1. Find the volume of the rectangular prism. Use the
volume formula V = L*W*H to justify your answer.
10 cm
L= 10cm
W= 8cm
H=12cm
Volume = 80cm
12 cm
8 cm
V=

Any three lengths that satisfy the triangle inequality theorem can form the sides of a triangle. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In order for three lengths to form a triangle, they must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's consider three lengths: a, b, and c.

To determine if they can form a triangle, we need to check the following conditions:

a + b > c

a + c > b

b + c > a

If all three conditions are true, then the lengths a, b, and c can form a triangle.

For example, let's consider the lengths 3, 4, and 5.

3 + 4 > 5 (True)

3 + 5 > 4 (True)

4 + 5 > 3 (True)

Since all three conditions are true, the lengths 3, 4, and 5 can form a triangle.

Therefore, any three lengths that satisfy the triangle inequality theorem can be the lengths of the sides of a triangle.

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The road trip will take approximately 5 hours.

Two one-step equations:

a) 2x + 5 = 13

In this equation, the variable 'x' represents an unknown number. By performing one step of subtraction, we can find the value of 'x' that makes the equation true.

The solution is x = 4.  

b) 3y - 7 = 16

Similar to the first equation, 'y' represents an unknown number.

By adding 7 to both sides of the equation, we can isolate the variable and solve for 'y.'  

The solution is y = 7.

Two equations with fractions:

a) (1/3)x + 2 = 5

Here, the variable 'x' is multiplied by a fraction.

To isolate 'x,' we can subtract 2 from both sides and then multiply both sides by the reciprocal of 1/3, which is 3/1.

The solution is x = 9.

b) (2/5)y - 3 = 1

In this equation, 'y' is multiplied by a fraction.

We can isolate 'y' by adding 3 to both sides and then multiplying both sides by the reciprocal of 2/5, which is 5/2.

The solution is y = 4.

One equation with the distributive property:

a) 2(x + 3) = 10

This equation demonstrates the distributive property.

By applying it, we multiply 2 by both x and 3, resulting in 2x + 6 = 10.

We can then solve for 'x' by subtracting 6 from both sides.

The solution is x = 2.

One equation with decimals:

a) 0.4x + 0.8 = 1.6

In this equation, 'x' is multiplied by a decimal.

To isolate 'x,' we subtract 0.8 from both sides and then divide both sides by 0.4.

The solution is x = 2.

Real-world problem:

Imagine you're planning a road trip.

The distance you'll be traveling is 250 miles, and your car's average speed is 50 miles per hour.

You want to determine how long the trip will take.

Let 't' represent the time in hours it will take to complete the trip.

The equation that represents this situation is:

50t = 250

By dividing both sides of the equation by 50, we find that t = 5.

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The solutions of the quadratic equation are: x = 1

The x-intercept is x = 1

The formula for the equation of a line in slope intercept form is:

y = mx + c

where:

m is slope

c is y-intercept

The y-intercept is the point at which the graph crosses the y-axis and in this case it is: y = 1

The x-intercepts are where the graph touches the x-axis and in this case, it is x = 1

The zeros of the quadratic equation are the x-intercepts and since the curve does not cross, then we can say that the zeros are x = 1 and x = 1 which signifies double root and as such the solution is x = 1

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Answer:

The missing variable θ is equal to (11π)/8.

Step-by-step explanation:

Given:

ω = π/8 radian per min

t = 11 min

Step 1: Identify the formula and the missing variable.

The formula is ω = (θ/t), and we are trying to find the value of θ.

Step 2: Rearrange the formula to solve for the missing variable.

To isolate θ, we can multiply both sides of the equation by t:

ω * t = θ

Step 3: Substitute the given values.

Substituting the given values into the rearranged formula, we have:

θ = (π/8) * 11

Step 4: Simplify the expression.

To multiply fractions, we multiply the numerators and multiply the denominators:

θ = (π * 11) / (8 * 1)

θ = (11π) / 8

Step 5: Finalize the answer.

The value of the missing variable θ is (11π)/8. This is the exact answer unless otherwise indicated.

Therefore, the step-by-step process shows that the missing variable θ is equal to (11π)/8.

Both the angles are supplementary angles hence the value of x is 8°.

To solve for the value of x in the given scenario, we can use the fact that the interior angles between two parallel lines are supplementary, meaning they add up to 180 degrees.

Given:

Angle 1: (8x + 1)

Angle 2: 115°

Since these two angles are supplementary, we can set up the equation:

(8x + 1) + 115 = 180

Now we can solve for x by simplifying and isolating the variable:

8x + 1 + 115 = 180

8x + 116 = 180

8x = 180 - 116

8x = 64

To isolate x, we divide both sides of the equation by 8:

8x/8 = 64/8

x = 8

Therefore, the value of x is 8.

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Answer: The answer is...

y = -2/3 x + 1

Step-by-step explanation:

The others would have crossed paths with y = -2/3 x + 10

The volume of the sphere, given that the sphere has a diameter of 14.6 mm is 1628.7 mm³

The following data were obtained from the question:

The volume of a sphere is giving by the following formula

Volume of sphere = 4/3πr³

Inputting the given parameters, we can obtain the volume of the sphere as follow:

Volume of sphere = (4/3) × 3.14 × 7.3³

Volume of sphere = (4/3) × 3.14 × 389.017

Volume of sphere = 1628.7 mm³

Thus, we can conclude from the above calculation that the volume of the sphere is 1628.7 mm³

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Two equivalent expressions for the given expression 14(8 - 16x) + 3x are 112 - 221x and 112 - 221x.

Equivalent expression 1:

Expanding the expression 14(8 - 16x) and combining like terms, we get:

112 - 224x + 3x

Simplifying further, we have:

112 - 221x

Equivalent expression 2:

Distributing the coefficient 14 to both terms inside the parentheses, we have:

112 - 224x + 3x

Combining the terms with the same variable, we get:

112 - 221x

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Answer:

A.  19.8

B.  15.9

Step-by-step explanation:

A.  

To find the geometric mean between two numbers, you multiply them together and then take the square root of the product.

28 × 14 = 392

[tex]\sqrt{392}[/tex] = 19.799 ≈ 19.8

B.

7 x 36 = 252

[tex]\sqrt{252}[/tex] = 15.8745 ≈ 15.9

The meaure of m∠KJL in the circle is:

m∠KJL = 25°

Since ΔJKL is inscribed in circle P with diameter JK and mJL = 130°. Thus, m∠JLK is an inscribed angle.

Since an angle inscribed in a semicircle is a right angle.

Thus, m∠DFE = 90°

Since the measure of inscribed angle is half the measure of its intercepted arc. Thus:

m∠JKL = 1/2 * mJL

m∠JKL = 1/2 * 130

m∠JKL = 65°

Therefore:

m∠KJL = 180 - 90 - 65 = 25° (sum of angles in a triangle)

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Answer: $561.63

Step-by-step explanation: To find the distance you subtract 500 by -61.63

You get the equation 500--61.63

The two negative signs cancel out to become a positive sign

So the new equation is 500+61.63

So the distance is $561.63

Answer:

$561.63

Step-by-step explanation:

To find out the distance between 500 and -61.63 it will be $500-$-61.63. So 500 - (-61.63) becomes 500 + 61.63 because 2 negatives in a row make a positive. So when you do 500+61.63, you'll get 561.63.

When dividing the polynomial[tex]P(x) = 2x^3 + 5x^2 - 3x + 4[/tex] by the binomial (x - 2), the quotient is [tex]5x^2 + 10x + 20.[/tex]

To find the quotient when dividing the polynomial [tex]P(x) = 2x^3 + 5x^2 - 3x + 4[/tex] by the binomial (x - 2), we can use synthetic division. Synthetic division is a method used to divide polynomials quickly and efficiently.

First, we set up the synthetic division table by writing the coefficients of the polynomial in descending order:

    2  |   5   -3   4

        |___________

Next, we bring down the first coefficient, which is 5:

    2  |   5   -3   4

        |___________

        |   5

To calculate the next row, we multiply the divisor (2) by the value in the previous row (5) and write the result below the next coefficient:

    2  |   5   -3   4

        |___________

        |   5

        |___________

             10

We add the values in the second and third rows:

    2  |   5   -3   4

        |___________

        |   5

        |___________

            10    7

We repeat this process until we reach the last coefficient:

    2  |   5   -3   4

        |___________

        |   5

        |___________

             10    7

             20  34

The quotient is given by the numbers in the bottom row: [tex]5x^2 + 10x + 20.[/tex]

Therefore, when dividing the polynomial[tex]P(x) = 2x^3 + 5x^2 - 3x + 4[/tex] by the binomial (x - 2), the quotient is [tex]5x^2 + 10x + 20.[/tex]

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The complete question may be like:

Using synthetic division, what is the quotient when dividing the polynomial [tex]P(x) = 2x^3 + 5x^2 - 3x + 4[/tex] by the binomial (x - 2)? human generated answer without plagiarism. 200 words.

The number of bags of rice that would be needed to make 80 recipes is 48.664 bags.

1 recipe = 3.5 Cups of rice

If

1 cup of rice = 158 grams

1 bag of rice = 2 pounds

2.2 lbs. = 1 kg,

1kg = 1000 g

80 recipes = 3.5 cups 80

= 280 cups

1 cup of rice = 158 grams

280 cups = 158 × 280

= 44,240 grams

1kg = 1000 g

44,240 grams = 44.24 kg

2.2 lbs. = 1 kg; x lbs = 44.24

2.2/1 = x/44.24

x = 2.2 × 44.24

x = 97.328 pounds

1 bag of rice = 2 pounds ; x bags = 97.328 pounds

1/2 = x/97.328

97.328 = 2x

x = 48.664 bags

Hence, 48.664 bags of rice is needed for 80 recipes.

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The statistics that always corresponds to the 75th percentile in a distribution include the following: B. Third Quartile.

In Mathematics and Statistics, IQR is an abbreviation for interquartile range and it can be defined as a measure of the middle 50% of data values when they are ordered from lowest to highest.

Mathematically, interquartile range (IQR) of a data set is the difference between third quartile (Q₃) and the first quartile (Q₁):

IQR = Q₃ - Q₁ = 75th percentile - 25th percentile.

In this context, we can reasonably infer and logically deduce that the 75th percentile in a distribution is always equal to the third quartile.

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A truck is being filled with cube-shaped packages that have side lengths of foot, the volume of the part of the truck that is being filled is 34,808 cubic feet.

To discover the finest range of applications that may in shape in the truck, we want to calculate the volume of the truck and divide it by using the quantity of every cube-formed bundle.

The quantity of the truck is given with the aid of the method:

Volume = Length x Width x Height

Volume = 8 ft x 61 ft x 71 ft

Volume = 34,808 cubic toes

The volume of every cube-shaped bundle is given with the aid of the method:

Volume = Side [tex]Length^3[/tex]

Volume = 1 [tex]ft^3[/tex]

Number of packages = Volume of truck / Volume of each package

Number of packages = 34,808 ft^3 / 1 ft^3

Number of packages = 34,808

Thus, the greatest number of packages that can fit in the truck is 34,808.

Volume of the part of the truck being filled = Number of packages x Volume of each package

Volume of the part of the truck being filled = 34,808 x 1 ft^3

Volume of the part of the truck being filled = 34,808 ft^3

Thus, the volume of the part of the truck that is being filled is 34,808 cubic feet.

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Answer:

13)  4.9 m

14)  0.9 m

Step-by-step explanation:

The given diagram shows the height of the same cactus plant a year apart:

We are told that the cactus continues to grow at the same percentage rate. To calculate the growth rate per year (percentage increase), use the percentage increase formula:

[tex]\begin{aligned}\sf Percentage \; increase &= \dfrac{\sf Final\; value - Initial \;value}{\sf Initial \;value}\\\\&=\dfrac{ 2-1.6}{1.6}\\\\&=\dfrac{0.4}{1.6}\\\\&=0.25\end{aligned}[/tex]

Therefore, the growth rate of the height of the cactus is 25% per year.

As the cactus grows at a constant rate, we can use the exponential growth formula to calculate its height in Year 6.

[tex]\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Growth Formula}\\\\$y=a(1+r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the growth factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}[/tex]

The initial value is the height in Year 1, so a = 1.6.

The growth factor is 25%, so r = 0.25.

As we wish to calculate its height in Year 6, the value of t is t = 5 (since there are 5 years between year 1 and year 6).

Substitute these values into the formula and solve for y (the height of the cactus):

[tex]\begin{aligned}y&=a(1+r)^t\\&=1.6(1+0.25)^5\\&=1.6(1.25)^5\\&=1.6(3.0517578125)\\&=4.8828125\\&=4.9\; \sf m\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, if the cactus continues to grow at the same rate, its height in Year 6 will be 4.9 meters (to the nearest tenth).

Check by multiplying the height each year by 1.25:

[tex]\hrulefill[/tex]

The given diagram shows the height of the same snowman an hour apart:

We are told that the snowman continues to melt at the same percentage rate. To calculate the decay rate per hour (percentage decrease), use the percentage decrease formula:

[tex]\begin{aligned}\sf Percentage \; decrease&= \dfrac{\sf Initial\; value - Final\;value}{\sf Initial \;value}\\\\&=\dfrac{1.8-1.53}{1.8}\\\\&=\dfrac{0.27}{1.8}\\\\&=0.15\end{aligned}[/tex]

Therefore, the decay rate of the snowman's height is 15% per hour.

As the snowman melts at a constant rate, we can use the exponential decay formula to calculate its height after another 3 hours.

[tex]\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Decay Formula}\\\\$y=a(1-r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the decay factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}[/tex]

The initial value is the snowman's initial height, so a = 1.8.

The decay factor is 15%, so r = 0.15.

As we wish to calculate the snowman's height after another 3 hours, the value of t is t = 4 (i.e. the first hour plus a further 3 hours).

Substitute these values into the formula and solve for y (the height of the snowman):

[tex]\begin{aligned}y&=a(1-r)^t\\&=1.8(1-0.15)^4\\&=1.8(0.85)^4\\&=1.8(0.5220065)\\&=0.93961125\\&=0.9\; \sf m\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, if the snowman continues to melt at the same rate, its height after another 3 hours will be 0.9 meters (to the nearest tenth).

Check by multiplying the height each hour by 0.85:

Answer:

x = 8 units

Step-by-step explanation:

Because this is a right triangle, we can find x using the Pythagorean theorem, which is given by:

a^2 + b^2 = c^2, where

In the triangle, the 6 unit side and the x unit sides are the legs while the 10 unit side is the hypotenuse.

Thus, we can solve for x by plugging in 6 for a and 10 for c in the theorem and solving for a (aka x, simply a in the theorem):

a^2 + 6^2 = 10^2

a^2 + 36 = 100

a^2 = 64

a = 8

x = 8

Thus, x is 8 units.

The mathematical statement that represents "17 more than a number n is 26" can be written as: n + 17 = 26

Given that a statement we need to convert it into a mathematical statement,

The mathematical statement that represents "17 more than a number n is 26" can be written as:

n + 17 = 26

To solve for n, we can subtract 17 from both sides of the equation:

n + 17 - 17 = 26 - 17

Simplifying the equation gives:

n = 9

Therefore, the number n is 9.

Hence the mathematical statement that represents "17 more than a number n is 26" can be written as: n + 17 = 26

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The total amount of money shared by Lee and Olivia is £132 and  Lee and Olivia shared £84 and £48, respectively.

Lee has 7/11 of the total money, which means Lee has (7/11) × x amount of money.

When Lee gives Olivia £18, they have the same amount of money.

So Olivia's share becomes equal to Lee's share.

Therefore, we can set up the following equation:

(7/11) × x - £18 = (1/2)×x

To solve this equation, we can simplify it:

7x/11 - £18 = x/2

Multiplying both sides of the equation by 22 (the least common multiple of 11 and 2) to eliminate the denominators:

14x - 22 × £18 = 11x

14x - 396 = 11x

Subtracting 11x from both sides:

14x - 11x - 396 = 0

3x - 396 = 0

Adding 396 to both sides:

3x = 396

Dividing both sides by 3:

x = 396/3

x = 132

To find Lee and Olivia's individual shares, we can substitute this value back into the initial conditions:

Lee's share = (7/11)× £132 = £84

Olivia's share = £132 - £84 = £48

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The value of the expression [tex]n^2 - 11n + 10[/tex] is equivalent to (n - 1)(n - 10).

To find the value of the expression [tex]n^2 - 11n + 10[/tex], we can follow these steps:

Start with the given expression: [tex]n^2 - 11n + 10.[/tex]

Look for any like terms that can be combined. In this case, there are no like terms.

Since there are no like terms, we can simplify further by factoring the expression. We need to find two numbers that multiply to give 10 (the constant term) and add up to -11 (the coefficient of the middle term, which is -11n).

The numbers that satisfy these conditions are -1 and -10, because

(-1) × (-10) = 10 and (-1) + (-10) = -11.

Now we can rewrite the expression using these numbers:

[tex]n^2 - 11n + 10 = (n - 1)(n - 10).[/tex]

So the factored form of the expression is (n - 1)(n - 10).

Therefore, the value of the expression [tex]n^2 - 11n + 10[/tex] is equivalent to (n - 1)(n - 10).

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The triangles are similar because they are both right triangles, so they both have an angle that is equals to 90º. They both share a common vertex, <C. So they are similar by the Angle-Angle criterion.

Two triangles are defined as similar triangles when they share these two features listed as follows:

For this problem, they have two equal angle measures, the right angle and < C, hence the third angle also must be equal, and they are similar by the Angle-Angle criterion.

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Answer:

3. Substitution because it said angle HKI = angle GKH, so we substitutioned that angle for the other one. I'm not sure about 4. If you provide us with the answer choices for that one, then I could help

The complete equation is x² + 1/2x -2 = 2 + 2

The given equation can be rewritten as:

x² + (1/2)x + () = 2 + ()

Since the equation involves two missing values, let's consider them one by one.

Solve for the first missing value indicated by (_):

To find the missing value indicated by (_), we equate the quadratic equation to 2:

x² + (1/2)x + (_) = 2

Comparing this with the standard form of a quadratic equation,

ax² + bx + c = 0, we have:

a = 1, b = 1/2, c = (_ - 2)

Using the quadratic formula, x = (-b ± √(b²- 4ac)) / (2a), we can substitute the values:

x = (-(1/2) ± √((1/2)² - 4(1)(_-2))) / (2(1))

x = (-1/2 ± √(1/4 + 8(1)(_-2))) / 2

x = (-1/2 ± √(1/4 + 8(_ - 2))) / 2

Therefore, the first missing value is represented by (_ - 2).

Solve for the second missing value indicated by (_):

To find the missing value indicated by (_), we equate the constant term to 2:

(_) = 2

This indicates that the second missing value is equal to 2.

Putting it all together, the solution to the equation x² + (1/2)x + _ = 2 + _ is:

x = (-1/2 ± √(1/4 + 8(_ - 2))) / 2

with (_ - 2) representing the first missing value, and the second missing value is 2.

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1. The area of composed figure is 52.26 cm².

2. The area of composed figure is 74 ft².

1. Area if Triangle

= 1/2 x b x h

= 1/2 x 6 x 8

= 24 cm²

Area of semicircle

= πr²/2

= 3.14 x 3 x 3

= 28.26 cm²

So, area of composed figure

= 28.26 + 24

= 52.26 cm²

2. Area of Trapezium

= 1/2 (7 + 13) x 5

= 1/2 x 20 x 5

= 50 ft²

Area of Triangle

= 1/2 x b x h

= 1/2 x 8 x 6

= 24 ft²

So, area of composed figure

= 24 + 50

= 74 ft²

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The solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.

Given the inequalities that include 3 in the following inequalities

x > 1.4, x < 2.6, x > 4.2, x < 5.1 and x < 8.2.

To find the solution set which include 3, write the solution set which consists of integer.

The solution set of x > 1.4 is { 2, 3, 4, 5, 6,     ........}

The solution set of x < 2.4 is { 2, 1. 0, -1, ...............}

The solution set of x > 4.2 is { 5, 6. 7, 8, ...............}

The solution set of x < 5.1 is { 5, 4, 3, 2, 1, ...............}

The solution set of x < 8.2 is { 8, 7, 6, 5, 4, 3, ...............}

Hence, the solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.

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The correct answer is 86.08, as the octagon is given and the apothem given here is 13 units. The calculation after putting the value in formula is  86.08.

An octagon is a polygon with eight sides and eight angles. It is a two-dimensional geometric shape.  Each angle in a regular octagon measures 135 degrees, and all sides of a regular octagon are of equal length.

The formula is given below,

P= side length ×n

Apothem of octagon =13 units,

side length is = tan (360° / (2 × 8)) = (n/2) ÷ 13

= tan (360° / (2 × 8)) = n/26

tan 22.5°= n/26

n/26 =  0.4142

n = 10.76

perimeter of octagon = 8 ×  10.76 = 86.08

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The amount of juice served this year is given as follows:

513 gallons.

The amount of juice served this year is obtained applying the proportions in the context of the problem.

The amount last year was given as follows:

540 gallons.

The percentage of this year's amount relative to last year's amount is given as follows:

95%, due to the decay of 5%, 100 - 5 = 95%.

Hence the amount of juice served this year is given as follows:

0.95 x 540 = 513 gallons.

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The number of degrees the Figure A must be rotated counterclockwise around the origin to line up with Figure B is = 270°

Given data ,

Let the number of degrees the Figure A must be rotated counterclockwise around the origin to line up with Figure B be represented as A

Now , the triangle is represented by the figure A with coordinates as

A ( -2 , 3 )

And , the coordinates of the rotated triangle is A' ( 3 , 2 )

270° clockwise rotation: (x,y) becomes (-y,x)

270° counterclockwise rotation: (x,y) becomes (y,-x)

So , the triangle is rotated 270° counterclockwise rotation

Hence , the rotation is 270° counterclockwise

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