For the following function, find the Taylor series centered at x= 2π​and then give the first 5 nonzero terms of the Taylor series and the open interval of convergence. f(x)=cos(x) .f(x)=∑ n=0[infinity]​f(x)=? The open interval of convergence is: (Give your answer in interval notation.)

1/2, -5/2, and 5/3 are the zeros of the given function.

To find the zeros of the polynomial f(x) = 6x³ - 31x² + 4x + 5, we can use various methods, such as factoring, synthetic division, or using the rational root theorem. Here, we will use the rational root theorem, which states that any rational zero of a polynomial must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.

In this case, the constant term is 5, and the leading coefficient is 6. Therefore, any rational zero must have the form of ±(factor of 5) / (factor of 6).

Possible factors of 5 are ±1 and ±5, and possible factors of 6 are ±1, ±2, ±3, and ±6. So, the possible rational zeros of f(x) are:

±1/1, ±5/1, ±1/2, ±5/2, ±1/3, ±5/3, ±1/6, ±5/6

Now, we can use synthetic division or substitute each of these values into f(x) to see which ones are actual zeros. Doing so, we find that f(1/2) = 0 and f(-5/2) = 0, so the zeros of f(x) are:

x = 1/2, -5/2, and 5/3.

Therefore, the zeros of f(x) are 1/2, -5/2, and 5/3.

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The correct formula for the revenue generated by the item is Revenue = 200p - 5p^2.

The revenue generated by the item can be calculated by multiplying the price (p) by the unit sales (200 - 5p). Therefore, the correct formula for the revenue generated by the item is:

Revenue = Price x Unit Sales
Revenue = p(200 - 5p)
Revenue = 200p - 5p^2

The revenue generated by the item is a quadratic function of the price (p). To find the maximum revenue, we need to differentiate the function with respect to p and set it equal to zero:

dRevenue/dp = 200 - 10p = 0
10p = 200
p = 20

Therefore, the maximum revenue is generated when the price of the item is $20. Substituting p = 20 in the revenue formula, we get:

Revenue = 200(20) - 5(20^2) = $2000

Hence, the correct formula for the revenue generated by the item is Revenue = 200p - 5p^2, and the maximum revenue is achieved when the price of the item is $20, generating a revenue of $2000.

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According to the given question we have The mass of the ball of radius 3 is approximately 15π (1 - e^(-27)) ≈ 65.2.

To find the mass of the ball, we need to integrate the density over the entire volume of the ball. We can use spherical coordinates to make this calculation easier.

First, let's set up the integral in terms of spherical coordinates. The density function is given in terms of (rho, phi, theta), where rho is the distance from the origin, phi is the angle between the positive z-axis and the vector, and theta is the angle between the positive x-axis and the projection of the vector onto the xy-plane. We can express the volume element in terms of these variables as:

dV = rho^2 sin(phi) d rho d phi d theta

Now, we can set up the integral:

m = ∭V (rho,φ,θ) dV
 = ∫0^2π ∫0^π ∫0^3 5e^(-rho^3) rho^2 sin(phi) d rho d phi d theta

We can solve this integral using u-substitution:

Let u = rho^3, then du = 3rho^2 d rho

The limits of integration also change:

When rho = 0, u = 0
When rho = 3, u = 27

Using these substitutions, the integral becomes:

m = 15π ∫0^27 e^(-u) du
 = 15π (-e^(-27) + 1)

Therefore, the mass of the ball is approximately 15π (1 - e^(-27)) ≈ 65.2.

In summary, the mass of the ball of radius 3 centered at the origin with a density of (rho, phi, theta) = 5e^(-rho^3) is approximately 65.2.

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(1)Elena should purchase a rainproof tarp that is at least 140 ft² in size. Option E (2)it would take 140 ft³ of water to fill the tent completely. Option A.

(1) To determine the size of the rainproof tarp Elena should purchase, we need to calculate the total surface area of the tent.

The tent has 4 sides, and each side has the same dimensions. Given that the height of the tent is 8.75 ft and the length and breadth are both 4 ft, the surface area of each side can be calculated as:

Side area = Length × Height = 4 ft × 8.75 ft = 35 ft²

Since there are 4 sides, the total surface area of the tent is:

Total tent surface area = Side area × 4 = 35 ft² × 4 = 140 ft²

Therefore, Elena should purchase a rainproof tarp that is at least 140 ft² in size.

The correct option would be e) X-Large: 144 ft², as it is the only size that is equal to or greater than the required size.

(2) If the tent fills up completely with water, we can calculate the volume of water it would hold using the formula:

Volume = Length × Breadth × Height

Given that the length and breadth of the tent are both 4 ft and the height is 8.75 ft, we can substitute these values into the formula:

Volume = 4 ft × 4 ft × 8.75 ft = 140 ft³

Therefore, it would take 140 ft³ of water to fill the tent completely.

Hence, the correct answer is 140 ft³. So Option E is correct  for 1. and Option A is correct for 2

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The Greates common factor of 3, 26, and 31 is 1.

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder.

In other words, it is the largest number that is a factor of two or more given numbers.

Here we have 3, 26, 31

To find the greatest common factor (GCF) of 3, 26, and 31,

We need to write the given numbers as product of prime numbers

=> 3 = 3 × 1

=> 26 = 2 × 13

=> 31 = 31 × 1

Now we can find the common factors of these three numbers

Here we can see that the only common factor is 1, since none of their prime factors are the same.

Therefore,

The Greates common factor of 3, 26, and 31 is 1.

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The points corresponding to the parameter values t = -2, -1, 0, 1, and 2 are:

(-2, -5), (0, -2), (0, 1), (6, 4), (18, 7).

What is a parametric equation?

A parametric equation is a mathematical representation of a curve or a set of coordinates in terms of one or more parameters. Instead of representing a curve or shape in the usual form of y = f(x), where y is expressed as a function of x, parametric equations express the x and y coordinates separately in terms of one or more parameters.

To find the points (x, y) corresponding to the parameter values of t = -2, -1, 0, 1, and 2, we can substitute these values into the given parametric equations and evaluate them. Let's calculate the points step by step:

For t = -2:

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

= 3[tex](-2)^2[/tex] + 3(-2)

= 12 - 6

= 6

y = 3t + 1

= 3(-2) + 1

= -6 + 1

= -5

So, when t = -2, the point is (x, y) = (6, -5).

For t = -1:

x = 3t² + 3t

= 3(-1)² + 3(-1)

= 3 - 3

= 0

y = 3t + 1

= 3(-1) + 1

= -3 + 1

= -2

When t = -1, the point is (x, y) = (0, -2).

For t = 0:

x = 3t² + 3t

= 3(0)² + 3(0)

= 0 + 0

= 0

y = 3t + 1

= 3(0) + 1

= 0 + 1

= 1

At t = 0, the point is (x, y) = (0, 1).

For t = 1:

x = 3t² + 3t

= 3(1)² + 3(1)

= 3 + 3

= 6

y = 3t + 1

= 3(1) + 1

= 3 + 1

= 4

At t = 1, the point is (x, y) = (6, 4).

For t = 2:

x = 3t² + 3t

= 3(2)² + 3(2)

= 12 + 6

= 18

y = 3t + 1

= 3(2) + 1

= 6 + 1

= 7

At t = 2, the point is (x, y) = (18, 7).

Therefore, the points corresponding to the parameter values t = -2, -1, 0, 1, and 2 are:

(-2, -5), (0, -2), (0, 1), (6, 4), (18, 7).

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Sydnie:

The solution is correct. The exact values of x are x = 1.62 and x = -0.62. Sydnie gets 5/5. Trevor: The solution is incorrect. To complete the square, z2 - 12z must add and subtract (12/2)2 = 36.

The correct steps are:

z2 - 12z - 27 = 0

z2 - 12z + 36 - 36 - 27 = 0

(z - 6)2 = 63

z = 6 + sqrt(63) and z = 6 - sqrt(63)

The exact values of z are z = 6 + sqrt(63) and z = 6 - sqrt(63). Trevor gets 3/5.

Brayden:

The solution is incorrect. The quadratic formula is being used correctly, but the square root of -56 should be simplified to 4i(sqrt(14)). The correct steps are:

x = (-2 ± sqrt(4 - 60)) / (2 * 3)

x = (-2 ± sqrt(-56)) / 6

x = (-2 ± 4i(sqrt(14))) / 6

x = (1 ± 2i(sqrt(14))) / 3

The exact values of x are x = (1 + 2i(sqrt(14))) / 3 and x = (1 - 2i(sqrt(14))) / 3. Brayden gets 3/5.

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sydnie The solution is correct. The exact values of x are x = 1.62 and x = -0.62. Sydnie gets 5/5. Trevor: The solution is incorrect. To complete the square, z2 - 12z must add and subtract (12/2)2 = 36.

The correct steps are:

z2 - 12z - 27 = 0

z2 - 12z + 36 - 36 - 27 = 0

(z - 6)2 = 63

z = 6 + sqrt(63) and z = 6 - sqrt(63)

The exact values of z are z = 6 + sqrt(63) and z = 6 - sqrt(63). Trevor gets 3/5.

Brayden:

The solution is incorrect. The quadratic formula is being used correctly, but the square root of -56 should be simplified to 4i(sqrt(14)). The correct steps are:

x = (-2 ± sqrt(4 - 60)) / (2 * 3)

x = (-2 ± sqrt(-56)) / 6

x = (-2 ± 4i(sqrt(14))) / 6

x = (1 ± 2i(sqrt(14))) / 3

The exact values of x are x = (1 + 2i(sqrt(14))) / 3 and x = (1 - 2i(sqrt(14))) / 3. Brayden gets 3/5.

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Answer:

8, -12

option c

Step-by-step explanation:

y= -4x

-4*-2 = 8

3* -4 = -12

The value of x is 8.

Angles that have a total angle of less than 90 degrees are said to be complementary. To put it another way, if two angles combine to form a right angle, that combination is said to be complementary. In this instance, we say that the two angles complement one another well.

In this given figure, we need to find what x is to add up to 90 degrees.

This means that [tex]\sf x^\circ + 82^\circ = 90^\circ[/tex]

[tex]\sf x^\circ=90^\circ-82^\circ[/tex]

[tex]\sf x^\circ=8^\circ[/tex]

Hence, The value of x is 8.

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Given that heart rate before an exam for STA 100 students is normally distributed with mean 100 bpm and standard deviation 20.2 bpm, the probability that a randomly selected student's heart rate is under 125 bpm given that it is over 100 bpm is approximately 0.6306.

We are given that the heart rate before an exam for STA 100 students follows a normal distribution with mean 100 bpm and standard deviation 20.2 bpm.

We want to find the conditional probability that the heart rate is under 125 bpm given that it is over 100 bpm.

Let A be the event that the heart rate is over 100 bpm, and let B be the event that the heart rate is under 125 bpm. Then we want to find P(B | A).

We can use Bayes' theorem to find P(B | A):

P(B | A) = P(A | B) * P(B) / P(A)

We know that P(A) = 1 - P(B), since the two events are complements. We also know that P(B) can be found using the standard normal distribution as follows:

P(B) = P(Z < (125 - 100) / 20.2) = P(Z < 1.2376) = 0.8917,

where Z is a standard normal random variable. To find P(A | B), we need to compute the conditional probability that the heart rate is over 100 bpm given that it is under 125 bpm:

P(A | B) = P(B | A) * P(A) / P(B) = P(B | A) * (1 - P(B)) / P(B).

Since the heart rate follows a normal distribution, we can standardize it by subtracting the mean and dividing by the standard deviation:

P(A | B) = P(Z > (100 - 100) / 20.2 | Z < (125 - 100) / 20.2)

= P(Z > 0 | Z < 1.2376)

= P(Z < 1.2376)

= 0.8917.

Finally, we can plug in the values we have found to obtain the probability we are looking for:

P(B | A) = P(A | B) * P(B) / P(A)

= 0.8917 * 0.8917 / (1 - 0.8917)

= 0.6306 (approximately).

Therefore, the probability that a randomly selected student's heart rate is under 125 bpm given that it is over 100 bpm is approximately 0.6306 or 63.06%.

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Complete Question

6. Assume that heart rate (in beats per minute, or bpm) before an exam for STA 100 students is distributed nor- mal, with a mean of 100 bpm and a standard deviation of 20.2 bpm. Assume all students in the following problem are selected from this population.

If we know a randomly selected students heart rate is over 100 (it is given), what is the probability that it is under 125?

"B: degrees of freedom > 10"  is not a way to check for the nearly normal condition.

To check the nearly normal condition, we can use several methods such as histogram, normal probability plot, and goodness of fit test. The central limit theorem can also be used when the sample size is large enough. However, checking the degrees of freedom is not a way to check for the nearly normal condition. Degrees of freedom refer to the number of independent pieces of information that are used to estimate a statistic. While it is important in hypothesis testing and confidence intervals, it is not related to checking for the nearly normal condition.

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Basically

3+(5+2)= 10

3+7= 10

10=10

will give the same answer as

(3+5)+2=10

8+2=10

10=10

Hence the order of adding didn't affect our answer

Elena's yearly cash inflows from the project after the payback period is $15,000. A correct answer is an option (b).

The payback period is the time it takes for the project's cash inflows to equal the initial investment. In this case, the payback period is 6 years, meaning that after 6 years, Elena will have received enough cash inflows to recover her initial investment of $150,000.

Since the project pays Elena an even amount per year for 10 years, and the payback period is 6 years, she will receive cash inflows for an additional 4 years after the payback period. Therefore, to calculate Elena's yearly cash inflows, we divide the remaining cash inflows by the number of years remaining:

Remaining cash inflows = $150,000 (initial investment) - cash inflows received during the payback period

= $150,000 - ($15,000 x 6)

= $60,000

Yearly cash inflows = Remaining cash inflows/number of years remaining

= $60,000 / 4

= $15,000

Hence, B is the correct option.

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Answer:

option B: (x+1)^2+(y+3)^2=5.

Step-by-step explanation:

Answer:

[tex](x-1)^{2} + (y+3)^2 = 25[/tex]

Step-by-step explanation:

[tex]-2x = -x^{2} - 6y -y^{2} + 15\\x^{2} - 2x + y^{2} + 6y = 15\\(x-1)^{2} - 1 + (y + 3)^{2} - 9 = 15\\(x-1)^{2} + (y+3)^2 = 25[/tex]

Based on the given radius of the circle, the area of the circle is approximately 26.41 square ft.

Area of a circle = πr²

Where,

π = 3.14

Radius, r = 2.9 ft

Area of a circle = πr²

= 3.14 × 2.9²

= 3.14 × 8.41

= 26.4074 square ft

Approximately,

26.41 square ft

Hence, 26.41 square ft is the area of the circle.

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Shape R and Shape S were put in the same group because they are the shapes with perpendicular line segments. Therefore the correct option is option D.

Shape S, a trapezium with one vertical side, and Shape R, a square, both have perpendicular line segments. All of the sides of a square are perpendicular, while one of the non-parallel sides of a trapezium with a vertical side is perpendicular to the base.

Reasons for ruling out other options

Option A, "Shapes without any right angles," is incorrect because Shape R (a square) has four right angles.

Option B, "Shapes with exactly one pair of parallel sides," is incorrect because Shape S (a trapezoid with one vertical side) has only one pair of parallel sides, while Shape R(a square) has two pairs of parallel sides.

Option C, "Shapes without parallel line segments," is also incorrect because all the other shapes P, Q, and T have parallel line segments.

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The surface area of the sphere is about 50.27 square feet.

The formula for the surface area of a sphere is:

Surface area = 4πr²

The circumference of the sphere as C = 4r ft.

The radius of the sphere as follows:

C = 2πr

4r = 2πr

r = 2 ft

The radius of the sphere can use the formula for surface area to find the answer:

Surface area = 4πr²

Surface area = 4π(2²)

Surface area ≈ 50.27 square feet

Rounding the answer to the nearest hundredth, we get:

Surface area ≈ 50.27 square feet (rounded to two decimal places)

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Answer:

The value for c would be 5

Step-by-step explanation:

$9.50 x 4 adults = $38

$65.50-$38= $27.50

$27.50/cost of children ($5.50)= 5

In a Cartesian join, also known as a cross join, each row from the first table is combined with each row from the second table.

If the first table has 5 rows and the second table has 3 rows, the result of the Cartesian join will consist of 5 * 3 = 15 rows.

As long as the data points used to calculate the time constant are representative of the decay process, the difference in the time constant value should be relatively small.

What is the slope?

The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as "rise over run" (change in y divided by change in x).

To estimate the number of data points collected during the decay, we would need more information such as the time interval between each data point and the total time elapsed during the decay.

Regarding the question about whether the time constant calculated from the decay would be different if other points were selected, it's possible that the time constant would be slightly different if different points were selected.

The time constant is determined by the slope of the decay curve, which can vary depending on which data points are used to calculate it.

Hence, as long as the data points used to calculate the time constant are representative of the decay process, the difference in the time constant value should be relatively small.

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The maximum number of moles of barium phosphate that can be formed is B) 0.15 mol, which corresponds to the amount of BaCl2 present. Therefore, the answer is (B) 0.15.

The balanced chemical equation for the reaction between sodium phosphate (Na3PO4) and barium chloride (BaCl2) is:

3 Na3PO4 + 2 BaCl2 → Ba3(PO4)2 + 6 NaCl

From the balanced equation, we can see that 2 moles of BaCl2 react with 3 moles of Na3PO4 to form 1 mole of Ba3(PO4)2.

Therefore, the limiting reactant in this reaction is the one that will be completely consumed first. To determine the limiting reactant, we need to compare the number of moles of each reactant with the stoichiometric ratio in the balanced equation.

For Na3PO4:

3 moles Na3PO4 = 1 mole Ba3(PO4)2

0.50 mol Na3PO4 = (1/3) × 0.50 mol Ba3(PO4)2 = 0.167 mol Ba3(PO4)2

For BaCl2:

2 moles BaCl2 = 1 mole Ba3(PO4)2

0.30 mol BaCl2 = (1/2) × 0.30 mol Ba3(PO4)2 = 0.15 mol Ba3(PO4)2

Therefore, the maximum number of moles of barium phosphate that can be formed is 0.15 mol, which corresponds to the amount of BaCl2 present.

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Therefore, the third, fourth, and fifth moments are:

E(X^3) = M'''(0) = 6λ^3

E(X^4) = M''''(0) = 24λ^4

E(X^5) = M^(5)(0) = 120λ^5

The third, fourth, and fifth moments of an exponential random variable with parameter λ can be found using the moment generating function.

The moment generating function (MGF) of an exponential distribution with parameter λ is:

M(t) = 1 / (1 - λt), for t < 1/λ

To find the nth moment of the distribution, we take the nth derivative of the MGF and evaluate it at t = 0. This gives:

E(X^n) = M^(n)(0)

Taking the derivatives of the MGF and evaluating at t = 0, we get:

M'(t) = λ / (1 - λt)^2

M''(t) = 2λ^2 / (1 - λt)^3

M'''(t) = 6λ^3 / (1 - λt)^4

Therefore, the third, fourth, and fifth moments are:

E(X^3) = M'''(0) = 6λ^3

E(X^4) = M''''(0) = 24λ^4

E(X^5) = M^(5)(0) = 120λ^5

Thus, the third moment is 6λ^3, the fourth moment is 24λ^4, and the fifth moment is 120λ^5

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The independent samples t-test is the appropriate statistic to use when comparing the mean number of hours worked per week for males and females.

To compare the mean number of hours worked per week for males and females, you would use the independent samples t-test. The independent samples t-test is a statistical test used to determine if there is a significant difference between the means of two independent groups. In this case, the independent groups are males and females.

The t-test allows you to compare the means of the two groups and determine if any observed difference is statistically significant or simply due to chance. It takes into account the sample means, sample sizes, and sample variances of both groups.

By conducting the independent samples t-test, you can assess whether there is evidence to suggest that the mean number of hours worked per week differs significantly between males and females. If the p-value associated with the t-test is below a predetermined significance level (commonly 0.05), it suggests that there is a statistically significant difference in the mean number of hours worked per week between the two groups.

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The area of the smaller sector or minor sector is  125.66 yd².

The area of the larger sector or major sector is  326.73 yd².

The areas of the minor and major sectors is calculated by applying the following formulas follow;

Area of sector is given as;

A = (θ/360) x πr²

where;

The area of the smaller sector or minor sector is calculated as follows;

A = ( 100 / 360 ) x π ( 12 yd)²

A = 125.66 yd²

The area of the larger sector or major sector is calculated as follows;

θ = 360 - 100

θ = 260⁰

A =  ( 260 / 360 ) x π ( 12 yd)²

A = 326.73 yd²

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Answer:

y = 4x + 10 y = 1 + 4^x

x y x y

0 10 0 2

1 14 1 5

2 18 2 17

3 22 3 65

4 26 4 257

Rate of change on [1, 3]:

For y = 4x + 10:

(22 - 14)/(3 - 1) = 8/2 = 4

For y = 1 + 4^x:

(65 - 5)/(3 - 1) = 60/2 = 30

The rate of change on [1, 3] is much greater on y = 1 + 4^x than on y = 4x + 10 because y = 1 + 4^x generally gives larger numbers than y = 4x + 10 as x gets larger.

The minimum and the maximum of the function are

Minimum = -1

Maximum = 18

From the question, we have the following parameters that can be used in our computation:

f(x) = ∛(x + 9)

The above function is a cubic function that has been transformed as follows

Shifted left by 9 units

Next, we plot the graph using a graphing tool by taking not of the above transformations rules

The graph of the function is added as an attachment

For the minimum, we set x = -10

So, we have

Minimum = ∛(-10 + 9)

Minimum = ∛-1

Minimum = -1

For the maximum, we set x = 108

So, we have

Maximum = ∛(18 + 9)

Maximum = ∛27

Maximum = 3

From the graph, we have confirm that the minimum is -1 and the maximum is 18

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The fraction of tomatoe recipe greater than 1 used is 7/4

The question requires that we Subtract the required value of tomato recipe from 1

The subtraction expression can be written thus ;

Amount of tomatoe recipe - 1

[tex]2 \frac{3}{4} - 1[/tex]

Converting to a proper fraction;

11/4 - 1/1

Take L.C.M of the denominator

L.C.M of 4 and 1 is 4

11/4 - 1/1 = (11 - 4)/4

11/4 - 1/1 = 7/4

Therefore, the fraction of tomatoe greater than 1 used is 7/4

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The inverse of α=(123)(456) is α^−1 = (456)(123). Option (B) is the correct answer.

The question requires finding the inverse of the given permutation α=(123)(456). To get the inverse of any permutation, the order of the cycles should be reversed, and the order of elements in each cycle should be reversed. Following this rule, the inverse of α is found to be α^−1 = (456)(123). Therefore, option (B) is the correct answer. In other words, if we apply α and then α^−1 to any set, we get back the original set. The inverse permutation of a permutation has useful applications in algebra and cryptography, among other fields.

Therefore, the inverse of α=(123)(456) is α^−1 = (456)(123). Thus, option (B) is the correct answer.

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Five and three hundred seven thousandths.

Or do you want it in number-expanded form...?

The margin of error is 0.1415, where z is the critical value for the desired confidence level,

The margin of error can be calculated using the formula: Margin of error = z * (standard deviation / sqrt(n))

where z is the critical value for the desired confidence level, standard deviation is the population standard deviation (which can be estimated using the sample standard deviation), and n is the sample size.

For a 99.8% confidence level, the critical value is 2.967. Using the given values of x and n, we can calculate the sample proportion as 46/98 = 0.4694.

To estimate the population standard deviation, we can assume that the sample proportion is a good estimate of the population proportion, and use the formula:

standard deviation = sqrt(p*(1-p)/n)

where p is the sample proportion. Substituting the values, we get: standard deviation = sqrt(0.4694*(1-0.4694)/98) = 0.0519

Now we can plug in the values into the margin of error formula to get: Margin of error = 2.967 * (0.0519 / sqrt(98)) = 0.1415

Therefore, the margin of error is 0.1415.

It is important to note that the margin of error represents the amount by which the sample proportion may differ from the population proportion with a certain level of confidence.

It is also important to remember that the margin of error is not the same as the sampling error, which is the difference between the sample mean and the population mean.

The margin of error can be used to determine the sample size required for a given level of precision, or to compare different sample sizes to determine which is more likely to yield a representative sample.

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