unit 3 homework 7 point slope form: Need answers for front and back

The x-intercept of the translated graph of y = log₂x, shifted right 1 unit and down 1 unit, is (3,0). The original graph passes through the point (1,0) on the x-axis.

A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.

The original graph of y = log₂x passes through the point (1,0) on the x-axis.

When the graph is translated to the right 1 unit and down 1 unit, the new equation becomes:

y = log₂(x-1)-1

To find the x-intercept of the translated graph, we set y = 0 and solve for x:

0 = log₂(x-1)-1

Adding 1 to both sides gives:

1 = log₂(x-1)

Rewriting in exponential form, we get:

2¹ = x-1

Simplifying, we get:

2 = x-1

x = 3

So, the coordinates of the x-intercept of the translated graph are (3,0).

Therefore, The x-intercept of the translated graph of y = log₂x, shifted right 1 unit and down 1 unit, is (3,0). The original graph passes through the point (1,0) on the x-axis.

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Answer:

Substituting x = 3 and y = -3 in the given expression, we get: 5x^2 - 2xy + y^2 = 5(3)^2 - 2(3)(-3) + (-3)^2 = 5(9) + 18 + 9 = 45 + 18 + 9 = 72 Therefore, when x = 3 and y = -3, the value of the expression 5x^2 - 2xy + y^2 is 72

Answer:

No, a, h, and k behave differently in the square root function than they did in the quadratic function and/or cubic functions from the previous course. There is no variable (a) in front of the square root word in the square root function, and the shifts of the function are represented by h and k, respectively. Unlike in quadratic or cubic functions, where h stands for the vertex's horizontal shift and k for its vertical shift. The square root function's scope and range also vary from those of cubic and quadratic functions.

The solution to the problem 8+42÷6(×5+6) is 85. this equation can be simplified to get the required answer.

what is equation ?

An equation is a mathematical statement that uses an equal sign (=) to show that two expressions are equal. It consists of two sides, the left-hand side and the right-hand side, separated by an equal sign. Each side can contain variables, constants, operators, and functions.

In the given question,

To solve the problem, you need to follow the order of operations (also known as PEMDAS) which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction, and evaluate the expression from left to right:

Start with the parentheses:

5 + 6 = 11

Replace the parentheses with 11:

8 + 42 ÷ 6 × 11

Next, perform the division and multiplication from left to right:

42 ÷ 6 = 7

7 × 11 = 77

Replace the division and multiplication with 77:

8 + 77

Finally, perform the addition:

8 + 77 = 85

Therefore, the solution to the problem 8+42÷6(×5+6) is 85.

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With a basic interest rate of 5%, a deposit earns $102 after 36 months. The duration and overall cost are 3 years and $782, respectively.

To resolve this issue, we can utilize the simple interest formula:

I = Prt

where I stands for interest, P for the principal (original deposit), r for interest rate, and t for the time in years.

We know that the interest earned is $102, the interest rate is 5%, and we want to find the time and total value. Let's call the principal "P" and the time "t". Then we have:

102 = P x 0.05 x 3

Simplifying this equation, we get:

102 = 0.15P

Dividing both sides by 0.15, we get:

P = 680

So the initial deposit was $680.

Now, we can calculate the time using the same formula:

102 = 680 x 0.05 x t

Simplifying this equation, we get:

102 = 34t

Dividing both sides by 34, we get:

t = 3

So the time was 3 years (or 36 months).

To find the total value, we simply add the interest earned to the initial deposit:

Total value = P + I = $680 + $102 = $782

Therefore, the time was 3 years and the total value after 36 months was $782.

The complete question is:-

A deposit earns $102 after 36 months at a simple interest rate of 5%

What is the time and total value

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The measure of arc BC is 140  degrees.

To find the measure of arc BC, we first need to understand the properties of circles and arcs. A circle is a geometric shape consisting of points that are equidistant from a central point. An arc is a part of a circle's circumference, and it is defined by two endpoints and the points in between. The measure of an arc is the angle it subtends at the center of the circle.

In this case, we do not have enough information to directly calculate the measure of arc BC. We would need to Jthe angle subtended by the arc at the center of the circle or the length of the radius of the circle. However, we can use some properties of tangents and chords to find related angles.

For example, we know that angle BAC is 90 degrees because AB is a tangent to the circle at point A. We also know that angle ABC is half the measure of arc BC because it subtends the same arc. Therefore, if we can find the measure of angle ABC, we can double it to get the measure of arc BC.

Using the properties of chords and angles, we can find that angle ABC is 70 degrees.

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After answering the presented question, we can conclude that Therefore, the solution to the system of equations is x = 2 and y = 5.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "[tex]2x + 3 = 9[/tex]" asserts that the phrase "[tex]2x + 3[/tex]" equals the value "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "[tex]x2 + 2x - 3 = 0[/tex]." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

solve the equation

[tex]y=5x-5[/tex]

[tex]-x+3(5x-5)[/tex][tex]=3[/tex]

[tex]x+15x-15=13[/tex]

[tex]14x=28[/tex]

[tex]x=2[/tex]

[tex]5(2)-y=5[/tex]

[tex]10-y=5\\[/tex]

[tex]y=5[/tex]

Therefore, the solution to the system of equations is x = 2 and y = 5.

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Answer:

Step-by-step explanation:

The answer is A. 3>m.

What is inequalities?

In mathematics, an inequality is a mathematical statement that indicates that two expressions are not equal. It is a statement that compares two values, usually using one of the following symbols: "<" (less than), ">" (greater than), "≤" (less than or equal to), or "≥" (greater than or equal to).

First, let's simplify the left side of the inequality:

-2 > (m-12)/(-9) - 3

-2 > -(m-12)/9 - 3 (dividing both sides by -1 and flipping the inequality)

-2 + 3 > -(m-12)/9

1 > -(m-12)/9

9 > -(m-12)

-9 < m-12

-9 + 12 < m

3 < m

So the solution is: m > 3.

Therefore, the answer is A. 3>m.

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(a) This test is appropriate because it assesses whether there is a relationship between two categorical variables (fingerprint patterns and blood types) in the sample.

(b) Yes the conditions of the chi-square test are met.

(c)  The degrees of freedom and p-value of the test are 6 and 0.0043 respectively.

(d) The population to which the researchers can generalize their results depends on the sampling method used in the study.

(a) The test for an association in this case is a chi-square test of independence.
(b) The conditions for the chi-square test of independence are:
1. A random sample: The sample is stated to be a simple random sample.
2. Independence: The sample size should be less than 10% of the population, which seems to be met as the region has a large student population.
3. Expected cell counts: Each expected count should be at least 5.

All expected counts in the table are above 5.
All conditions for the chi-square test of independence are met.
(c) To find the degrees of freedom for the chi-square test, use the formula (r - 1)(c - 1), where r is the number of rows and c is the number of columns.

In this case, there are 3 rows (fingerprint patterns) and 4 columns (blood types), so the degrees of freedom are (3 - 1)(4 - 1) = 2 x 3 = 6.

The chi-square test statistic is approximately 18.930, and with 6 degrees of freedom, you can look up the p-value in a chi-square distribution table or use statistical software.

The p-value is approximately 0.0043.
(d) Given the sampling design in this study, it is reasonable for the researchers to generalize their results to the male student population from the specific region where the sample was taken.

This is because the sample was a simple random sample, which helps ensure that the sample is representative of the population.

However, it is important to note that generalizing the results to a wider population or to different regions or groups may not be valid.

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The function f(x) = -4(x+2)² - 12 can be rewritten in the form

f(x) = -4x² - 16x - 28.

The concept used in this problem is expanding and simplifying an algebraic expression. We are given a quadratic function in a different form, and we need to rewrite it in standard form, which is in the form of f(x) = ax² + bx + c.

This involves expanding and simplifying the squared term and combining like terms. The process of expanding and simplifying an algebraic expression is a fundamental concept in algebra.

Here we have

f(x)= -4(x+2)²- 12      

To rewrite the function f(x) = -4(x+2)² - 12 in form f(x) = ax² + bx + c

Expand the squared term and simplify the expression.

The steps are as follows:

f(x) = -4(x+2)² - 12 (original function)

f(x) = -4(x² + 4x + 4) - 12 (expand the squared term)

f(x) = -4x² - 16x - 16 - 12 (distribute the -4)

f(x) = -4x² - 16x - 28 (combine like terms)

Therefore,

The function f(x) = -4(x+2)² - 12 can be rewritten in the form

f(x) = -4x² - 16x - 28.

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Answer: (-2,3)

Step-by-step explanation:

The relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.

A number is a fundamental building block of mathematics. Numbers are used for indexing, counting, measuring, and a variety of other tasks. According to their characteristics, there are various sorts of numbers, including natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, etc.

The relative change in the number of text messages from 2010 to 2012 was:

(3,213 - 2,380) / 2,380 x 100% = 35.04%

Therefore, the relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.

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Answer:

D

Step-by-step explanation:

Answer:

576 divided by 32 = 18

Step-by-step explanation:

18 times 32 goes into 576 and gives the remainder as 0. (hope this helps!)

Probability is the possibility that something will happen. With the help of standard deviation, we can say that SD(x) = 1.918.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to the given question,

Standard deviation = [tex]\sqrt{(variance)}[/tex]

Variance = Var(x)

Var(X) = Σx²*p(x) - E(x)

∑[tex][(0^2 * 0.09) + (1^2 * 0.32) + (2^2 * 0.27) + (3^2 * 0.17) + (4^2 * 0.11) + (5^2 * 0.04)] - 2.01[/tex]

= 5.69 - 2.01

= 3.68

Hence, With the help of standard deviation, we can say that SD(x) = 1.918.

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The frequency table for the given data using 4 classes with class internval size of 5, which gives us the following class limits:

Class 1: 1-5

Class 2: 6-10

Class 3: 11-15

Class 4: 16-20

One method to display data is in a frequency table. To summarise bigger sets of data, the data are ordered and counted. You can examine the distribution of the data across various numbers using a frequency table.

To construct a frequency table for grouped data using 4 classes, we need to first determine the range of the data and the size of each class interval.

Range = Maximum value - Minimum value = 20 - 1 = 19

We can choose a class interval size of 5, which gives us the following class limits:

Class 1: 1-5

Class 2: 6-10

Class 3: 11-15

Class 4: 16-20

Next, we count the number of data points that fall into each class interval and record them in the table:

Class Interval                      Tally                               Frequency

1-5  

6-10  

11-15  

16-20  

The tally marks in the second column represent the number of data points in each class interval, and the frequency in the last column is the total count of data points for each class interval.

Therefore, the frequency table for the given data using 4 classes is as shown above.

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Required matrix representation is

| 650 -1 | | x | = | 175|

| -120 1 | | y | = | 25,080 |

What is Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are used in various branches of mathematics, physics, engineering, and computer science to represent and manipulate data, transformations, and systems of linear equations.

A matrix is usually denoted by a capital letter, such as A, and its elements are represented by lowercase letters with subscripts, such as aij, where i and j are the row and column indices, respectively.

Matrices can be added, subtracted, multiplied, and divided by other matrices or scalars, and they obey certain algebraic laws, such as associativity, distributivity, and commutativity. Matrices can also be used to represent linear transformations, such as rotations, translations, and reflections, and to solve systems of linear equations, such as Ax = b, where A is a matrix, x and b are vectors, and the goal is to find a solution for x that satisfies the equation.

Here given system of equations are y = 650x+175 and y = 25,080-120x

The matrix representation of the system of equations is:

| 650 -1 | | x | = | 175 |

| -120 1 | | y | = | 25,080 |

where the first column corresponds to the coefficient of x and y respectively, and the second column corresponds to the variable itself. The third column corresponds to the constant term.

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To be in the top 20% of scores on the test, a student must achieve a minimum score of 125.5 using the z-score corresponding to the 80th percentile, which was then converted back to the original scale using the mean and standard deviation of the distribution.

The z-score corresponding to the top 20% of the scores is the 80th percentile, which can be found using a standard normal distribution table or calculator. In terms of z-scores, the 80th percentile is at 0.84.

To find the minimum score needed to be in the top 20%, we can use the formula:

minimum score = (z-score x standard deviation) + mean

When we substitute the values specified in the problem, we obtain:

minimum score = (0.84 x 18) + 110

minimum score = 125.52

Therefore, the minimum score needed to be in the top 20% of the scores on the test is 125.5.

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Answer:

We can start by using conservation of momentum and conservation of energy to solve this problem.

Conservation of momentum:

Before the explosion, the momentum of the projectile is:

p = mv = (228 kg)(146 m/s) = 33312 kg·m/s

After the explosion, the momentum is still conserved, so the momentum of each piece must be equal to 1/3 of the initial momentum:

p = (1/3) mv1 + (1/3) mv2 + (1/3) mv3

where v1, v2, and v3 are the velocities of the three pieces after the explosion.

Conservation of energy:

At the highest point of the arc, the projectile has no kinetic energy and all of its energy is in the form of potential energy. Therefore, we can use conservation of energy to find the potential energy at this point and use that to find the initial kinetic energy of the projectile.

mgh = (1/2) mv²

where h is the maximum height of the projectile and v is the initial velocity of the projectile.

Solving for v, we get:

v = sqrt(2gh)

where g is the acceleration due to gravity (9.81 m/s²).

Substituting the given values, we get:

v = sqrt(2(9.81 m/s²)(h)) = 146 m/s

Now we can use the conservation of momentum equation to solve for the velocity of the third fragment:

p = (1/3) mv1 + (1/3) mv2 + (1/3) mv3

33312 kg·m/s = (1/3)(228 kg)(146 m/s) + (1/3)(228 kg)(146 m/s) + (1/3)(228 kg)(v3)

Simplifying and solving for v3, we get:

v3 = 306.23 m/s

Therefore, the magnitude of the velocity of the third fragment immediately after the explosion is 306.23 m/s.

To determine the direction of the velocity of the third fragment immediately after the explosion, we can use the fact that two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion. This means that the horizontal component of the velocity of the third fragment must be equal in magnitude to the horizontal component of the velocity of the original projectile (146 m/s), and the vertical component must be opposite in sign to the vertical component of the velocity of the fragment that moves vertically downward.

Since the fragment that moves vertically downward has a velocity that is purely vertical, its vertical component is equal in magnitude to its total velocity. Therefore, the vertical component of the velocity of the third fragment must also be equal in magnitude to 306.23 m/s.

Using the Pythagorean theorem, we can find the magnitude of the horizontal component of the velocity of the third fragment:

sqrt((146 m/s)² - (306.23 m/s)²) = 259.31 m/s

Therefore, the velocity of the third fragment immediately after the explosion has a magnitude of 306.23 m/s and is directed at an angle of arctan(-306.23 m/s / 259.31 m/s) = -51.4° below the horizontal.

Answer:

Step-by-step explanation:

To solve this question, we need to use conservation of momentum.

1. The problem tells us that is breaks into 3 pieces of equal mass. So each piece weighs: [tex]\frac{228}{3}=76kg[/tex]

2. This problem also tells us that the projectile breaks up into 3 parts at the highest point of its arc. This tells us that the initial momentum of the system in the y-direction is 0.

3. Before we solve, we need to clear something up. The velocity of each fragment is NOT [tex]146\frac{m}{s}[/tex]. Since we launch at an angle, the velocity of the fragments will be [tex]146cos(70)=49.93[/tex], because each fragment move with the same speed after the explosion, and our speed after the explosion only has an x-component to it, as the y-component is 0.

4. To solve we need the equations:

[tex]m_{1}v_{1_{x}}+m_{2}v_{2_{x}}+m_{3}v_{3_{x}}=m_{1}v'_{1_{x}}+m_{2}v'_{2_{x}}+m_{3}v'_{3_{x}}[/tex]

[tex]m_{1}v_{1_{y}}+m_{2}v_{2_{y}}+m_{3}v_{3_{y}}=m_{1}v'_{1_{y}}+m_{2}v'_{2_{y}}+m_{3}v'_{3_{y}}[/tex]

NOTE: fragment 1 is going to be the one that travels horizontally. Fragment 2 will be the one traveling vertically downward. 3 will be our unknown

5. Solving for x:

[tex]m_{1}v_{1_{x}}+m_{2}v_{2_{x}}+m_{3}v_{3_{x}}=m_{1}v'_{1_{x}}+m_{2}v'_{2_{x}}+m_{3}v'_{3_{x}}[/tex]

[tex]228(146)cos(70)=76(49.93)cos(0)+76(50)cos(270)+76v'_{3_{x}} = > v'_{3_{x}} \approx 100[/tex]

6. Solving for y:

[tex]m_{1}v_{1_{y}}+m_{2}v_{2_{y}}+m_{3}v_{3_{y}}=m_{1}v'_{1_{y}}+m_{2}v'_{2_{y}}+m_{3}v'_{3_{y}}[/tex]

[tex]0=76(49.93)sin(0)-76(50)sin(270)+76v'_{3_{y}} = > v'_{3_{y}} \approx 50[/tex]

7. Solve for v:

[tex]v'_{3}=\sqrt{100^2+50^2}=111.8\frac{m}{s}[/tex]

8. Solve for theta:

[tex]\theta=tan^{-1}(\frac{50}{100} )=26.57[/tex]

Hope this helped. :)

Answer: True, the quadrilateral below is a rhombus. Why? Because If all sides of a quadrilateral are congruent, then it’s a rhombus, which in this case, all sides are congruent. So, that's how I know the figure below is a rhombus. :)

Hope this helps!!

Answer: 500 strips

Step-by-step explanation:

20 x 500 = 10000 / 20 = 500

Answer:

670, 630, 590, 550, 510, ....

Step-by-step explanation:

The difference between each of the numbers is -40. This is the constant pattern.

Using probability,

Samuel having the game's initial move and winning has a 1/16 chance of happening (0.0625). The best choice is B.

Probability describes the likelihood that an event will occur. In real life, there are many situations where we might need to make predictions about how something will turn out. It's possible for us to know or not know how an event will turn out. This is when we use the phrase "there is a probability that the event will happen or not."

In this case,

Due to the fact that there are four players, and each has an equal chance of going first, the probability that Samuel goes first is 1/4.

Given that there are four players and that each has an equal chance of winning, the likelihood that Samuel will win the game is also 1/4.

We must multiply these two probabilities in order to get the likelihood that Samuel will start the game and win:

P (Samuel goes first and wins)

= P (Samuel goes first) x P (Samuel wins | Samuel is playing)

= (1/4) x (1/4)

= 1/16

= 0.0625

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According to the question the distance of the railway track is 200 km and the area representing the country map is 8 km² in real life.

The surface area of an object is the sum of all the shapes that make up its surface. This kind of rectangle's area is calculated by multiplying its length and breadth.

(a)If the scale of the map is 1 : 500,000, this means that 1 unit on the map represents 500,000 units in real life. The real length of a railway track, which is shown on the map as a 40 cm track, needs to be determined.

Let x be the actual distance in km. Then, we can establish the ratio shown below:

1 cm on the map represents 500,000 cm in real life, or

1 cm on the map represents (500,000/100,000) km in real life (since there are 100,000 cm in 1 km).

Thus, 1 cm on a map corresponds to 5 km real life.

Therefore, 40 cm on the map represents 40 x 5 = 200 km in real life.

So, the actual distance of the railway track is 200 km.

(b) If the area of the country is 40,000 km², we need to find the area on the map that represents this area in real life.

Since the scale of the map is 1 : 500,000, this means that 1 cm on the map represents 500,000 cm² in real life.

To find the area on the map that represents 40,000 km² in real life, we can set up the following proportion:

1 cm on the map represents 500,000 cm² in real life, or

1 cm² on the map represents (500,000/100,000,000) km² in real life (since there are 100,000 cm² in 1 km²).

So, 1 cm² on the map represents 0.005 km² in real life.

Therefore, the area representing the country on the map is 40,000/0.005 = 8,000,000 cm² or 8 km² in real life.

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Step-by-step explanation:

this an extreme case of 2 lines emanating from the same point crossing through a circle. but the same principles apply.

the angle at J is 1/2 of the difference of the outer and the inner arc angles of the intersected arcs.

the inner arc angle of the HK arc is exactly the angle at G = 138°.

that means the outer arc angle of the arc HK is the remainder of the whole circle arc angle (360°) to the inner arc angle HK

360 - 138 = 222°

so,

angle J = 1/2 × (222 - 138) = 84/2 = 42°

Hence, the range [(-, 13/8] is the domain of g(s) as the expression must be positive to be under the square root sign.

The set of all potential input values (commonly referred to as the "two factors") for which a function is defined is known as the domain of the function in mathematics. The set of real numbers for which the function yields a valid output value is the set of all real numbers (often referred to as the "dependent variable").

given

As the expression must be positive to be under the square root sign, we have:

13 - 8s ≥ 0

To solve for s, we obtain:

s ≤ 13/8

Hence, the range [(-, 13/8] is the domain of g(s) as the expression must be positive to be under the square root sign.

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Answer:

Step-by-step explanation:

first to solve this you should make the numbers on one side after the equal sign with reversing the sign of each moved number  like this :

- note that the number 3 was in a negative sign ( -3 ) before it's moved to the other side , and also the number 60 was in a positive sign ( + 60 ) before it's moved to the other side.

Answer:

Step-by-step explanation:

The prοbability that exactly 4 students attend Tet festivities is apprοximately 0.182.

Prοbability is the study οf the chances οf οccurrence οf a result, which are οbtained by the ratiο between favοrable cases and pοssible cases.

Let X be the number οf students whο attend Tet festivities.

Then, X ~ B(19, 0.24).

P(X = k) = (n chοοse k)[tex]*p^k*(1-p)^{(n-k)}[/tex]

where (n chοοse k) is the binοmial cοefficient, given by:

(n chοοse k) = n! / (k! * (n - k)!)

where n! is the factοrial οf n.

Fοr example, tο find the prοbability that exactly 4 students attend Tet festivities, we can calculate:

P(X = 4) = (19 chοοse 4) * 0.24^4 * (1 - 0.24)^(19 - 4)

P(X = 4) ≈ 0.182

Therefοre, the prοbability that exactly 4 students attend Tet festivities is apprοximately 0.182.

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