Your new computer cost $1500 but it depreciates in value by about 18% each year, how long will it take before your computer is worth $50?

The limit of the trigonometric function f(x) = (1 - cos x) / x is equal to 0.

In this problem we need to determine the limit of a trigonometric function for x → 0. This can be done by simplifying the expression by trigonometric formulas. First, write the trigonometric function:

f(x) = (1 - cos x) / x

Second, modify the expression by means of algebra properties and trigonometric formulas:

f(x) = (2 / x) · (1 - cos x) / 2

f(x) = sin² (x / 2) / (x / 2)

f(x) = sin (x / 2) · [sin (x / 2) / (x / 2)]

For u = x / 2:

f(u) = sin u · (sin u / u)

Third, use limits to evaluate the trigonometric function:

f(u) = 0 · 1

f(u) = 0

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The probability that the number of times Luke will hit the inner ring of the target out of the 5 attempts is less than the mean of X is 0.951.

A discrete random variable with a countable number of potential values is said to have a discrete probability distribution. Each possible value of the random variable is given a probability by the probability distribution, and the sum of these probabilities is 1. The number of heads you get while flipping a coin or the number of cars that pass through a specific crossroads in a given hour are both examples of discrete random variables.

The mean that Luke will hit the inner ring is given as:

E(X) = np

Now, n = 5 and p = 0.90.

So, E(X) = 5 x 0.90 = 4.5

Now, the probability of less than 4.5 is given as:

P(X < 4.5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

P(X < 4.5) = 0.0005 + 0.0144 + 0.1361 + 0.4095 + 0.3915

P(X < 4.5) = 0.951

Hence, the probability that the number of times Luke will hit the inner ring of the target out of the 5 attempts is less than the mean of X is 0.951.

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The work required to pump the oil out through a hole to the top of the tank is approximately 6,476,160π/3 foot-pounds.

We can find the work required to pump the oil out of the tank by using the formula:

W = ∫[V1, V2]ρgh dV

Where

First, we need to find the density of the oil. We are told that the oil weighs 50 pounds per cubic foot, so:

ρ = 50 lb/ft^3

Next, we need to find the height of the oil column being pumped. The tank is half full, so the height of the oil column is:

h = r - (r/2) = r/2 = 8/2 = 4 feet

Now, we need to find the volume of oil being pumped. Since the tank is half full, the volume of oil is:

V = (1/2)(4/3)πr^3 = (1/2)(4/3)π(8)^3 = 1,024π/3 cubic feet

Finally, we can integrate the work formula to find the total work required:

W = ∫[V1, V2]ρgh dV

W = ∫[0, 1,024π/3] (50 lb/ft^3)(32.2 ft/s^2)(4 ft) dV

W = (6,476,160π/3) ft-lb

Therefore, the work required to pump the oil out through a hole to the top of the tank is approximately 6,476,160π/3 foot-pounds.

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Answer:

Using the logarithmic identity log(a) + log(b) = log(ab), we can simplify the left-hand side of the equation:

log2(x-1) + log2(x+5) = log2((x-1)(x+5))

So the equation becomes:

log2((x-1)(x+5)) = 4

Using the exponential form of logarithms, we can rewrite the equation as:

2^4 = (x-1)(x+5)

Simplifying:

16 = x^2 + 4x - 5

Rearranging:

x^2 + 4x - 21 = 0

Using the quadratic formula:

x = (-4 ± sqrt(4^2 - 4(1)(-21))) / (2(1))

x = (-4 ± sqrt(100)) / 2

x = (-4 ± 10) / 2

So x = -7 or x = 3.

However, we need to check whether these solutions satisfy the original equation. We can see that x = -7 does not work, because both terms inside the logarithms would be negative. Therefore, the only solution is x = 3.

Answer:

Using the properties of logarithms, we can simplify the left-hand side of the equation:

log2(x-1) + log2(x+5) = log2((x-1)(x+5))

Therefore, the equation becomes:

log2((x-1)(x+5)) = 4

Using the definition of logarithms, we can rewrite this equation as:

2^4 = (x-1)(x+5)

16 = x^2 + 4x - 5

Simplifying further:

x^2 + 4x - 21 = 0

We can now use the quadratic formula to solve for x:

x = (-4 ± sqrt(4^2 - 4(1)(-21))) / (2*1)

x = (-4 ± sqrt(100)) / 2

x = (-4 ± 10) / 2

x = -7 or x = 3

However, we need to check if these solutions satisfy the original equation.

When x = -7:

log2(x-1) + log2(x+5) = log2((-7-1)(-7+5)) = log2(16) = 4

So x = -7 is a valid solution.

When x = 3:

log2(x-1) + log2(x+5) = log2((3-1)(3+5)) = log2(16) = 4

So x = 3 is also a valid solution.

Therefore, the solutions to the equation log2(x-1) + log2(x+5) = 4 are x = -7 and x = 3.

Step-by-step explanation:

Answer:

4. n + 24

5. b - 11

6. d/5

The answers are explained in the solution.

Considering the triangle, ABC,

BC = √AC²-AB² [Pythagoras theorem]

BC = √126.4²-126²

BC = 10 ⇒ Height of the ramp at B,

Slope = tanBC/AB = 10/126

The slope is less than 1/12, hence, it will get ADA approval,

Let θ be angle of elevation,

θ = tan⁻¹(10/126)

= 4.5° < 5°

Hence the town met the given requirement.

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The real distance between City X and City Y is 17 miles.

We know that the scale of the drawing is:

1 inch = 17 miles.

Now, if you look at the diagram for cities X and Y, you can see that the distance between City X and City Y is exactly 1 inch.

And we know that 1 inch is equivalent to 17 miles, then we can conclude that the actual distance between the two cities is exactly 17 miles.

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The probability of selecting an R or an E without replacement, and then selecting an S is 5/36

Because the word SEPTEMBER has 9 letters, there are 9 different alternatives for the initial letter.

The probability of selecting a R or an E without replacing is 2+3=5.

The odds of picking a R or an E on the initial draw are 5/9.

After the first letter is drawn, the bag contains eight letters, including one S. If the first letter is not replaced, there are only four letters that fit the requirement.

Given that a R or an E was selected without replacement on the first draw, the probability of selecting a S on the second draw is 4/8.

When we multiply these probability together, we get:

P(R or E, not replacing) * P(S after R or E, not replacing) = (5/9) * (4/8) = 10/72 = 5/36

Hence, the probability of selecting an R or an E without replacement, and then selecting an S is 5/36

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The equations solved for the variables T₁ and P₁ are:

We start with the equation:

(P₁*V₁)/T₁ = (P₂*V₂)/T₂

And we want to solve this for T₁, we can multiply both sides by T₁ and divide both sides by the expression in the right side.

(P₁*V₁) = T₁*[ (P₂*V₂)/T₂]

(P₁*V₁)*[T₂/(P₂*V₂)] = T₁

That is the equation solved for T₁.

34: Now we have the same equation but we want to solve it for P₁, to do so, just multiply both sides by T₁/V₁

We will get:

(T₁/V₁)*(P₁*V₁)/T₁= (T₁/V₁)*(P₂*V₂)/T₂

P₁ = (T₁/V₁)*(P₂*V₂)/T₂

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Answer:

I hope this helps...

Please mark me brainliest

Answer:

Step-by-step explanation:

1 out of 100 is 1 percent. This is because percentage is always out of 100 so you don’t have to change anything. That means 1 is always 1 percent of 100. It is also 1 part of 100.

1). There are 5,245,786 different number combinations that could win. The likelihood of taking home the lottery's grand prize is 1 in 5,245,786 or roughly 0.000019%.

2). There are 42,467,328,000 different winning number combinations that a gambler can choose from.

Combinations are the various ways, independent of their sequence, in which a group of things or objects can be chosen.

The formula n! / (r! * (n-r)! can be used to determine the number of potential combinations of r items from a collection of n items, which is symbolised by the symbol C(n,r).

1. Six numbers are chosen at random from a pool of 42 numbers in a 6/42 lottery. The formula for combinations can be used to determine how many winning number combinations a gambler has the option of choosing:

C(42, 6) = 42! / (6! * (42-6)!)

= 5,245,786

2. The number of winning number combinations that a bettor may choose to select can be determined using the formula for permutations with repetition if the 6/42 Lottery permits repeat of numbers:

[tex]42^6[/tex] = 42 * 42 * 42 * 42 * 42 * 42 = 42,467,328,000

There are therefore 42,467,328,000 different ways to pick winning numbers. The odds of taking home the lottery's grand prize are 1 in 42,467,328,000, or roughly 0.000000002%.

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Answer:

Step-by-step explanation:

False.

The correct identity is:

cos^2(x/3) = 1/2[1+cos(2x/3)]

To see why, use the double angle formula for cosine:

cos(2x/3) = 2cos^2(x/3) - 1

Substitute this into the original equation:

cos(x/3)cos(x/3) = 1/2[1+2cos^2(x/3)-1]

Simplify:

cos^2(x/3) = 1/2[1+cos(2x/3)]

Answer:

Statement is true!

Step-by-step explanation:

Required to Prove:

[tex]\Large \textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}[/tex]

This is a special property, used in integral calculus, that can be derived and hence proved, from the double angle formula of cosine.

[tex]\large \textsf{Given that cos(A+B) = cosA\,cosB $-$ sinA\,sinB,}\\ \\\large \textsf{Hence cos(A+A) = cosA\,cosA $-$ sinA\,sinA}\\ \\\large \textsf{$\therefore$ cos2A = cos$^2$A $-$ sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 1 $-$ 2sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 2cos$^2$A $-$ 1 (using Pythagorean Identity $\Rightarrow cos^2A+sin^2A = 1$)}[/tex]

This property, can be quoted in exams and only has to be derived, not proved. Now using the Cos2A property, we can manipulate the formula:

[tex]\large \textsf{$\cos2\rm A = \cos^2A - \sin^2A$}\\ \\ \large \textsf{$\rm \phantom{\cos 2A}=2\cos^2A-1$}\\ \\ \large \textsf{$\rm \therefore \cos2A+1 = 2\cos^2A$}\\ \\ \large \textsf{$\rm \cos^2A=\frac{\cos2A+1}{2}$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(\cos2A+1)$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(1+\cos2A)$}[/tex]

And since:

[tex]\large \textsf{$\cos \left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)=\cos^2\left(\frac{x}{3}\right)$}[/tex]

Therefore, inputting the value of A = [tex]\Large \textsf{$\frac{x}{3}$}[/tex] into the formula we derived above, hence:

[tex]\Large \boxed{\boxed{\textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}}} \Large \textsf{ , as required}[/tex]

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true

Step-by-step explanation:

all square have the same features and properties like

all side are equal

The Amplitude is 1/2 and period is π/2.

We have the function as

y= 1/2 cos π/4 x

As, The general equation of a Cosine function is

y=A cos (B(x−D))+C

where A is Amplitude , D is the shift.

So, the amplitude is 1/2

Period = 2π / 4= π/2

and, the phase shift is not possible to determine.

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The missing sides of each geometric system are summarized below:

Case 9: (x, y) = (9, 12)

Case 12: (x, y) = (√51, 7)

Case 15: (x, y) = (8, 15)

In this question we find three cases of geometric systems formed by two right triangles, all missing sides can be found by means of Pythagorean theorem:

r = √(x² + y²)

Where:

Now we proceed to determine all missing sides:

Case 9

y = √(15² - 9²)

y = 12

x = 9

Case 12

x = √(10² - 7²)

x = √51

y = 7

Case 15

y = √(17² - 15²)

y = 8

x = 15

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Answer:

The image you provided appears to be a rectangular prism. To find the surface area of a rectangular prism, we need to add up the areas of all of its faces.

The rectangular prism has dimensions of 4 cm x 6 cm x 8 cm.

Each face of the rectangular prism is a rectangle, so the area of each face can be found by multiplying the length by the width.

The surface area of the rectangular prism is:

2(4 cm x 6 cm) + 2(4 cm x 8 cm) + 2(6 cm x 8 cm)

= 48 cm^2 + 64 cm^2 + 96 cm^2

= 208 cm^2

Therefore, the surface area of the rectangular prism is 208 square centimeters.

have a good day and stay safe

The equation which represents the rule for the variable values is given by option c. y = 2x + 2

The values in the table are,

x  -2   -1    0   1   2

y   -2    0   2   4  6

let us consider two coordinates of the given values in the table .

( x₁ , y₁ ) = ( -2 , -2 )

( x₂ , y₂ ) = ( 0 , 2 )

Using the formula for the slope intercept form of the line we get the equation,

( y - y₁ ) / ( x - x₁ ) = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substitute the values to get the equation of the line we have,

⇒ ( y - ( - 2 ) ) / ( x -  ( -2 ) ) = ( 2 - ( - 2 ) ) / ( 0 - ( - 2 ) )

⇒  ( y + 2 ) / ( x + 2 ) = ( 2 + 2 ) / ( 0 + 2)

⇒  ( y + 2 ) / ( x + 2 ) = 4  / 2

⇒  ( y + 2 ) / ( x + 2 ) = 2

⇒ y + 2 = 2 ( x + 2)

⇒ y + 2 = 2x + 4

⇒ y = 2x + 4 - 2

⇒ y = 2x + 2

Therefore, the equation which represents the rule for the given values of the variable is equal to option c. y = 2x + 2

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the answer is the fourth option

a < -4 1/2

Answer: ∠DGB or ∠EGA

Step-by-step explanation:

Supplementary: Either of two angles whose sum is 180°.

Starting Angle: ∠DGE

Possible Supplements: ∠DGB or ∠EGA

I hope this helps ^^

The probability that exactly 19 men and 7 women are chosen is 0.053107%.

Probability:

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.

[tex]C_n_,_x[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_n_,_x=\frac{n!}{x!(n-x)!}[/tex]

Desired outcomes:

19 men, from a set of 21

7 women, from a set of 11

[tex]D= C_2_1_,_1_9[/tex]  × [tex]C_1_1_,_7[/tex] [tex]=\frac{21!}{19!2!}[/tex] ×  [tex]\frac{11!}{7!4!}[/tex][tex]=69,300[/tex]

Total outcomes:

26 people from a set of 21 + 11 = 32.

[tex]T=C_3_2_,_2_6=\frac{32!}{26!6!}[/tex][tex]=13,049,164,800[/tex]

The probability is :

P = [tex]\frac{D}{T}= \frac{69,300}{13,049,164,800} = 5.3107[/tex]

0.053107% probability that exactly 19 men and 7 women are chosen.

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The answer choice that matches the calculated confidence interval is 62.9 to 83.1

In statistics, the mean is a measure of central tendency that represents the average value of a set of numerical data. It is also known as the arithmetic mean, and it is calculated by adding up all the values in the dataset and dividing the sum by the number of values.

The mean is a useful measure of central tendency because it is easy to calculate, and it provides a single value that represents the center of the dataset. It is affected by outliers, which are extreme values that are far from the other values in the dataset, so it may not accurately represent the typical value of the data if there are outliers present.

To construct a 95.44% confidence interval for the mean score, u, of all students taking the test, we can use the formula:

CI = x ± t(alpha/2, n-1) * (s / √(n))

where CI is the confidence interval, x is the sample mean (73), t(alpha/2, n-1) is the t-value for the given alpha level (0.0278) and degrees of freedom (n-1=4) from the t-distribution table, s is the sample standard deviation, and n is the sample size.

The sample standard deviation is not given, so we will assume that it is the same as the population standard deviation, which is 15. Thus, s = 15.

Using the t-distribution table with 4 degrees of freedom and an alpha level of 0.0278, we find that the t-value is approximately 3.747.

Plugging in the values into the formula, we get:

CI = 73 ± 3.747 * (15 / √(5))

Simplifying, we get:

CI = 73 ± 16.27

Therefore, the 95.44% confidence interval for the mean score, u, of all students taking the test is:

CI = (73 - 16.27, 73 + 16.27)

CI = (56.73, 89.27)

Rounding to one decimal place, we get:

CI = (56.7, 89.3)

Therefore, the answer choice that matches the calculated confidence interval is:

62.9 to 83.1

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The complete question is:

The mean score, x on an aptitude test for a random sample of 5 students was 73, assuming that 0 = 15, construct a 95.44% confidence interval for the mean score, u of all students taking the test. answer choices, 43 to 103, 59.6 to 86.4, 62.9 to 83.1, and 67.0 to 79.0.

a z-test for the proportional difference is the proper hypothesis test.

A hypothesis test can be used to find whether the deviation in proportions impacts the medication's effectivity. We may compare the secondary hypothesis—that the proportions are different—to the null hypothesis.

which states that the dimension of patients who show cut down blood pressure is the same for the two doses of the drug (50 mg and 100 mg).

Popular test statistics like the z-test can be applied to this hypothesis test and other statistical analyses.

[tex]z = (p1 - p2) / SE[/tex]

where p1 and p2, for the 50 mg and 100 mg doses, respectively, are the sample proportions of patients who show fallen blood pressure, and SE is the standard error of the difference between the proportions.

the samples are assumed to be independent or dependent, impacts the SE formula. The samples in this instance are presumed to be independent because they came from various patients. Consequently, the equation for SE is:

[tex]SE = \sqrt(p1 \times (1 - p1)/n1 + p2 *\times(1 - p2)/n2)[/tex]

here, the sample sizes for two doses is n1 and n2.

We can compute the z-test statistic based on the sample sizes and proportions and compare the result to a critical value or p-value to decide whether to accept or reject the null hypothesis.

Therefore, a z-test for the proportional difference is the proper hypothesis test.

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The volume of the of box is derived to be 12 cubic inches, which makes option B correct.

The volume of the box also known as a cuboid can be calculated using the formula:

V = l x w x h

where:

V is the volume of the cuboid

l is the length of the cuboid

w is the width of the cuboid

h is the height of the cuboid

We shall evaluate for the volume of the box as follows:

Volume of the box = 3 in × 2 in × 2 in

Volume of the box = 12 in²

Therefore, the volume of the of box is derived to be 12 cubic inches.

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The volume of the of box is derived to be 12 cubic inches, which makes option B correct.

The volume of the box also known as a cuboid can be calculated using the formula:

V = l x w x h

where:

V is the volume of the cuboid

l is the length of the cuboid

w is the width of the cuboid

h is the height of the cuboid

We shall evaluate for the volume of the box as follows:

Volume of the box = 3 in × 2 in × 2 in

Volume of the box = 12 in²

Therefore, the volume of the of box is derived to be 12 cubic inches.

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The missing angle C is 103, length of a is 9 m, and length b is 16 m.

option B.

The missing angle C is calculated as follows;

A + B + C = 180 (sum of angles in a triangle)

26 + 51 + C = 180

C = 180 - 77

C = 103

The value of length a and length b is calculated as follows;

sin 26/a = sin 103/20

0.438/a = 0.0487

a = 0.438/0.0487

a = 9 m

b/sin51 = 20/sin103

b = 16 m

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The value of x, based on the Alternate Interior Angles Theorem, is calculated as: x = 5.

The Alternate Interior Angles Theorem states that if two parallel lines are intersected by a transversal, then the pairs of alternate interior angles formed are congruent. In other words, if two lines are parallel and a third line intersects them, then the angles that are inside (or "interior" to) the two parallel lines and on opposite sides of the transversal are congruent.

Therefore, we have:

3x - 8 = x + 2 [based on the Alternate Interior Angles Theorem]

Combine like terms:

3x - x = 8 + 2

2x = 10

2x/2 = 10/2

x = 5

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Area of the two right triangles:

A = 1/2(b)(h)

A = 1/2(10)(24)

A = 120

Total area = 240

Area of the left-most rectangle:

A = (b)(h)

A = (24)(25)

A = 600

Area of the right-most rectangle:

A = (b)(h)

A = (25)(26)

A = 650

Area of the base rectangle:

A = (b)(h)

A = (10)(25)

A = 250

Surface Area:

240 + 600 + 650 + 250

1740

Answer: 1740 cm^2

Hope this helps!

[tex]\sf SA=\boxed{\sf 1740cm^{2} }.[/tex]

Check attached 1 to see what parts we're referring to in this step.

This part forms a right triangle. Therefore, the formula to use to find it's area is the following:

[tex]\sf A=\dfrac{bh}{2}[/tex]; where "b" is the length of the base of the triangle, and "h" is its height.

Since we have another section identical to this part at the back, we multiply this area by 2 and calculate:

[tex]\sf A=2\dfrac{bh}{2}=(10cm)(24cm)=240cm^{2}[/tex]

Check image 2 to see this part highlighted.

This shape forms a rectangle. Therefore, use the following formula to calculate:

[tex]\sf A=lw[/tex]; where "l" is length, and "w" is width.

[tex]\sf A=(25cm)(10cm)=250cm^{2}[/tex]

Check image 3.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(24cm)(25cm)=600cm^{2}[/tex]

Check image 4.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(26cm)(25cm)=650cm^{2}[/tex]

The total surface area of this prism is given by the addition of all of its individual areas that we just calculated.

[tex]\sf SA=240cm^{2} +250cm^{2} +600cm^{2} +650cm^{2} =\boxed{\sf 1740cm^{2} }.[/tex]

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The probability of rolling a number less than 9 is 1

The probability of rolling a number less than 9 is 1, since all the possible outcomes are 1, 2, 3, 4, 5, 6 and all of them are less than 9.

The set of equally likely outcomes when rolling a 6-sided die is {1, 2, 3, 4, 5, 6}.

There are no outcomes greater than 6 since that is the maximum number on the die.

Therefore, the probability of rolling a number less than 9 is equal to the probability of rolling any number on the die, which is 1, since all outcomes in the sample space are equally likely.

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