please help! Thank you!

The square root of 50 is the distance between the points i.e. 5√2.

It is possible to measure a distance between two points or objects quantitatively or sporadically qualitatively. The word "distance" can be used to describe a physical length or an estimate based on other factors in physics or everyday speech.

To do this we know that a² +b²  = x²

for b:

The distance between -1 and -4 is 5. we will use 5²

for a:

The distance between2 and 1 is also 5. we we use 5²

put into the formula

5² +5² = c²

25+25 = c²

50 = c²

so the square root of 50 is the distance between the points i.e. 5√2.

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Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

The derivative is what calculus is all about. The instantaneous pace at which a function changes with regard to one of its variables is known as the derivative. Finding the slope of the tangent line to the function at a certain location is equal to doing this.

Calculus defines a derivative as the rate at which one quantity, y, changes in relation to another, x. The differential coefficient of y with respect to x is another name for it. Finding a function's derivative is the process of differentiation.

The typical representation of a function's derivative is d/dx (f(x)) (or) df/dx (or) Df(x) (or) f' (x). Let's examine the technical definition of a derivative. Two points on a function f(x) curve are (x, f(x)) and ((x + h), f(x + h)), respectively. If so, [f(x + h) - f(x)]/(x + h - x) = [f(x + h) - f(x) / h] will be the slope of the secant line across these locations. The second point overlaps the first point in the image below when the distance between two points is almost equal to 0 (i.e., when h approaches 0), and the secant line turns into the tangent line.

Given that;

f(x) = 8√(10x⁴+x²-1)⁹

So, First derivatives is ;

[tex]\begin{aligned}& \frac{d}{d x}[f(x)]=f^{\prime}(x) \\& \frac{d}{d x}\left[8\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& =8 \cdot \frac{d}{d x}\left[\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& \left.=8 \cdot \frac{9}{2}\left(10 x^4+x^2-1\right)^{\left(\frac{9}{2}\right.}-1\right) \cdot \frac{d}{d x}\left[10 x^4+x^2-1\right] \\\end{aligned}[/tex]

[tex]\begin{aligned} & =36\left(10 \cdot \frac{d}{d x}\left[x^4\right]+\frac{d}{d x}\left[x^2\right]+\frac{d}{d x}[-1]\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& =36\left(10 \cdot 4 x^3+2 x+0\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& \left.=36\left(40 x^3+2 x\right)\left(10 x^4\right)+x^2-1\right)^{\frac{7}{2}} \\& =\left(1440 x^3+72 x\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}}\\\end{aligned}[/tex]

So, the Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

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The domain and the range of the composite function are given as follows:

a) Domain: {3, 5, 6, 7}.

b) Range: {0, 9}.

The composite function of f(x) and g(x) is given by the following rule:

(f ∘ g)(x) = f(g(x)).

It means that the output of the inside function serves as the input for the outside function.

The points on the composite function are given as follows:

The domain is the set of input values, hence it is given by:

{3, 5, 6, 7}.

The range is the set of output values, hence it is given by:

{0, 9}.

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a. The point slope form of the equation is y-4=-3(x+5)/10.

b. The y-intercept of the line is 2.5.

c. The line crosses the y-axis halfway between the given points so the y-intercept is halfway between the y-coordinate of the point on the left, and that of the point on the right. The number halfway between those values is 2.5.

d. The slope intercept form is y = (-3/10)x + (5/2).

The solution has been obtained using linear equations.

What is linear equation?

A linear equation is one that has the highest degree of 1 possible. This indicates that there are no variables in a linear equation with exponents greater than 1. Such an equation on the graph, forms a straight line.

a. We know that general form of point- slope form of equation is

y-y₁=m(x-x₁)

Here, y₁= 4 and x₁= -5, y₂= 1 and x₂=5

Now, m = (y₂ - y₁) / (x₂ - x₁)

⇒m= (1-4)/(5-(-5))

⇒m= -3/10

Thus, we get the equation as y-4=-3(x+5)/10

b. At the y-intercept, the x coordinate is 0.

So, putting x coordinate as 0 in the equation, we get

⇒y-4=-3(0+5)/10

⇒y-4=-15/10

⇒y=-1.5 + 4

⇒y= 2.5

Thus, the y-intercept of the line is 2.5.

c. The line crosses the y-axis halfway between the given points so the y-intercept is halfway between the y-coordinate of the point on the left, and that of the point on the right. The number halfway between those values is 2.5.

d. The general equation for slope intercept form is y=mx+b

m= -3/10

b = y₁ - x₁(y₂ - y₁)/(x₂ - x₁)

⇒ b = 4 - (-5)(1 - 4)/(5 - (-5))

⇒ b = 4 - 15/10

⇒ b = 25/10

⇒ b = 5/2

So, we get y = (-3/10)x + (5/2).

Hence, the required solutions have been obtained.

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According to the given  expression There will be 910 women among the 1,560 runners.

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself.

Using mathematical operations like addition, subtraction, multiplication, and division, a combination of numbers and integers can be expressed numerically in mathematics.

I'm looking for 7/12 of 1560. I really do mean, "Divide 1560 among 12 groups and count how many there are in each category overall."

1560 divided into 12 groups.

1560/12

=130.

There are 130 people in each group. We multiply to determine how many there are in the seven groupings.

130/7

=910

Among the 1,560 runners, 910 of them will be female.

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The length of a circle's radius where the diameter's endpoints are at (2,4) and (-3,-1) is 5/2.

Then,

[tex]$D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}[/tex]

As a result, the diameter's end points are (2,4) and (-3,-1)

[tex]d=\sqrt{(-3-2)^2+(-1-4)^2}[/tex]

[tex]d=\sqrt{(-5)^2+(-5)^2}[/tex]

[tex]\mathrm{d}=\sqrt{25+25}=\sqrt{50}=5 \sqrt{2}[/tex]

[tex]$\text { We know Radius } & =\frac{\text { Diameter }}{2} \\[/tex]

[tex]$& =\frac{5 \sqrt{2}}{2}=\frac{5}{\sqrt{2}}[/tex]

Hence, the radius of given circle is [tex]$} \Rightarrow \frac{5}{\sqrt{2}}[/tex]

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The label of the triangles based on the center are;

A) Incenter

B) Orthocenter

C) Circumcenter

D) Centroid

1) The centroid of a triangle is defined as the point in which the three medians of the triangle intersect.

2) The orthocenter of a triangle is defined as the point where all the three altitudes of the triangle cut or intersect each other.

3) The incenter of a triangle is defined as the intersection point of all the three interior angle bisectors of the triangle.

4) The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.

Looking at the given triangles, we can conclude that triangle A shows the incenter, triangle B shows the orthocenter, triangle C shows the circumcenter while triangle D shows the centroid,

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Net income is the amount of accounting profit a business has left over after covering all of its expenses. Sales revenue is multiplied by COGS, SG&A, depreciation and amortisation, interest expenditure, taxes, and any other expenses to arrive at net income.

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Answer:

[tex]\mathrm{Flagpole\;Height = 17 \;feet}[/tex]

Step-by-step explanation:

Here we have two similar right triangles

One right triangle has base of 8 ft and Jeremy's height of 68" as the height

The other right triangle has the base (8 + 16 = 24) and the height is the height of the flagpole. This is the large triangle

Since the triangles are similar,

The ratios of the sides must be the same

[tex]\mathrm{\dfrac{Jeremy's \;Height}{Flagpole \; Height}} = \dfrac{8}{24}= \dfrac{1}{3}\\\\\\68" = \dfrac{68}{12} \; feet\\\\\textrm{Dividing both numerator and denominator by 4:}\\\\= \dfrac{17}{3}[/tex]

Therefore

[tex]\mathrm{\dfrac{17/3}{Flagpole\;height} = \dfrac{1}{3}}\\\\[/tex]

Cross -multiplying we get
[tex]\mathrm{\dfrac{17}{3} \times 3 = Flagpole\; Height}\\\\\mathrm{Flagpole\;Height = 17 \;feet}[/tex]

The error in writing the polynomial 3m²-5m⁵+m⁴ in standard form

as -5m³+3m²+m⁴ is, Power of  -5m is 5 not 3 and powers should be in descending order

The polynomials are the algebraic expressions, which have multiple powers. Polynomials are always written in a form of descending powers.

And based on powers of polynomials, we can categorize polynomials into following categories-

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

The polynomial-        3m²-5m⁵+m⁴

The standard form-    -5m³+3m²+m⁴

the errors are in standard form of given polynomial,

 3m²-5m⁵+m⁴

For converting it into standard form,

So, it can be arranged in descending powers

-5m⁵+m⁴+ 3m²

By comparing it with given standard form

It can be seen that there are two errors in given form

1.   Power of -5m is 5 not

2.  Variables should be arranged in descending powers

Hence, the correction is power -5 and descending order of powers

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The golfer's average performance over the two-year period was [mean] out of 10,with a standard deviation of [standard deviation].

The mean is a measure of central tendency, which refers to a single value that is representative of a series of values. In this case, the mean is the average performance the golfer achieved over the two-year period. To calculate the mean, add up all the performance scores and divide by the total number of scores. The mean can then be rounded to the nearest whole number. The standard deviation is a measure of variability, which indicates how much the scores vary from the mean. To calculate the standard deviation, subtract each performance score from the mean, square the differences, and then add them up. Finally, divide the sum by the number of scores, and then take the square root. The standard deviation can then be reported to the nearest decimal.

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locating a number's component. Doing percentage and decimal calculations proportional relationships.So,  7/6 times taller is Sally than May.

Why is it important to use fractions?

Identifying the portion of a number is one application for fractions. figuring out percentages and decimals. Verhältnis und Proportion.

Learning more complex mathematical concepts requires a strong foundation in fractions. The greatest way to introduce algebra is through fractions because they are a student's initial exposure to mathematical abstraction.

Let's call Jenny's height H. Next, we have this:

Sally's height is equal to 7/8*H.

May's height is equal to 6/8"H.

We only need to divide Sally's height by May's height to compare their heights:

Sally's height divided by May's height equals (7/8)*H / (6/8)*H.

= (7/8) / (6/8) = (7/8) * (8/6) = 7/6

Sally therefore outranks May by 7/6.

Therefore, the answer is 7/6.

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Type 540,000,000 as a floating-point literal using scientific notation with a single digit before and after the decimal point is 5.4e8

Type 540,000,000 as a floating-point literal using scientific notation with a single digit before and after the decimal point is 5.4e8.

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The amount in the bank after 2.5 years will be $241.875.

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. Mathematically -

[tex]$A = P (1 + rt)[/tex]

{A} = final amount

{P} = initial principal balance

{r} = annual interest rate

{t} = time (in years)

Given is the amount deposited by Velvet in a bank for 2.5 years at a rate of 3%.

The amount in the bank after 2.5 years will be -

A = 225(1 + 0.03 x 2.5)

A = 225(1 + 0.075)

A = 225 x 1.075

A = 241.875

Therefore, the amount in the bank after 2.5 years will be $241.875.

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This figure's surface measures 2332 square inches.

The area or region that an object’s surface occupies is known as its surface area. Volume, on the other hand, refers to how much room an object has. There are numerous shapes and dimensions in geometry, including spheres, cubes, cuboids, cones, cylinders, etc. Each form has its own volume and surface area.

Given, The figure below is a net for a triangular prism. Side a = 46inches, side b = 12 inches, side c = 32inches, and altitude d = 22inches

From the figure,

The surface area of triangle = 1/2* d * a +  1/2* d * a

The surface area of the rectangle = b *a  +2 * c * b

Thus the complete surface area of Shape =  The surface area of the rectangle + The surface area of a triangle

Total surface area =d*a + b*a + 2*c*b

Total surface area =  22 * 46 + 12 * 46 + 2 * 32 * 12

Total surface area = 2332

Therefore, The surface area of this figure is 2332 square inches.

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Complete question:

The expected value of a random variable is the sum of the probabilities of all possible outcomes which is 10.083

In this case, since the random variable is the sum of the three independent rolls of a fair die, the expected value, E[X], is the sum of all the possible outcomes of three die rolls, each multiplied by the probability of 1/6 that each outcome will occur:

E[X] = (1 + 2 + 3 + 4 + 5 + 6) × (1/6)³

= 3.5

The variance of a random variable is the expected value of the squared deviation from the mean. In this case, the variance for X is:

Var(X) = E[X^2] - (E[X])^2

= (1² + 2² + 3² + 4²+ 5² + 6²) × (1/6)³ - (3.5)²

= 10.0833

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Answer:

Step-by-step explanation:

(1+1/a)^2-(1-1/a)^2

= (1+1/a)^2 - (1-1/a)(1+1/a)

= (1+1/a)^2 - 1 + 1/a^2

= (1+1/a+1/a^2+1/a^3) - 1 +1/a^2

= 1/a^3+2/a^2+2/a+1

So the final answer is 1/a^3+2/a^2+2/a+1

11.9% of students have height less than 136.4cm. The solution has been obtained using probability.

What is probability?

Probability is the ability to happen. The subject of this area of mathematics is the occurrence of random events. From 0 to 1 is used to express the value.

Let the height be 'd' cm.

We are given mean as 160 cm and standard deviation of 20 cm.

Probability(x>d) = 0.12

Probability(x<d) = 0.881

Probability(z ≤ d-160/20) = 0.881

⇒d-160/20 = 1.1800  (using table)

⇒d-160 = 23.6

⇒d = 183.6

Therefore, d-mean = -23.6

⇒d-160 = -23.6

⇒d = 160-23.6

⇒d = 136.4cm

Hence, 11.9% of students have height less than 136.4cm.

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Answer:

5 oranges and 10 apples were sold  

Step-by-step explanation:

Let x be the number of oranges sold and y be the number of apples sold

Since 15 fruits were sold in all we get the following equation

x + y = 15 ....................[1]

Each orange costs $1 so if x oranges were sold that would fetch x dollars in sales = x dollars in sales

Each apple costs $2 so if y apples were sold, that would result in 2y dollars in sales

Total sales in $ = x + 2y and we know this is $25 so we get the second equation as

x + 2y = 25 ..................[2]

Look at the two equations. The coefficient of the x term is the same. We can eliminate the x term by subtracting equation [1] from equation [2] as follows:

x + 2y = 25
                    -
x  +   y = 15
-----------------
        y = 10

(2y - y = y and 25-15 = 10)

Now that we know y = 10, we can use equation [1] to find x

x + 10 = 15

which makes x = 5

ANSWER

5 oranges and 10 apples were sold  

The next number in the list would be -11.

What is Inductive reasoning?

Inductive reasoning is a form of reasoning that derives a general principle from a set of observations. It entails drawing broad conclusions from specific observations. Deductive reasoning differs from inductive reasoning.

Based on the pattern in the list, the next number in the sequence using inductive reasoning would likely be -11.

The pattern appears to be that each number is the previous number minus 3.

Hence, By using inductive reasoning the next number in the list would be -11.

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The graph of the y = x³ function is given by the image given at the end of the answer.

The function for this problem is defined as follows:

y = x³.

The domain and the range of the function are given as follows:

Some points on the graph of the function are given as follows:

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To distinguish between linear growth and exponential growth, you need to look at the equation of the function and the shape of the graph.

Linear functions have a constant rate of change, meaning that the slope of the line is always the same. The equation for a linear function is y = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line.

Exponential functions, on the other hand, have a variable rate of change, meaning that the slope of the line changes as the x-value increases. The equation for an exponential function is y = ab^x, where a and b are constants. The graph of an exponential function is a curve that increases or decreases rapidly.

If the graph is a straight line it is linear, if it is curved and increases or decreases rapidly then it is exponential, otherwise, it would be considered as other.

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Constructors, instance variables, accessors/mutators, and methods the user needs to manipulate the number of objects of the class should all be public.

Public members of a class are those that are visible and accessible to users of the class. These include constructors, instance variables, accessors/mutators, and methods that the user needs to manipulate the objects of the class. These members should be designed and implemented carefully, since they form the interface between the class and its users. Properly designed public members can ensure that the class is easy to use and understand. Making the wrong members public can lead to confusion, bugs, and difficult-to-maintain code.

Constructors, instance variables, accessors/mutators, and methods the user needs to manipulate the number of objects of the class should all be public.

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simplest polynomial function with zeroes -2 and 3 is expanded form is f(x) = [tex]x^2 - x - 6[/tex].

To write a polynomial function with given zeros, we can use the factorization method. The zeros of a polynomial function are the solutions of the equation f(x) = 0. So, if we know the zeros of a polynomial function, we can use them to write the function in factored form.

For example, if the zeros of the polynomial function are -2 and 3, we can write the function as:

f(x) = (x + 2)(x - 3)

This function can be expanded into:

f(x) = [tex]x^2 - x - 6[/tex]

It's important to notice that the degree of the polynomial function is determined by the highest exponent of the variable in the equation, in this case 2.

It's also important to notice that the zeroes of the function are -2 and 3, as when you substitute them in the equation, the result is zero.

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The complete question is:

What is the simplest polynomial function with zeroes -2 and 3? Write the equation in the expanded form.

The maximum area of rectangle is 8450 m².

Given that there is fence only on 3 sides and one length side have river flowing trough it

so the perimeter of the plot is given by:

l + w + w

=> 2w+l

where w is width and l is length of the plot

given , perimeter = 260

=> 2w+l = 260  ---- (i)

area of the plot is = l*w

putting the value of w from equation (i) we get

(260-2w)*w

=> 260w-2w²

so A = 260w-2w²

To maximise A we have dA/dw=0

=> d(260w-2w²)/dw

=> 260-4w

so 260 - 4w=0

=> 4w = 260

=> w = 65

l = 260 -2 w

=> 260 -2 *65

=> 260-130

=> 130

so Maximum area of the rectangle = 130*65 = 8450

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The roots of the given quadratic equation [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]   after solving are  [tex]\alpha = \frac{5 + 47i }{2}[/tex]  and  [tex]\beta = \frac{5 - 47i }{2}[/tex] .

Given,

          Quadratic Equation is   [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]

Multiply L.H.S. and R.H.S. by 3, and we get:

            [tex]x^{2} -5x - 18 = 0[/tex]

Here,  [tex]a= 1, b= -5 ,c= -18[/tex]

Let's find out the determinant,

         [tex]D = b^{2} - 4ac[/tex]

      [tex]D= (-5)^{2} - 4(1)(-18)\\D = 25 - 72\\D = -47\\[/tex]

Therefore, the roots of the quadratic equation will be :

       [tex]\alpha = \frac{-b + \sqrt{D} }{2a}[/tex]     and    [tex]\beta= \frac{-b - \sqrt{D} }{2a}[/tex]

Now, Substitute the value of [tex]a,b,c[/tex]

      [tex]\alpha = \frac{5 + 47i }{2}[/tex]     and    [tex]\beta = \frac{5 - 47i }{2}[/tex]

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The  height of her eye level above the sea is 45.6 km

Height talks about how high the persons eye is above te sea level.

The equation so formed is

3.6√h = 13.5 Kilometers

Making √h the subject of the relation we have

√h  = 3.75 Kilometers

Making h the subject by squaring both sides of the equation we have

h = √3.75 Kilometers above sea level

h = 1.9 Kilometers above sea level rounded

Taking these various steps, the height of the eye is 1.9 kilometers above sea level.

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Correct question:

If a person's eye level is h meters above sea level and she can see d kilometers to the horizon, then d = 3.6√h . Suppose the person can see 13..5 kilometers to the horizon

The solid S is a triangular prism whose base is a triangle with vertices (0, 0), (1,0), and (0, 1). The cross-sections perpendicular to the x-axis are equilateral triangles of the same height.

The solid S is a three-dimensional shape with a triangular base. The base is a triangle with vertices (0, 0), (1,0), and (0, 1). This triangle has sides of length 1, and the angles at each vertex are equal. The equilateral triangles are the cross-sections of the solid perpendicular to the x-axis. This means that each side of each triangle is the same length, and all the angles are equal. Each cross-section has the same height, determined by the height of the solid. The shape of the solid is a triangular prism, with triangular sides that connect the three vertices. This shape has a three-dimensional volume, and the volume can be calculated by multiplying the area of the base by the height of the solid.

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The orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.

To find this, we can take the derivative of the original equation 3x^2 − y^2 = C with respect to x, which will give us 6x. Then, we can take the negative reciprocal of this derivative, which is -1/6x. This is the slope of the orthogonal trajectories.

We then can use the point-slope form of a line to get y - y1 = -1/6x (x - x1). Since the slope of the orthogonal trajectories is always -1/6x, we can just set the equation y - y1 = -1/6x (x - x1) equal to 0 and solve for x to get x1 = -y1/6.

We can then plug this value back into the point-slope equation to get y - y1 = -1/6(-y1/6) (x - (-y1/6)), which simplifies to y - y1 = y1/6 + x. Now, if we let y1 = -C/3, then we get y - (-C/3) = C/6 + x, which can be rewritten as x^2 + y^2 = C/3.

Therefore, the orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.

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