( 7 X 18 + 45) divided by three x two

A group of student organization who wants to study about voting preferences in its students during presidential election in 2012. So, they selected a sample of 120, is an example of stratified random sampling.

Stratified random sampling is a widely used statistical technique in which a population is divided into different subgroups, or strata, based on some shared characteristics. The purpose of stratification is to ensure that each stratum in the sample and to make inferences about specific population subgroups, that is they share (e.g., race, gender, educational attainment).

Therefore, the stratified random sample involves dividing the population into two or more strata (groups). These strata are expressed as H. A stratified random sampling because a random sample has been taken from each different strata (Freshmen through seniors).

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Answer:

P(multiple of 3) = 1/3 (fraction)

P(multiple of 3) = (1/3) * 100 = 33.33% (percentage rounded to the nearest percent)

Step-by-step explanation:

There are two numbers on a cube labeled 1 through 6 that are multiples of 3: 3 and 6.

The total number of possible outcomes is 6 (since there are 6 sides on the cube). So, the probability of rolling a multiple of 3 is calculated as follows:

P(multiple of 3) = (number of favorable outcomes) / (total number of possible outcomes)

P(multiple of 3) = 2/6

P(multiple of 3) = 1/3

To express this probability as a percentage rounded to the nearest percent, multiply the fraction by 100:

P(multiple of 3) = (1/3) * 100 = 33.33%

Rounded to the nearest percent, the probability of rolling a multiple of 3 on a number cube labeled 1 through 6 is 33%.

Answer:

Step-by-step explanation:

17

The point estimate of the population variance in this case would be: 400. The standard deviation is a measure of how spread out the data is from the mean, so a high standard deviation indicates that there is a lot of variability in the number of rooms occupied per day.

By calculating the point estimate of the population variance, the managers can better understand the variability of their data and make more informed staffing decisions.

To calculate the point estimate of the population variance, we use the formula:

Point estimate of population variance = Sample standard deviation squared

Therefore, the point estimate of the population variance in this case would be:

Point estimate of population variance = 20^2 = 400

Managers of the hotel are interested in the variability in the number of rooms occupied per day during a particular season of the year because it helps them make staffing decisions. If they know that the variability is high, they may need to schedule more staff to handle the influx of guests, while if the variability is low, they can get by with fewer staff.

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The number of seconds until both planes are at the same altitude would be 50 seconds.

Assum that after t seconds, both planes will be at the same altitude.

The formula for plane A would be:

= 2, 775 + 25. 25t

The formula for Plane B would be :

= 80.75 t

We can find t by equating both formulas :

2, 775 + 25. 25t = 80. 75t

55. 5t = 2, 775

t = 2, 775 / 55. 5

t = 50 seconds

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Question is:

How many seconds will pass before the planes are at the same altitude?

At approximately 4.1724 years, there will be about 412.78 foxes and 412.78 rabbits present on the game reserve. However, since we are dealing with whole animals, we can say that there will be 413 foxes and 413 rabbits present at that time.

Let's denote the current number of foxes as [tex]F_0[/tex] = 265 and the current number of rabbits as [tex]R_0[/tex] = 104. We want to know how long it takes for the number of rabbits to catch up with the number of foxes, which means that we want to find the time t when R(t) = F(t).

The number of foxes after t years can be represented as F(t) =[tex]F_0[/tex]+ 36t, and the number of rabbits after t years can be represented as R(t) = R_0 + 65t. Therefore, we can set up the following equation:

R(t) = F(t)

[tex]R_0[/tex] + 65t =[tex]F_0[/tex] + 36t

Simplifying and solving for t, we get:

29t =[tex]F_0 - R_0[/tex]

t = (F_0 - R_0) / 29

Substituting the values, we get:

t = (265 - 104) / 29

t = 4.1724

Therefore, it takes approximately 4.1724 years for the number of rabbits to catch up with the number of foxes.

To find the number of foxes and rabbits at that time, we can substitute t = 4.1724 into the equations for F(t) and R(t):

F(4.1724) = 265 + 36(4.1724) ≈ 412.78

R(4.1724) = 104 + 65(4.1724) ≈ 412.78

Therefore, at approximately 4.1724 years, there will be about 412.78 foxes and 412.78 rabbits present on the game reserve. However, since we are dealing with whole animals, we can say that there will be 413 foxes and 413 rabbits present at that time.

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Step-by-step explanation:

The product of the form method is a technique used to factorize a quadratic expression of the form ax^2 + bx + c. Here are the steps to follow:

1. Write down the quadratic expression in the standard form ax^2 + bx + c, where a, b, and c are constants.

2. Multiply the coefficient a by the constant c to get the product ac.

3. Find two factors of ac that add up to the coefficient b. In other words, find two numbers p and q such that pq = ac and p + q = b.

4. Rewrite the quadratic expression by replacing the middle term bx with the two terms px and qx. This is done by splitting the middle term of the quadratic expression using the two numbers p and q found in step 3. So the quadratic expression becomes ax^2 + px + qx + c.

5. Factor the first two terms of the expression ax^2 + px using the greatest common factor (GCF). This gives us a(x + p/a)x + qx + c.

6. Factor the last two terms qx + c using the GCF. This gives us a(x + p/a)(x + c/q).

7. Simplify the expression by combining any like terms and check that the factors obtained in step 6 can be expanded back into the original quadratic expression.

8. Write down the factored form of the quadratic expression, which is (x + p/a)(x + c/q).

These are the steps of the product of the form method.

P( x > n) = P(y < r)

What is binomial distribution?

In probability theory and statistics, the discrete probability distribution of the number of successes in a series of n separate experiments, each asking a yes-or-no question and each with its own Boolean-valued outcome: success or failure, is known as the binomial distribution with parameters n and p.

Here, we have

Given: let x be a negative binomial random variable with parameters r and p, and let y be a binomial random variable with parameters n and p.

We have to show that P(x >n) = P(y<r)

We are going to prove that events x >n and y<r are equivalent. As a consequence, these events will have the same probabilistic measure.

If x >n that means that we needed more than r attempts to reach successes that happens with probability p.

That implies that in n attempts we made strictly less than r successes, which is exactly y < r.

If y < r, that means that in n attempts we made strictly less than r successes.

The total number of trials, until we reach r successes, will be strictly greater than n.

That is exactly x > n.

So, we have proved that { x > n} = { y < r}

Hence,  P( x > n) = P(y < r)

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Answer:

There are 12 different ways.

There are six different ways that Kim, Dan, and Pat can finish in first and second place in the contest

Kim, Dan, and Pat can place first and second in the competition in six different scenarios. This is an example of a permutation problem, which involves determining the number of ways that a set of objects can be arranged in a specific order. In this case, there are three finalists (Kim, Dan, and Pat) and two prizes (first and second place).

The number of ways to arrange three objects in a specific order is given by the formula

P(3,2) = 3!/(3-2)!

= 3 × 2 × 1

= 6

Therefore, there are six different ways that Kim, Dan, and Pat can finish in first and second place in the contest.

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The probability that event will not occur is 0.68 (1-0.32). The odds for event are 32:68 or simplified to 8:17 (divide both sides by 4). The odds against event are 68:32 or simplified to 17:8 (divide both sides by 4).

Given that the probability of the event occurring is 0.32, we can find the probability of the event not occurring by subtracting this value from 1:

Probability (Event Not Occurring) = 1 - Probability (Event Occurring) = 1 - 0.32 = 0.68

So, the probability that the event will not occur is 0.68.

Now, let's find the odds for the event. Odds for an event is calculated as:

Odds For = Probability (Event Occurring) / Probability (Event Not Occurring) = 0.32 / 0.68 ≈ 0.47

So, the odds for the event are approximately 0.47 to 1.

Lastly, let's calculate the odds against the event:

Odds Against = Probability (Event Not Occurring) / Probability (Event Occurring) = 0.68 / 0.32 ≈ 2.13

Therefore, the odds against the event are approximately 2.13 to 1.

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According to the information in the table, "The French government gathers €543,141,128,984 more than the German government".

Hence, the correct option is B.

Based on the information in the table, we can compare the annual tax revenues of Germany and France as follows.

The annual tax revenue of Germany is €705,129,000,000, while that of France is €1,248,270,128,984. This indicates that the French government gathers a significantly larger amount of tax revenue than the German government.

To find the difference between the two, we can subtract the annual tax revenue of Germany from that of France

€1,248,270,128,984 - €705,129,000,000 = €543,141,128,984.

Therefore, the French government gathers €543,141,128,984 more than the German government. Thus, the correct answer is (b) The French government gathers €543,141,128,984 more than the German government.

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-- The given question is incomplete, the complete question is attached below. --

a. True, the determinant of A is 0, because the determinant is 0 if two columns (or rows) are linearly dependent.

b. True, the determinant is just the product of the diagonal entries, since all other terms are 0.

c. True, it is exactly the expression given in the statement.

A matrix is a rectangular array made up of numbers, equations, or symbols. With an order of number of rows x number of columns, this arrangement is made up of horizontal rows and vertical columns.

(a) True. If the rank of A is 1, then A has only one linearly independent column, and all other columns are linearly dependent on the first column. Therefore, the determinant of A is 0, because the determinant is 0 if two columns (or rows) are linearly dependent.

(b) True. If A is a triangular matrix, then the determinant of A is the product of the diagonal entries of A. This is because when finding the determinant of a triangular matrix, the determinant is just the product of the diagonal entries, since all other terms are 0.

(c) True. We know that [tex]det(A) = det(A^T)[/tex], so we can work with [tex]A^T[/tex] instead of A. Let B be the matrix obtained by replacing c2 with x and y, respectively, in the second column of [tex]A^T[/tex]. Then, we have [tex]A^T[/tex] = [c1 | B], where | denotes concatenation of matrices. By expanding the determinant of [tex]A^T[/tex] along the second column, we get det([tex]A^T[/tex]) = det([c1 | x | c3 | ... | c10]) + det([c1 | y | c3 | ... | c10]). This is exactly the expression given in the statement. Therefore, the statement is true.

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Out of the given choices, the closest value to this is 2.94.  the value of the test statistic is approximately 2.94.

To compute the test statistic for the difference between two proportions, we can use the following formula:

z = (p1 - p2) / sqrt(p * (1 - p) * ((1 / n1) + (1 / n2)))

where p1 is the proportion in the first sample, p2 is the proportion in the second sample, p is the pooled proportion (calculated by combining the two samples), n1 is the sample size of the first sample, and n2 is the sample size of the second sample.

From the given information, we have:

n1 = (first sample size)

n2 = (second sample size)

x1 = (number of events in the first sample)

x2 = (number of events in the second sample)

We can calculate the sample proportions as:

p1 = x1 / n1

p2 = x2 / n2

We can calculate the pooled proportion as:

p = (x1 + x2) / (n1 + n2)

We can now substitute these values into the formula to calculate the test statistic:

z = (p1 - p2) / sqrt(p * (1 - p) * ((1 / n1) + (1 / n2)))

= ((x1 / n1) - (x2 / n2)) / sqrt(((x1 + x2) / (n1 + n2)) * (1 - ((x1 + x2) / (n1 + n2))) * ((1 / n1) + (1 / n2)))

We can now plug in the values for the sample sizes and numbers of events and simplify the expression to obtain the test statistic:

Rounding off to two decimal places, we get:

z =  2.94

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Decoding Forecast starting at 17:00Z:

Wind:

24014G23KT

Visibility:

5 statute miles

Weather:

Light rain showers

Clouds:

Overcast at 1500 feet

Incomplete report.

The decoded report is:

Location:

KJAX (Jacksonville International Airport)

Date/Time: 10th at 23:20Z

Wind:

00000KT

Visibility:

More than 6 statute miles

Clouds:

Scattered at 3500 feet

Forecast starting at 11:00Z:

Wind:

00000KT

Visibility:

5 statute miles

Weather:

Mist

Clouds:

Broken at 1000 feet, Broken at 2000 feet

Forecast starting at 06:00Z:

Wind:

16003KT

Visibility:

2 statute miles

Weather:

Mist

Clouds:

Broken at 500 feet, Overcast at 1000 feet

Temporary condition between 08:00Z and 12:00Z:

Visibility:

1 statute mile

Weather:

Mist

Clouds:

Overcast at 300 feet

Forecast starting at 14:00Z:

Wind:

20010G18KT

Visibility:

More than 6 statute miles

Weather:

Vicinity showers

Clouds:

Broken at 1500 feet, Overcast at 2500 feet

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Population in 1998 = 3000 + 105 = 3105 people. We can calculate it in the following manner.

To find the population in the year 1998, we need to first find the value of t when t=7 (since we want to find the population in the year 1998, which is 7 years after 1991).

So, we plug in t=7 into the function r(t) = 36 - 3t:

r(7) = 36 - 3(7)

r(7) = 36 - 21

r(7) = 15

This means that the change in population in the year 1998 was 15 (i.e. there were 15 fewer people in the town in 1998 compared to 1991).

To find the population in the year 1998, we need to subtract this change from the population in 1991:

Population in 1998 = 3000 - 15

Population in 1998 = 2985

Therefore, the population in the year 1998 was 2985.
To find the population in 1998, we first need to determine the change in population from 1991 to 1998 using the given function r(t) = 36 - 3t, where t represents the number of years since 1991. In this case, t = 1998 - 1991 = 7 years.

Now, we can plug t into the function:
r(7) = 36 - 3(7) = 36 - 21 = 15

This tells us that the population increased by 15 people per year during the 7 years between 1991 and 1998. To find the total population change, we can multiply this rate by the number of years:
Total population change = 15 people/year × 7 years = 105 people

Finally, we can add this change to the initial population in 1991 (3000 people) to find the population in 1998:
Population in 1998 = 3000 + 105 = 3105 people

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A particular solution and the fundamental solution of a nonhomogeneous differential equation cannot be the same.

1. Nonhomogeneous differential equation: A differential equation that has a non-zero term independent of the dependent variable (the function you are trying to find). It can be represented as L(y) = f(x), where L is the differential operator, y is the dependent variable, and f(x) is a non-zero function of the independent variable x.

2. Particular solution: A specific solution to a nonhomogeneous differential equation that satisfies both the differential equation and the initial or boundary conditions. It represents a single instance of the infinite possible solutions.

3. Fundamental solution: A set of linearly independent solutions to the corresponding homogeneous differential equation, i.e., the equation with the non-zero term set to zero (L(y) = 0). These solutions form a basis to construct the complementary function, which, when added to the particular solution, provides the general solution of the nonhomogeneous differential equation.

Since the fundamental solution refers to solutions of the homogeneous equation, and the particular solution is a specific solution to the nonhomogeneous equation, they cannot be the same. The general solution to the nonhomogeneous differential equation is obtained by combining the complementary function derived from the fundamental solutions and the particular solution.

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The 8th term of the sequence is 4374. A geometric sequence is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

To find the 8th term of the geometric sequence, we can use the formula for the nth term of a geometric sequence:

an = a1 x r^(n-1)

where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

Given that the first term is 2 and the common ratio is 3, we have:

a1 = 2
r = 3

Plugging in n = 8, we get:

a8 = 2 x 3^(8-1)
a8 = 2 x 3^7
a8 = 2 x 2187
a8 = 4374


In summary, a geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant called the common ratio. In this case, the first term is 2 and the common ratio is 3. We can use the formula an = a1 x r^(n-1) to find the nth term of the sequence. By plugging in n = 8, we get the 8th term of the sequence as 4374.

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The decision to use a univariable or multivariable regression model depends on the research question, data availability, model complexity, and goodness of fit.

What is the linear regression equation?

The formula for simple linear regression is Y = mX + b, where Y is the response (dependent) variable, X is the predictor (independent) variable, m is the estimated slope, and b is the estimated intercept.

When deciding whether to use a univariable or multivariable linear regression model, there are several factors to consider:

Research question: Consider the research question you are trying to answer. If you are interested in understanding the relationship between a single independent variable and a dependent variable, then a univariable regression model may be sufficient. However, if you want to explore the effect of multiple independent variables on a dependent variable, then a multivariable regression model may be more appropriate.

Data availability: Look at the data you have available. If you have only one independent variable that you believe is relevant to your research question, then a univariable regression model may be appropriate. However, if you have multiple independent variables that could potentially influence the dependent variable, then a multivariable regression model may be necessary.

Model complexity: Consider the complexity of the model you want to build. If you are interested in a simple linear relationship between an independent variable and a dependent variable, then a univariable regression model may be sufficient. However, if you believe that there are interactions between multiple independent variables that could affect the dependent variable, then a multivariable regression model may be necessary.

Model fit: Evaluate the goodness of fit of both univariable and multivariable models. Compare the R-squared values of each model to determine which model provides a better fit to the data. A higher R-squared value indicates a better fit between the independent and dependent variables.

Hence, the decision to use a univariable or multivariable regression model depends on the research question, data availability, model complexity, and goodness of fit.

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The appropriate null hypothesis is that there is no significant difference in preference for in-person or remote learning across the four grade levels.

The alternative hypothesis is that there is a significant difference in preference for in-person or remote learning across the four grade levels.

a) Null and alternative hypotheses:
H0 (null hypothesis): There is no association between grade level and preference for remote or in-person learning.
Ha (alternative hypothesis): There is an association between grade level and preference for remote or in-person learning.

b) Expected count calculation for the remote/senior cell:
To find the expected count, you'll use the formula: (Row total * Column total) / Grand total

Row total for remote learning: 3 + 12 + 14 + 15 = 44
Column total for seniors: 15 + 10 = 25
Grand total: 25 freshmen + 25 sophomores + 25 juniors + 25 seniors = 100 students

Expected count for remote/senior cell = (44 * 25) / 100 = 11

Complete table of expected counts:

              | Remote | In-person
---------------
Freshmen | 11      | 14
Sophomores | 11      | 14
Juniors   | 11      | 14
Seniors   | 11      | 14

c) Calculation of the chi-square test statistic:
Chi-square (X²) = Σ [(O - E)² / E], where O is the observed count, and E is the expected count.

X² = ( (3-11)²/11 + (12-11)²/11 + (14-11)²/11 + (15-11)²/11 + (22-14)²/14 + (13-14)²/14 + (11-14)²/14 + (10-14)²/14 )

X² = ( 64/11 + 1/11 + 9/11 + 16/11 + 64/14 + 1/14 + 9/14 + 16/14 ) = 32.73

The chi-square test statistic is approximately 32.73.

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Answer:

9.36

Step-by-step explanation:

26% of 36 is what number

Change the percent to decimal form

.26 * 36 = 9.36

Answer:

9.36

Step-by-step explanation:

26% of 36

= [tex]\frac{26}{100}[/tex] × 36

= 0.26 × 36

= 9.36

Answer: It approaches negative infinity as x approaches negative infinity and approaches positive infinity as x approaches positive infinity.

Step-by-step explanation:

As x approaches negative infinity, the expression inside the absolute value bars becomes more and more negative, so the function becomes 1/2 times a large negative number plus 2, which approaches negative infinity.

As x approaches positive infinity, the expression inside the absolute value bars becomes more and more positive, so the function becomes 1/2 times a large positive number plus 2, which approaches positive infinity.

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The probability of selecting exactly 2 broken calculators out of 4 when randomly selecting 4 calculators from a box of 50 calculators is 0.255.

The probability formula allows us to determine the likelihood of an event by dividing the number of favorable outcomes by the total number of possible outcomes. The probability of an event occurring can range from 0 to 1, as the number of favorable outcomes can never be greater than the total number of outcomes.

Using this formula, we can calculate the probability of getting exactly 2 broken calculators:

P(X=2) = C(4,2) * (4/50)² * (46/50)²
where C(4,2) is the number of ways we can select 2 broken calculators from a total of 4 broken calculators, which is equal to 6.
Therefore, plugging in the values, we get:
P(X=2) = 6 * (4/50)² * (46/50)²

P(X=2) = 0.255
So the probability of selecting exactly 2 broken calculators out of 4 when randomly selecting 4 calculators from a box of 50 calculators is 0.255.

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The standard deviation for the sample mean speed in a random sample of 100 cars is 0.5 mph.Therefore, the standard deviation for the sample mean speed in a random sample of n = 100 cars is 0.5 mph.

The standard deviation for the sample mean speed in a random sample of n = 100 cars can be calculated using the formula:

σ/√n

where σ is the population standard deviation (given as 5 mph) and n is the sample size (given as 100 cars).

Plugging in the values, we get:

σ/√n = 5 mph/√100 = 5 mph/10 = 0.5 mph

Therefore, the standard deviation for the sample mean speed in a random sample of n = 100 cars is 0.5 mph.

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If Quadrilateral ABED is inscribed in a circle with AE is a diameter, the measure of angle DEB is 2 degrees.

Since AE is a diameter of the circle, angle AEB is a right angle (90°).

Using the fact that the opposite angles of an inscribed quadrilateral are supplementary, we can find the measure of arc DE:

m(arc DE) = 180° - m(∠ABE) - m(∠AED)

m(arc DE) = 180° - (90° + 86°)

m(arc DE) = 4°

Since arc DE is a central angle, it is twice the measure of angle DEB:

m(arc DE) = 2m(∠DEB)

4° = 2m(∠DEB)

m(∠DEB) = 2°

In conclusion, using the properties of inscribed quadrilaterals and central angles in circles, we can determine that the measure of angle DEB in quadrilateral ABED is 2 degrees.

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Based on the given side lengths a=3 cm, b=5 cm, and c=6 cm, the triangle is a(n) scalene triangle. A scalene triangle has all sides of different lengths, which applies to this triangle with sides 3 cm, 5 cm, and 6 cm.

Triangles are described in terms of their sides and angles in geometry. A closed planar three-sided polygon shape with three sides and three angles is known as a triangle. The lengths of the sides of a scalene triangle vary. They are not equal, and the angles have three measurements. However, it still has a 180° angle sum, just like all triangles.

A scalene triangle is a triangle with three different side lengths and three different angle measurements. The total of all internal angles, however, is always equal to 180 degrees. As a result, it satisfies the triangle's condition of angle sum.

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The rate of the first bus is 15 mph. The solution involves using the formula distance = rate x time for both buses and setting them equal to each other to solve for the unknown rate of the first bus.

Let's assume that the first bus is traveling at a rate of x miles per hour.

We know that the second bus is traveling at a rate of 35 miles per hour.

When they meet, they will have traveled a total distance of 350 miles.

Using the formula distance = rate x time, we can set up the following equation

x(7) + 35(7) = 350

Simplifying this equation

7x + 245 = 350

7x = 105

x = 15

Therefore, the rate of the first bus is 15 miles per hour.

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Answer:

Step-by-step explanation:

15t+8t-2t=16

23t - 2t = 16

21t = 16

t= 16/21
t≈0,762

The factored form of the polynomial is: P(x) = 2(x - 1/2)(x - 2)(2x² + x + 2)

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

To find the rational zeros of the polynomial, we can use the rational root theorem, which states that any rational root of the polynomial must have the form p/q, where p is a factor of the constant term (-4 in this case) and q is a factor of the leading coefficient (2 in this case).

The factors of -4 are ±1, ±2, and ±4, and the factors of 2 are ±1 and ±2. Therefore, the possible rational zeros of the polynomial are:

±1/2, ±1, ±2, ±4

We can now test these values using synthetic division or long division to see which ones are actually zeros of the polynomial. After trying these values, we find that the polynomial has two rational zeros:

x = 1/2 and x = 2

To write the polynomial in factored form, we can use these zeros to factor it as follows:

P(x) = [tex]2x^4[/tex] − 7x³ + 3x² + 8x − 4

= 2(x - 1/2)(x - 2)(2x² + x + 2)

Therefore, the factored form of the polynomial is:

P(x) = 2(x - 1/2)(x - 2)(2x² + x + 2)

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From the logarithmic differentiation, function [tex]f(x) = \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}[/tex],

a) [tex] ln (f(x)) = \frac{1}{x} ln( 1 + 2x) - ln(sin(x))\\ [/tex]

b) [tex] \frac{f'(x)}{f(x)} = \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x) \\ [/tex]

c ) The derivative of function, f(x) is

[tex]f'(x) = \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}( \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x)) \\ [/tex]

A logarithmic differentiation calculator is one of online tool used to calculate the derivative of a function using logarithm.

We have a function, [tex]f(x) = \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}[/tex].

We have to use logarithmic differentiation to determine the derivative and other values of the function.

a) Taking natural logarithm both sides in f(x), [tex]ln (f(x)) = ln( \frac{( 1 + 2x)^{\frac{1}{2}}}{ sin(x)})[/tex]

Now, using the logarithm property,

[tex]ln(\frac{m}{n}) = ln(m) - ln(n) [/tex]

[tex]ln (f(x)) = ln( 1 + 2x)^{\frac{1}{x}} - ln(sin(x)) \\ [/tex]. Also use power property, ln(p)² = 2ln(p),

[tex] ln (f(x)) = \frac{1}{x} ln( 1 + 2x) - ln(sin(x)) - - (1) \\ [/tex]

b) Now, we determine the ratio of f'(x)/f(x)

Take a derivative of equation (1), we have

[tex]\frac{f'(x)}{f (x) } = \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - \frac{cos(x)}{sin(x)}\\ [/tex]

[tex]= \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x) \\ [/tex]

c) Now, we determine the derivative of f(x), Substitute original value of f(x) in previous equation,[tex] \frac{f'(x)}{ \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}} = \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x) \\ [/tex]

f'(x) [tex] = \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}( \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x)) \\ [/tex]. Hence, required value is [tex] \frac{( 1 + 2x)^{\frac{1}{x}}}{ sin(x)}[ \frac{2}{x( 1 + 2x)} - \frac{1}{x²} ln( 1 + 2x) - cot(x)] \\ [/tex].

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Answer: x^2 + 2x - 2 = 0

Step-by-step explanation:

subtract 2x from both sides to get -2 + 2x + x^2 = 0

arrange terms to get x^2 + 2x - 2 = 0

Note that the cardinality of the sets are given below.

A) 0
B) 1
C) 2
4) 3

(a) The cardinality of ∅ is 0.

Because it is an empty set, there are no or 0 elements.

(a) The Cardinality of  {∅} is 1.

It has one element, which is a set enclosing an empty set.

(c) The Cardinality of {∅, {∅}} is 2.

It has two elements: an empty set (∅) and a set that includes an empty set (∅).

(d) The cardinality of ) {∅, {∅}, {∅, {∅}}} is three.

It has three elements: an empty set (∅), a set containing an empty set (∅), and a set containing a set containing an empty set (∅).

set containing {∅,{∅}}. The whole set is regarded as one in the third element.

A set S = a, b, c, d, e, for example, has a cardinality of three. The first element is an in this case, while the second is a.

The second element is b, while the third element is a set of c, d, e.

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