Find the particular solution for:
f"(x) = 0.25x⁻³/², f'(4) = - 1/8 and f(0) = 2.

The area of the regular pentagon is approximately 392.2 square centimeters.

The area of a regular polygon can be calculated using the formula:

A = (1/2) * apothem * perimeter

where perimeter = number of sides * side length.

For a pentagon with side length s = 15.1 cm, the perimeter is:

perimeter = 5 * 15.1 = 75.5 cm

The apothem is a = 10.4 cm.

Using the formula, we get:

A = (1/2) * 10.4 * 75.5

A = 392.2 cm^2

Therefore, the area of the regular pentagon is approximately 392.2 square centimeters.

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We can read 91 books in a year if you increase your reading speed so that each page takes you 30 seconds less than it did before.

Now If each page now takes 30 seconds less to read than before,

then you will save 30*200 = 6000 seconds (or 100 minutes) on each book.

So, the time it will take you to read a 200-page book will be

20 minutes - 100 minutes = -80 minutes,

which means you will finish a 200-page book in 80 minutes (or 1 hour and 20 minutes).

In a year, there are 365 days. If you read for 20 minutes per day, then you will read for a total of

365 * 20 = 7300 minutes (or 121.67 hours) in a year.

Since you can finish a 200-page book in 80 minutes, you can read 7300/80 = 91.25 books in a year.

However, since you cannot read a fraction of a book, the maximum number of 200-page books you can read in a year is 91 books.

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Complete question is :

If you increase your reading speed so that each page takes you 30 seconds less than it did before,

and you begin reading 20 minutes per day,

How many 200 page books can you now read in a year?

A

Either divide each by one fourth or multiply each by 0.25. Then turn the answer to a fraction.

The local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.

To find the local minimum and maximum of the given function, we need to find its critical points, where the derivative is zero or undefined.

[tex]f(x) = 2x^3 + 30x^2 - 54x + 5[/tex]

[tex]f'(x) = 6x^2 + 60x - 54[/tex]

Setting f'(x) = 0, we get:

[tex]6x^2 + 60x - 54 = 0[/tex]

[tex]x^2 + 10x - 9 = 0[/tex]

(x + 9)(x - 1) = 0

x = -9 or x = 1

Now, we need to determine if these critical points correspond to local minimum or maximum.

To do so, we can use the second derivative test. We calculate the second derivative of f(x):

f''(x) = 12x + 60

At x = -9:

f''(-9) = 12(-9) + 60 = -48 < 0

This means that f(x) has a local maximum at x = -9.

At x = 1:

f''(1) = 12(1) + 60 = 72 > 0

This means that f(x) has a local minimum at x = 1.

To find the values of the local minimum and maximum, we plug in the corresponding x-values into the original function:

[tex]f(-9) = 2(-9)^3 + 30(-9)^2 - 54(-9) + 5 = 2575[/tex]

[tex]f(1) = 2(1)^3 + 30(1)^2 - 54(1) + 5 = -17[/tex]

Therefore, the local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.

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The system of inequalities that models the situation is given as follows:

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

The perimeter of a rectangle of width w and length l is given as follows:

P = 2w + 2l.

You want the run to be at least 10 ft wide, hence:

w ≥ 10.

The run can be at most 50 ft long, hence:

0 < l ≤ 50.

(length has to be greater than zero).

You have 126 ft of fencing, hence the perimeter is represented as follows:

2w + 2l ≤ 126.

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The system of inequalities in the graph is the one in option A.

y ≥ (-2/5)x - 2/5

y ≥ (3/2)*x - 1

Here we can see the graph of a system of inequalities, on the graph we can see two lines.

The first one is a line with a positive slope, it has an y-intercept of -1, the shaded region is above that line,  and it is a solid line, so one of the inequalities is:

y ≥ a*x - 1

Where a is positive.

The second line has a negative slope, and we can see that the shaded region is also above the line, so this second inequality is like:

y ≥ line with negative slope.

It is easy to identify the correct option because there is only one with these properties, which is the first option:

y ≥ (-2/5)x - 2/5

y ≥ (3/2)*x - 1

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To find the maximum concentration of an antibiotic between hours 1 and 7, first find the critical points of the function C(t), then evaluate C(t) at the critical points and endpoints to choose the highest value.

To determine the maximum concentration of the antibiotic between hours 1 and 7, follow these steps:
1. Find the critical points of the function C(t) = 4te^(-397). To do this, find the first derivative of the function, C'(t), and set it equal to 0.
2. Check the value of C(t) at the critical points and the endpoints of the interval, t=1 and t=7.
3. Choose the highest value of C(t) among the critical points and the endpoints.
1: Find the first derivative, C'(t).
C(t) = 4te^(-397)
C'(t) = 4e^(-397)(1-397t)
2: Set the first derivative equal to 0 and solve for t.
4e^(-397)(1-397t) = 0
1 - 397t = 0
t = 1/397
3: Evaluate C(t) at the critical point t = 1/397 and the interval endpoints t = 1 and t = 7.
C(1/397) = 4(1/397)e^(-397(1/397)) ≈ 0.01 ag/mg
C(1) = 4(1)e^(-397(1)) ≈ 0.00 ag/mg
C(7) = 4(7)e^(-397(7)) ≈ 0.00 ag/mg
The maximum concentration of the antibiotic occurs at t = 1/397 hours, with a concentration of approximately 0.01 ag/mg. What is Titration: Titration is a technique by which we know the concentration of unknown solution using titration of this solution with solution whose concentration is known.

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Answer:

That's the correct order.

Step-by-step explanation:

2x+20+5=55

2x+25=55

2x=30

x=15

The expression equation y = sin x + cos x / csc x can be simplified to y = cos x sin^2 x + cos^2 x.

To simplify the expression, we can first replace csc x with 1/sin x. This gives us y = sin x cos x + cos^2 x / sin x.

Next, we can factor out cos x from the numerator of the second term to get y = cos x (sin x + cos x) / sin x.

Using the identity sin^2 x + cos^2 x = 1, we can replace sin^2 x with 1 - cos^2 x in the numerator of the first term. This gives us y = cos x (1 - cos^2 x) / sin x + cos x (sin x + cos x) / sin x.

Simplifying the expression further, we get y = cos x (1 - cos^2 x + sin x + cos x) / sin x.

Finally, we can combine the terms in the numerator to get y = cos x (sin^2 x + 2cos x sin x + 1) / sin x.

Using the identity sin^2 x = 1 - cos^2 x, we get y = cos x (3cos^2 x + 2cos x) / sin x.

Simplifying the expression, we arrive at y = cos x (cos x + 2) (3cos x + 2) / sin x.

Therefore, the simplified expression is y = cos x sin^2 x + cos^2 x, which can also be written as y = cos x (sin x)^2 + cos^2 x.

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The value of x and y for the angles are 4 and 9 respectively.

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

10x - 4 = 6(x + 2)     (opposite angles are equal to each other)

10x - 4 = 6x + 12

4x = 16

x = 4

Also:

18y - 18 = 16y

2y = 18

y = 9

The value of x and y are 4 and 9 respectively.

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The volume of the gift bag is given as 1152 cubic inches.

Since it is shaped like a rectangular prism, we can write the formula for its volume as V = l × w × h, where l, w, and h are the length, width, and height of the rectangular prism, respectively.

To determine the dimensions of the gift bag, we need more information such as the ratio of its length, width, and height or any one of its dimensions. If we assume one of the dimensions, say, the length is L inches, then we can write the volume as V = L × w × h. Solving for w × h, we get w × h = V/L = 1152/L.

We can then use this equation along with the fact that the gift bag is a rectangular prism to find the other dimensions. For example, if the width is W inches, then we have h = 1152/(L × W) and the volume can be expressed as V = L × W × 1152/(L × W) = 1152.

Similarly, if the height is H inches, then we have w = 1152/(L × H) and the volume can be expressed as V = L × 1152/(L × H) × H = 1152.

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The following is the value of x and mBD: x = 83/6 and mBD⌢ = 83°.

Given circle E (⊙E) with diameter AC, we have the following information:

1. m∠CED = (3x)°
2. m∠BEC = (3x + 2)°
3. m∠AEB = (6x + 7)°

Since AC is the diameter of the circle, the inscribed angle ∠AEB is subtended by the diameter and thus forms a semicircle. In a semicircle, the inscribed angle is always a right angle. Therefore, m∠AEB = 90°. We can now set up the equation:

(6x + 7)° = 90°

Solving for x:
6x = 83
x = 83/6

Now, we need to find the measure of arc BD (mBD⌢). We can do this by finding the measure of angle BEC, as the measure of an inscribed angle is half the measure of its intercepted arc.

m∠BEC = (3x + 2)° = (3(83/6) + 2)° = (83/2)°

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

mBD⌢ = 2(m∠BEC) = 2(83/2)° = 83°

So, x = 83/6 and mBD⌢ = 83°.

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The relationship between these two variables can be described as inversely proportional.

The variation between the distance that a man walks in one hour and the rate at which the man is walking can be described as inversely proportional.

This means that as the rate at which the man is walking increases, the distance that he walks in one hour decreases. Similarly, as the rate at which he is walking decreases, the distance that he walks in one hour increases.

Therefore, the relationship between these two variables can be described as inversely proportional.

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Answer:

the probability of selecting a blue sock from the laundry basket is 2/7 or approximately 0.2857.

Step-by-step explanation:

The probability of selecting a blue sock can be found by dividing the number of blue socks by the total number of socks in the basket:

Probability of selecting a blue sock = Number of blue socks / Total number of socks

Probability of selecting a blue sock = 4 / 14

Simplifying the fraction by dividing both the numerator and denominator by 2 gives:

Probability of selecting a blue sock = 2 / 7

Limiting space heating would help decrease the greatest carbon-releasing activity in US homes.

From the table, we can see that space heating accounts for the largest percentage of energy use in US homes at 45%. Therefore, by limiting space heating, we can significantly reduce the amount of carbon released into the atmosphere, thus decreasing our carbon footprint.

Other activities like limiting time in hot showers, wearing layers of clothing, and turning off lights and electronics when not in use can also help reduce our carbon footprint, but they are not as effective as limiting space heating.

Additionally, we can consider using energy-efficient heating systems, improving insulation, and reducing air leaks to further reduce our energy use and carbon footprint.

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Answer:

1. x = 5

2. x = [tex]\sqrt 7 \approx[/tex] 2.65

Step-by-step explanation:

To solve these problems, we can use the diagram below, as well as Pythagoras's Theorem, which states:

[tex]\boxed{\mathrm{a^2 = b^2 + c^2}}[/tex],

where a is the hypotenuse (longest side) of a right-angled triangle, and b and c are the legs of the triangle.

1. x is the hypotenuse:

From the diagram below, we can see that, if x is the hypotenuse of the triangle, then 3 and four are the legs of the triangle. Therefore, using the above equation:

[tex]x^2 = 3^2 + 4^2[/tex]

⇒ [tex]x = \sqrt{3^2 + 4^2}[/tex]      [Taking the square root of both sides of the equation]

⇒ [tex]x = \sqrt{25}[/tex]

⇒ [tex]x = \bf 5[/tex]

2. x is one of the legs:

From the diagram below, we can see that when x is one of the legs, the hypotenuse is 4, and the other leg is 3.

The hypotenuse isn't 3 because the hypotenuse is the longest side in a right-angled triangle, and 4 is longer than 3.

Therefore,

[tex]4^2 = x^2 + 3^2[/tex]

⇒ [tex]x^2 = 4^2 - 3^2[/tex]             [Subtracting 3² from both sides]

⇒ [tex]x = \sqrt{4^2 - 3^2}[/tex]          [Taking the square root of both sides of the equation]

⇒ [tex]x = \sqrt{7}[/tex]

       [tex]\approx \bf 2.65[/tex]

Yes, the points that fit the rule x→y=−2x can be plotted or marked on a coordinate plane, and will form a straight line passing through the origin with a slope of -2.

The rule x→y=−2x means that for every value of x, the corresponding value of y is equal to -2 times x. To plot the points that fit this rule, we can choose some values of x, substitute them into the equation, and then plot the resulting points on the coordinate plane.

For example, if we choose x=1, then y=−2x=−2(1)=−2. So the point that fits the rule is (1, −2). Similarly, if we choose x=2, then y=−2x=−2(2)=−4. So the point that fits the rule is (2, −4).

We can continue this process for other values of x, and plot all the resulting points on the coordinate plane. The resulting graph will be a straight line that passes through the origin, with a slope of -2. This means that all the points that fit the rule will lie on this line.

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The first hotel is approximately 0.6246 meters away from the dock, and the second hotel is approximately 0.4931 meters away from the dock.

To solve this problem, we can use trigonometry and create a system of equations based on the given angles and distances. Let's assume the distance between the dock and the first hotel is x meters and the distance between the dock and the second hotel is y meters.

Using the law of sines, we can relate the angles and distances:

For the first hotel:

sin(49°) = y / x     ...(Equation 1)

For the second hotel:

sin(56°) = x / y     ...(Equation 2)

We can rearrange Equation 1 to solve for y:

y = x * sin(49°)

Substituting this value of y into Equation 2:

sin(56°) = x / (x * sin(49°))

sin(56°) = 1 / sin(49°)

Now, we can solve for x by isolating it:

x = sin(56°) * sin(49°)

Plugging in the values and evaluating the equation:

x = 0.8290 * 0.7539

x ≈ 0.6246

Therefore, the distance between the dock and the first hotel (x) is approximately 0.6246 meters.

To find the distance between the dock and the second hotel (y), we can substitute this value back into Equation 1:

y = 0.6246 * sin(49°)

y ≈ 0.4931

Hence, the distance between the dock and the second hotel (y) is approximately 0.4931 meters.

In summary, the first hotel is approximately 0.6246 meters away from the dock, and the second hotel is approximately 0.4931 meters away from the dock.

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Ms. Yamagata needs to buy at least 2628 6-inch tiles to cover the floor of her rectangular bathroom.

To determine the least number of 6-inch tiles Ms. Yamagata needs to buy to cover her 9 feet long and 73 feet wide bathroom floor, follow these steps:

1. Convert the dimensions of the bathroom to inches, as the tiles are measured in inches:
  9 feet * 12 inches/foot = 108 inches long
  73 feet * 12 inches/foot = 876 inches wide

2. Determine the total area of the bathroom in square inches:
  Area = length * width = 108 inches * 876 inches = 94,608 square inches

3. Calculate the area of a single 6-inch tile:
  Area = length * width = 6 inches * 6 inches = 36 square inches

4. Divide the total area of the bathroom by the area of a single tile to find the least number of tiles needed:
  Number of tiles = total area / tile area = 94,608 square inches / 36 square inches ≈ 2,628.56

Since Ms. Yamagata cannot buy a fraction of a tile, she needs to buy at least 2,629 6-inch tiles to cover her bathroom floor.

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The Experimental probability of  rolling the color green is 0.15

The experimental probability of rolling the color green can be calculated by dividing the number of times the die lands on green by the total number of rolls:

Experimental probability of green = Number of green rolls / Total number of rolls

Experimental probability of green = 18 / 120

Experimental probability of green = 0.15 or 15%

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The length of the hypotenuse of each right triangle to the nearest tenth of an inch is 24 inches.

Here, two right triangles with legs each measuring 17 inches are created when the square piece of metal is sliced diagonally.

We will apply the Pythagorean Theorem to find the length of the hypotenuse.

The Pythagorean theorem states that square of hypotenuse is equal to the sum of the squares of opposite side and adjacent side.

Now,

[tex]Hypotenuse^{2} = 17^{2}+ 17^{2} \\Hypotenuse^{2} =289+289\\Hypotenuse^{2} =578\\Hypotenuse=\sqrt{ 578}\\Hypotenuse=24.04[/tex]

⇒ Hypotenuse ≈ 24 inches (to the nearest tenth of an inch)

Therefore, the length of the hypotenuse of each right triangle to the nearest tenth of an inch is 24 inches.

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question 8.

It is expected to see precipitation on approximately 1.15 days in any given week in Raleigh, NC based on the data from January 1, 2022, to March 26, 2022.

question 9.

The probability that exactly 90 of the plants will successfully grow is approximately 0.0860.

Option  A is correct.

0(6/13) + 1(4/13) + 2(0) + 3(2/13) + 4(0) + 5(1/13) + 6(0) + 7(0) = 1.1538

binomial distribution with n = 100 (the number of trials) and

p = 0.87 (the probability of success on each trial).

we use the binomial probability formula to find the probability that exactly 90 plants will grow,

P(X = 90) = (100 choose 90) * (0.87)^90 * (0.13)^10

P(X = 90)  =  0.0860.

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The equation of curve passing through the point (9,8) with slope [tex]6\sqrt{x}[/tex] is [tex]y = 4\times x^{(3/2)} -100[/tex] .

Slope of curve is given by [tex]6\sqrt{x}[/tex] and curve is passing through point (9,8) .

A curve can be represented in a graph using the standard form of equations. Equation will represent the slope of curve and point through which the curve is passing.

Equation of curve :

Differentiate the slope equation,  

dy/dx =  [tex]6\sqrt{x}[/tex]

dy = [tex]6\sqrt{x}[/tex] dx

Integrating both sides,

Integration rule : [tex]\int\ {x^n} \, dx = x^{n+1}/n+1 + c[/tex]

[tex]y = 6 \times (x^{3/2})/(3/2) +c[/tex]

[tex]y = 4 x^{(3/2)} + c[/tex]

Now substitute (9,8) in the equation of y,

[tex]8 = 4\times (9)^{3/2} + c[/tex]

[tex]c = -100[/tex]

Substitute the value to get the equation of curve,

The equation of curve is [tex]y = 4\times x^{(3/2)} -100[/tex] .

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The contents of the box, by weight, consists of 14.29 percent caramels.

We need to find the percentage of caramels in the box, given the weights of caramels and coconut candies.

Step 1: Determine the total weight of the candies in the box.
The box contains 0.6 pounds of caramels and 3.6 pounds of coconut candies. Add these two weights together:

Total weight = 0.6 (caramels) + 3.6 (coconut)
Total weight = 4.2 pounds

Step 2: Calculate the percentage of caramels in the box.
To find the percentage, divide the weight of caramels by the total weight of the box and then multiply by 100:

Percentage of caramels = (Weight of caramels / Total weight) x 100
Percentage of caramels = (0.6 / 4.2) x 100

Step 3: Solve the equation.
Percentage of caramels = (0.6 / 4.2) x 100 ≈ 14.29%

So, approximately 14.29% of the contents of the box, by weight, consists of caramels.

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Using a hammer and nail instead of a screwdriver and screw would change the type of force applied (linear versus rotational) and the distance covered (shorter linear distance versus longer turning distance). These differences can affect the efficiency, holding power, and ease of use when connecting materials.

When using a screwdriver and screw, the applied force of 20 newtons and turning distance of 0.5 meters involve rotational motion to insert the screw into the wood. The screwdriver acts as a lever, and the screw's threads translate the rotational force into linear motion, increasing the grip strength and holding power.

In contrast, when using a hammer and nail, the force applied would be different because the action is linear instead of rotational. The hammer delivers a series of high-impact, short-duration forces to drive the nail into the wood. The amount of force required would depend on factors like the size of the nail, the hardness of the wood, and the user's strength.

Additionally, the distance covered during hammering would be different. Unlike the screwdriver's 0.5-meter turning distance, the hammer's motion covers a shorter linear distance as it strikes the nail head repeatedly. The total distance depends on the number of hammer strikes and the length of the nail.

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Complete Question:

A person uses a screwdriver to turn a screw and insert it into a piece of wood. The person applies a force of 20 newtons to the screwdriver and turns the handle of the screwdriver a total distance of 0.5 meter. How would these numbers be different if the person inserted a nail with a hammer instead of the screw with the screwdriver?

A. The force applied would be greater, but the distance would be shorter.

B. The force applied would be less, but the distance would be greater.

C. The force applied would be the same, but the distance would be shorter.

D. The force applied would be the same, but the distance would be greater.

The most appropriate conclusion that can be made when the null hypothesis is rejected for the chi-square test of independence is: There is a significant association between fear of crime and one's perception of their neighborhood as a high crime area or not.

If the null hypothesis is rejected in a chi-square test of independence, it means that there is a significant association between the two variables being studied. In this case, the fear of crime is dependent on one's perception of whether their neighborhood is a high crime area or not.

Therefore, the most appropriate conclusion that can be made is that there is a significant relationship between the two variables, and one's perception of their neighborhood being a high crime area or not is a predictor of fear of crime. However, the chi-square test of independence does not determine causality, so it is not possible to conclude which variable is causing the other. Further research would be required to determine the direction and nature of the relationship between the two variables.

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The measure of ∠N to the nearest degree is 82°.

To find the measure of ∠N, we can use the Pythagorean theorem and trigonometric functions.

First, we can use the Pythagorean theorem to find the length of MN:

MN² = NO² - OM²

MN² = (6.7 feet)² - (1.7 feet)²

MN² = 44.56 feet²

MN = 6.67 feet

Now, we can use the sine function to find the measure of ∠N:

sin(N) = MN/NO

sin(N) = 6.67 feet / 6.7 feet

sin(N) ≈ 0.994

N ≈ sin⁻¹(0.994)

N ≈ 82.4°

The measure of ∠N to the nearest degree is 82°.

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Answer: To solve for x in the equation X/y=z-8, we need to isolate x on one side of the equation.

Multiplying both sides by y, we get:

X = y(z-8)

Therefore, the solution for x is:

X = y(z-8)

Step-by-step explanation:

The system of linear inequalities representing the fishing restrictions is a ≤ 15, y ≤ 10, a + y ≤ 20.

Let's define the variables as follows:

a = Number of surfperch caught per day.

y = Number of rockfish caught per day.

Based on the given information, we can write the following system of linear inequalities:

Surfperch inequality: a ≤ 15

This inequality represents the restriction that you cannot catch more than 15 surfperch per day.

Rockfish inequality: y ≤ 10

This inequality represents the restriction that you cannot catch more than 10 rockfish per day.

Bottomfish inequality: a + y ≤ 20

This inequality represents the overall restriction that the total number of bottom fish (which includes both surfperch and rockfish) cannot exceed 20 per day.

Therefore, the system of linear inequalities is:

a ≤ 15

y ≤ 10

a + y ≤ 20

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