a study is run to estimate the mean total cholesterol level in children 9-11 years of age. a random sample of 169 participants is selected and their mean total cholesterol levels is 161.5. assume the population standard deviation is 19.5. give the following information for a 95% confidence interval for the mean cholesterol.
Sample mean =
Standard Deviation =
Sample size =
Do you use Z or t?
Z or t =
Standard Error (Rounded to nearest tenth)=
Margin of Error (Rounded to nearest tenth) =
Lower limit (Rounded to nearest tenth) =
Upper limit (Rounded to nearest tenth) =

Answer:

5/37

Step-by-step explanation:

There are 37 possible outcomes when rolling a 37-sided die, so the probability of rolling any one specific number is 1/37.

To find the probability of rolling any of the given numbers (35, 25, 33, 9, or 19), we need to add the probabilities of rolling each individual number.

Probability of rolling 35: 1/37

Probability of rolling 25: 1/37

Probability of rolling 33: 1/37

Probability of rolling 9: 1/37

Probability of rolling 19: 1/37

The probability of rolling any one of these numbers is the sum of these probabilities:

1/37 + 1/37 + 1/37 + 1/37 + 1/37 = 5/37

So the probability of rolling any of the given numbers is 5/37, which is approximately 0.1351 when rounded to four decimal places.

The difference between the numbers whose sum is of 96 and have a ratio of 11:1 is of 80.

The difference between the two amounts is obtained applying the proportions in the context of the problem.

The variables that are going to represent the two amounts are given as follows:

x and y.

The sum of two numbers is 96, hence:

x + y = 96.

The ratio between them is 11:1, hence:

x/y = 11.

x = 11y.

Replacing the second equation into the first, the value of y is given as follows:

y + 11y = 96

12y = 96

y = 8.

The value of x is given as follows:

x = 11 x 8 = 88.

The difference is then given as follows:

x - y = 88 - 8 = 80.

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For equation a, x = 3

For equation b, x = -11/4.

For equation c, x = 1.

For equation d, x = -7/3.

For equation e, x = 9.

For equation f, x = 1/2.

To solve this equation, we need to isolate the variable x on one side of the equation.

7x - 6 = 6x + 3

Subtracting 6x from both sides:

x - 6 = 3

Adding 6 to both sides:

x = 9

Therefore, the solution to the equation is x = 9.

In the other equations:

a) 4x + 7 = 2x + 13

Subtracting 2x from both sides:

2x + 7 = 13

Subtracting 7 from both sides:

2x = 6

Dividing by 2:

x = 3

Therefore, the solution to the equation is x = 3.

b) x - 2 = 10 + 5x

Subtracting x from both sides:

-2 = 9 + 4x

Subtracting 9 from both sides:

-11 = 4x

Dividing by 4:

x = -11/4

Therefore, the solution to the equation is x = -11/4.

c) -3x - 8 = -7x - 4

Adding 7x to both sides:

4x - 8 = -4

Adding 8 to both sides:

4x = 4

Dividing by 4:

x = 1

Therefore, the solution to the equation is x = 1.

d) 2t + 5 = 5t + 12

Subtracting 2t from both sides:

5 = 3t + 12

Subtracting 12 from both sides:

-7 = 3t

Dividing by 3:

t = -7/3

Therefore, the solution to the equation is t = -7/3.

f) 15x = 7x + 4

Subtracting 7x from both sides:

8x = 4

Dividing by 8:

x = 1/2

Therefore, the solution to the equation is x = 1/2.

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Complete Question:

Find X for each equation.

A) 4 x + 7 = 2 x + 13 ;

b) x – 2 = 10 + 5 x ;

c) – 3 x – 8 = – 7 x – 4 ;

d) 2 t + 5 = 5 t + 12 ;

e) 7 x – 6 = 6 x + 3

f) 15 x = 7 x + 4

Answer: Different sample proportions, ^p, could result for each intern, but for either sample size, they would be centered (have their mean) at the true population proportion, p.

Step-by-step explanation:

The x ≤ 32Putting x = 24 in the expression we get RC = 96Therefore, the value of RC is 96.

In order to find RC, we will make use of the given information in the following manner: Given that ABDC is a parallelogram. Hence, AB = DC. We have also been given that PQ = 735 and QR = 112.Now, extend PQ to meet DC at S.Let PS = x; then DS = DC - x = AB - x. (Since AB = DC)We have that PS/SP = QR/RB (Since PQR is similar to DBR)Therefore, we getx/SP = 112/(AB - x)We can cross multiply and simplify to getSP = (112* x)/(AB - x)......(1)Further, we have that AQ/QB = SP/BR (Since PQR is similar to AQB)Therefore, we getx/(AB - x) = SP/BROn substituting the value of SP from equation (1) above, we getx/(AB - x) = (112* x)/(BR*(AB - x))Therefore, we getBR = (x*(AB - x)*QR)/[PQ*(AB - 2*x)]BR = (x*(32 - x)*112)/(735*(32 - 2*x))BR = (56*x*(16 - x))/(245*(16 - x))BR = (56*x/245)Since the sum of all sides of a parallelogram is equal to the sum of its opposite sides, we have thatRC + QR = AB + AQ - QBRearranging the terms we getRC = AB + AQ - QB - QR......(2)Now, AQ = PQ - APSubstituting the values of PQ and AP we get AQ = 735 - (DC - x) = 735 - 32 + x = x + 703Also, QB = AB - AQ = (32 - x) - (x + 703) = -x - 671Substituting the values of AQ and QB in equation (2) above we getRC = 32 + (x + 703) + (x + 671) - 112RC = 48 + 2xRC = 2(x + 24)We know that AB = 32, hence, x ≤ 32Putting x = 24 in the expression we get RC = 96Therefore, the value of RC is 96.

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The ratio of gold to silver to bronze medals won by Belarus in its simplest form is 4:0:1, which means that Belarus won four times as many gold medals as bronze medals, and no silver medals.

In mathematics, the simplest form refers to the most reduced or compact form of an expression.

The simplest form can be achieved by simplifying, reducing, or condensing a mathematical expression or ratio to its smallest possible form using various techniques such as factorization, cancellation, and distribution.

To find the ratio of gold to silver to bronze medals won by Belarus, we need to look at the row corresponding to Belarus in the table:

Country Gold Silver Bronze

Belarus    8             0              2

The ratio of gold to silver to bronze medals won by Belarus is: 8:0:2

However, this ratio is not in its simplest form, because we can simplify it by dividing all of the numbers by the greatest common factor of the three numbers. In this case, the greatest common factor of 8, 0, and 2 is 2, so we can simplify the ratio by dividing each number by 2:

8 ÷ 2 : 0 ÷ 2 : 2 ÷ 2

4 : 0 : 1

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Answer:

.

Step-by-step explanation:

...................

Hi!
Let's write this out.
Her limit is $900, and she's used $450. So, we subtract 450 from 900.
900-450 = 450.

So, she has $450 available credit left.
Hope this helps!
~~~PicklePoppers~~~

Answer: your credit utilization ratio on that card would be 50% but the answer is 450

Step-by-step explanation:

900-450 = 450

Answer:

the ribbon costs $0.80 per meter.

Step-by-step explanation:

To calculate the cost per meter of ribbon, we can divide the total cost of the ribbon by the length of the ribbon:

Cost per meter = Total cost of ribbon / Length of ribbon

In this case, the total cost of the ribbon is $4, and the length of the ribbon is 5 meters:

Cost per meter = $4 / 5 meters = $0.80/meter

To determine the cost of each meter of the ribbon, we divide the cost for 5 meters of ribbon by 5. That is,

                                       [tex]x = \$4 / 5[/tex]

where x is the cost of each meter. Simplifying will give us an answer of $0.8/m. Converting this to per mm.

                                  [tex]($0.8/m) \times (1 m/ 1000 mm)[/tex]

                              [tex]= \$0.0008/mm[/tex]

Step-by-step explanation:

You need solve for y:

[tex]y-4=3(x-1)\\y-4=3x-3\\y=3x-3+4\\\therfore \quad y=3x+1[/tex]

now, evaluate in each x's value, for example:

[tex]y=3(-2)+1\\\therefore \quad y=-5[/tex]

This, is the value of [tex]y[/tex], when [tex]x=-2[/tex]

Therefore:

[tex]\begin{tabular}{|c|c|} \cline{0-1}x & y \\ \cline{0-1}-2 & -5 \\ -1 & -2\\ 0& 1 \\ 1& 4\\ 2& 7\\ \cline{0-1}\end{tabular}[/tex]

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

x³ + 343 = (x + 7) (x² - 7x + 49)    is the function.

A relation is any subset of a Cartesian product.

As an illustration, a subset of is referred to as a "binary connection from A to B," and more specifically, a "relation on A."

A binary relation from A to B is made up of these ordered pairs (a,b), where the first component is from A and the second component is from B.

Every item in a set X is connected to one item in a different set Y through a connection known as a function (possibly the same set).

A function is only represented by a graph, which is a collection of all ordered pairs (x, f (x)).

Every function, as you can see from these definitions,

is a relation from X

Hence, x³ + 343 = (x + 7) (x² - 7x + 49)    is the function.

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Answer:

x = 45.

Step-by-step explanation:

Given a straight line with the equation.

First collect like terms:

2x + x = 180 - 45

Then calculate:

3x = 135

Finally after dividing both sides by 3:

x = 45

The required value of the base of the triangle is   [tex]\frac{7\sqrt{3} }{3}[/tex].

Right-angle triangles are formed when the angle formed by two of their edges is exactly 90 degrees. Obtuse angle triangle: An obtuse angle triangle is one in which the angle formed by two sides is larger than 90 degrees.

In the given triangle, we will use tangent function to find the value of x.

tan(∅) = Perpendicular/ base

tan(60°) = 7/x

x = 7/tan(60°)

x = 7/√3                         ∴ (tan(60°) =√3  )

On rationalizing

x = [tex]\frac{7\sqrt{3} }{3}[/tex]

Thus, required value of x is [tex]\frac{7\sqrt{3} }{3}[/tex].

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The probability that the response indicated that the dog is small-sized given that they enjoyed the treat is 0.286 (or 2/7) in fraction in the lowest terms.

Bayes' theorem is used to update probabilities of a hypothesis or an event in light of new data or evidence. It is used to calculate the conditional probability of an event based on prior knowledge of the conditions that might be relevant to the event.In the given problem, we have to find the probability that the response indicated that the dog is small-sized given that they enjoyed the treat.

The probability that the dog is small-sized given that they enjoyed the treat is the conditional probability P(S|T), where S is the event that the dog is small-sized and T is the event that they enjoyed the treat. To find the value of P(S|T), we will use Bayes' theorem. Bayes' theorem states that P(S|T) = P(T|S) * P(S) / P(T) where P(T|S) is the probability that they enjoyed the treat given that the dog is small-sized, P(S) is the prior probability that the dog is small-sized, and P(T) is the probability that they enjoyed the treat.

P(S) = 3/7P(T|S) = 2/3P(T) = (2/3 * 3/7) + (1/4 * 4/7) = 18/84 + 4/28 = 1/3

(adding the probabilities of T given S and T given L)Therefore, P(S|T) = (2/3 * 3/7) / (1/3) = 2/7 = 0.285714...Rounding off to the nearest millionth, the probability is 0.286. Therefore, the probability that the response indicated that the dog is small-sized given that they enjoyed the treat is 0.286 (or 2/7) in fraction in the lowest terms.

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The function is a set of ordered pairs (x, y), where x is an element of the domain and y is the corresponding element of the range. The notation f(x) is commonly used to denote the output value of the function for a given input value x.

At x=3, the derivative is negative, f'(3)=-12, so the function is decreasing at x=3.

The function is always decreasing since its derivative is constant and negative. Therefore, the interval where f(x) is decreasing is the entire real line, or (-∞, ∞).

Since the function is always decreasing, it does not have a relative maximum.

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Therefore, 0.02 times 100 is equal to 2.

Multiplication is a mathematical operation that involves combining two or more quantities to find their product. The quantities being multiplied are called factors, and the result of the multiplication is called the product. Multiplication is denoted by the "×" or "*" symbol, and is read as "times" or "multiplied by". Multiplication has many applications in mathematics and everyday life. It is used for calculating areas and volumes, scaling objects, calculating distances and speeds, and much more. It is also a fundamental operation in algebra, calculus, and other branches of mathematics.

Here,

To find out what number is equal to 0.02 times 100, we can use the formula:

0.02 * 100 = x

where "x" is the number we want to find. To solve for "x", we can multiply 0.02 and 100 using the distributive property of multiplication, which states that:

a * (b + c) = a * b + a * c

Using this property, we can write:

0.02 * 100 = 0.02 * (50 + 50)

Then, we can distribute the 0.02 to each term inside the parentheses:

0.02 * 100 = 0.02 * 50 + 0.02 * 50

Next, we can simplify each multiplication:

0.02 * 100 = 1 + 1

Finally, we can add the two numbers to get the result:

0.02 * 100 = 2

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Meredith will take 8 hours 45 minutes to file 70% of the 10 reports given.

The time taken by Meredith to file 70% of the 10 reports given, given that each report takes an average of 75 minutes is 6 hours, 30 minutes. Therefore, the correct option is B. 6 hours, 30 minutes. How long will it take Meredith to file 70% of 10 reports, given that each report takes an average of 75 minutes to file?

Here, the total number of reports = 10

Average time to file one report = 75 minutes

To find out the time taken to file 70% of 10 reports, we will need to multiply the average time taken to file one report by the number of reports to be filed, which is 7 in this case:

75 × 7 = 525 minutes

We have calculated the time it will take Meredith to file 7 reports. Now, to convert this time to hours and minutes, we divide the total minutes by 60 to get the hours and then find out the remainder for the minutes:

525 ÷ 60 = 8 hours 45 minutes

Therefore, Meredith will take 8 hours 45 minutes to file 70% of the 10 reports given.

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Answer:

Step-by-step explanation:

The percentage of 27 miles in 23 miles is 117.39%.

Percentage is a way of expressing a fraction or a portion of a whole number in relation to 100. It is denoted by the symbol "%". A percentage is calculated by dividing the part (numerator) by the whole (denominator) and multiplying the result by 100. A percentage can also be used to represent a rate or a percentage change over time.

In this case, we are asked to find what percentage of 23 miles is 27 miles. The part is 27 miles and the whole is 23 miles. Therefore, the percentage is:

% = part/whole = 27/23 x 100 = 117.39%

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The area of the composite figure with the given subshapes is 99.13 square feet

Given the following parameters:

The composite figure with the following shapes

Semi-circle with diameter 8 ft

Triangle with base of 8 ft and height of 6 feet

Rectangle of 10 by 5 feet

The area of the composite figure is the sum of the individual areas

So, we have

Area = 1/2 * π(8/2)² + 1/2 * 8 * 6 + 10 * 5

Evaluate

Area = 99.13

Hence, the area is 99.13 square feet

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There are, 6300 different possibilities for the researcher’s study.

Total number of class I railroads = 10Number of class I railroads selected = 3Total number of class II railroads = 6Number of class II railroads selected = 2Total number of class III railroads = 5Number of class III railroads selected = 1Number of different possibilities for selecting 3 class I railroads from 10 class I railroads = 10C3 = (10 x 9 x 8)/(3 x 2 x 1) = 120

Number of different possibilities for selecting 2 class II railroads from 6 class II railroads = 6C2 = (6 x 5)/(2 x 1) = 15Number of different possibilities for selecting 1 class III railroad from 5 class III railroads = 5C1 = 5Total number of different possibilities for selecting 3 class I railroads from 10 class I railroads, 2 class II railroads from 6 class II railroads, and 1 class III railroad from 5 class III railroads = 10C3 x 6C2 x 5C1= 120 x 15 x 5= 6300Therefore, there are 6300 different possibilities for the researcher’s study.

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Therefore, we estimate that about 7 sixth graders should be chosen to create a representative sample, given that 9 seventh graders were chosen.

Statistics is a branch of mathematics that involves the collection, analysis, interpretation, presentation, and organization of data. It provides methods for summarizing and describing data, making inferences about populations based on samples, and testing hypotheses. Statistics is used in many fields, such as business, economics, health care, social sciences, engineering, and more, to make decisions and draw conclusions based on data. Some common techniques used in statistics include descriptive statistics (e.g., mean, median, mode, standard deviation), probability theory, hypothesis testing, regression analysis, and sampling methods.

Here,

To estimate the number of sixth graders that should be chosen to create a representative sample, we can use the proportion of seventh graders in the sample and assume that the proportions of each grade in the sample should match the proportions in the population. We know that the sample contains 9 seventh graders, but we don't know the total size of the sample yet. We can use the following formula to estimate the sample size:

sample proportion = population proportion

where

sample proportion = number of seventh graders in the sample / total sample size

population proportion = number of seventh graders in the population / total population size

We can plug in the values we know:

9 / total sample size = 180 / (160 + 180 + 140)

Simplifying the right-hand side:

9 / total sample size = 0.4

Multiplying both sides by total sample size and simplifying:

total sample size = 9 / 0.4

total sample size ≈ 22.5

This means that the sample size should be about 22.5 students. Since we can't choose a fractional number of students, we should round up to the nearest whole number, giving a total sample size of 23 students.

To estimate the number of sixth graders in the sample, we can use the proportion of sixth graders in the population and assume that the proportion should be the same in the sample:

number of sixth graders in the sample / total sample size = 160 / (160 + 180 + 140)

Simplifying the right-hand side:

number of sixth graders in the sample / total sample size = 0.32

Multiplying both sides by total sample size and simplifying:

number of sixth graders in the sample = 0.32 * 23

number of sixth graders in the sample ≈ 7

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Complete question:

A population of middle school students contains 160 sixth graders, 180 seventh graders, and 140 eighth graders. Nine seventh graders were part of a random sample of the population chosen to participate in a survey. For the sample to accurately represent the population, about how many sixth graders should be chosen for each students favorite restaurant at my school?

According to the question the value of the 5 digit in 4,597 is 9.

Digit is a numerical symbol used to represent numbers in the decimal system. It is a single symbol used to represent a number ranging from 0 to 9. Digits are used to represent numbers in all forms of mathematical calculations and equations. The most common digits used in calculations are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit has an associated numerical value, which can be used to create larger numbers or to perform complex mathematical operations.

The answer is D) 9. The value of the 5 digit in 4,597 is 5. To find the 10 times its value, you must multiply 5 by 10, which equals 50. Therefore, the value of the 5 digit in 4,597 is 9.

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To perform matrix multiplication in MIPS, we can use nested loops to iterate over the rows and columns of the matrices.

The outer loop iterates over the rows of the first matrix, while the inner loop iterates over the columns of the second matrix. We then perform the dot product of the corresponding row and column, which involves multiplying the elements and summing the products.

To perform multiplication efficiently, we can use MIPS registers to store intermediate values and avoid accessing memory unnecessarily. We can also use assembly instructions like "lw" and "sw" to load and store values from memory, and "add" and "mul" to perform arithmetic operations.

In summary, matrix multiplication in MIPS involves nested loops, efficient use of registers and assembly instructions, and arithmetic operations.

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Answer:

yes

Step-by-step explanation:

To estimate the difference between the giving rates of school A and school B, we must first identify the missing value. It is not given in the statement.

1. However, it can be assumed that the giving rate of school B is 0%, as the difference between the two figures should be expressed as a percentage.

2. If we learn that schools A and B have identical student-to-faculty ratios, the answer to the previous question would not change. The student-to-faculty ratio has no bearing on the graduation and giving rates of the school.

3. The question does not provide a list of 123 schools to choose from. It is not possible to determine the school with the highest or lowest giving rate without this information.

4. To calculate whether the school's giving rate is greater or lesser than 8%, we must first estimate the value of the giving rate. The problem statement does not provide a clue to the school's giving rate.

Therefore, it is not possible to calculate the school's giving rate without more information.

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The sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$

The given problem can be solved using algebraic and geometric methods. We can use algebraic methods, such as the equations given in the problem, and we can use geometric methods by visualizing what the problem is asking. To start, let's translate the given problem into mathematical equations. Let $z_0$ be the original complex number. We want to rotate this point by 90 degrees counter-clockwise about some complex number $c$ to get $z_2$. Thus,$$z_2 = c + i(z_0 - c)$$$$=c + iz_0 - ic$$$$= (1-i)c + iz_0.$$We also know that this transformation will rotate the point $z_1 = (z_0 + z_2)/2$ by 45 degrees. Thus, using similar logic,$$z_1 = (1-i/2)c + iz_0/2.$$Now let's use the formula for rotating a point about the origin by $\theta$ degrees (where $\theta$ is measured in radians) to find a relationship between $z_1$ and $z_0$.$$z_1 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c + iz_0/2 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c = (e^{i\theta/2} - 1)z_0/2.$$We can solve for $c$ by dividing both sides by $1-i/2$.$$c = \frac{e^{i\theta/2} - 1}{1-i/2}\cdot\frac{z_0}{2}.$$We can now use the information given in the problem to solve for the sum of the real and imaginary parts of $c$. We know that rotating $z_0$ by 90 degrees counter-clockwise will result in the complex number $z_2$. Visually, this means that $c$ is located at the midpoint between $z_0$ and $z_2$ on the line that is perpendicular to the line segment connecting $z_0$ and $z_2$. We can use this geometric interpretation to solve for $c$. The midpoint of the line segment connecting $z_0$ and $z_2$ is$$\frac{z_0+z_2}{2} = c + i\frac{z_0-c}{2}.$$Solving for $c$, we get$$c = \frac{z_0+z_2}{2} - \frac{i}{2}(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0- (e^{i\theta/2} - 1)(z_0/2)/(1-i/2)).$$We can now find the real and imaginary parts of $c$ and add them together to get the desired answer. Let's first simplify the expression for $c$.$$2c = z_0+z_2 - i(z_0 - (e^{i\theta/2} - 1)\cdot(z_0/2)\cdot(1+i)/2)$$$$= z_0 + z_2 - i(z_0 - z_0(e^{i\theta/2} - 1)(1+i)/4)$$$$= z_0 + z_2 - i(z_0 - z_0e^{i\theta/2}(1+i)/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0(1-e^{i\theta/2})/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0/4(1-e^{i\theta/2} + 1 - i))$$$$= z_0 + z_2 - i(z_0/2(1-\cos(\theta/2) - i\sin(\theta/2)))$$$$= z_0 + z_2 - i(z_0(1-\cos(\theta/2)) + z_0\sin(\theta/2) - i(z_0\cos(\theta/2))/2.$$Now we can find the real and imaginary parts of $2c$ and divide by 2 to get the real and imaginary parts of $c$. We have$$\operatorname{Re}(2c) = \operatorname{Re}(z_0+z_2) - \operatorname{Im}(z_0)(1-\cos(\theta/2)) - \operatorname{Re}(z_0)\sin(\theta/2)$$$$\operatorname{Im}(2c) = \operatorname{Im}(z_0+z_2) - \operatorname{Re}(z_0)(1-\cos(\theta/2)) + \operatorname{Im}(z_0)\sin(\theta/2).$$Thus, the sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$

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Since 7 is the smallest integer that divides 173/168, we can conclude that the sum is not a prime number.

How to solve?

To add the fractions 4/7, 1/8, and 1/3, we need to find a common denominator.

The prime factorization of 7 is 7, the prime factorization of 8 is 2²3, and the prime factorization of 3 is 3. The least common multiple (LCM) of these three numbers is 7× 2²3× 3 = 168.

So, we can rewrite the fractions with the common denominator of 168:

4/7 = 96/168

1/8 = 21/168

1/3 = 56/168

Now we can add these fractions:

96/168 + 21/168 + 56/168 = 173/168

To check if this sum is a prime number, we can use trial division by checking all the integers between 2 and the√ of 173/168 (which is approximately 1.053):

2 does not divide 173/168

3 does not divide 173/168

4 does not divide 173/168

5 does not divide 173/168

6 does not divide 173/168

7 divides 173/168 (24 times)

8 does not divide 173/168

9 does not divide 173/168

...

Since 7 is the smallest integer that divides 173/168, we can conclude that the sum is not a prime number.

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Complete question:

What is the result of adding 4/7, 1/8, and 1/3, and is the sum a prime number?

The  [tex]$\vec{a}=\langle2,3\rangle$[/tex] and [tex]\vec{b}=\langle-4,\frac{8}{3}\rangle$[/tex]  are orthogonal.So, the value of k is 8/3.

A quantity or phenomenon with separate characteristics for both magnitude and direction is called a vector. The word can also refer to a quantity's mathematical or geometrical representation. Velocity, momentum, energy, electromagnetic fields, and weight are a few examples of vectors in nature.

Two vectors are orthogonal if and only if their dot product is zero. Therefore, we need to find the value of k such that the dot product of [tex]$\vec{a}$[/tex] and[tex]$\vec{b}$[/tex] [tex]$\vec{b}$[/tex][tex]\vec{b}[/tex] is zero.

The dot product of [tex]$\vec{a}$[/tex] and [tex]$\vec{b}$[/tex] is given by:

[tex]$$\vec{a} \cdot \vec{b} = (2)(-4) + (3)(k) = -8 + 3k$$[/tex]

For the vectors to be orthogonal, their dot product must be zero, so we set -8 + 3k = 0 and solve for k:

-8 + 3k = 0

3k = 8

[tex]$k = \frac{8}{3}$$[/tex]

Therefore, [tex]$\vec{a}=\langle2,3\rangle$[/tex] and [tex]\vec{b}=\langle-4,\frac{8}{3}\rangle$[/tex]  are orthogonal.So, the value of k is 8/3.

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