Consider the following system of first order linear differential equations
x' (t) = Ax(t).
Suppose the solution to the system can be written as
x(t) = 5e2t - 3et
Which of the following A allows for the above equation to be a solution to the system?

The differential   [tex]dy=2[/tex] when [tex]x = 4[/tex]  and [tex]dx = 0. 2.[/tex]

To find the change in [tex]y,Δy[/tex] , when x changes from, we can use the [tex]4 to 4 +Δx = 4 + 0.2 = 4.2[/tex] formula:

[tex]Δy = y(x + Δx) - y(x)[/tex]

where[tex]y(x) = 5x^2.[/tex]

So, plugging in[tex]x = 4[/tex] and[tex]x + Δx = 4.2[/tex] , we get:

[tex]Δy = y(4.2) - y(4)[/tex]

[tex]= 5(4.2)^2 - 5(4)^2[/tex]

[tex]= 44.2[/tex]

Therefore, the change in y is [tex]44.2[/tex] when x changes from [tex]4 to 4.2.[/tex]

To find the differential dy when [tex]x = 4[/tex] and [tex]dx = 0.2,[/tex]  we can use the formula:

[tex]dy = f'(x) × dx[/tex]

where the derivative of y with respect to x, which is:

[tex]f'(x) = 10x[/tex]

Plugging in [tex]x = 4[/tex]   we get:

[tex]= 2[/tex]

Therefore, the differential [tex]dy = 2[/tex] when [tex]x = 4[/tex] and [tex]dx = 0.2.[/tex]

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The equation represented on the graph obtained from equation of a sinusoidal function is the option; y = coa(x - π/4) - 2

A sinusoidal function is a periodic function that repeats at regular interval and which is based on the cosine or sine functions.

The coordinate of the points on the graph indicates that we get;

The period, T = π/4 - (-7·π/4) = 8·π/4 = 2·π

Therefore, B = 2·π/(2·π) = 1

B = 1

The amplitude, A = (-1 - (-3))/2 = 2/2 = 1

The vertical shift, D = (-1 + (-3))/2 = -4/2 = -2

The vertical shift, D = -2

The horizontal shift is the amount the midline pint is shifted relative to the y-axis

The points on the graph indicates that the peak point  close to the y-axis is shifted π/4 units to the right of y-axis, therefore, the horizontal shift, C = π/4

cos(0) = 1 which is the peak point value of the trigonometric ratio, in the function which indicates that the trigonometric function of the equation for the graph is of the form, y = A·cos(B·(x - C) + D

Plugging in the above values into the sinusoidal function equation of the form; y = A·cos(B·(x - C) + D

We get;

A = 1, B = 1, C = π/4, and D = -2

The function representing the graph is therefore;

y = cos(x - π/4) - 2

The correct option is therefore;

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Answer:no it doesn’t it makes a trapezoid which doesn’t have 90 degree angles or right angles

Step-by-step explanation:

The total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax is approximately $219.50.

To calculate the total cost, we first need to find the amount of sales tax. We do this by multiplying the cost of the item by the sales tax rate:

$200.00 x 0.0975 = $19.50

Then, we add the sales tax amount to the cost of the item to get the total cost:

$200.00 + $19.50 = $219.50

Therefore, the total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax, is $219.50.

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Complete Question:

If the City of Dallas has a 9.75% sales tax on all online purchases, what is the total cost when you buy an item online that costs $200.00?

The angles are 11°, 42° and 35°.

Given are circles, we need to find the missing angles,

1) ∠1 = 1/2 [119° - (360° - (119°+174°)]

= 1/2 [119° - 97°]

∠1 = 11°

2) ∠1 = 1/2[360°-138°-138°]

∠1 = 1/2 x 84

∠1 = 42°

3) ∠1 = 1/2[111°-360°-(111°+104°+104°)]

∠1 = 1/2 x 70

∠1 = 35°

Hence the angles are 11°, 42° and 35°.

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Answer:

0.2

Step-by-step explanation:

Step-by-step explanation:

Area of base ( 1/2 *  L1  *  L2 )     *  height = volume

            1/2 ( 7)(8)     *   5 =  140 in^3

The table should be completed with the correct key features as follows;

Axis of symmetry (1st graph): x = 1.

Vertex (1st graph): (1, -9).

Minimum (1st graph): -9.

y-intercept (1st graph): (0, -8).

Axis of symmetry (2nd graph): x = 2.

Vertex (2nd graph): (2, 16).

Maximum (2nd graph): 16.

y-intercept (2nd graph): (0, 12).

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first graph of a quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Based on the second graph of a quadratic function, we can logically deduce that the graph is a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).

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Answer:

Step-by-step explanation:

Answer:

The range of a set of data is the difference between the highest and lowest values. In this case, the highest value is 98 and the lowest value is 65, so the range is 98-65 = 33. This means that the scores on the test ranged from 65 to 98, a difference of 33 points.

The range is a measure of the spread of the data. In this case, the range is relatively large, which means that the scores were spread out over a wide range of values. This suggests that the test was challenging and that there was a wide range of student abilities.

Answer:

-13.57142857142857

Step-by-step explanation:

According to the given box plot:

Most of the response times were under 13 minutes is true

Fewer than 30 of the response times were over 13 minutes is true

More than half of the response times were 11 minutes or greater is true

About 75% of the response times were 13 minutes or less is true

In box plot the difference between a dataset's first and third quartiles is used to determine the interquartile range (IQR), a measure of variability.

The interquartile range (IQR) depicts the range of values centred on the data's median. Because it is more resistant to outliers than other measures of variability like the range or the standard deviation, it is valuable in statistics.

Observations that are more than 1.5 times the IQR from the adjacent quartile are considered probable outliers and can be located using the IQR.

Box plots and the IQR are frequently combined to compare and summarise data from several groups or samples.

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This is the equation in terms of w that models the situation:[tex]w^2 - 10w + 18.75 = 0[/tex].

Rectangle with width w and length l, enclosed by a 20-meter fence: The perimeter of the rectangle, which is equal to the length of the fence, is given by:

2w + 2l = 20

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

w + l = 10

We also know that the area of the rectangle is 18.75 square meters:

w * l = 18.75

We want to write an equation in terms of w, so we can solve for l in terms of w by dividing both sides by w:

l = 18.75 / w

This expression for l into the equation for the perimeter, we get:

w + (18.75 / w) = 10

Multiplying both sides by w, we get:

[tex]w^2 + 18.75 = 10w[/tex]

Rearranging this equation, we get:

[tex]w^2 - 10w + 18.75 = 0[/tex]

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Answer:

1. a

2. a

3. b

Step-by-step explanation:

Q1. The equation is 3x+6=30. b c d are all good answers so it has to be a. If you want more details on why a is wrong, if you expand it becomes 3x+18=30 which is wrong

Q2. This one is 4(x+6) = 40 because you have x+6 4 times and it tells you the total is 40. So the one that matches is a.

Q3.

You have 6 plus 4 x which makes a total of 40.

So 6 + 4x = 40. The equation that matches is b.

Hmu if you need more explanation

P(x > 76 and -2.86 < z < 0) = 0.5000 - 0.3665 = 0.1335.

(a) P(x < 21 and z < 3)

Using standardization, we get:

z = (21 - 25)/3 = -4/3

Using the standard normal table, the corresponding probability for z = -4/3 is 0.0912.

Therefore, P(x < 21 and z < 3) = 0.0912.

(b) P(24 < x < 30 and a < z < 8)

Using standardization, we get:

z1 = (24 - 26)/3 = -2/3

z2 = (30 - 26)/3 = 4/3

Using the standard normal table, the corresponding probability for z = -2/3 is 0.2514 and for z = 4/3 is 0.4082.

Therefore, P(24 < x < 30 and a < z < 8) = 0.4082 - 0.2514 = 0.1568.

(c) P(x > 25 and z < 5)

Using standardization, we get:

z = (25 - 30)/5 = -1

Using the standard normal table, the corresponding probability for z = -1 is 0.1587.

Therefore, P(x > 25 and z < 5) = 0.1587.

(d) P(17 < x < 21)

Using standardization, we get:

z1 = (17 - 20)/3 = -1

z2 = (21 - 20)/3 = 1/3

Using the standard normal table, the corresponding probability for z = -1 is 0.1587 and for z = 1/3 is 0.3707.

Therefore, P(17 < x < 21) = 0.3707 - 0.1587 = 0.2120.

(e) P(x > 76 and -2.86 < z < 0)

Using standardization, we get:

z1 = (76 - 80)/12 = -1/3

z2 = 0

Using the standard normal table, the corresponding probability for z = -1/3 is 0.3665 and for z = 0 is 0.5000.

Therefore, P(x > 76 and -2.86 < z < 0) = 0.5000 - 0.3665 = 0.1335.

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According to the figure of a circle, x is equal to 17

The total arc length in a circle is equal to 360 degrees

hence arc QR + arc RS + arc QS = 360 degrees

Where

arc QR = 120

arc RS = 2 * 67 = 134

arc QS = 5x + 20

plugging in the values

120 + 134 + 5x + 21 = 360

5x = 360 - 120 - 134 - 21

5x = 85

x = 17

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1.26 more than the scarf originally so 8.26 for each scarf and 430 total made from the markup and the total profit is 63 dollars.

The description of the cross section tells us that it is a rectangle

Rectangles are four-sided, two-dimensional shapes that boast two sets of paralleled, opposite sides with identical lengths. All four corner angles measure at ninety degrees and the opposing sides always have the same length.

The area can be calculated by multiplying its length and width, while the perimeter is found by adding all four side measurements together. Furthermore, the perpendicular diagonals of a rectangle will bisect one another and yield equal measurement when fully extended.

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(a) The given statement, "If A is linearly dependent, then Sp{u, v} = Sp{u, w}" is false because there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

(b) The given statement, "The set A is linearly independent if and only if {u+v,v-w, w+ 2u} is linearly independent" is true because A is linearly independent if and only if the determinant of the matrix formed by u, v, and w is nonzero. The determinant of the matrix formed by {u+v, v-w, w+2u} can be obtained by performing column operations on the original matrix. Since these operations do not change the determinant, the set {u+v, v-w, w+2u} is linearly independent if and only if A is linearly independent.

(c)The given statement, "Assume that A is linearly dependent. We define u₁ = 2u, v₁ = -3u + 4v, and W₁ = u + 2v - tw for some t E R. Then, there exists t E R such that {u₁, v₁, w₁} is linearly independent" is true because  A is linearly dependent, there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0.

Let us discuss this in detail.

(a) False. If A is linearly dependent, then there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0. Without loss of generality, assume α ≠ 0. Then we can solve for u: u = (-β/α)v + (-γ/α)w. Therefore, u is a linear combination of v and w, which means Sp{u, v} = Sp{u, w}.

(b) True. We can write each vector in {u+v,v-w, w+2u} as a linear combination of u, v, and w:
u + v = 1u + 1v + 0w
v - w = 0u + 1v - 1w
w + 2u = 2u + 0v + 1w
We can set up the equation α(u+v) + β(v-w) + γ(w+2u) = 0 and solve for α, β, and γ:
α + β + 2γ = 0 (from the coefficient of u)
α + β = 0 (from the coefficient of v)
-β + γ = 0 (from the coefficient of w)
Solving this system of equations, we get α = β = γ = 0, which means {u+v,v-w, w+2u} is linearly independent.

(c) True. Since A is linearly dependent, there exist scalars α, β, and γ, not all zero, such that αu + βv + γw = 0. Without loss of generality, assume α ≠ 0. Then we can solve for u: u = (-β/α)v + (-γ/α)w. Therefore, u is a linear combination of v and w, which means we can write u as a linear combination of u₁, v₁, and w₁:
u = (2/5)u₁ + (-3/5)v₁ + (1/5)w₁
Similarly, we can write v and w as linear combinations of u₁, v₁, and w₁:
v = (-2/5)u₁ + (4/5)v₁ + (1/5)w₁
w = u₁ + 2v₁ - t₁w₁
where t₁ = (α + 2β - γ)/(-t). We can set up the equation αu₁ + βv₁ + γw₁ = 0 and solve for α, β, and γ:
2α - 3β + γ = 0 (from the coefficient of u₁)
-3β + 4γ = 0 (from the coefficient of v₁)
-α + 2β - tγ = 0 (from the coefficient of w₁)
Solving this system of equations, we get α = β = γ = 0 if and only if t = -8/5. Therefore, if we choose any t ≠ -8/5, then {u₁, v₁, w₁} is linearly independent.

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Step-by-step explanation:

120 = 1000 e^(-.05t)

120/1000 = e^(-.05t)   take natural LN of both sides

-2.12 = - .05t

t = 42.4 days

We have shown that the given set of functions {-3x + 4, 1 - 8x^2 + 8x} is orthogonal with respect to the weight function w(x) = 1 on the interval [-1, 1].

To show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval, we need to show that the integral of the product of any two functions in the set, multiplied by the weight function, over the interval is equal to zero, except when the two functions are the same.

Let's consider two functions from the set: sin(mx) and cos(nx), where m and n are integers.

∫₀²π sin(mx) cos(nx) dx = 0

We can use the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) to rewrite the integral as:

∫₀²π (1/2)[sin((m+n)x) + sin((m-n)x)] dx

Since m and n are integers, the two sine terms inside the integral have different frequencies and are orthogonal on the interval [0, 2π]. Therefore, their integral over this interval is zero. Thus, we have:

∫₀²π sin(mx) cos(nx) dx = 0, for any integers m and n

Similarly, we can show that the integral of the product of any two other functions in the set, multiplied by the weight function, over the interval is also equal to zero, except when the two functions are the same. Therefore, we have shown that the given set of functions {1, sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x), ...} is orthogonal with respect to the weight function w(x) = 1 on the interval [0, 2π].

To show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval, we need to show that the integral of the product of any two functions in the set, multiplied by the weight function, over the interval is equal to zero, except when the two functions are the same.

Let's consider two functions from the set: -3x + 4 and 1 - 8x^2 + 8x.

∫₋₁¹ (-3x + 4)(1 - 8x^2 + 8x) dx = 0

Expanding the product and integrating, we get:

∫₋₁¹ (-3x + 4)(1 - 8x^2 + 8x) dx = ∫₋₁¹ (-3x + 4) dx - 8∫₋₁¹ x^3 dx + 8∫₋₁¹ x^2 dx

Evaluating the integrals, we get:

∫₋₁¹ (-3x + 4)(1 - 8x^2 + 8x) dx = 0

Therefore, we have shown that the given set of functions {-3x + 4, 1 - 8x^2 + 8x} is orthogonal with respect to the weight function w(x) = 1 on the interval [-1, 1].

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The radius of convergence comes out as 1/4. To get the radius of convergence, we can use the ratio test.

Step:1. Let's call the nth term in the series a_n, where a_n = 2^(2n)/n^2.
Step:2. Using the ratio test, we take the limit as n approaches infinity of |a_(n+1)/a_n|:
|a_(n+1)/a_n| = (2^(2(n+1))/(n+1)^2) * (n^2/2^(2n))
Step:3. Simplifying this expression, we can cancel out the 2^n terms and get: |a_(n+1)/a_n| = 4((n^2)/(n+1)^2)
Step:4. Taking the limit as n approaches infinity, we get:
lim n→∞ |a_(n+1)/a_n| = 4
Since this limit is less than 1, the series converges.
Step:5. Now we just need to find the radius of convergence, which is given by:
R = 1/lim sup n→∞ |a_n|^(1/n)
Step:6. Taking the limit superior of |a_n|^(1/n), we get:
lim sup n→∞ |a_n|^(1/n) = lim sup n→∞ (2^(2n)/n^(2n/n))^(1/n)
= lim sup n→∞ 2^2 = 4
So the radius of convergence is:
R = 1/lim sup n→∞ |a_n|^(1/n) = 1/4
Therefore, the radius of convergence is 1/4.

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For problem 1, the solution is an = n.

For problem 2, the solution is an = 3n - 1.

For problem 3 (the hard problem), we can solve for the values of a, b, and c in the quadratic equation: [tex]an^2 + bn + c = 0[/tex], where a = 5, b = 6n - 1, and c = -2.

Using the quadratic formula, we get:

[tex]n= \frac{-b±\sqrt{b^{2}-4ac }  }{2a}[/tex]

Substituting the values of a, b, and c, we get:

[tex]n= \frac{-(6n-1)±\sqrt{(6n-1)^{2}-4(5)(-2) }  }{2(5)}[/tex]

Simplifying, we get:

[tex]n = \frac{(-6n+1 ± \sqrt{36n^{2}-48n+49 } ) }{10}[/tex]

Therefore, the solution for problem 3 is:

[tex]an= 5n^{2} + \frac{-6n+1 + \sqrt{36n^{2}-48n+49 } }{10}[/tex]

or

[tex]an= 5n^{2} + \frac{-6n+1 - \sqrt{36n^{2}-48n+49 } }{10}[/tex]

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The calculated value of the probability P(head) is 0.5 i.e. one half

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

Sections = 2

Sections = head and tail

Using the above as a guide, we have the following:

Head = 1

When the head section is flipped, we have

P(head) = head/section

The required probability is

P(head) = 1/2

Evaluate

P(head) = 0.5

Hence, the value is 0.5

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The hours and minutes elapsed from 8:00 a.m. to 2:30 p.m is 6 hours and 30 minutes.

The hours and minutes elapsed from 7:40 p.m. to 1:10 a.m. is 5 hours and 30 minutes.

The hours and minutes elapsed from 12:00 noon to 4:59 p.m is 4 hours and 59 minutes.

The hours and minutes that elapsed from 1:23 a.m. to 7:35 a.m is 6 hours and 12 minutes.

The hours and minutes that belapsed from 11:28 p.m. to 5:30 a.m is 6 hours and 2 minutes.

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The  probability that the person chosen belonged to Group Y is 69/164.

As, Out of 200 persons in the sample, those having at least one dream are 200− those who had no dream are

= 200−36

=164

Now, out of 164 people belonged to group Y

= 100−21

=79

So, the probability that the person chosen belonged to Group Y become

= 69/164

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The circumference of 1/8th of a circle with a radius of 30 centimeters is 11.78 centimeters.

The formula for the circumference of a circle is C = 2πr, where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

To find the circumference of 1/8th of a circle, we need to divide the circumference of the full circle by 8. So, the formula becomes:

C = (1/8) * 2πr

Substituting the given value of the radius, we get:

C = (1/8) * 2π(30)

Simplifying, we get:

C = (1/4) * π(30)

C = (1/4) * 30π

C = 7.5π

Approximating π as 3.14, we get:

C = 7.5 * 3.14

C = 23.55/2

C = 11.78

Therefore, the circumference of 1/8th of a circle with a radius of 30 centimeters is 11.78 centimeters.

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Complete Question:

Circumference of 1/8th of a circle. There is 1/8 of a circle. The radius of the circle is 30 centimeters. The radius of the circle is 30 centimeters. Find the distance of the figure. Give steps.

Answer:

246 boys like to play outdoors

Step-by-step explanation:

82% = 0.82

0.82 x 300 = 246

Answer:

246 boys like to play outdoors

Step-by-step explanation:

82% = 0.82

0.82 x 300 = 246

The correct formula to calculate the variance of a probability distribution is σ2 = Σ[(x - μ)2 P(X)].

The formula is σ2 = Σ[(x - μ)2 P(X)],

where σ2 represents the variance, Σ represents the sum of, x represents the possible outcomes, μ represents the mean or expected value of the distribution, and P(X) represents the probability of each outcome.

The number of trials and the probability of success are not directly involved in this formula, but they may be used to calculate the probabilities of each outcome.

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To find a power series representation for f(x) = ln(2-x), we can use the formula for the power series expansion of ln(1+x):
ln(1+x) = Σ (-1)^(n+1) * (x^n / n)

We can use this formula by setting x = -x/2 and multiplying by -1 to get:
ln(2-x) = ln(1 + (-x/2 - 1)) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))
Therefore, the power series representation for f(x) is:
f(x) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))

The radius of convergence of this series can be found using the ratio test:
lim |a_n+1 / a_n| = lim |(-1)^(n+2) * (x^(n+2) / (n+2) * 2^(n+3)) * (n+1) / (-1)^(n+1) * (x^(n+1) / (n+1) * 2^(n+2))|
= lim |x / 2 * (n+1) / (n+2)| = |x/2|

Therefore, the radius of convergence is R = 2. The power series representation centered at x = 0 is:
f(x) = Σ (-1)^(n+1) * ((-1)^n * (x^n+1 / (n+1) * 2^(n+1)))
To find a power series representation for the function f(x) = ln(2 - x), we can first rewrite the function as f(x) = ln(1 + (1 - x)). Now, we'll use the Taylor series formula for ln(1 + u) centered at x = 0:

ln(1 + u) = u - (1/2)u^2 + (1/3)u^3 - (1/4)u^4 + ... + (-1)^n(1/n)u^n + ...
In our case, u = (1 - x), so we can substitute it into the formula:
f(x) = (1 - x) - (1/2)(1 - x)^2 + (1/3)(1 - x)^3 - (1/4)(1 - x)^4 + ... + (-1)^n(1/n)(1 - x)^n + ...

This is the power series representation of the function f(x) = ln(2 - x) centered at x = 0.
Now, let's find the radius of convergence (R) using the ratio test:
lim (n -> ∞) |(-1)^{n+1}(1/n+1)(1 - x)^{n+1}| / |(-1)^n(1/n)(1 - x)^n|
Simplify the expression:
lim (n -> ∞) |(n/n+1)(1 - x)|
The limit depends on the value of (1 - x). To ensure convergence, the limit should be less than 1:
|(1 - x)| < 1

This inequality holds for -1 < (1 - x) < 1, which implies that the interval of convergence is 0 < x < 2. Therefore, the radius of convergence, R, is 1.

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