Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station.

The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.

(1, 4), (2, 8), (3, 16), (4, 32)

Part A: Is this data modeling a linear function or an exponential function? Explain your answer. (2 points)

Part B: Write a function to represent the data. Show your work. (4 points)

Part C: Determine the average rate of change between station 2 and station 4. Show your work. (4 points)

do not copy and paste from a different question because those aren't right.

The ratio of total points scored to games played for each player are  Kiara 7:1,  Cecilia = 6:1, and Kayleigh= 7:1. (b) The ratios of Kiara and Kayleigh are proportional

You should know that two ratios are proportional to each other if they are equal.  also, you should be aware that ratios show the number of times a number appears in another number when divided.

The parameters that will guide us to get these ratios are

Players             Number of games     Total points             Ratios

Kiara                               25                      175                         1:7

cecilia                             20                       120                       1:6

Kayleigh                          15                        105                       1:7

Therefore from the table shown above, the ratios of Kiara and Kayleigh are proportional with 1:7 respective.

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The graph of equation is attached.

According to the mathematical definition of inequality, neither side is equal. When a relationship leads to a non-equal comparison between two expressions or numbers, it is known as an inequality in mathematics. The equal sign "=" in this statement can be replaced with any of the inequality symbols, such as greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤), or not equal to (≠). The different types of mathematical inequality include absolute value inequality, rational inequality, and polynomial inequality.

Given inequality,

y > 4x + 1

table for x and y

x    0     2       -1

y   >1    >9    >-3

with these points we can plot the graph and shaded region shows the inequality.

Hence the graph is attached.

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a. The intervals of equal width must be used because it allows for a fair comparison.

b. Some of the data can be left out of the frequency polygon if it is not relevant to the analysis.

a) For a frequency polygon to accurately display the distribution of data, intervals of equal width must be used because it allows for a fair comparison between different parts of the data. If the intervals are not equal in width, it can give the impression that there are more or fewer data points in a particular interval, which can be misleading.

For example, if we have data on the heights of 100 people and we want to create a frequency polygon to display this data, we would first need to decide on the intervals that we want to use. If we use intervals of equal width, such as 1 inch, it would allow us to fairly compare the number of people who are between 5'0" and 5'1", 5'1" and 5'2", and so on. However, if we use intervals of unequal width, such as 1 inch for heights between 5'0" and 5'3", and 2 inches for heights between 5'3" and 5'6", it would give the impression that there are fewer people in the second interval than in the first, even if that is not the case.

b) Some of the data can be left out of the frequency polygon if it is not relevant to the analysis that we are trying to perform. For example, if we are interested in understanding the distribution of heights among people who are between the ages of 18 and 25, we could exclude data on the heights of people who are outside of this age range from our frequency polygon.

However, it is important to be careful about which data we choose to exclude, as it can significantly affect the interpretation of the results. If we exclude too much data, it could lead to a distorted view of the distribution of the data.

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Answer:

Step-by-step explanation:

b

The length of DF along the new border is 41 feet. The value of x is 8.5. The length of XD is 95 feet.

Similar Triangles are defined as two triangles with the same shape, equal pair of corresponding angles, and the same ratio of the corresponding sides.

Triangular garden patch in his backyard, represented by AEXY.

Border DF that is parallel to XY, where XY/DE = 0.64.

The length of XY on the existing border is 16 feet.

EY = (14x-7.5), YF = (6x), and EX = (12x-7).

We can find the length of DF by using the ratio given in the problem. Since XY/DE = 0.64, we have DE = XY / 0.64 = 16 / 0.64 = 25.

Therefore, the length of DF is 25 + 16 = 41 feet.

To find the value of x, we can use the equations given in the problem. We are told that EY = 14x - 7.5 and YF = 6x, so adding these equations gives us the length of EF = EY + YF = 14x - 7.5 + 6x = 20x - 7.5.

We are also told that EX = 12x - 7, so adding this to EF gives us the length of FX = EX + EF = 12x - 7 + 20x - 7.5 = 32x - 7.5.

We know that FX is the same as DF, so 32x - 7.5 = 41.

Solving this equation for x gives us x = 8.5.

Finally, to find the length of XD, we can use the equation EX = 12x - 7. Substituting in the value of x that we found above, we get :

EX = 12(8.5) - 7 = 102 - 7 = 95.

Therefore, the length of XD is 95 feet.

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Let [r, s] denote the least common multiple of positive integers r and s. Find the number of ordered triples (a, b, c) of positive integers for which [a, b] = 1000, [b, c] = 2000, and [c, a] = 2000.

Solution 1

It's clear that we must have a = 2^j5^k, b = 2^m^6 and c = 2^p5^q for some nonnegative integers j, k, m, n, p, q. Dealing first with the powers of 2: from the given conditions, max(j, m)=3, max(m, p) = max(p, j) = 4. Thus we must have p = 4 and at least one of m, j equal to 3. This gives 7 possible triples (j,m,p): (0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1, 4) and (3, 0, 4).

Now, for the powers of 5: we have max (k, n) = max(n, q) = max(q, k) = 3. Thus, at least two of k, n, g must be equal to 3, and the other can take any value between 0 and 3. This gives us a total of 10 possible triples: (3, 3, 3) and three possibilities of each of the forms (3, 3, n), (3, n, 3) and (n, 3, 3).

Since the exponents of 2 and 5 must satisfy these conditions independently, we have a total of 7. 10= 70 possible valid triples.

Solution 2

1000=2³5³ and 2000= 2⁴5³. By looking at the prime factorization of 2000, c must have a factor of 2¹. If c has a factor of 53, then there are two cases: either (1) a or b = 5³2³, or (2) one of a and has a factor of 5³ and the other a factor of 2³. For case 1, the other number will be in the form of 2^x5^y, so there are 4 4 16 possible such numbers; since this can be either a orb there are a total of 2(16)-1=31 possibilities. For case 2, a and b are in the form of 2³5^x and 2^y5³, with a x<3 and y<3 (if they were equal to 3, it would overlap with case 1). Thus, there are 2(3-3)= 18 cases.

If c does not have a factor of 5³, then at least one of a and b must be 2³5³, and both must have a factor of 5³. Then, there are 4 solutions possible just considering a = 2353, and a total of 4.2-17 possibilities. Multiplying by three, as 0<=c<=2, there are 7.3 = 21.

Together, that makes 31 +18+21= 70 solutions for (a, b, c).

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There will be approximately 1.9 billion rabbits after 30 days. It will take approximately 9.8 days for the population of rabbits to reach 10,000.

An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.

a) To find the number of rabbits after 30 days, we can substitute 30 for t in the equation [tex]P(t) = 18(2)^t[/tex]:

[tex]P(30) = 18(2)^{30}[/tex]

= 18 × 1073741824

= 1934917632

So there will be approximately 1.9 billion rabbits after 30 days.

b) To find the time it takes for the population to reach 10,000 rabbits, we need to solve the equation P(t) = 10,000 for t. We can do this by first dividing both sides of the equation by 18 to get:

(1/18)P(t) = (1/18)(10,000)

[tex](2)^t = (10,000/18)[/tex]

We can then take the logarithm of both sides of the equation with base 2:

t = log₂((10,000/18))

= log₂(555.56)

= 9.8

Therefore, tt will take around 9.8 days for the rabbit population to reach 10,000.

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After the discount and the tax is applied, the final cost of the shoes is $42.21

Remember that if the original price is P, and we have a discount of X, the new price will be:

P' = P*(1 - X/100%)

If instead we have an increase, like a tax, the new price will be:

P' = P*(1 - x/100%)

Here the original price is $67, first we apply a discount of 40% so we get:

P' = $67*(1 - 40%/100%)

P' = $67*(1 - 0.4)

P' = $67*(0.6)

And now we apply a tax of 5%, then the final price will be:

P'' = $67*(0.6)*(1 + 5%/100%)

P'' = $67*(0.6)*(1.05) = $42.21

The final cost of the shoes is $42.21

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The range of a data-set is given by the difference between the highest value of the data-set by the smallest value of the data-set.

The highest and smallest high temperatures for the city are given as follows:

Hence the range is obtained as follows:

105 - 57 = 48º.

80.2º would be neither the highest nor the smallest value in the data-set, which would then remain constant.

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations.

As it involves all values, it would be changed the most by changing the value of 58º to 101º, and would then assume a value of 84º.

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Replacing y(t) into the initial value, we reach an identity, meaning that it can be verified that both y(t) = 1 - t and y(t) = -t²/4 are solutions to the initial value problem.

The initial value problem is defined as follows:

[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]

The first solution is given as follows:

y(t) = 1 - t.

Hence the derivative is given as follows:

y'(t) = -1.

Replacing into the initial value problem, we have that:

[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]

[tex]-1 = \frac{-t + \sqrt{t^2 + 4(1 - t)}}{2}[/tex]

[tex]-1 = \frac{-t + \sqrt{t^2 - 4t + 4}}{2}[/tex]

[tex]-1 = \frac{-t + \sqrt{(t - 2)^2}}{2}[/tex]

[tex]-1 = \frac{-t + t - 2}}{2}[/tex]

-1 = -1.

Identity, verifying the solution.

The second solution is given as follows:

y(t) = -t²/4.

Hence the derivative is of:

y' = -0.5t.

Replacing into the initial value problem, we have that:

[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]

[tex]-0.5t = \frac{-t + \sqrt{t^2 - 4\frac{t^2}{4}}}{2}[/tex]

[tex]-0.5t = \frac{-t + \sqrt{0}}{2}[/tex]

-0.5t = -0.5t.

Identity, verifying the solution.

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Explanation:

The special symbol [tex]\varnothing[/tex] represents the empty set. Another way to write the empty set is to use curly braces with nothing inside them. Not even zero is inside the empty set.

One property of set mathematics is that the empty set is a subset of every set, even itself.

Therefore, the claim [tex]\varnothing \subseteq \left\{\text{Charlie Chaplin},\text{Groucho Marx},\text{Woody Allen}\right\}[/tex] is a true statement.

Answer:

B can

Step-by-step explanation:

126 and 79 are not divisible by 5, only numbers that end in 0 or 5 can

The mean absolute deviation (MAD) of the data set is equal to 180.625.

The mean absolute deviation of a data set is to find the average distance of each observation of the data set and the mean of the data set.

Mean absolute deviation = 1/n ∑ absolute value of (xi - x bar)

x bar =average value of the given data set

n= total number of data set

xi = data value in a given set

According to the question,

x bar = sum of all the data set / total number of the data set

x bar = 2052/16 = 128.25

Substitute the value in the formula, we have

Mean absolute deviation = 2890/16 = 180.625

Hence, the Mean absolute deviation of the data set is equal to 180.625.

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As per the given data, the Mean absolute deviation (MAD) of the data set is equals to 17.5

The term Mean absolute deviation in math is defined as the average distance of each observation of data set  and the mean of the data set.

Here we have given the following data.

=> 13, 0 , 14 , 36, 18, 9

Then the mean of the data is calculated as,

=> (13 + 0 + 14 + 36 + 18 + 9)/6

=> 90/6

=> 15

Now, we have to the Substitute the value in the formula, we have

And then the Mean absolute deviation is calculated as,

=> (90 + 15)/6

=> 105/6

=> 17.5

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The 95% confidence interval for the amount spent on the child's last birthday gift is between  $44.5 and $55.5.

To construct a 95% confidence interval, we first calculate the margin of error, which is equal to 1.96 times the standard deviation divided by the square root of the sample size.

In this case, the margin of error is 1.96 x 13/√14 = $5.5. Then, we add the margin of error to the mean to obtain the upper bound of the confidence interval, which is 50 + 5.5 = $55.5. We subtract the margin of error from the mean to obtain the lower bound of the confidence interval, which is 50 - 5.5 = $44.5.

Finally, we round the upper and lower bounds to the nearest decimal place, which gives us the 95% confidence interval of (36.4, 63.6).

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A like radical to the one shown is Option C.

We know that we can be able to perform mathematical operations on the radicals such as adding and subtracting but this must be performed when we have like radicals.

When we talk about like radicals we mean that the radicals must be such that it easy for the radical to be worked upon as we have it in the question that is before us here to locate a like radical among the options.

The radical in option C has the same 2x and power as the given radical hence they are like radicals.

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The equation of the perpendicular line is y= -5/4x +2/5 when the equation of the line is -4x+5y=2.

Given that,

The equation of the line is -4x+5y=2.

We have to find the equation of the perpendicular line passing through the equation.

We know that,

Take the equation of the line is -4x+5y=2.

5y=2+4x

y= 4/5x+2/5

The equation is in the form of y=mx+b

We flip the number and alter the sign to determine a perpendicular line.

m is 4/5

So, for perpendicular line equation

m =  -5/4

Therefore, The equation of the perpendicular line is y= -5/4x +2/5 when the equation of the line is -4x+5y=2.

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When the square sheet of cardboard is cut and folded to create a rectangular box with a lid, the maximum possible volume is achieved when x = 20 inches. All other values will result in a smaller volume for the box.

1. Calculate the side lengths of the box and lid:

Box: 40 inches - 2x = 40 - 2x

Lid: 2x

2. Calculate the volume of the box with a lid when x = 20 inches:

Box Volume:

(40 - 2x) x (40 - 2x) x (40 - 2x) = (40 - 2*20) x (40 - 2*20) x (40 - 2*20) = (20) x (20) x (20)

= 8000 in^3

Lid Volume: 2x x 2x x 2x = 2*20 x 2*20 x 2*20

= 8000 in^3

Total Volume: 8000 in^3 + 8000 in^3 = 16000 in^3

3. Calculate the volume of the box with a lid when x = 20/3 inches:

Box Volume: (40 - 2x) x (40 - 2x) x (40 - 2x) = (40 - 2*20/3) x (40 - 2*20/3) x (40 - 2*20/3) = (33 1/3) x (33 1/3) x (33 1/3)

= 7157.7 in^3

Lid Volume: 2x x 2x x 2x = 2*20/3 x 2*20/3 x 2*20/3

= 1365.3 in^3

Total Volume: 7157.7 in^3 + 1365.3 in^3

= 8523 in^3

4. Comparison: When x = 20 inches, the box has a minimum possible volume of 16000 in^3, while when x = 20/3 inches, the box has a volume of 8523 in^3. Therefore, the statement is true

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The expression that represents how much money that will be in the account is A = 800.00(1 + 0.015)⁴.

The expression that represents how much money you will have in one year if 6% is compounded quarterly on a 800$ for one year is as follows:

Hence, using the compound interest formula,

A = P + I

where

Hence,

r = 6/100

r = 0.06 rate per year

Then solve the equation for A

[tex]A = p(1 + \frac{r}{n} )^{nt}[/tex]

where

Therefore,

A = 800(1 + 0.06/4)⁴ˣ¹

Therefore, the expression is  A = 800(1 + 0.015)⁴

A = $849.09

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The equation that can be used to find the number of peaches Muriel has is; 2 + 3·x = 17

A mathematical equation is a statement that has two expressions, joined by the equals to, '=' sign.

The number of peaches Frances has = 17

Let x represent the number of peaches Muriel has

Therefore;

The number of peaches Frances has = 2 + 3·x

Which indicates through substitution property, that we get;

17 = 2 + 3·x

The equation that can be used to find the number of peaches Muriel has, x is; 17 = 2 + 3·x

Solving for x in the above equation, we get;

17 = 2 + 3·x

3·x = 17 - 2 = 15

x = 15 ÷ 3 = 5

The number of peaches Muriel has, x = 5 peaches

The 17 peaches Frances has and the indication that Frances has 2 more than 3 times the number of peaches Muriel has, indicates that the equation that could be used to find the the number of peaches Muriel has is; 17 = 2 + 3·x

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According to the given statement  if Divisor = 5,  Dividend = 3075 then The quotient is 615.

Division is an arithmetic operation that is the inverse of multiplication. It is used to determine how many times one number, called the divisor, is contained within another number, called the dividend. The result of the division is called the quotient.The three most widely discussed are the Commutative, Associative, and Distributive Laws.

The quotient is 615.

You can find this by dividing the dividend (3075) by the divisor (5) using long division or by using the formula: Quotient = Dividend / Divisor

In this case, 3075 / 5 = 615

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Since AB || CD and m∠2 = 35° in Quadrilateral ABCD, the measure of angle 5 (m∠5) is equal to 35°.

In Mathematics, the Alternate Interior Angles Theorem states that when two (2) parallel lines are cut through by a transversal, the alternate interior angles that are formed are congruent.

Additionally, these angles are generally referred to as alternate interior angles. In this scenario, we can logically deduce the following information and parameters from Quadrilateral ABCD:

Therefore, we can logically deduce and conclude that m∠5 in Quadrilateral ABCD is equal to 35 degrees in accordance with the Alternate Interior Angles Theorem.

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Answer:

34

Step-by-step explanation:

substitute the given values for x, y and z into the expression

(x + y)² + 6z

= (0 + (- 4) )² + 6(3)

= (0 - 4)² + 18

= (- 4)² + 18

= 16 + 18

= 34

The percentage he has chopped is 25%

From the question, we have the following parameters that can be used in our computation:

Total number of tomatoes = 60

Total number of chopped tomatoes = 15

The percentage he has chopped is then calculated as

Percentage = Total number of chopped tomatoes/Total number of tomatoes * 100%

Substitute the known values in the above equation, so, we have the following representation

Percentage = 15/60 * 100%

Evaluate

Percentage = 25%

Hence, the percentage is 25%

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Answer:

$50

Step-by-step explanation:

900 - 350 - 500

50

The estimated answer for 212 divided by 18 is 12.

Estimation means approximating a quantity to the required accuracy.

This is obtained by rounding off the numbers involved in the calculation and getting a quick and rough answer. It is used to find a value that is close enough to the right answer.

Estimation in math is done by rounding off the numbers to their closest whole value to get a quick and simple rough answer. It saves time and effort.

Dividing 212 by 18

= 212/18

= 11.77777

Estimated to be 12

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The polynomial in the standard form is 4m³ - 9m² - m + 25. Then the polynomial is a cubic polynomial.

A polynomial expression is an algebraic expression with variables and coefficients. Unknown variables are what they're termed. We can use addition, subtraction, and other mathematical operations. However, a variable is not divisible.

The polynomial is given below.

⇒ 25 - 9m² - m + 4m³

Convert the polynomial into its standard form. Then we have

⇒ 25 - 9m² - m + 4m³

⇒ 4m³ - 9m² - m + 25

The degree of the polynomial is 3. Then the polynomial is a cubic polynomial.

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The maximum area for each stall can be achieved when the width and length are equal, so the dimensions must be any value greater than 0.

a) Let x = width of each stall, y = length of each stall

b) Area of each stall = xy

c) Domain: x > 0, y > 0

Range: all real numbers

d) To maximize the area of each stall, we must maximize the product of x and y. Taking the derivative of xy with respect to both x and y, we get y = x. Therefore, the maximum area will be achieved when the width = length = x. Therefore, the dimensions that will produce the maximum area for each stall is x = x = any value greater than 0 so the dimensions must be any value greater than 0.

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The complete question is

pat has of feneing to enclose three identical stalls behind the barn. as shown. what dimensions will produce a maximum area for each stall? a) label correctly the given diagram at the side with chosen variables that must be identified. use let statements. b) express the area of the stalls as a function of its width. c) determine the domain and range of this area function. show your work. domain: range: d) determine the dimensions thst would produce a maximum area for each stall.

A pair of alternate interior angles given that g || h include the following: D. <5 and <7.

In Mathematics, parallel lines can be defined as two (2) lines that are always the same (equal) distance apart and never meet.

The Alternate Interior Angles Theorem states that when two (2) parallel lines are cut through by a transversal, the alternate interior angles that are formed are congruent. Additionally, these angles are generally referred to as alternate interior angles.

In this scenario, we can logically conclude that angle 5 and angle 7 forms a pair of alternate interior angles because line g is parallel to line h.

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Evaluating and simplifying -4 5/6 + (-1 2/3) gives the value of = 6 1/2

The equation involves -4 5/6 + (-1 2/3) addition of fraction

The fraction required are:

-4 5/6  and -1 2/3

converting the mixed numbers to improper fractions

= -29/6 + (-5/3)

making the numbers have similar denominator by multiplying by 2/2

= -29/6 - 5/3 * 2/2

= -29/6 - 10/6

The addition is done by

= (-29 - 10) / 6

= -39/6

simplifying gives

= -39/6

= -13/2

= 6 1/2

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B. Students who did not complete the review sheet were less likely to have passed the exam than students who did complete the review sheet.

The given two-way frequency table shows the number of students who passed the exam and completed the review sheet (20 students), the number of students who passed the exam but did not complete the review sheet (10 students), the number of students who did not pass the exam but completed the review sheet (5 students), and the number of students who did not pass the exam and did not complete the review sheet (5 students). From this information, it can be seen that 20 out of 25 students who completed the review sheet passed the exam, while only 10 out of 15 students who did not complete the review sheet passed the exam. This indicates that students who completed the review sheet were more likely to pass the exam than those who did not complete the review sheet. Therefore, statement B is true.

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According to the concept of frequency table, the correct statement is Students who did not complete the review sheet were less likely to have passed the exam than students who did complete the review sheet.

The term frequency table in math is defined as  a chart that summarizes all the data under two columns that is known as variables/categories, and their frequency.

Here we know that Khaled's eighth-grade class took an exam on two-way frequency tables.

At the time of exam, Khaled's teacher made the two-way frequency table below to show the data on completing the exam review sheet and passing the exam.

Here we know that the number of students who have passed the exam is one who are not completed the review sheet.

Therefore, option (b) is correct.

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