula for simple interest:

[tex]\boxed{I = PRT}[/tex],

where:

• I ⇒ interest = $168

• P ⇒ principal (original) amount = $?

• R ⇒ interest rate = 6% = 0.06

• T ⇒ time for which the money is invested = 4 years.

Using the above formula and rearranging it to make P the subject of the equation, we can calculate the principal amount:

[tex]I = PRT[/tex]

⇒  [tex]P = \frac {I}{RT}[/tex]

⇒ [tex]P = \frac{168}{0.06 \times 4}[/tex]

⇒ P = 700

Therefore, the amount originally invested was $700.

Northern Dancer will take 2 2/5 minutes to run the Belmont stakes

An equation is the equality between two algebraic expressions, which have at least one unknown or variable.

2 minutes / (1 1/4 miles) = x / (1 1/2 miles)

We transform the mixed fractions into improper fractions and we have:

1 1/4 =

[(4*1)+1])/ 4=

(4+1) / 4 =

5/4

1 1/2 =

[(2*1)+1] / 2 =

(2+1) / 2 =

3/2

Substituting and clearing the value we get:

2 minutes /5/4 miles = x / 3/2 miles

8/5 = 2x/3

2x = 8/5 * 3

2x = 24/5

x = 24/5 / 2

x = 24/10

x = 12/5 = 2 2/5 minutes

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Correctly written question:

A Horse named Northern dancer won the Kentucky derby with a time of exactly 2 minutes. At this rate, how long would it take Northern dancer to run the Belmont stakes? The Kentucky Derby is  1 1/4 miles in the Belmont stakes is 1 1/2 miles.

False, t < 0 expresses that t is negative in symbols.

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.

Given, t < 0 this explains that the symbolic value of the t will be less than zero in the number line.

Since A horizontal line with evenly spaced numerical increments is called a number line. How the number on the line can be answered depends on the numbers present. The use of the number is determined by the question that it corresponds to, such as when graphing a point.

Therefore, We can say t < 0 expresses that t is negative in symbols.

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the two roots of the equation are x1 = -2.5 and x2 = -1.5.

x1 = -2.5;

x2 = -1.5.

Step 1: Calculate the discriminant

b2 - 4ac = (-2)2 - 4(2)(-3) = 4 + 24 = 28

Step 2: Calculate the roots

x1 = (-b + √D) / 2a = (-(-2) + √28) / 4 = (-2 + 5.29) / 4 = -2.5

x2 = (-b - √D) / 2a = (-(-2) - √28) / 4 = (-2 - 5.29) / 4 = -1.5

We start by calculating the discriminant, which is b2 - 4ac. In this case we have b2 = (-2)2 = 4 and ac = 2(-3) = -6, so the discriminant is 4 + 24 = 28. Then we calculate the two roots, x1 and x2, using the formula x1 = (-b + √D) / 2a and x2 = (-b - √D) / 2a. Substituting in the values we have, x1 = (-(-2) + √28) / 4 = (-2 + 5.29) / 4 = -2.5 and x2 = (-(-2) - √28) / 4 = (-2 - 5.29) / 4 = -1.5. Therefore, the two roots of the equation are x1 = -2.5 and x2 = -1.5.

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First, let's determine what (1/3) to the power of 4 is:

1/3 x 1/3 x 1/3 x 1/3 = 1/81

(1 x 1 x 1 x 1 = 1; 3 x 3 x 3 x 3 = 81)

1/81

Now what is 3 as a decimal?

We have to use the same denominator as 1/81 to be able to do a subtraction:

81 x 3 = 243

3 = 243/81

243/81 - 1/81 = 242/81

The answer is D.

Hope this helps! :) If you have any questions, let me know.

The area of the composite figure that consists of a half circle and a triangle is calculated as  86.24 cm^2.

The composite figure shown is composed of a half circle and a triangle. Therefore:

The area of the composite figure = area of semicircle + area of triangle = pi*r^2 + 1/2(base * height)

r = 4 cm

Base = 2(4) = 8 cm

Height = 9 cm

Area = 3.14 * 4^2 + 1/2(8 * 9)

= 50.24 + 36

Area of composite figure = 86.24 cm^2

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Jessie and Bob's balloon payment will be $529,229.56.

A balloon mortgage is a type of mortgage in which the borrower makes smaller monthly payments over the term of the loan, with a large balloon payment due at the end.

To calculate the balloon payment, we need to know the amount of the loan, the interest rate, and the term of the loan.

Given that Jessie and Bob are financing $425,500 to purchase a house, the interest rate is 6.55%, and the term of the loan is 30/8 (or 3.75) years, we can use the below formula to calculate the balloon payment:

Balloon payment = Loan amount * (1 + Interest rate * Loan term)

Balloon payment = $425,500 * (1 + 6.55% * 3.75)

Balloon payment = $425,500 * 1.249625

Balloon payment = $529,229.56

Therefore, Jessie and Bob's balloon payment will be $529,229.56.

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(D) Statistical significance is achieved when the result is very unlikely to occur by chance. Practical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use. It is possible for a treatment to have statistical significance, but not practical significance is correct.

Statistical significance is achieved when the result is very unlikely to occur by chance, and is usually determined by a p-value. Practical significance is related to whether common sense suggests that the treatment makes enough of a difference to justify its use, and is determined by the magnitude of the effect, such as the size of the difference between the treatment and control groups.

It is possible for a treatment to have statistical significance, but not practical significance. This means that the result is unlikely to occur by chance, but the magnitude of the effect may be small and not clinically meaningful. Conversely, a treatment can also have practical significance, but not be statistically significant, meaning that the magnitude of the effect is large enough to be considered important, but the result may not be statistically significant due to a large sample size, for example.

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If the p-value is less than α in a two-tailed test then the null hypothesis should be rejected.

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution.

A method for drawing statistical conclusions from population data is hypothesis testing. It is a technique for analysis that evaluates presumptions and establishes the likelihood of something based on a predetermined level of accuracy. Testing hypotheses offers a technique to determine whether the outcomes of an experiment are reliable.

Before doing the hypothesis testing, a null hypothesis and an alternative hypothesis are established.

Thus, If the p-value is less than α in a two-tailed test then the null hypothesis should be rejected.

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The solution to the inequality equation -3p > -6 or 4p + 1 > 13 is p < 2 or p > 3

-3p > -6

p < -6/-3

p < 2

or

4p + 1 > 13

Subtract 1 from both sides

4p > 13 - 1

4p > 12

divide both sides by 4

p > 12/4

p > 3

Hence, the solution to the given inequality 3p > -6 or 4p + 1 > 13 is p < 2 or p > 3

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If T is a linear transformation then the standard matrix of T is [tex]\left[\begin{array}{ccc}1&-19\\0&1\end{array}\right][/tex].

What is a matrix?

A matrix is a rectangular array or table with numbers or other objects arranged in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. Matrix operations include addition, scalar multiplication, multiplication, transposition, and many others.

The transformation is given as - T : IR² → IR²

Take the (x,y) values as x(1,0) and y(0,1)

(x,y) = x(1,0) + y(0,1)

Apply the transformation -

T(x,y) = T[x(1,0) + y(0,1)]

Since T is a linear map then -

T(x,y) = x T((1,0)) + y T((0,1))

Now, substitute the values -

T(x,y) = x Te1 + y Te2

T(x,y) = x [e1 - 19e2] + y [e2]

T(x,y) = x e1 + e2(-19x + y)

Resubstitute the values back -

T(x,y) = x (1,0) + (0,1)(-19x + y)

T(x,y) = (x , -19x+y)

For T : IR² → IR²

T(x,y) = (ax+by , cx+dy)

Then A = [tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex]

So, now the matrix A is -

A = [tex]\left[\begin{array}{ccc}1&-19\\0&1\end{array}\right][/tex]

Therefore, the matrix is  [tex]\left[\begin{array}{ccc}1&-19\\0&1\end{array}\right][/tex].

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Answer:

36 is 18% of 200

So you will just want to divide 36 by 200, and multiply by 100.

36/200 = 0.18

When multiplying a decimal by 100, you can just move the decimal point 2 digits to the right.

0.18 * 100 = 18

18%

Step-by-step explanation:

Hope it helps! =D

The value represents the content of the current iteration, while the position represents the index or iteration number of that value.

In a for loop, the "value" refers to the current iteration value assigned to a variable, while the "position" refers to the iteration number or index.

For example, in a for loop that iterates over a list of numbers, the value could be each individual number in the list, while the position could be the index of that number in the list.

For example, consider the following code:

numbers = [1, 2, 3, 4, 5]

for i, value in enumerate(numbers):

   print("Position:", i, "Value:", value)

The output of this code would be:

Position: 0 Value: 1

Position: 1 Value: 2

Position: 2 Value: 3

Position: 3 Value: 4

Position: 4 Value: 5

Here, i represents the position, starting from 0 and increasing by 1 for each iteration, and value represents the value assigned to each number in the numbers list.

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Best data tools to represent each information with the options from the image;

Scatter plot - 7th, 9th, 10th and 12th

stem-and-leaf plot - 1st, 3rd, 5th

histogram - 4th, 8th

box-and-whisker plot - 2nd, 6th, 11th

The form in which data is stored, processed, and transmitted is referred to as data representation. Scatter plot are graphs that show the relationship between two variables in a data set eg. to plot heights and weights of a group of people.

Stem-and-leaf plot is a table in which each data value is divided into a "stem" (the first digit or digits) and a "leaf" (usually the last digit) like; To show how many A's, B's, C's, D's and F's there were on a test.

Histogram are graphs that uses rectangles to show the frequency of numerical data such as to plot how many students brought lunch at school each month

Box-and-whisker plot is a graphical method of representing variation in a set of data such as to show the medians of quartiles of the grades on the chapter 5 test for 3rd periods and to show the median score of the basket ball games.

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The equation that represents the number of fourth-grade students in the group will be 4 x 8 - 27 = f.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

The capacity of four vans is given as,

⇒ 4 x 8

There are 27 third-grade students in the group. Then the number of fourth-grade students in the group will be given as,

4 x 8 - 27 = f

The equation that represents the number of fourth-grade students in the group will be 4 x 8 - 27 = f.

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The mathematical operation is as follow:

12/ 3 + 7 x 1 - 11 = 0

BODMAS rule is an acronym that is used to remember the order of operations to be followed while solving expressions in mathematics. It stands for B - Brackets, O - Order of powers or roots, (in some cases, 'of'), D - Division, M - Multiplication A - Addition, and S - Subtraction.

Given:

12___3___7___1___11 = 0

We have to put the sign such that the resultant value 0.

First take division operation then addition then multiplication  and then subtraction.

12 / 3 = 4

and, 4 + 7 = 11

and, 11 x 1= 11  

and, 11 - 11 = 0

So, 12/ 3 + 7 x 1 - 11 = 0

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The rate at which the population of the city was growing was an average of 1.5 million people per year, which can be expressed as an integer or reduced fraction as 3/2.

To break this down further, this means that the city's population was increasing by an average of 416,666 people each month or 14,053 people each day.

This is an impressive rate of growth, especially when considering that the city was already quite large at 5.2 million people. This growth can be attributed to a number of factors, including high levels of immigration, a strong economy, and a booming job market.

Moreover, the growth in population can be seen as a testament to the city's quality of life and its ability to attract new residents.

The population of the city in 1980 was 5.2 million and by 1992 it had grown to 7.6 million. This shows that the rate at which the population of the city was growing was an average of 1.5 million people per year, which can be expressed as an integer or reduced fraction as 3/2.

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Team A will need to sell at least 256 bags of popcorn to avoid losing money.

When an algebraic equation is graphed, it always results in a straight line since each term in a linear equation has an exponent of 1. For this reason, it is referred to as a "linear equation."

There are both one-variable and two-variable linear equations.

Given total amount to be raised by each team is $100,

Team A wants to sell popcorn,

rent of popcorn machine = $15

total cost to be earned for team B = $100 + $15 = $115

let the number of candies sold be x

cost used in popcorn bag = $0.05 per stick

cost used in x popcorn bag  = $0.05x

the selling price of each popcorn bag  = $0.50

the selling price of x popcorn bag  = $0.50x

total gain = $0.50x - $0.05x

total gain = $0.45x

minimum number of popcorn bags to be sold for no loss

$0.45x = $115

x = 115/0.45

x = 255.55

x = 256

Hence for no loss, at least 256 popcorn bags are to be sold.

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On solving the provided question, we can say that Hence, there are 26 terms in given arithmetic progression.

A series of numbers known as an arithmetic progression is one in which the difference between each term and the term before it stays constant over the course of the sequence. In this mathematical progression, the constant difference is referred to as the tolerance. A series of numbers is arranged in an arithmetic progression (AP) if the difference between any two consecutive integers is a constant value. additionally known as mathematical order. The gap between succeeding terms in an arithmetic progression is always the same. Because the difference between succeeding phrases is always 2, episodes 3, 5, 7, and 9 are examples of arithmetic.

Clearly, the given sequence is an AP with first term a=5 and common difference d=3.

Let there are n terms in this AP. Therefore,

⇒5+(n−1)3=80

⇒(n−1)3=75

⇒n−1=25

⇒n=26

Hence, there are 26 terms in given AP

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Answer:

h = -12

Step-by-step explanation:

Here's a step-by-step explanation to solve the equation:

h/6 - 1 = -3

Start by adding 1 to both sides of the equation to isolate h/6 on one side.

h/6 - 1 + 1 = -3 + 1

h/6 = -2

Next, multiply both sides of the equation by 6 to isolate h on one side.

6 * (h/6) = 6 * (-2)

h = -12

So the solution is: h = -12.

The parametrization of the line of intersection of the two planes is <12, -6, 12>

A parametrization is a way of representing a line or a curve using a set of parameters.

First, we can find the normal vectors of the two planes.

For the first plane, the normal vector is given by <3, -6, -2>, and for the second plane, the normal vector is given by <2, 1, -2>.

The line of intersection will be parallel to the cross product of these two normal vectors, which is <12, -6, 12>.

Plugging in the values from the first plane, we get:

3x - 6y - 2z = 3

3x = 6y + 2z + 3

x = 2y + z + 1

Now that we have a point on the line, we can use this to parametrize the line.

We can use the direction vector <12, -6, 12> and the point <1, 0, 0> to find the parametrization of the line:

x = 1 + t * 12

y = 0 - t * 6

z = 0 + t * 12

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The domains of the relation are (-3, -2, -1, 2, 4)

The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.

The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.

We have,

The independent variables are the domain of the relation.

Now,

The coordinates from the graph are:

(2, 0), (4, -1), (-1, 1), (-3, 4), and (-2, 0)

Now,

2, 4, -1, and -3 are the independent variables.

The domains are (-3, -2, -1, 2, 4)

Thus,

The domains are (-3, -2, -1, 2, 4)

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Answer:

It would be a undefined slope. This is because the x value of a straight vertical line can go infinite in both ways without ever changing its y value.

Distinct prime factors does the product of these numbers have 2,3 , and 5 are the three distinct prime factors

Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers.

Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors.

As we know, a composite number has more than two factors, therefore, this method is applicable only for composite numbers and not for prime numbers.

The numbers must be 4,5 , and 6 .

4+5+6=15 and

4*5*6=120

=2^3*3*5

Distinct prime factors does the product of these numbers have 2,3 , and 5 are the three distinct prime factors.

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Distinct prime factors do the product of these numbers have 2,3, and 5 are the three distinct prime factors

Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers.

Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors.

As we know, a composite number has more than two factors, therefore, this method is applicable only to composite numbers and not to prime numbers.

The numbers must be 4,5, and 6.

4+5+6=15 and

4*5*6=120

=2^3*3*5

Distinct prime factors the product of these numbers have 2,3, and 5 are the three distinct prime factors.

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The ladder makes a 60° angle with the wall.

What makes a right triangle?

A right triangle consists of two legs and a hypotenuse. The two legs meet at a 90° angle and the hypotenuse is the longest side of the right triangle and is the side opposite the right angle.

In this case, we can draw a diagram. What we get is a right triangle with one leg as 8 and the hypotenuse as 16.

This is a 30-60-90 triangle (because the leg is half the hypotenuse).

They want the angle the ladder makes with the wall, so the angle between hypotenuse and leg, which is 60 degrees.

cos (angle) = adjacent / hypotenuse = 8/16

arcos(8/16) = 60°

x = 60°

Hence, The ladder makes a 60° angle with the wall.

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The ladder makes an angle of 60° with the wall . The angle made between the ladder and the wall is 60°.

Here, these values are given,

Height  of ladder = 16 foot

Height of wall = 8 foot

So, We have to find the angle made between the ladder and the brick wall.

Let, the angle made between the ladder and the wall be x °.

So, According to question ,

We use trigonometry for finding the value of angle between the ladder and the wall.

cos∅ °= base/height ( b/h)

Because, base = height of wall and hypotenuse = height of ladder.

Then, cos x° = 1/2

 cos x ° = cos 60°

 So, x° = 60°

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Answer:

Answer: 1/6

Step-by-step explanation:

Total number of numbers = 12Total number of 4  or 6 = 2P( 4 or 6) = 2/12 = 1/6Answer: 1/6

The cumulative percent frequency for students working less than 20 hours per week is 25%.

In statistics, the frequency of the first-class interval is added to the frequency of the second class, and this sum is added to the third class and so on then, frequencies that are obtained this way are known as cumulative frequency (c.f.). A table that displays the cumulative frequencies that are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table. There are two types of cumulative frequency - lesser than type and greater than type. Cumulative frequency is used to know the number of observations that lie above (or below) a particular frequency in a given data set. Cumulative frequency is the total frequencies showcased in the form of a table distributed in class intervals. There are two types of cumulative frequency i.e. lesser than and greater than, let us learn more about both types.

The number of hours frequency is given as follows:

Number of hours   Frequency  Cumulative frequency

0 - 9                          20                20

10 - 19                       80                100

20 - 29                     200               300

30 - 39                     100                400

So, the cumulative percent frequency  for students working less than 20 hours per week is

= (No. of students working less than 20 hours/ total number of students)* 100

= (100/400)*100

= 25%

Thus, the cumulative percent frequency for students working less than 20 hours per week is 25%.

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Todd has 48 words to spell for a puzzle, then he earned $7.35 and he spell correctly 39 words and 9 words missed.

Given that,

Todd has 48 words to spell for a puzzle

His mother offers to pay him 20 cents for each word

If Todd will pay her 5 cents for each word he spells incorrectly,

Let x= number of words Todd spells correctly

48-x = number of words that he misses

In order to avoid the decimals, let's solve this in "cents". Does that make "cents" to you, It means that he gets 20 cents for correct words, he pays 5 cents for words missed, and there is a total income of 735 cents. Here is the equation:

[tex]20*x[/tex] -5(48-x) = 735

[tex]20x[/tex] - 240 + [tex]5x[/tex] = 735

[tex]25x[/tex] - 240 = 735

[tex]25x[/tex] = 735 + 240

[tex]25x[/tex] = 975

[tex]x[/tex] = 39 words correct

48-39 = 9 words missed

So,

39*$.20 -9*$.5 =  $7.35

$780-$45 = $735

Therefore,

Todd has 48 words to spell for a puzzle, then he earned $7.35 and he spell correctly 39 words and 9 words missed.

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Todd has 48 words to spell for a puzzle, then he earned $7.35 and he spells correctly 39 words and 9 words missed.

Given that,

Todd has 48 words to spell for a puzzle

His mother offers to pay him 20 cents for each word

If Todd will pay her 5 cents for each word he spells incorrectly,

Let x= number of words Todd spells correctly

48-x = number of words that he misses

In order to avoid the decimals, let's solve this in "cents". Does that make "cents" to you, It means that he gets 20 cents for correct words, he pays 5 cents for words missed, and there is a total income of 735 cents. Here is the equation:

20x -5(48-x) = 735

20x - 240 + 5x  = 735

25x - 240 = 735

25x = 735 + 240

25x = 975

x = 39 words correct

48-39 = 9 words missed

So,

39*$.20 -9*$.5 =  $7.35

$780-$45 = $735

Therefore,

Todd has 48 words to spell for a puzzle, then he earned $7.35 and he spells correctly 39 words and 9 words missed.

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Multiplying by 0.05 is the same as decreasing by 95%.

Multiplying a number by 0.05 is equivalent to decreasing that number by a certain percentage. To determine this percentage, we can use the formula: 100 - (100 * 0.05) = 100 - 5 = 95. This means that multiplying a number by 0.05 is the same as decreasing it by 95%. For example, if a person has $100 and multiplies it by 0.05, the result will be $5, which is a decrease of 95% from the original amount.

In other words, 0.05 is equivalent to 95% of a number, and multiplying by 0.05 is the same as decreasing the number by 95%. This relationship between multiplication by 0.05 and decreasing by a certain percentage is important in various mathematical and real-world applications.

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