i need some help on this problem please

Answer:

Step-by-step explanation:

197.97

Answer:

[tex]\textsf{Increasing}: \quad (- \infty, -1) \cup (1, \infty)[/tex]

[tex]\textsf{Decreasing}: \quad(-1, 1)[/tex]

[tex]\textsf{Minimum}: \quad (1, -4)[/tex]

[tex]\textsf{Maximum}: \quad (-1, 0)[/tex]

Step-by-step explanation:

[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}}\implies f'(x) > 0[/tex]

[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies f'(x) < 0[/tex]

Given function:

[tex]f(x)=x^3-3x-2[/tex]

Differentiate the given function:

[tex]\implies f'(x)=3x^2-3[/tex]

To find the interval where f(x) is increasing, set the differentiated function to more than zero:

[tex]\implies 3x^2-3 > 0[/tex]

[tex]\implies 3(x^2-1) > 0[/tex]

[tex]\implies x^2-1 > 0[/tex]

[tex]\implies x^2 > 1[/tex]

[tex]\implies x < -1, \;\;\; x > 1[/tex]

So the interval on which function f(x) is increasing is:

[tex](- \infty, -1) \cup (1, \infty)[/tex]

To find the interval where f(x) is decreasing, set the differentiated function to less than zero:

[tex]\implies 3x^2-3 < 0[/tex]

[tex]\implies 3(x^2-1) < 0[/tex]

[tex]\implies x^2-1 < 0[/tex]

[tex]\implies x^2 < 1[/tex]

[tex]\implies -1 < x < 1[/tex]

So the interval on which function f(x) is decreasing is:

[tex](-1, 1)[/tex]

To find the relative extrema, set the differentiated function to zero and solve for x:

[tex]\implies 3x^2-3=0[/tex]

[tex]\implies 3(x^2-1)=0[/tex]

[tex]\implies x^2-1=0[/tex]

[tex]\implies x^2=1[/tex]

[tex]\implies x=-1,\;\;x=1[/tex]

To find the y-coordinates of the relative extrema, substitute the found values of x into the function and solve for y:

[tex]\implies f(-1)=(-1)^3-3(-1)-2=0[/tex]

[tex]\implies f(1)=(1)^3-3(1)-2=-4[/tex]

Therefore the minimum and maximum points of the function are:

[tex]\textsf{Minimum}: \quad (1, -4)[/tex]

[tex]\textsf{Maximum}: \quad (-1, 0)[/tex]

The directional derivative of f(x, y, z) = [tex]xy+z^{3}[/tex] at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point is √2 (3cos(3π/4) + 2sin(3π/4)).

To find the directional derivative of f(x, y, z) = xy + z^3 at the point (2, 3, 1) in the direction of a vector making an angle of 3π/4 with the gradient of f at that point, we need to calculate the gradient of f at that point and then use it to find the dot product with the given direction vector.

The gradient of f at the point (2, 3, 1) is given by ∇f(2, 3, 1) = (y, x, 3z^2) = (3, 2, 3).

The direction vector in the direction of 3π/4 with the gradient can be found as follows:

Let [tex]d = (cos(3\pi /4), sin(3\pi /4), 0)[/tex]

The directional derivative of f in the direction d at the point (2, 3, 1) is given by

∇f(2, 3, 1) . d = [tex](3, 2,3).(cos(3\pi /4), sin(3\pi /4),0)[/tex]

= [tex]3cos(3\pi /4)+2sin(3\pi /4)[/tex]

= √2 (3cos(3π/4) + 2sin(3π/4))

So the directional derivative of f in the direction of a vector making an angle of 3π/4 with the gradient of f at (2, 3, 1) is √2 (3cos(3π/4) + 2sin(3π/4)).

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The sample standard deviation of murder rates is approximately 2.2, and the sample variance is approximately 4.9. The standard deviation and variance are measured in the same units as the original data (murders per 100,000 inhabitants), squared and unsquared.

We can use a TI-83, TI-83 Plus, or TI-84 to calculate the sample standard deviation and variance of the murder rates:

We get the following results using this method:

2.209 is the sample standard deviation (s).

4.877 sample variance (s2) (rounded to three decimal places)

As a result, the sample standard deviation of the murder rates is about 2.2, and the sample variance is about 4.9. It should be noted that the standard deviation and variance are measured in the same units as the original data (murders per 100,000 inhabitants), squared and unsquared.

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Answer option 5) 0.77 is correct. The power of the test against this alternative is 0.77, given the calculations of the Scientist.

The power of a statistical test is the probability of correctly rejecting a false null hypothesis. In this case, the null hypothesis is rejected in favor of a specific alternative, and the scientist has calculated that the probability of making a type II error (failing to reject the null hypothesis when the alternative is actually true) is 0.23.

Therefore, the power of the test is the complement of the type II error probability, which is 1 - 0.23 = 0.77. This means that the test has a 77% chance of correctly rejecting the null hypothesis when the alternative is actually true. Answer option 5) 0.77 is correct.

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The slopes of the lines are listed below:

Case 1: m = 1

Case 2: m = - 4 / 5

Case 3: m = - 4

Case 4: m = 5

Case 5: m = 2 / 7

Case 6: m = - 1

Case 7: m = 0

Case 8: m = NaN (Not a Number)

Case 9: m = 3 / 5

Case 10: m = - 8 / 3

In this question we find ten cases of lines, whose slopes must be determined according to the following rules:

Now we proceed to determine the slope for each case:

Case 1

m = (5 - 0) / [0 - (- 5)]

m = 1

Case 2

m = (- 3 - 1) / [0 - (- 5)]

m = - 4 / 5

Case 3

m = (- 3 - 1) / (1 - 0)

m = - 4

Case 4

m = [1 - (- 4)] / (1 - 0)

m = 5

Case 5

m = (3 - 1) / [0 - (- 7)]

m = 2 / 7

Case 6

m = (0 - 3) / (3 - 0)

m = - 1

Case 7

m = 0

Case 8

m = NaN (Not a Number)

Case 9

m = [- 1 - (- 4)] / (5 - 0)

m = 3 / 5

Case 10

m = (- 4 - 0) / (1.5 - 0)

m = - 4 / 1.5

m = - 8 / 3

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With a mean of $32,500 and a standard deviation of $2,500. The z-score for a sales associate who earns $37,500 is 2.

The z-score, also known as a standard score, is a measure of how many standard deviations a given value is from the mean. It is used to standardize the data and make it easier to compare and interpret. The formula for the z-score is:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

In this case, the mean annual salary for sales associates from a particular store is $32,500 and the standard deviation is $2,500. We want to find the z-score for a sales associate who earns $37,500. Plugging in the values, we get:

z = ($37,500 - $32,500) / $2,500 = 2

This means that the sales associate's salary is 2 standard deviations above the mean.

Interpreting the z-score can help us understand how unusual or typical a given value is in comparison to the mean. In this case, a z-score of 2 indicates that the sales associate's salary is relatively high compared to the mean, but it is not an extremely high salary.

We can use a standard normal table to find the proportion of the data that falls below a given z-score, or to find the corresponding value for a given proportion of the data.

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Answer: so u do 1 + n = 21

Step-by-step explanation:

21

Answer:

Step-by-step explanation:

SRS

The missing statement from his process;

RM = RL and MA = LA

Radii of the constructed arc are some length.

Angle bisectors are lines that cut across the angle under consideration. The term "bisect" means to divide into two equal portions. The considered angle is therefore divided in half by the bisected halves.

Given:

Oleg is writing proof of the construction of an angle bisector.

The missing statement from his process;

RM = RL and MA = LA

Radii of the constructed arc are some length.

From the reflexive property;

RA ≅ RA.

And ΔRMA ≅ ΔRLA, by SSS criterion.

And corresponding angles of congruent angles

∠MRA ≅ ∠LRA.

Therefore, ∠MRA ≅ ∠LRA.

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The amount freedome ride charge more than Envo car in d miles is $20.05

word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem.

The total charge for Enzo car =$50.50 +8¢ = $50.50 + $0.08 = $50.58

The total charge for Freedom = $70.50 + 12¢

= $70.50 + $0.12

= $70.62

therefore in d miles ;

Enzo will charge $50.58d

freedome will charge

$70.62d

therefore the amount freedome will charge more than Enzo car is $70.62 - $50.58

= $20.05

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The average number of likes will be 16.5. the correct option is D.

Mean is defined as the ratio of the sum of the number of data sets to the total number of data. The mean is calculated by adding all the data points and dividing it by the total number of the data.

Given that Bethany has a photo website where she places photos that she has taken so others can see them. Below is how many likes each of her current photos was given: 10, 13, 15, 15, 17, 29.

The average will be calculated as:-

Mean = ( 10 + 13 + 15 + 15 + 17 + 29 ) / 6

Mean = 99 / 6

Mean = 16.5

Therefore, the average number of likes will be 16.5. the correct option is D.

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The value of the sum notation is 729

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

The sum notation

From the notation, we have the following values

First term, a = 972

Common ratio, r = -1/3

The sum is then represented as

Sum = a/(1 -r)

So, we have

Sum = 972/(1 + 1/3)

Evaluate

Sum = 729

Hence. the sum is 729

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1) The Area of the first compound shape is given as  = 52cm

2) The area of the second shape is given as = 47cm

Note that the method for deriving this area of the shapes is by dividing them into "regular" shapes as shown in the attached images.

A compound form is created by combining simple shapes. It's also known as a "composite" or "complex" form.

For Shape A:

We have a triangle and a rectangle when the compound shape is de-regularized.

The dimensions of the triangle are:

Base 8cm

Height 3cm

Since the area of a triangle is given as:

1/2 B * H,

The area of the triangle is = 1/2 * 8 * 3cm

= 12cm

For the rectangle, the area of a rectangle is given as L * B. Since

L = 8cm and

B = 5cm

The area = 8* 5 = 40cm

The total area of Shape A = 40cm + 12cm

= 52cm

For Shape B:

We have two rectangles and one right-angled triangle when the shape is "converted".

The dimensions of rectangle 1 are:

L = 3

B = 2

Area = 6

The dimensions of rectangle 2  =
L = 8cm
B = 4cm

Area = 32cm

The dimension of the Right angled triangle is:

Base = 6cm
Height = 3cm (that is 7-4)

Thus:

The area of the triangle is:

1/2 * 6* 3
= 9cm

Thus the total area of the second shape is:

= 32 + 6 + 9

= 47cm

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½ of a 12 inches pizza represents the bigger portion.

It is the close curve of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

Let r be the radius of the circle. Then the area of the circle will be

A = πr² square units

½ of a 12-inch pizza, then the area is given as,

A = (π / 2) x (12 / 2)²

A = 56.52 square inches

30% of a 9-inch pizza. Then the area is given as,

A = 0.30 x π (9 / 2)²

A = 19.0755 square inches

½ of a 12 inches pizza represents the bigger portion.

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Answer:

4(1) - 5 = -1

4(2) - 5 = 3

4(3) - 5 = 7

Step-by-step explanation:

Replace variable "n" with values of "n".

4n-5 when n = 1 is 4(1) - 5

4(1) - 5 = -1

4(2) - 5 = 3

4(3) - 5 = 7

The formula for the area of a circle is A = πr^2, where r is the radius.

The area of the circle with a radius of 3.5 centimeters is approximately 38.5 square centimeters.

Substituting r = 3.5 centimeters and using π ≈ 3.14, we get:

A = 3.14 × (3.5)^2

A = 38.465 square centimeters (rounded to the nearest tenth)

Consequently, the radius of the circle is 3.5 centimetres, and its area is roughly 38.5 square centimeters.

The area of a circle is the measure of the region enclosed by the circle in a two-dimensional space. It is the amount of space inside the circle and is typically measured in square units, such as square centimeters (cm²) or square meters (m²). The formula to calculate the area of a circle is:

Area = π x radius²

Where π (pi) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter, which is approximately 3.14159, and radius is the distance from the center of the circle to any point on the circumference. The area formula can also be expressed in terms of the diameter:

Area = π x (diameter/2)²

This formula tells us that the area of a circle depends on the length of its radius or diameter, and it allows us to calculate the area of any circle given either of these two measurements.

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Answer: y = 5x - 3

Step-by-step explanation:

The lines need the same slope

2 = 5(1) + b

2 = 5 + b

-5   -5

-3 = b

y = 5x - 3

The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.

Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.

We know that the speed formula

Speed = Distance/Time

The light travels at a speed of 1.86 × 10⁵ miles per second. And it takes about 15,000 seconds for the light to reach Neptune.

Then the number of miles is given as,

1.86 × 10⁵ = d / 15,000

d = 1.86 × 10⁵ × 15,000

d = 2.79 × 10⁹ miles

The distance that the light travels in the scientific notation will be 2.79 × 10⁹ miles.

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Answer:

The z-score is a statistical measure that describes how far a sample estimate is from the population parameter in standard deviation units. It is calculated by taking the difference between the sample estimate and the population parameter, and dividing it by the standard deviation of the sampling distribution of the estimate. The resulting z-score can be used to calculate the probability of observing a sample estimate as extreme or more extreme than the one obtained, assuming the null hypothesis is true.

In this case, the sample estimate is the proportion of individuals who watch television primarily through streaming services, which is 70%, and the population parameter is the proportion of all individuals who watch television primarily through streaming services, which is 56%. The sample size is 350. The null hypothesis is that there is no difference between the sample and population proportions, and the alternative hypothesis is that the sample proportion is greater than the population proportion.

To calculate the z-score, we need to determine the standard error of the sampling distribution of the proportion. The standard error is a measure of the variability of sample estimates due to random sampling variation. It is calculated by taking the square root of the population proportion times one minus the population proportion, divided by the sample size. In this case, the standard error is:

standard error = sqrt[(0.56 * (1 - 0.56)) / 350] = 0.028

The z-score can now be calculated as:

z-score = (0.7 - 0.56) / 0.028 = 5.19

This z-score is quite large, indicating that the sample estimate is more than 5 standard errors away from the population parameter. This suggests strong evidence against the null hypothesis, as the probability of observing a sample estimate as extreme or more extreme than the one obtained is very low, assuming the null hypothesis is true.

In other words, the result of observing a sample proportion of 70% or above is highly unlikely to have occurred by chance alone, assuming that the true population proportion is 56%. This result provides evidence in support of the alternative hypothesis, which states that the sample proportion is greater than the population proportion.

Answer:12.90

Step-by-step explanation:

The value of f(0) is -2.

Option A is the correct answer.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

From the graph, we see that,

f(x) = -1 for -1 ≤ x < 0

Similarly,

f(x) = -2 for 0 ≤ x < 1 _____(1)

Now,

The value of f(x) at x = 0.

From (1),

f(0) = -2

Thus,

The value of f(0) is -2.

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Answer:

44 ÷ 2,876 = 0.01528 (rounded to five decimal places)

Step-by-step explanation:

To divide 44 by 2,876, we can use long division. Here are the steps:

Write 44 under the long division symbol (÷) and 2,876 outside the symbol.

      ___

   2,876 | 44

Since 2 is smaller than 44, we can't divide 2,876 by 44, so we need to add a decimal point and a zero to the quotient and bring down the next digit of the dividend (4).

      ___

   2,876 | 44.0

      ___

   2,876 | 44.0

               4

We see that 28 is the largest multiple of 2,876 that is less than or equal to 440. We write 28 above the line, and multiply 2,876 by 28 to get 80,528. We subtract 80,528 from 440, which gives us a remainder of 4.

       28

      ___

   2,876 | 44.0

              4 4 0

         -    3 4 2 8

              -------

                8 4

Since we have a remainder of 4, we can't bring down any more digits from the dividend, so we add a decimal point and a zero to the quotient and bring down a zero to make the dividend 40. We repeat the process of finding the largest multiple of 2,876 that is less than or equal to 400, writing it above the line, multiplying it by 2,876, subtracting the result from the dividend, and bringing down the next digit.

       28. 0

      ___

   2,876 | 44.0

          4 4 0

        - 3 4 2 8

             -------

            8 4

            8 3 6

          - 8 0 4

             -------

              3 2

We have a remainder of 32, and we can't bring down any more digits from the dividend, so we add a decimal point and a zero to the quotient and bring down another zero to make the dividend 320. We repeat the process of finding the largest multiple of 2,876 that is less than or equal to 3,200, writing it above the line, multiplying it by 2,876, subtracting the result from the dividend, and bringing down the next digit.

       28. 00

      ___

   2,876 | 44.0

               4 4 0

             - 3 4 2 8

               -------

               8 4

               8 3 6

             - 8 0 4

              -------

              3 2

              2 8 2

            - 2 8 7 6

            -------

                  6

We have a remainder of 6, and we can't bring down any more digits from the dividend, so we have our final answer:

44 ÷ 2,876 = 0.01528 (rounded to five decimal places)

Answer:

Below

Step-by-step explanation:

For this , I will assume  i = [tex]\sqrt{-1}[/tex]

i^{11} + i^{16} + i^{21} + i^{26} + i^{31}  =

-  i         +1        +i            -1        - i       =     - i  

The probability that a newly arrived email which does mention "free gift card", is not spam is 0.0165.

Let S be the event that an email is spam, and let F be the event that an email mentions "free gift card". We want to find P(S' | F), the probability that the email is not spam given that it mentions "free gift card".

We can use Bayes' theorem to find P(S' | F):

P(S' | F) = P(F | S') P(S') / P(F)

We can calculate each of these probabilities as follows:

P(F | S') = the probability that a non-spam email mentions "free gift card". From the problem, we know that this probability is 0.005 (i.e., 99.5% of non-spam emails do not mention "free gift card").

P(S') = the probability that an email is not spam. From the problem, we know that this probability is 0.4 (i.e., 60% of emails are spam, so 40% are not spam).

P(F) = the probability that an email mentions "free gift card". This can be calculated using the law of total probability:

P(F) = P(F | S) P(S) + P(F | S') P(S')

= 0.20 × 0.60 + 0.005 × 0.40

= 0.121

In the first term, we use the given probability that 20% of spam emails mention "free gift card". In the second term, we use the probability that 0.5% of non-spam emails mention "free gift card".

Plugging these values into Bayes' theorem, we get:

P(S' | F) = 0.005 × 0.4 / 0.121 ≈ 0.0165

Therefore, the probability that a newly arrived email which does mention "free gift card" is not spam is approximately 0.0165 or 1.65%.

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All the sample tests are stated below:

1) A two-independent-sample t test.

2) A one-sample t test.

3) A related-samples t test.

4)  A related-samples t test.

It is a subgroup of people with traits from a wider population. When population sizes are too big for the test to include all potential participants or observations, samples are utilized in statistical testing. A sample must be representative of the entire population and must not be biased toward any one trait.

A professor runs an experiment to see if front-row students perform better on an exam than back-row pupils.

This is a two-independent-sample t test, as there are two distinct groups (students in the front row and students in the back row) and the professor is interested in comparing the mean exam scores between them.

To determine whether students spend more than 30 minutes on an activity on average, a graduate student chooses a sample of 25 participants.

This is a one-sample t test, as there is only one group of participants and the graduate student is interested in comparing the mean time students attend to a task to a specific value (30 minutes).

To determine whether right-handed siblings exhibit higher emotional intelligence than left-handed siblings, a researcher pairs up right- and left-handed siblings.

This is a related-samples t test using a matched-pairs design, as the researcher is interested in comparing emotional intelligence scores of right-handed and left-handed siblings who are matched based on certain criteria (in this case, being siblings).

A local school's principal is curious to discover how much an honors class benefits the children there. He administers tests to honors English students before the start of the school year and again at the conclusion of the year to gauge their progress.

This is a related-samples t test using a repeated-measures design, as the principal is interested in comparing the test scores of the same group of students before and after taking the honors English class to measure how much they learned during the year.

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The water station would be located at 2.92 miles. Round to the nearest tenth of a mile, the answer is 2.9 miles.

Distance is a numerical value that quantifies "how much ground an object has traveled" while in motion. Distance is calculated as the sum of a person's movement and time.

Call the distance between the starting point and the water station x. 5 - x meters separate from the water station from the finish line.

We may write the following equation because we are aware that the ratio between these two lengths is 7:5.

x / (5 - x) = 7/5

Expanding and solving for x, we get:

5x = 7(5) - 7x

12x = 35

x = 35/12 = 2.92 miles

Therefore, the distance of the water station would be 2.9 miles.

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The number of bags of potatoes and carrots did the farmer sell in all is 75 bags.

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

Given that, a farmer is selling  potatoes and carrots into bags, 3/5 were bags of potatoes and the rest were bags of carrots. He sold 45 bags of potatoes and 3/4 of the bags of carrots, and had 1/4 of the bags left.

The bags of potato = 3x/5

The bags of carrot = 2x/5

Therefore, according to question,

x-45-3x/5×2x/5 = x/4

7x/10-x/4 = 45

18x/40 = 45

x = 45×40/18

x = 100

Total bags sold = 100×2/5+3/5+45 = 75

Hence, the number of bags of potatoes and carrots did the farmer sell in all is 75 bags.

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The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

here, we have,

24 students went to the zoo and 3 students went to the aquarium.

i.e. we get,

24 students went to the zoo

3 went to the aquarium

so, total no. of students = 24+ 3

                                       = 27

then, we have,

the ratio of the number of students who went to the aquarium to the total number of students is = 3 : 27

                                    = 1: 9

Hence, The solution is, the ratio of the number of students who went to the aquarium to the total number of students is 1:9.

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The question which could not be answered based solely on the information in this data set is How many registered republicans voted for Gary Johnson in the 2016 presidential election? So, correct answer is option (d).

We have a large data consists of information on all registered republicans in the United States. The information of voters which present in data are Name, Gender, Age, Home address, Whether they voted or not in 2016 presidential election. We have to determine the question which answer is not possible on basis of this data. Let's consider the options one by one.

a) Here, gender and voters who vote or not in 2016 election, so it is easy to determine the percent of registered republican voters are male.

b) Address details and the information of voters who vote or not, helps to determine the state which had the most registered republican voters vote in the 2016 election.

c) The Age and registered voter information that he vote or not is used to determine the average age of registered republican voters.

d) Now, there is no information regarding the voters choice for voting their votes. So, it not determined that how many voters voted for Gary Johnson in the 2016 presidential election. This is write choice here.

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[tex] \boxed{\sf Option \: A\text{:} \: \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1}{4} + \sqrt{2}i} [/tex]

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We are asked to express the given number in the form a + ib, also known as the algebraic form.

[tex] \\ \\ [/tex]

[tex] \\ \\ [/tex]

[tex] \sf z = a + ib \\[/tex]

Where:

✰ a and b are real numbers and are the complex number's real part and imaginary part, respectively.

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✰ i is the imaginary unit number and can be expressed as follows:

[tex] \\ \sf i= \sqrt{-1}[/tex]

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[tex] \sf z = \dfrac{1 + \sqrt{-32}}{4} = \dfrac{1 + \sqrt{32 \times (-1)}}{4} \\ \\ \\ \sf \diamond \: We \: know \: that \: \sqrt{ab} = \sqrt{a} \times \sqrt{b}. \\ \\ \implies \dfrac{1 + \sqrt{32 \times (-1)}}{4} = \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4}[/tex]

[tex] \\ \\ \sf \implies \dfrac{1 + \sqrt{32}\times \sqrt{-1}}{4} = \dfrac{1 + \sqrt{16 \times 2} \times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times \sqrt{-1}}{4}[/tex]

[tex] \\ \\[/tex]

[tex] \diamond \: \sf Substitute \: \sqrt{-1} = i. \\ \\ \implies \sf \dfrac{1 + 4\sqrt{2}\times \sqrt{-1}}{4} = \dfrac{1 + 4\sqrt{2} \times i}{4} = \dfrac{1 + 4i\sqrt{2}}{4} [/tex]

[tex] \\ \\ \: \sf \diamond \: Simplify \: knowing \: that \: \dfrac{a + b}{c} = \dfrac{a}{c}+\dfrac{b}{c}. [/tex]

[tex] \\ \\ \\ \implies \: \sf \dfrac{1 + 4i\sqrt{2}}{4} = \dfrac{1}{4} + \dfrac{4i\sqrt{2}}{4} = \boxed{\sf \dfrac{1}{4} + \sqrt{2}i} [/tex]

[tex] \\ [/tex]

Therefore, the first option is correct.

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