linear equations in one variable

a. Time-invariant (no time-variable terms)

b. Time-variable (t² is time-variable)

c. Time-invariant (no time-variable terms)

d. Time-invariant (no time-variable terms)

e. Time-invariant (no time-variable terms)

A linear differential equation is one that involves only linear combinations of the dependent variable and its derivatives, as well as any coefficients that are functions of the independent variable (time in this case).

In the first equation, +2y=0, there are no terms that involve the independent variable, so this is a time-invariant equation.

In the second equation, (t²y)'=0, there is a term involving the independent variable t, specifically t². Therefore, this equation is time-variable.

In the third equation, y''+y'=-t+1, there are two terms involving the independent variable, namely -t and 1. Therefore, this equation is time-variable.

In the fourth equation, (cos(t)y)'=0, there is a term involving the independent variable t, specifically cos(t). Therefore, this equation is time-variable.

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Answer:

A

Step-by-step explanation:

The coordinate of the vertices (x,y) after a reflection over the x-axis is (x, -y). It is also known as a rule.

Coordinates are two numbers (Cartesian coordinates) or a letter and a number that point to a specific point on a grid known as a coordinate plane. A coordinate plane has four quadrants and two axes: x (horizontal) and y (vertical).

here, we have,

A reflection of a point, line, or figure in the x-axis entailed mirroring the image over the x-axis. In this case, the x-axis is referred to as the axis of reflection.

The rule for reflecting over the x-axis is to negate the value of each point's y-coordinate while keeping the x-value constant.

For instance, when point P with coordinates (7,3) is reflected across the x-axis and mapped onto point P', P"s coordinates are (7,-3). The x-coordinate for both points remained unchanged, but the y-coordinate changed from 3 to -3.

Hence, The coordinate of the vertices (x,y) after a reflection over the x-axis is (x, -y). It is also known as a rule.

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The area of rectangle B is 450 centimeters squared.

How to find the area of rectangle B?

The area of rectangle B can be found by using the property of similar rectangles: the ratio of their corresponding lengths is equal to the ratio of their corresponding areas.

Therefore, the ratio of the lengths of rectangles A and B is 1:3.

Given:

Rectangle A

Length = 10 centimeters

Width = 5 centimeters

Then,

Rectangle B

Length = 30 centimeters

Width = (5 x 3) centimeters = 15 centimeters

Area of rectangle = Length x Width

Area of rectangle B = (30 x 15) centimeters

Area of rectangle B = 450 centimeters

Thus, the correct option is B. 450 centimeters squared.

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In the given scenario, the average current produced within the loop is approximately 2.13 A.

We can begin by computing the change in magnetic flux across the loop as it expands to determine the average current generated within the loop.

The following equation provides the magnetic flux across a loop:

Φ = B * A * cos(θ)

ΔΦ = B * ΔA

ΔA = A₂ - A₁ = π * (1.00 m)² - π * (0.40 m)² = π * (1.00² - 0.40²) = π * (1.00 + 0.40)(1.00 - 0.40) = π * (1.40)(0.60) = 0.84π m²

So,

ΔΦ = B * ΔA = (24 mT) * (0.84π m²) = 20.25π m²·T

emf = ΔΦ / Δt = (20.25π m²·T) / (1.0 s) = 20.25π V

As:

emf = I * R

So, again

I = emf / R = (20.25π V) / (30 Ω) ≈ 2.13 A

Thus, the average current produced within the loop is approximately 2.13 A.

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Steps of Construction:

1. Draw a line segment BC = 6 cm.

2. Draw a perpendicular bisector of BC that intersects the line BC at Q.

3. Mark A on the line such that OA = 4 cm.

4. Join A to B and C.

5. Draw a ray BX making an acute angle with BC.

6. Mark four points B1,B2, B3, and B4 on the ray BX. such that BB1 = B1B2 = B2B3 = B3B4.

7. Join B4C. Draw a line parallel to B4C through B3 intersecting line segment AB at A'.

Hence ΔA'BC' is the required triangle.

An isosceles triangle is a type of triangle that has two equal sides and two equal angles. The third angle is called the base angle and is typically different from the other two angles. The equal sides are called legs, and the third side is called the base.

Isosceles triangles have some interesting properties. One of them is that the base angles are always equal. This means that if you know the measure of one of the base angles, you can find the measure of the other one by subtracting it from 180 degrees and dividing by 2. Another property is that the altitude from the apex (the point opposite the base) always bisects the base, meaning that it cuts the base into two equal parts.

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Complete Question:

Construct an isosceles triangle whose base is 6 cm and altitude 4 cm. Then construct another triangle with sides are 3/4 the corresponding sides  of the isosceles triangle.

The equation of the circle is (x - 5)² + (y - 4)² = (24.4)².

To write the equation of a circle with endpoints, we need to determine the center and radius of the circle.

Step 1: Determine the center of the circle by using the endpoints of the diameter. The midpoint of the diameter is the center of the circle. The midpoint formula is as follows. (x1 + x2/2, y1 + y2/2) = (center)

Use the given endpoints to find the center of the circle. (-5 + 15)/2, (-3 + 11)/2 = (5, 4)

Thus, the center of the circle is (5, 4).

Step 2: Determine the radius of the circle. The radius of the circle is half of the diameter. The distance formula is used to determine the distance between the two endpoints of the diameter. √((x2 - x1)² + (y2 - y1)²) = radius

Use the given endpoints to find the radius.√((15 - (-5))² + (11 - (-3))²) = √(20² + 14²) = √(400 + 196) = √596 ≈ 24.4

Thus, the radius of the circle is ≈24.4.

Step 3: Use the center and radius to write the equation of the circle. (x - 5)² + (y - 4)² = (24.4)². Therefore, the equation of the circle is (x - 5)² + (y - 4)² = (24.4)².

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The value of b =  42°  and the value of c = 138°

In geometry, the intersection of lines is the point where two or more lines cross each other. The intersection of two lines occurs when they have a common point, which satisfies both of their equations simultaneously.

b° = 42° (Vertically opposite angles)

c°:

42° + 42° + c° + c° = 360° (Sum of angles at a point)

84° + 2c° = 360°

2c° = 360° - 84° = 276°

∴ c° = 276° ÷ 2 = 138°

c° = 138°

The intersection of lines is an important concept in geometry and is used in various applications such as solving systems of linear equations, finding the point of collision of two moving objects, and more.

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Answer: $1,071.54

Step-by-step explanation:

To find the interest refund, first we need to calculate the total interest charged on the loan. We can do this by multiplying the monthly interest by the number of months in the loan:

Monthly interest = Total interest / Number of months

Monthly interest = $2,802 / 35

Monthly interest = $80.06

Total interest charged on the loan = Monthly interest x Number of months

Total interest charged on the loan = $80.06 x 35

Total interest charged on the loan = $2,802.10

Now we need to calculate the interest that would have been charged for the remaining 13 months of the loan:

Interest for remaining 13 months = Monthly interest x Remaining months

Interest for remaining 13 months = $80.06 x 13

Interest for remaining 13 months = $1,040.78

Finally, we can find the interest refund by subtracting the interest for the remaining 13 months from the total interest charged on the loan:

Interest refund = Total interest charged - Interest for remaining months

Interest refund = $2,802.10 - $1,040.78

Interest refund = $1,074.32

Therefore, the interest refund on the loan is $1,074.30.

Answer:

20.25 m²

Step-by-step explanation:

Area of triangle = (1/2) · b · h

b = 9m

h = 4.5m

Let's solve

(1/2) · 9 · 4.5 = 20.25 m²

So, the triangle area is 20.25 m²

If f(x,y) > 0 and is a continuous function defined over a rectangle R=[a,b]x[c,d], then the double integral over R of f(x,y) can be interpreted as the volume of a solid that lies in the first octant and under the graph of the function f(x,y) over the region R.

The geometric interpretation of the double integral over R of f(x,y) if f(x,y) > 0, where f is a continuous function defined on a rectangle R = [a,b] × [c,d] is given as follows:

The double integral of f(x,y) over R, if f(x,y) > 0, gives the volume under the graph of the function f(x,y) over the region R in the first octant.

Consider a point P (x, y, z) on the graph of f(x, y) that is over the region R, and let us say that z = f(x,y). If f(x,y) > 0, then P is in the first octant (i.e. all its coordinates are positive).

As a result, the volume of the solid that lies under the graph of f(x,y) over the region R in the first octant can be found by integrating the function f(x,y) over the rectangle R in the xy-plane, which yields the double integral.

The following formula represents the double integral over R of f(x,y) if f(x,y) > 0:

∬Rf(x,y)dydx

The geometric interpretation of the double integral over R of f(x,y) if f(x,y) > 0 is given by the volume of the solid that lies under the graph of the function f(x,y) over the region R in the first octant.

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Therefore , the solution of the given problem of surface area comes out to be the box's surface size is 304 in².

Its total size can be determined by figuring out how much room would be required to completely cover the outside. When choosing comparable substance with a rectangular shape, the surroundings are taken into account. Something's total dimensions are determined by its surface area. The volume of water that a cuboid can contain depends on the number of edges that are present in the region between its four trapezoidal angles.

Here,

Six faces make up the box: the top, bottom, two sides, and both extremities. Given that both the top and lower faces are rectangles with 10 by 8-inch measurements, the area of each face is:

=>  80 in²= 10 in * 8 in

The region of each side face is thus:

=> 10 in * 4 in = 40 in²

=>  8 in * 4 in = 32 in²

As a result, the box's surface area equals the total of the areas of its six faces:

=>  Surface area = 2(80 in²) + 2(40 in²) + 2(32 in²)

=>   Surface area = 160 in² + 80 in² + 64 in²

=>   Surface area = 304 in²

Consequently, the box's surface size is 304 in².

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If the average of three numbers is 16 and one of the numbers is 18, then the sum of the other two numbers is option (d) 30

Let's use algebra to solve this problem. Let x and y be the other two numbers we are looking for. We know that the average of the three numbers is 16, so we can write:

(18 + x + y) / 3 = 16

Multiplying both sides by 3, we get,

[(18 + x + y) / 3] × 3 = 16 ×3

18 + x + y = 48

Subtracting 18 from both sides, we get,

18 + x + y - 18 = 48

x + y = 30

Therefore, the correct option is (d) 30

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We can represent the decimal number 0.0009765625 as a binary floating point number with an 8-bit mantissa. The mantissa would be: 00000001

To represent the decimal number 0.0009765625 as a binary floating point number with an 8-bit mantissa, follow these steps:

Step 1: Convert the given decimal number into binary

[tex]0.0009765625 = 0.00000001111101000100100001111111[/tex] (approx.)

Step 2: Normalize the binary number and represent it in scientific notation

0.00000001111101000100100001111111 = 1.111101000100100001111111 x 2^-15

Step 3: Separate the sign, mantissa, and exponent. The sign will be 0, as the given number is positive. The exponent will be -15.

And the mantissa will be the 8 bits from the binary number (excluding the first bit, which is 1)1.11101000 (mantissa)

Step 4: Round the last bit of the mantissa if necessary, which is not needed here.

Hence the mantissa would be 00000001 (since we have to represent the mantissa with an 8-bit number).

Hence, the binary floating-point number with an 8-bit mantissa representing the decimal number 0.0009765625 is:

[tex]0 01111000 00000001[/tex]

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Around 9.49% more population increase live in Staverton every year.

Calculating the overall percentage change as a result of multiple successive percentage changes is required for repeated percentage change questions.

Let P be the town's current population, and let r represent the growth rate in decimal form. In six years, the population can be stated as follows:

P * (1 + r)⁶

The problem states that this population is 60% more than the present population, or

P * (1 + 0.6) = 1.6P

Therefore:

1.6P = P * (1 + r)⁶

Dividing both sides by P:

1.6 = (1 + r)⁶

Taking the sixth root of both sides:

(1 + r) = 1.6(1/6)

Subtracting 1 from both sides:

r = 1.6(1/6) - 1

Using a calculator, we find that:

r ≈ 0.0949 or about 9.49%

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By answering the presented question, we may conclude that Therefore, equation the cost per pound of turkey is $1.99 and the cost per pound of ham is [tex]$2.39[/tex]  .

In mathematics, an equation is an assertion that affirms the equivalence of two factors. An algebraic equation (=) separates two sides of an equation. For instance, the assertion [tex]"2x + 3 = 9"[/tex] states that the word "2x + 3" corresponds to the number "9".

The goal of solution solving is to figure out which variable(s) must still be adjusted for the equations to be true. It is possible to have simple or intricate equations, recurring or complex equations, and equations with one or more components.

For example, in the equations  [tex]"x2 + 2x - 3 = 0,"[/tex] the variable x is lifted to the powercell. Lines are utilised in many areas of mathematics, include algebra, arithmetic, and geometry.

Therefore,Let's denote the cost per pound of turkey as $t, and the cost per pound of ham as $h. Then we can write the following system of linear equations.

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To find the volume under a surface and above a rectangle, use a double integral. Integrate the surface function over the rectangle and approximate using Riemann sums or numerical methods to obtain an estimate of the volume

To find the volume under a surface and above a rectangle in three-dimensional space using a double integral, we can follow these steps:

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For a bounded region, R between the curves y = 4x² and y = 4 and the volume of a solid has base R, and cross sections perpendicular to the y-axis is equals to the π square units.

We have a region R is bounded by the curves y = 4x² and y = 4. Solid has base R and cross sections perpendicular to the y-axis are semicircles with the diameter lying in region R. When solving the volume using slicing method we use the basic formula which is the area times the length V = A×l, where A is the area of the cross-section. Also, the formula for the area of cross section for semicircle

= (1/2)π(d/2)²

= (π/8)d², where d is the diameter

Based on the graph the volume of a single semicircle strip is, dV = (π/8)x²dy

=> dV = (π/8) (y/4) dy ( since, y = 4x² , x²

= y/4 )

=> dV = (π/32)4y dy

=> dV = (π/8)ydy --(1)

Now, the limits are, y = 0, 4

Integrating equation (1), with limits 0 to 4.

[tex]∫dV = \frac{π}{8} ∫_{0}^{4}y dy[/tex]

[tex]V= \frac{π}{8} [ \frac{y^{2} }{2} ]_{0}^{4} [/tex]

[tex]V = \frac{π}{8} [ \frac{16}{2} - 0] = 8(\frac{π}{8}) = π[/tex]

Hence, the required volume is π square units.

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The eigenvalues of A are λ = 0 (with multiplicity 1) and λ = 5 (with multiplicity 2), and the corresponding eigenvectors are [1, 0, -1], [0, 1, -1], and [1, -1, 1].

The matrix A = [5 5 5; 5 5 5; 5 5 5] is a 3x3 matrix with all entries equal to 5.

First, we can calculate the determinant of A - λI, where I is the identity matrix and λ is an unknown eigenvalue:

A - λI = [5-λ 5 5; 5 5-λ 5; 5 5 5-λ]

det(A - λI) = (5-λ)[(5-λ)(5-λ)-25] - 5[5(5-λ)-25] + 5[5-25]

= (5-λ)(λ^2 - 15λ) = -λ(λ-5)^2

From this equation, we can see that the eigenvalues are λ = 0 and λ = 5 (with multiplicity 2).

To find the eigenvectors, we can substitute each eigenvalue into the equation (A - λI)x = 0 and solve for x.

For λ = 0, we have:

A - 0I = A = [5 5 5; 5 5 5; 5 5 5]

(A - 0I)x = 0x = [0 0 0]

This implies that any vector of the form [a, b, -a-b] is an eigenvector for λ = 0. For example, we can choose [1, 0, -1] and [0, 1, -1] as linearly independent eigenvectors corresponding to λ = 0.

For λ = 5, we have:

A - 5I = [0 5 5; 5 0 5; 5 5 0]

(A - 5I)x = 0

⇒ 5x2 + 5x3 = 0

⇒ 5x1 + 5x3 = 0

⇒ 5x1 + 5x2 = 0

This implies that any vector of the form [1, -1, 1] is an eigenvector for λ = 5. Therefore, we can choose [1, -1, 1] as another linearly independent eigenvector corresponding to λ = 5.

Eigenvectors are a fundamental concept in linear algebra. They are essentially special vectors that remain in the same direction when a linear transformation is applied to them, only changing in magnitude. In other words, an eigenvector of a linear transformation is a vector that when multiplied by the transformation matrix, results in a scalar multiple of itself.

Eigenvectors play a crucial role in diagonalizing matrices, which can simplify calculations involving matrix operations. They are also useful for solving differential equations and understanding the behavior of dynamic systems. In addition, eigenvectors are often used for data analysis, such as in principal component analysis (PCA), which is a technique for reducing the dimensionality of data.

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Answer:

y=-3x -8

Step-by-step explanation:

4= -3(-4) = b

b=4-12 = -8

y=-3x -8

Answer:

y = -5x - 3

Step-by-step explanation:

A linear equation is y = mx + b

m = the slope

b = y-intercept

We know

Points (-2,7) (0,-3)

Slope = rise/run or (y2 - y1) / (x2 - x1)

We see the y decrease by 10 and the x increase by 2, so the slope is

m = -10/2 = -5

Y-intercept is located at (0, -3)

So, the equation is y = -5x - 3

The number of sales a promoter can expect if he spends $10,000 in advertising is $80,000 that is option B.

While going through a word problem like this, read it a few times to comprehend the context without focusing too much on the numbers..... Something along the lines of....

"If a promoter spends money on advertising, he will get more customers."

Short sentences should be used to summarise the content.

$10,000 is spent by the promoter.

"For every $100 spent, 20 new customers are drawn."

"Tickets are $40."

In one calculation this would be:

Using ratio of,

income/expenditure = 800/100 = income/10,000

= 10000 x 20 x 40 / 100

= 80,000

Therefore, the value of sale increase should be $80,000.

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we can see that the small box offers the best value for money with a cost of £0.34 per coffee pod. Therefore, the answer isThe small box gives the best value for money.

To determine the best value for money among the three sizes of coffee pod boxes, we need to calculate the cost per coffee pod for each size of the box.

For a small box with 12 coffee pods costing £4.08, the cost per coffee pod can be calculated as:

Cost per coffee pod = £4.08 ÷ 12 = £0.34

For a medium box with 20 coffee pods costing £7.80, the cost per coffee pod can be calculated as:

Cost per coffee pod = £7.80 ÷ 20 = £0.39

For a large box with 35 coffee pods costing £12.95, the cost per coffee pod can be calculated as:

Cost per coffee pod = £12.95 ÷ 35 = £0.37

Comparing the cost per coffee pod for each box size, we can see that the small box offers the best value for money with a cost of £0.34 per coffee pod. Therefore, the answer is:

The small box gives the best value for money.

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The probability that it is 1 or 2 on the first roll, 3 or 4 on the second roll, or 5 or 6 on the third roll when a fair die is rolled three times is 1/4.

Probability is the study of random events. It is a measure of the likelihood of an event occurring. Probability can be expressed as a decimal, a fraction, or a percentage. There are two types of probability - empirical probability and theoretical probability.

Empirical probability is calculated by conducting experiments or collecting data. It is calculated using the following formula:

Empirical probability = Number of favourable outcomes/Total number of outcomes

Theoretical probability is calculated using probability formulas. It is calculated using the following formula:

Theoretical probability = Number of favourable outcomes/Total number of possible outcomes

In the given problem, a fair die is rolled three times. We need to find the probability that it is 1 or 2 on the first roll, 3 or 4 on the second roll, or 5 or 6 on the third roll. There are 2 favourable outcomes for the first roll, 2 favourable outcomes for the second roll, and 2 favourable outcomes for the third roll.

Total number of outcomes = 6×6×6 = 216

Number of favourable outcomes = 2×2×2 = 8

Probability = Number of favourable outcomes/Total number of outcomes

Probability = 8/216

Probability = 1/27

Therefore, the probability that it is 1 or 2 on the first roll, 3 or 4 on the second roll, or 5 or 6 on the third roll when a fair die is rolled three times is 1/4.

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The statistical test that a researcher could use to test the null hypothesis that the mean body temperature of residents in a nursing home is 98.6°F is a one-sample t-test.

A statistical test is a method that enables the comparison of the collected data with the assumed distribution of the data. A statistical test aids in determining if the outcomes of the experiment or research are caused by the treatment or if they are due to the random variation in the data.

A null hypothesis is a type of hypothesis that predicts the absence of a relationship between variables or groups. The null hypothesis claims that no difference exists between two variables or groups, and that any observed differences are due to chance.

Alternative hypotheses are used to reject null hypotheses, as they predict the presence of a relationship between variables or groups.

The significance level, which is the probability of committing a Type I error, is often used to set the null hypothesis. The statistical test that a researcher could use to test the null hypothesis that the mean body temperature of residents in a nursing home is 98.6°F is a one-sample t-test.

The t-test will aid in determining if the difference between the mean body temperature of residents in the nursing home and 98.6°F is statistically significant.

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Answer:

Let the height of the parallelogram be h

From :the formula of finding Area of the parallelogram

a. The expected value of the sample mean is 70 and the standard deviation of the sample mean is 2.857.

b. The probability that a randomly selected customer spends less than 65 minutes at the gym is approximately 0.4013.

c. The probability that the average time spent at the gym exceeds 75 minutes is approximately 0.0070.

a. To answer this question, we need to standardize the value of 65 using the formula:

z = (x - μ) / σ

where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

z = (65 - 70) / 20

z = -0.25

Using a standard normal distribution table, we find that the probability of a z-score less than -0.25 is approximately 0.4013.

b. The expected value of the sample mean can be calculated using the formula:

E(x) = μ

where μ is the population mean.

E(x) = 70

The standard deviation of the sample mean can be calculated using the formula:

σ(x) = σ / √(n)

where σ is the population standard deviation and n is the sample size.

σ(x) = 20 / √(49)

σ(x) = 2.857

c. To answer this question, we need to standardize the value of 75 using the formula:

z = (x - μ) / (σ / √(n))

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

z = (75 - 70) / (20 / √(49))

z = 2.45

Using a standard normal distribution table, we find that the probability of a z-score greater than 2.45 is approximately 0.0070.

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The solution for the simultaneous of equations 4x + 7y = 40 and 4x + 4y = 4 by elimination are x = -11, y = 12

we shall write the equations as:

4x + 7y = 40...(1)

4x + 4y = 4...(2)

subtract equation (2) from (1) to eliminate x

4x + 7y - 4x - 4y = 40 - 4

3y = 36

divide through by 3

y = 12

put the value 12 for y in equation (1) to get

4x + 7(12) = 40

4x + 84 = 40

4x = 40 - 84 {subtract 84 from both sides}

4x = -44

divide through by 4;

x = -11

Therefore, the solution for the simultaneous of equations 4x + 7y = 40 and 4x + 4y = 4 by elimination are x = -11, y = 12

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