A square mirror is framed with stained glass as shown. Each corner of the frame began as a square with a side length of d inches before it was cut to it the mirror. The mirror has a side length of 3 inches. The area of the stained glass frame is 91 square inches. a. Write a polynomial that represents the area of the stained glass frame.What is the side length of the frame?

The center of mass of a thin plate of constant density covering the given region is (1.8, 3.6).

To find the center of mass, we must calculate the weighted average of all the points in the region. The region is bounded by the parabola y = 3x - x² and the line y = -3x.

We must calculate the integral of the region and divide by the total mass. The mass is equal to the area times the density, .

The integral of the region is calculated using the limits of the two curves, yielding a final integral of 32/15. Dividing this integral by the density gives the total mass, and multiplying by the density gives us the center of mass, (1.8, 3.6).

We can also find the center of mass by calculating the moments of the plate about the x-axis and y-axis.

The moment about the x-axis is calculated by finding the integral of the parabola and line using the x-coordinate, and the moment about the y-axis is calculated by finding the integral of the parabola and line using the y-coordinate. Once the moments are found, we can divide each moment by the total mass to get the center of mass.

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a) The approximate probability that X is at most 30 is 0.0111.

b) The approximate probability that X is less than 30 is 0.0073.

c) The (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

(a) Approximate probability that X is at most 30:

To find the probability that X is less than or equal to 30, the binomial distribution formula must be utilised. The Binomial Distribution is a distribution of a discrete random variable. It is used to obtain the probabilities of different values of n independent trials with two possible outcomes: success and failure.

The formula is shown below:

P(X = k) = (nCk)(pᵏ)(q^⁽ⁿ⁻ᵏ⁾), where P is the probability of a specific event occurring, X is the random variable, k is the number of events that occurred, p is the probability of success, q is the probability of failure, and n is the number of trials.

Using the formula:

P(X ≤ 30) = P(X < 30 + 0.5)≈ P(X < 30.5) = F(30.5), where F denotes the cumulative binomial probability function.

Using the binomial distribution formula:

P(X ≤ 30) = F(30.5)≈ F(30.5)= 0.0111.

Hence, the approximate probability that X is at most 30 is 0.0111.

(b) Approximate probability that X is less than 30:

To find the probability that X is less than 30, the binomial distribution formula must be utilized.

Using the formula:

P(X < 30) = P(X ≤ 29 + 0.5)≈ P(X < 29.5) = F(29.5)

Using the binomial distribution formula:

P(X < 30) = F(29.5)≈ F(29.5) = 0.0073

Therefore, the approximate probability that X is less than 30 is 0.0073.

(c) Approximate probability that X is between 15 and 25 (inclusive):

To calculate the probability that X is between 15 and 25 inclusive, use the formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)

Using the binomial distribution formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)≈ F(25.5) - F(14.5) = 0.0644 - 0.0003 = 0.0641

Hence, the (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

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The correct answer is A. y = 6.50x +11. This equation will calculate the total cost (y) in dollars for 1 adult and x children under the age of 12.

I think the answer is 10,560 feet

Answer:

2.1875 miles

Step-by-step explanation:

1 mile = 5,280 feet

990 feet ÷ 5,280 feet = 0.1875 in miles

The false statement is B. 50% of houses in Ames are smaller than 1,499.69 square feet.

Statistics uses both the mean and median as gauges of central tendency, although their definitions and methods of computation vary. A number's mean is determined by adding together all of the values and dividing by the total number of values. A group of numbers has a median, which is the midpoint value, with half of the values above and below.

The median size of a home in Ames is 1,499.69 square feet, thus this claim is untrue. Therefore, 50% of the homes are smaller and 50% are larger than 1,499.69 square feet. Thus, assertion B is untrue.

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The number of bacteria in a second study is modeled by the function, the growth rate r for the equation is 0.017, since the equation is [tex]A(t) = 2500e^{(0.017t)}[/tex].

To find the growth rate r of an exponential function, use the following formula:[tex]A = Pe^{(rt)}[/tex] Where:

To determine r, divide both sides by P and take the natural logarithm of both sides. It yields: ln(A/P) = rt Therefore: r = ln(A/P) / tNow, given that the number of bacteria in a second study is modeled by the equation: [tex]A(t) = 2500e^{(0.017t)}[/tex] Compare the given equation with [tex]A = Pe^{(rt)}[/tex]. The initial amount (P) is 2500, since that is the starting amount. The final amount (A) is [tex]2500e^{(0.017t)}[/tex], since that is the amount after a certain period of time (t).Thus, [tex]r = ln(A/P) / t= ln(2500e^{(0.017t)} / 2500) / t= ln(e^{(0.017t))} / t= 0.017[/tex] (rounded to 3 decimal places)Therefore, the growth rate r for the equation is 0.017.

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The height of the rectangular prism can be calculated using the following formula:

Volume (V) = Length (L) * Width (W) * Height (H)

Therefore, rearranging the formula, we can calculate the height of the prism:

Height (H) = Volume (V) / (Length (L) * Width (W))

For this problem, plugging in the known values, we get:

Height (H) = 490 cm³ / (10 cm * 7 cm)

Therefore,

Height (H) = 8.14 cm

According to the graph, the balloon ascends between seconds 0 and 2; it remains stable between seconds 2 and 3; drops rapidly between 3 and 4 seconds; it descends slowly between seconds 4 and 6. Additionally, the natural thing is that it does not ascend again because gravity will not allow it to ascend.

To describe the movement of the balloon we must analyze the relationship between the height of the bomb and time. Based on the above, we see that it ascends, holds, descends rapidly, and then slows its rate of descent as described below:

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b a

Step-by-step explanation:

The first quartile of the data set is 48.

Here is a list of 27 scores on a Statistics midterm exam: 20, 30, 31, 32, 46, 48, 49, 52, 54, 59, 61, 69, 71, 73, 74, 79, 81, 81, 81, 85, 86, 87, 88, 91, 94, 96, 97. Find Q2 and Q1.The median is the middle number of a data set arranged in ascending or descending order. There are 27 numbers in this data set. As a result, the median will be the 14th value when sorted in ascending order. The data set is given in ascending order. As a result, the median of the data set is 81. To find the first quartile or Q1 of this data set, the formula below will be used: Q1 = (n+1)/4th term Q1 = (27+1)/4th termQ1 = 7th terTo find the 7th term, the data set must be arranged in ascending order. The 7th term of the data set is 48.Therefore, the first quartile of the data set is 48.

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Answer:

w = 28 miles

Step-by-step explanation:

The perimeter is the sum of the lengths of the sides.

The sum of the lengths of the sides is w + 28 + 28.

The perimeter is 84.

w + 28 + 28 must equal 84.

w + 28 + 28 = 84

Now we solve for w.

Add 28 and 28.

w + 56 = 84

Subtract 56 from both sides.

w = 28

Answer: w = 28 miles

The direction of the greatest decrease is< -840, 2 >The directional derivative in the direction of the greatest decrease is given by

[tex]∇f∙(-840,2) = (-840(6) + 2(-1))/(√(840^2 + 2^2))∇f∙(-840,2) = -3,997.6[/tex]

Therefore, the most negative rate of change is -3,997.69.

The temperature at any point in the plane is given by [tex]T(x,y)=140x^2+y^2+2.F[/tex].

The minimum value of the directional derivative at (3,−1)

The directional derivative of a function is the rate at which the function changes, i.e., its rate of change, in a specific direction.

The maximum and minimum directional derivatives of a function are crucial concepts that are frequently used to describe the properties of a function's surface.

A direction vector of an equation, i.e., the slope of the equation, is the direction of the greatest increase. If the negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

Let’s find the direction of the greatest decrease in temperature at the point (3,−1)

The gradient vector is,[tex]∇T(x, y) = < dT/dx, dT/dy >∇T(x, y)[/tex] = [tex]< 280x, 2y >∇T(3, -1) = < 840, -2 >[/tex]The negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

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The given figure is made up of a square, rectangle and an arc as shown in the figure. The perimeter will be 38 units and the area will be 95 square units.

Perimeter the the total length of the boundaries. Here we know all the lengths except the length of the arc.

Length of an arc = θ/360 × 2πr

 Here θ = 90

So length = 90/360 × 2× 3.14× 5 = 7.85 units

So perimeter = 10 + 5 + 7.85 + 5 + 10 = 37.85 ≈ 38 units

Area is formed by a rectangle, square and an arc

Area of the given rectangle = l×b = 10× 5 = 50 sq. units

Area of the square = s² = 5² = 25 sq. units

Area of the sector =  θ/360 × πr² = 90/360× 3.14× 5² = 19.64 sq. units

So the total area = 50 + 25 + 19.64 = 94.64 ≈ 95 sq. units

So perimeter is 38 units and area is 95 units².

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The complete question is:

This figure is made from a part of a square and a part of a circle.

What is the perimeter of this figure, to the nearest unit?

What is the area of this figure, to the nearest square unit?

Diagram provided as image.

The probability that the coin lands on heads if one of them is flipped and lands on heads with probability 0.6 is 0.6 × 1/2 + 0.3 × 1/2 = 0.45. Therefore, the probability that the coin lands on heads is 0.45.  

a) Let A be the event that the chosen coin is the one that lands on heads with probability 0.6 and B be the event that the coin lands on heads. Then, the required probability is P(A | B) = P(A and B) / P(B) .

Here, P(A and B) = probability that the chosen coin is the one that lands on heads with probability 0.6 and it actually lands on heads.

Since the probability that the coin lands on heads are 0.45 and the probability that the chosen coin is the one that lands on heads with a probability of 0.6 is 1/2, we have P(A and B) = 0.6 × 1/2 = 0.3. The probability that the coin lands on heads is 0.45.

So, P(B) = probability that the coin lands on heads = 0.45.P(A | B) = P(A and B) / P(B) = 0.3 / 0.45 = 2/3.

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Answer: 15

Step-by-step explanation:

13--2 = 13 + 2 = 15

As per the given APR, the sum of amount after 18 years is $66,373. 60, and the total deposits made over the time period is  $43,200. (option d).

To calculate this, we can use the formula for future value of an annuity:

FV = PMT x (((1 + r)⁻¹) / r)

where FV is the future value, PMT is the monthly payment, r is the monthly interest rate (which is calculated by dividing the APR by 12), and n is the number of payments (which is 18 x 12 = 216 in this case).

Plugging in the numbers, we get:

FV = $200 x (((1 + 0.045/12)²¹⁶ - 1) / (0.045/12)) = $66,373.60

Therefore, you would have approximately $66,373.60 in your investment plan after 18 years.

Now let's compare this amount to the total deposits made over the time period. In this case, the total deposits would be:

$200 x 12 x 18 = $43,200

Hence the correct option is (d).

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The triangle LMN, with vertices L(6, −8), M(4, −4), and N(−12, 2), dilated with respect to the origin by a scale factor of 1/2, results in triangle L'M'N', with vertices L'(3, -4), M'(2, -2), and N'(-6, 1)

To dilate a triangle with respect to the origin, we need to multiply the coordinates of each vertex by the scale factor. In this case, the scale factor is 1/2, so we multiply each coordinate by 1/2.

The coordinates of L' are obtained by multiplying the coordinates of L by 1/2:

L'((1/2)6, (1/2)(-8)) = (3, -4)

The coordinates of M' are obtained by multiplying the coordinates of M by 1/2:

M'((1/2)4, (1/2)(-4)) = (2, -2)

The coordinates of N' are obtained by multiplying the coordinates of N by scale factor 1/2:

N'((1/2)×(-12), (1/2)×2) = (-6, 1)

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The given question is incomplete, the complete question is:

Triangle LMN with vertices L(6, −8), M(4, −4), and N(−12, 2) is dilated with respect to origin by a scale factor of 2 to obtain triangle L′M′N′. What are the new coordinates of  L′M′N′ ?

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

Cost computation helps in deciding on pricing, manufacturing output, and sales. It also helps in figuring out the costs of the products and services the company sells.

Let's assume that a cup of tea costs t and a cup of coffee costs c.

We can infer the following based on the initial piece of knowledge:

3t + 1c = 5.10 --------------(1)

We learn the following from of the second piece of information:

1t + 4c = 8.30 --------------(2)

We can find the solutions to t and c because we have two equations with two variables.

Equation (1) is multiplied by 4 and equation (2) is taken away to yield the following result:

9t = 5.90

Therefore:

t = 0.65

Using t = 0.65 as the replacement in equation (1), we get:

3(0.65) + 1c = 5.10

1c = 5.10 - 1.95

1c = 3.15

Therefore:

c = 3.15

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

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The length of the rectangle is 45 yards and the width is 9 yards whose perimeter is 108yd.

A rectangle is a four-sided flat shape with four right angles (90-degree angles) between the adjacent sides. The perimeter of a rectangle is the sum of the lengths of its sides, and the area of a rectangle is the product of its length and width.

Let L be the length.

Let W be the width.

From the problem, we know that L = 5W (since the length is five times the width).

P = 2L + 2W.

Substituting L = 5W into this formula, we get:

P = 2(5W) + 2W = 10W + 2W = 12W

We're also given that the perimeter of the rectangle is 108 yards, so we can set up the equation:

12W = 108

Solving for W, we get:

W = 9

Now that we know the width is 9 yards, we can use the equation L = 5W to find the length:

L = 5(9) = 45

Therefore, the length of the rectangle is 45 yards and the width is 9 yards.

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Amy completes one lap in four minutes. She won't complete more than 7 laps today. The possible numbers of minutes she will run today are 4, 8, 12, 16, 20, or 24 minutes.

Amy runs each lap for 4 minutes. She will run less than 7 laps today. To find the possible number of minutes she will run today, we need to find the minimum and maximum number of minutes she can run.

If Amy runs only one lap, she will take 4 minutes. If she runs two laps, it will take her 8 minutes (2 laps x 4 minutes per lap). Similarly, three laps will take 12 minutes, four laps will take 16 minutes, five laps will take 20 minutes, and six laps will take 24 minutes.

Since Amy is running less than 7 laps, the minimum number of minutes she can run is 4 minutes (for one lap) and the maximum number of minutes she can run is 24 minutes (for six laps). Therefore, the possible numbers of minutes she will run today are 4, 8, 12, 16, 20, or 24 minutes.

It is important to note that the actual number of minutes Amy will run today will depend on the number of laps she decides to run.

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The area of the replica is approximately 1.33 square feet, Rounded to the nearest hundredth.

If the original artwork has an area of 12 square feet, then a replica created by dilating the original by a scale factor of 1/3 will have an area that is (1/3)^2 = 1/9 of the original area. Therefore, the area of the replica will be:

12 square feet × 1/9 = 4/3 square feet ≈ 1.33 square feet

A scale factor is a multiplier that scales or resizes a shape, figure, or object. It is used to determine the size of a new shape or figure when it is scaled up or down by a certain factor. The scale factor can be expressed as a ratio, a fraction, or a decimal. For example, if a rectangle has a length of 10 units and a width of 5 units, and it is scaled up by a factor of 2, the new rectangle will have a length of 20 units and a width of 10 units. The scale factor, in this case, is 2.

Scale factors are commonly used in geometry, especially in transformations such as dilation, which involves scaling a shape by a certain factor without changing its shape. Scale factors can also be used in real-world applications such as maps, where a scale factor is used to represent the ratio between the distance on the map and the actual distance on the ground.

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Complete Question:

an art gallery is selling replicas of some famous artworks. a copy of the artwork is dilated by a scale factor of [tex]\frac{1}{3}[/tex] to create a replica of the original piece. if the area of some original artwork was 12 square feet, what will be the area of the replica? enter the answer in the box, rounded to the nearest hundredth.

Answer :

1. We are given with a parallelogram whose opposite sides are :-

We know that opposite sides and angles of the parallelogram are equal.

PS = RQ

[tex] \implies \sf \: (-1 + 4x) = (3x + 3) \\ [/tex]

[tex] \implies \sf \: 4x - 3x = 3 + 1\\ [/tex]

[tex] \implies \sf \: x = 4 \\ [/tex]

Length of RQ is (3x + 3)

Substituting value of x = 4 in RQ.

[tex] \implies \sf \: (3x +3) \\ [/tex]

[tex] \implies \sf \: 3(4) + 3 \\ [/tex]

[tex] \implies \sf \: 12 + 3 \\ [/tex]

[tex] \implies \sf \: 15 \\ [/tex]

Therefore, Length of RQ will be 15

2. Find [tex] \sf \angle G [/tex]

We are given with :-

Since, Opposite angles of parallelogram are also equal :

[tex]\angle G = \angle E \\ [/tex]

[tex]\implies \sf \: 5x -9 = 3x + 11 \\ [/tex]

[tex]\implies \sf \: 5x - 3x = 11 +9 \\ [/tex]

[tex] \implies \sf \: 2x = 20 \\ [/tex]

[tex] \implies \sf \: x = \dfrac{20}{10} \\ [/tex]

[tex] \implies \sf \: x = 10 \\ [/tex]

Since [tex] \sf \angle G [/tex] is 5x - 9.

Substituting value of (x = 10) in [tex] \sf \angle G [/tex]

[tex] \implies \sf \: 5(10) - 9 \\ [/tex]

[tex] \implies \sf \: 50 -9 \\ [/tex]

[tex] \implies \sf \: 41 \\ [/tex]

Therefore, [tex] \sf \angle G [/tex] is 41.

Answer:

19. QR = 15 units.

20. m ∡G=41°

Step-by-step explanation:

The properties of a parallelogram are:

Question No. 19.

We know that the opposite side of parallelogram is equal, So

PS=QR

-1+4x=3x+3

First of all we will find the value of x.

-1 + 4x = 3x + 3

Subtracting 3x from both sides, we get:

x - 1 = 3

Adding 1 to both sides, we get:

x = 4

SO, QR=3*4+3=15

Side QR is 15 units.

Question No. 20.

We know that the opposite angle of parallelogram is equal, So

EF=GD

3x+11=5x-9
Subtracting 3x from both sides:

3x + 11 - 3x = 5x - 9 - 3x

11 = 2x - 9

Next, we can add 9 to both sides to isolate the variable term on one side:

11 + 9 = 2x - 9 + 9

20 = 2x

Finally, we can solve for x by dividing both sides by 2:

20/2 = 2x/2

x=10

Now

m ∡G=5x-9=5*10-9=41°

therefore m ∡G=41°

Step-by-step explanation:

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

A 3-ary tree is a tree in which each internal node has exactly three children, there are 1086 leaf nodes in a 3-ary tree with 6 internal nodes.

For a 3-ary tree with 6 internal nodes, there will be a total of 7 levels (0 to 6). Consider the following formula for the number of nodes in a tree of height[tex]h: n = 1 + 3 + 3^2 + 3^ + ... + 3^h[/tex] The given 3-ary tree has a height of 6, so its number of nodes is:[tex]n = 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6[/tex]

Using the geometric series formula, we can simplify this: [tex]n = (3^7 - 1) / 2n = (2186 - 1) / 2n = 1092[/tex] The number of leaf nodes in a 3-ary tree with 6 internal nodes is:[tex]n - 6n - 6 = 1092 - 6n - 6 = 1086[/tex] Therefore, there are 1086 leaf nodes in a 3-ary tree with 6 internal nodes.

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The height of the building is 10.39 foot based on the length and distance between the ladder.

The height of the building will be calculated using Pythagoras theorem. The formula that will be used is -

Hypotenuse² = Base² + Perpendicular², where Hypotenuse is ladder, base is the distance between the two and perpendicular is the height of the building. Let us represent height of building as h.

12² = 6² + h²

144 = 36 + h²

h² = 144 - 36

h² = 108

h = ✓108

h = 10.39 foot

Hence the building is 10.39 foot tall.

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The probability that both A and B occur is given by P(A and B) = P(B | A) * P(A) = 0.2 * 0.5 = 0.1.

This can be visualized using a Venn diagram, where the intersection of A and B represents the probability of both events occurring, which is equal to 0.1 in this case.

The probability of B occurring and A not occurring is given by P(B and not A) = P(B) - P(B | A) * P(A') = 0.3 - 0.2 * 0.5 = 0.2. This represents the area of the B circle outside of the A circle.

Given that P(A) = 0.6 and P(A and B) = 0.5, we can use Bayes' theorem to find P(B | A) as follows: P(B | A) = P(A and B) / P(A) = 0.5 / 0.6 = 0.83. This means that the probability of B occurring given that A has occurred is 0.83.

We can also visualize this using a Venn diagram, where the overlap between A and B represents the probability of both events occurring, and the B circle represents the probability of B occurring given that A has occurred.

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Jordy's first error in his proof is option (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

Jordy's first error in his proof is that he used an inappropriate congruence criterion to prove that the two triangles are congruent. He established all necessary conditions, but AAA congruence is only valid for proving congruence of triangles in certain special cases, such as when the triangles are similar.

In general, AAA congruence is not a valid congruence criterion. This highlights the importance of choosing the correct congruence criterion when proving that two triangles are congruent, as using an invalid or inappropriate criterion can lead to an incorrect conclusion.

Therefore, the correct option is (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

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Two circles intersect at points A and B and have a common external tangent that is tangent to the circles at points T and U, M be the intersection of line AB and TU. If AB = 9 and BM = 3, then TU will be equal to 9.6.

In order to find the length of TU, We can use the Pythagorean theorem to find the length of TU. We know that AB = 9, and BM = 3, so the length of AM must be 6. We can then use the Pythagorean theorem to solve for TU:

TU = √(62 + 92) = √93 = 9.6.

Therefore, the length of TU is 9.6.

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The distribution of C) X+Y is a Poisson RV with parameter 9.

This is because the sum of two independent Poisson distributions with parameters λ1 and λ2 is also a Poisson distribution with parameter λ1 + λ2. Therefore, X+Y follows a Poisson distribution with parameter 4+5 = 9.

Option A is incorrect because an exponential distribution cannot arise from the sum of two Poisson distributions. Option B is also incorrect because the parameter of X+Y is not the average of the parameters of X and Y. Option C is the correct answer as explained above.

In summary, the distribution of X+Y is Poisson with parameter 9.

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