find the general solutions to the following inhomogeneous first-order linear differential equations using the particular solution method: i. y 0 3y

The zeroes of the factored polynomial as required to be determined in the task content are: x = -6 and x = 8.

It follows from the task content that the zeroes of the completely factorised polynomial are to be determined.

Since the given polynomial is; x² - 2x - 48 which has been factorised completely to; (x + 6) (x - 8).

Ultimately, the zeroes of the Polynomial are as follows;

x + 6 = 0; x = -6.

and

x - 8 = 0; x = 8.

Conclusively, the zeroes of the polynomial are; x = -6 and x = 8.

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a) Yes, the Cauchy-Schwarz Inequality holds for the given vectors and inner product. Since 74 ≤ 76.84, the Cauchy-Schwarz Inequality holds.

b) Yes, the triangle inequality holds for the given vectors and inner product. Since 20.25 ≤ 23.21, the triangle inequality holds.

Cauchy-Schwarz Inequality:

Using the given inner product, we have:

where ||u|| and ||v|| denote the norms of the vectors u and v, respectively.

We can calculate:

Then, applying the Cauchy-Schwarz Inequality, we have:

Triangle Inequality:

The triangle inequality states that for any vectors u, v, and w, we have:

Using the given vectors, we can calculate:

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The height of each shoebox is 5.5 inches and  the clerk can stack 7 shoe boxes (6 full and 1 partially filled) on the shelf.

A store clerk wants to stack shoe boxes on a shelf that is 3 ft tall.

A shoebox has a volume that is 528 cubic inches and the area of the base is 96 square inches.

Find the height of each shoebox and determine how many shoe boxes the clerk can stack on the shelf.

The volume of each shoebox.

To find the height of each shoebox, we have to know its volume and base area.

Let h be the height of the shoebox.

Volume of the shoebox is V = 528 cubic inches.

And area of the base is A = 96 square inches.

Therefore, Volume of the shoebox,

V = Ah ⇒ 528 = 96h ⇒ h = 528/96 ⇒ h = 5.5 inches.

Hence, the height of each shoebox is 5.5 inches.

We need to find how many shoe boxes the clerk can stack on the shelf.

The height of the shelf is 3 ft = 36 inches.

If the height of each shoe box is 5.5 inches, then the number of shoe boxes that the clerk can stack on the shelf can be found by dividing the total height of the shelf by the height of each box.

Therefore, the number of shoeboxes the clerk can stack on the shelf is:

No. of shoeboxes = Height of the shelf / Height of each shoebox

⇒ No. of shoeboxes = 36/5.5 = 6.54.

So, the clerk can stack 6 full shoe boxes and the 7th box can be filled partially (5.5 inches of height).

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The angle of rotation in standard form is 251.5651 degrees.

The length of arc S is approximately 20.94 feet.

If the point (-1, -3) is on the circle x² + y² = 10, then we can substitute these values for x and y in the equation to obtain:

(-1)² + (-3)² = 10

1 + 9 = 10

So, the point (-1, -3) satisfies the equation of the circle.

To find the angle of rotation in standard form, we need to use the formula:

tan(θ) = y/x

where θ is the angle of rotation.

In this case, x = -1 and y = -3, so we have:

tan(θ) = (-3)/(-1)

tan(θ) = 3

θ = tan⁻¹(3)

Using a calculator, we find:

θ ≈ 1.2490 radians or 71.5651 degrees

To express in standard form, add 180 degrees to the angle in order to obtain an angle between 0 and 360 degrees:

θ ≈ 71.5651 + 180 ≈ 251.5651 degrees

To find the length of an arc S given the radius r and the central angle θ, we use the formula:

S = (θ/360) x 2πr

In this case, r = 15 ft and θ = 1.396 radians

Substituting the values:

S = (1.396/2π) x 2π(15 ft)

S ≈ 1.396 x 15 ft

S ≈ 20.94 ft

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Complete question:

Consider the point (-1, -3), which is on the circle x² + y² = 10. Find the angle of rotation in standard form

Find the length of the arc S when r = 15 ft and θ = 13.96°

Answer:

7/20

Step-by-step explanation:

ok so it has 30 members, 7 are seniors it's possible. if there is multiple choice just let me know if it is not on there.

The value of x for the given inequality to justify the equation to get the desired result of the equation is x < - 4 .

Define inequality:

In mathematics, inequality refers to a statement that compares two values or expressions, indicating that one value or expression is greater than or less than the other. An inequality can be represented using symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to). For example, "2x + 3 > 5" is an inequality that states that the expression "2x + 3" is greater than "5". Inequalities are often used in algebra, calculus, and other branches of mathematics to represent relationships between variables or to solve equations.

What about equation in the relation?

In mathematics, an equation is a statement that asserts the equality of two expressions. An equation consists of two expressions, separated by an equals sign "=" and it states that the value of one expression is equal to the value of the other expression. The expressions in an equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Equations are used in many areas of mathematics to describe relationships between variables, to solve problems, and to make predictions. For example, the equation "x + 3 = 7" states that the value of the expression "x + 3" is equal to "7".

According to the given information:

For the following equation we have that,

⇒ - 8x < 32

⇒ -x < [tex]\frac{32}{8}[/tex]

⇒ -x < [tex]4[/tex]

⇒ x < [tex]- 4[/tex]

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At exactly 2:00, the distance between the tips of the two hands is changing at the rate of 4 cm/min.Therefore, at exactly 2:00, the distance between the tips of the two hands is changing at the rate of [tex]2\sqrt{13}[/tex] cm/min.

The phrase "at exactly 2:00" means that at 2:00:00, the hour hand and the minute hand overlap. Both hands are together pointing upwards in the vertical direction. At this moment, the angle between the minute hand and the hour hand is 0 degrees.How do we determine the rate of change of the distance between the tips of the two hands?We start by calculating the angle between the two hands. The angle is given by:

θ(t) = 30h - 11/2m

where h is the number of hours past midnight and m is the number of minutes past the most recent hour. At exactly 2:00, h = 2 and m = 0. Thus:

θ(t) = 30(2) - 11/2(0) = 60 degrees

Next, we use the formula for the distance between two points (the tips of the two hands). Let the hour hand be at the point (0,0) and let the minute hand be at the point (x,y).

The distance between the two points is given by:

[tex]d(t) = \sqrt{(x^2 + y^2)}[/tex]

To find x and y, we use the formulas:

x(t) = r cos θ(t)y(t) = r sin θ(t)

where r is the length of each hand. At exactly 2:00:

[tex]r = 4 cmx(t) = 4 cos 60 = 2 my(t) = 4 sin 60 = 2\sqrt{(3) m}[/tex]

Therefore: [tex]d(t) = \sqrt{(x^2 + y^2)}  = \sqrt{(2 m)^2 + (2sqrt(3) m)^2)}  = 2\sqrt{(13) m}[/tex]

Finally, we differentiate with respect to time t to find the rate of change of [tex]d(t):d'(t) = 2\sqrt{13}[/tex] cm/min

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We have shown that sin3A.cos2A - cos3A.sin2A / cos3A.cos2A + sin3A.sin2A = tanA is true for all values of A where the trigonometric functions are defined.

What is trigonometry?

The area of mathematics concerned with how triangles' sides and angles relate to one another as well as with the pertinent uses of any angle.

We can start by using trigonometric identities to simplify the left-hand side of the equation:

sin3A.cos2A - cos3A.sin2A = sin(3A + 2A) - sin(3A - 2A)

(using sum and difference formulae)

= 2 sin5A cosA

cos3A.sin2A = cos(3A - 2A) - cos(3A + 2A)

= -2 sinA cos5A

Substituting these into the original equation, we get:

(2 sin5A cosA) / (cos3A cos2A + sin3A sin2A) - (-2 sinA cos5A) / (cos3A cos2A + sin3A sin2A)

Simplifying further:

2 sin5A cosA + 2 sinA cos5A / (cos3A cos2A + sin3A sin2A)

We can use the identity tanA = sinA / cosA to write this as:

2 sinA cosA (sin4A + cos4A) / (cos3A cos2A + sin3A sin2A)

Using the identity sin2A + cos2A = 1 and rearranging, we get:

2 tanA (1 - sin2A cosA) / (cos2A + sin2A cosA)

= 2 tanA cosA / (cos2A + sin2A cosA)

= 2 sinA / (1 + cosA)

= (2 sinA / (1 + cosA)) * ((1 - cosA) / (1 - cosA))

= 2 sinA (1 - cosA) / (1 - cos^2A)

= 2 sinA (1 - cosA) / sin^2A

= 2 (1 - cosA) / sinA

= 2 sinA / sinA

= 2

Therefore, we have shown that sin3A.cos2A - cos3A.sin2A / cos3A.cos2A + sin3A.sin2A = tanA is true for all values of A where the trigonometric functions are defined.

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Answer: 32

Step-by-step explanation:

Answer:

Step-by-step explanation:

2 is the base and 5 is the exponent, so we have to multiply 2 5 times.

2*2*2*2*2=32

Answer: Proper fraction

Step-by-step explanation:

A proper fraction is a fraction that has its numerator value less than the denominator.

For example, ⅔, 6/7, 8/9, etc. are proper fractions.

Answer:

Step-by-step explanation: is 13 bc 13-8 is 5 and 5+1313is

Answer: -4

Step-by-step explanation:

If homogenous equation ax^2+2hxy+by^2=0---------------(a)

represent two perpendicular lines then, a+b=0

Calculation :

Given 3x² - 8xy + my² = 0

on comparing with equation (a) we get, a=3 and b=k

given lines are mutually perpendicular then

m+3= 0

m=-3

(a) The wave function (r,, d) = Ae(-r/a), with A and a constants. To determine the expected value of energy E, we must first compute |H|>, where H is the Hamiltonian operator.

H = -(2/2m) 2 + V(r) is the Hamiltonian operator for a particle in a spherically symmetric potential, where is the reduced Planck constant, m is the particle's mass, 2 is the Laplacian operator, and V(r) is the potential energy. We may assume that V(r) = 0 as r approaches infinity since the potential energy V(r) is given to be zero at infinity. As a result, the Laplacian operator in spherical coordinates is reduced to 1/r2(d/dr)(r2(d/dr)), and we obtain:[tex]< ψ|H|ψ > = ∫∫∫ ψ*(r,θ,φ) [-2/2m (1/r2)(d/dr)(r2(d/dr)) + V(r) (r,,)] dτ where d = r2 sin dr d d d[/tex] is the volume element. Using the given wave function and the simplification of the Laplacian operator,

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From the given information provided, Susan's percentile rank in class is approximately 6.9%.

To find the percentile rank for Susan who is 5th in a class of 58 students, we need to use the following formula:

Percentile rank = (Number of students below Susan / Total number of students) x 100

Since Susan is the 5th student in the class, there are 4 students with higher ranks than her. Therefore, the number of students below Susan is 4.

Using the formula, we can calculate the percentile rank as:

Percentile rank = (4 / 58) x 100

Percentile rank = 6.9 (rounded to one decimal place)

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We can claim that after answering the above question, the  This function would simulate the car's resale value if it depreciated at a rate of 15% each year.

In mathematics, a function is a relationship between two sets of numbers in which each member of the first set (known as the domain) corresponds to a single element in the second set (called the range). In other words, a function takes inputs from one set and produces outputs from another. Inputs are commonly represented by the variable x, while outputs are represented by the variable y. A function can be described using an equation or a graph. The equation y = 2x + 1 represents a linear function in which each value of x yields a distinct value of y.

1.1) Both Kala and Remy are incorrect. The given function, [tex]p(x) = 30,000(0.68)^x[/tex], models the resale value of an automobile after x years, with a starting value of $30,000. The function's exponent x reflects the number of years since the car was purchased, and the base 0.68 is the percentage of the original value that remains after each year.

1.2) If the car's value fell by 15% per year, the function would be[tex]p(x) = 30,000(1 - 0.15)^x[/tex]. This means that the car loses 15% of its worth each year, leaving 85% of the value. This function would simulate the car's resale value if it depreciated at a rate of 15% each year.

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For the above question by using inequalities she must buy atleast 8 tickets.

What are inequalities?

Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal.

So according to question

    One plan costs =$110

    Extra of $25 per ticket

 let the tickets be x, so the quation will be

                                   =>          110 + 25x ≤ 40x

                                    =>         110 ≤ 15x

                                     =>7 1/3 ≤ x

So, if she must  at least buy 8 tickets, the first plan saves money.

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The expression 24x = 55.51 -  3.19 the given condition and the number of bottle is 2.18 or 2 option D is correct

Given that,

She bought a bag of apple for $3.19 and a 24-pack of juice bottles. The total cost before tax was $55.51.

To determine the diagram could be used to represent the context if x represents how much each bottle of juice costs.

What is arithmetic?

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication, and division,

Here,

Let the cost of each juice bottle be x.

Now,

According to the question,

the expression will be

24x = 55.51 - 3.19

x= 2.18

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The Probability of that Lila selects a black ribbon and Jessica selects a green ribbon = 6/35. What is Probability? Probability is possibility that deals with "occurrence of random event. It values varies from 0 to 1". According to the question,

Number of black ribbon = 3

Number of green ribbon = 12

Total number of ribbon  = 3+12 =15

Formula for Probability of an event P(E) = (Number of favorable outcomes) /(Total number of outcomes)

Probability of Lila selecting black ribbon =3/15

Jessica selects a ribbon at random from the remaining ribbons, therefore total number of ribbons is reduced to 14.

Probability of Jessica selecting Green ribbon = 12/14

The Probability of that Lila selects a black ribbon and Jessica selects a green ribbon =(3/15)(12/14)=6/35.Hence, the Probability of that Lila selects a black ribbon and Jessica selects a green ribbon is 6/35.

To show that $n$ infinitely long straight lines, arranged on a plane in such a way that no two lines are parallel and no three lines intersect at a single point, divide the plane into $\frac{n^2 + n}{2}$ regions, follow these steps:

Solution:

1. Start with one straight line on the plane. This line divides the plane into two regions.
2. Add a second straight line that is not parallel to the first line and doesn't intersect at the same point. This second line will divide each of the two existing regions in half, creating two additional regions, for a total of four regions.
3. Now, add a third straight line that is not parallel to any of the existing lines and doesn't intersect at the same point as any other two lines. This line will intersect the two previous lines and divide each of the four existing regions in half, creating three additional regions for a total of seven regions.
4. Notice a pattern: with each new straight line added, the number of new regions it creates is equal to the line's order (i.e., the 1st line creates 1 new region, the 2nd line creates 2 new regions, the 3rd line creates 3 new regions, and so on).
5. To find the total number of regions for $n$ straight lines, sum the number of new regions created by each line. This is a simple arithmetic progression, so you can use the formula for the sum of an arithmetic series:

  $$\frac{n(n+1)}{2}$$

6. Thus, when $n$ infinitely long straight lines lie on a plane in such a way that no two of the lines are parallel and no three of the lines intersect at a single point, the arrangement divides the plane into $\frac{n^2 + n}{2}$ regions.

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Answer:

The total volume of the space is 55,000 cm³

Step-by-step explanation:

First, figure out the length for each of the two prisms. The prism on the right has a length of 25 cm, so we can subtract that from 50 cm (the combined length of both prisms) to get 25 cm.

Next, figure out the volume of each area. For a rectangular prism, Volume = length · width · height. The left prism's volume is 30,000 cm³ (do 25 · 60 · 20) while the right prism's volume is 25,000 cm³ (do 25 · 25 · 40).

Lastly, add the volumes together to get the total volume, which is 55,000 cm³

Answer:

circle K has greater area that circle J by 2000.18 cm sq.

Step-by-step explanation:

area of circle => π r^2

area of circle  j = 3.14 x 18 x 18 = 1017.36

area of circle K = 3.14 x 31 x 31 =3017.54

the difference in their areas : 3017.54- 1017.36  = 2000.18

Answer:

7 pounds

Step-by-step explanation:

3.5×8=28 pounds

35-28=7 pounds

so 7 pounds will b left

The probability that the speed is less than 80 kmph is 0.1587 (rounded to four decimal places).

The average speed of a car on the highway is 85 kmph with a standard deviation of 5 kmph.

Assume the speed of the car, x, is normally distributed.

The probability that the speed is less than 80 kmph is required.

We will use the Z-score formula to solve this problem.

Z-score formula:

Z = (X - μ) / σWhereX = 80 (the speed we are interested in)

μ = 85 (mean of the normal distribution)

σ = 5 (standard deviation of the normal distribution)

Z = (80 - 85) / 5 = -1

The Z-score for the speed of 80 kmph is -1.

Using the standard normal distribution table,

we find that the probability that the Z-score is less than -1 is 0.1587 (rounded to four decimal places).

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Diana's average speed for the entire race was approximately 0.583 meters per second.

We can start by using the formula

average speed = total distance / total time

We know that Diana ran a total distance of 700 meters, and we can find the total time by adding the time for the first lap and the time for the second lap. Let's call the distance of each lap "x"

total distance = 700 meters

distance for each lap = x meters

So, the total time is

total time = time for first lap + time for second lap

To find the time for each lap, we can use the formula

time = distance / speed

For the first lap, we have

time for first lap = x / 7

For the second lap, we have

time for second lap = x / 5

So, the total time is

total time = (x / 7) + (x / 5)

We can simplify this by finding a common denominator

total time = (5x + 7x) / (35)

total time = (12x) / 35

Now, we can substitute the values we have into the formula for average speed

average speed = total distance / total time

average speed = 700 / [(12x) / 35]

average speed = (700 × 35) / (12x)

average speed = 204.166... / x

To find the average speed for the entire race, we need to find the value of "x" that makes this expression true. We know that the two laps are equal in distance, so we can set

x + x = 700

2x = 700

x = 350

Substituting this value of "x" into the expression for average speed, we get

average speed = 204.166... / 350

average speed ≈ 0.583 meters per second

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1. Inverse of F=9/5C+32 is C=5/9(F-32)

2. 100 F = 37.8 C

3. 30 C = 86 F

To find the inverse of

F = 9/5C+32,

we need to solve for C. First, subtract 32 from both sides of the equation to get

F-32 = 9/5C.

Then, multiply both sides by 5/9 to isolate C, giving us

C = 5/9(F-32).

This is the inverse function.

To find the temperature in Celsius that corresponds to 100 degrees Fahrenheit, we can use the inverse function. Substituting F=100 into

C = 5/9(F-32), we get

C = 5/9(100-32) = 37.8 degrees Celsius.

To find the temperature in Fahrenheit that corresponds to 30 degrees Celsius, we can use the original equation. Substituting C=30 into

F=9/5C+32,

we get

F = 9/5(30)+32 = 86 degrees Fahrenheit.

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Complete Question:

F=9/5C+32, what would be its inverse? What temperature in C corresponds to 100 F? What temperature in F corresponds to 30 C?

Answer:

16/36 = 12/27

The ladder reaches the height of approximately 16.03 feet up the wall. (Rounded to the nearest hundredth of a foot)

We can use trigonometry to solve this problem. Let h be the height up the wall that the ladder reaches, and let θ be the angle that the ladder makes with the ground. We know that the length of the ladder (hypotenuse) is 16 feet and the angle θ is 78 degrees.

We can use the trigonometric function tangent (tan) to find the height h:

tan(θ) = opposite/adjacent

In this case, the opposite side is the height h, the adjacent side is the distance from the ladder to the wall (which is the same as the distance from the wall to Lauren), and the angle θ is 78 degrees. So we have:

tan(78°) = h/x

where x is the distance from the ladder to the wall. Solving for h, we get:

h = x * tan(78°)

We don't know the value of x, but we can use the fact that the ladder is 16 feet long to find it. Using the Pythagorean theorem, we have:

x^2 + h^2 = 16^2

Substituting h = x * tan(78°), we get:

x^2 + (x * tan(78°))^2 = 16^2

Simplifying, we get:

x = 16 / √(1 + tan^2(78°))

x ≈ 3.83 feet

Finally, we can substitute x into our equation for h:

h = x * tan(78°)

h ≈ 16.03 feet

Therefore, the ladder reaches approximately 16.03 feet up the wall. Rounded to the nearest hundredth of a foot, the answer is 16.03 feet.

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Answer:

The shape of each cut surface is a rectangle. The dimensions of each cut surface are 8 inches by 10 inches.

The area of each cut surface is 8 inches x 10 inches = 80 square inches.

Since there are two cut surfaces, the total area that Jenny needs to paint is 2 x 80 square inches = 160 square inches.

Answer: Each cut surface is a rectangle with dimensions of 8 inches by 10 inches.

The area of each cut surface is:

A = l * w

A = 8 in * 10 in

A = 80 sq in

So each cut surface has an area of 80 square inches.

To find the total area Jenny needs to paint, we need to find the combined area of both cut surfaces. Since there are two cut surfaces, we simply need to multiply the area of one cut surface by 2:

Total area = 2 * area of one cut surface

Total area = 2 * 80 sq in

Total area = 160 sq in

Therefore, Jenny needs to paint a total area of 160 square inches.

Step-by-step explanation:

The school can purchase 91 adult tickets and 41 children tickets, allowing for a total of 132 teachers and students to go on the trip within the budget of $12000.

Let's start by defining some variables to represent the quantities we're interested in: Let A be the number of adult tickets purchased. Let C be the number of children tickets purchased. Let T be the total number of teachers and students who go on the trip. To maximize the number of teachers and students while staying within budget, we can use linear programming. The objective function is N = T - A - C, where N is the number of teachers and students. The inequalities are 68A + 47C ≤ 12000 and T ≤ 225. Solving this linear program, we find that the school can purchase 91 adult tickets and 41 children tickets, allowing for a total of 132 teachers and students to go on the trip within the budget of $12000.

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the complete question :

A middle school wants to take students and teachers on a field trip to Six Flags Great America for spring break. The school has a budget of spending at most $12,000. They can also only take no more than 225 teachers and students. The adult tickets cost $68 each, and children tickets cost $47 each. How many adults and children can the school take to Six Flags within the given budget and attendance limit?

Joe is incorrect in claiming that he scored better than 80% of the people who took the test.

The 20th percentile means that Joe's score was equal to or greater than 20% of the other test takers, but lower than 80%. The percentile is not a percentage, so the statement Joe made is false.

It is important to understand that percentile rankings are not the same as percentages. Percentile rankings measure how one’s score compares to others who have taken the same test. For example, if Joe scored in the 20th percentile, it means that 20% of the other test takers had the same or lower scores. On the other hand, percentages measure a proportion of the whole. In Joe's case, 80% would mean that 80 out of 100 test takers had a higher score than Joe.

To be more accurate, Joe could have said that he scored better than 20% of the people who took the test. Percentile rankings are often used to measure an individual's performance in comparison to a larger group of peers. Although Joe might have performed well, it is important to understand the difference between percentile and percentage.

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