Suppose that mean retail price per gallon of regular grade gasoline is $3.49 with a standard deviation of $0.10 and that the retail price per gallon has a bell-shaped distribution. NOTE: Please use empirical rule approximations for this problem. a. What percentage of regular grade gasoline sells for between $3.29 and $3.69 per gallon (to 1 decimal)? % b. What percentage of regular grade gasoline sells for between $3.29 and $3.59 per gallon (to 1 decimal)? % c. What percentage of regular grade gasoline sells for more than $3.59 per gallon (to 1 decimal)? %

Answer:

area = 113.04 cm²

Step-by-step explanation:

radius = 12/2 = 6

area = πr² = 3.14(6²) = 113.04 cm²

After answering the presented question, we can conclude that We know from the graph of f(x) that the vertex is at (0, 2). This means that f(x) takes on its minimum value of 2 when x = 0.

A function's domain is the range of potential values that it can accept. These numbers indicate the x-values of a function like f. (x). The range of potential values that can be utilised with a function is known as its domain. The value that the method returns after inserting the x value belongs to this set. The formula for a function with y as the dependent variable and x as the independent variable is y = f. (x). When a single value of y can be successfully produced from a value of x, that value of x is said to be in the domain of the function.

To find the value of k, we can use the fact that g(x) = f(x + k).

Let's consider the point (-1, 5) on the graph of g(x). This point corresponds to the point (-1 - k, f(-1 - k)) on the graph of f(x), since g(x) = f(x + k).

Similarly, the point (0, 4) on the graph of g(x) corresponds to the point (0 - k, f(0 - k)) on the graph of f(x), and the point (1, 3) on the graph of g(x) corresponds to the point (1 - k, f(1 - k)) on the graph of f(x).

We know from the graph of f(x) that the vertex is at (0, 2). This means that f(x) takes on its minimum value of 2 when x = 0.

The complete question is as follows:

The graph shows f(x). The absolute value function g(x) is described in the table. The graph shows a v-shaped graph, labeled f of x, with a vertex at 0 comma 2, a point at negative 1 comma 3, and a point at 1 comma 3. x g(x) −1 5 0 4 1 3 2 2 3 3 If g(x) = f(x + k), what is the value of k? k = −2 k is equal to negative one half k is equal to one half k = 2

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To verify if each given function is a solution of the differential equation, we need to substitute the function into the differential equation and check if the equation holds true.

For the differential equation y" - 4y = 0:

a) Substitute y(t) = e^(2t):

y" = (e^(2t))'' = 4e^(2t)

4y = 4e^(2t)

The equation y" - 4y = 0 holds true.

b) Substitute y(t) = cosh(t):

y" = (cosh(t))'' = cosh(t)

4y = 4cosh(t)

The equation y" - 4y = 0 holds true.

For the differential equation y" + 2y - 3y = 0:

a) Substitute y(t) = e^(-3t):

y" = (-3e^(-3t))

2y = 2e^(-3t)

The equation y" + 2y - 3y = 0 holds true.

b) Substitute y(t) = e^(t):

y" = e^(t)

2y = 2e^(t)

The equation y" + 2y - 3y = 0 holds true.

For the differential equation 12y" + 5ty' + 4y = 0, t > 0:

a) Substitute y(t) = t^2:

y" = 2

y' = 0

5ty' = 0

12y" + 5ty' + 4y = 12(2) + 0 + 4(t^2) = 24 + 4t^2

The equation 12y" + 5ty' + 4y = 0 holds true.

b) Substitute y(t) = t^2:

y" = 2

y' = 0

5ty' = 0

12y" + 5ty' + 4y = 12(2) + 0 + 4(t^2) = 24 + 4t^2

The equation 12y" + 5ty' + 4y = 0 holds true.

For the differential equation y" + y = sec(t):

a) Substitute y(t) = sec(t):

y" = sec(t)tan(t)

y = sec(t)

The equation y" + y = sec(t) holds true.

b) Substitute y(t) = sec(t):

y" = sec(t)tan(t)

y = sec(t)

The equation y" + y = sec(t) holds true.

Therefore, each given function is a solution to its respective differential equation.

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Answer:

sin(B) = cos(90 - B).

Step-by-step explanation:

To answer this question, you must understand SOH CAH TOA.

SOH = Sine; Opposite divided by Hypotenuse

CAH = Cosine; Adjacent divided by Hypotenuse

TOA = Tangent; Opposite divided by Adjacent

I roughly drew a triangle for reference. Let's say we have a 3-4-5 triangle.

As you can see, sin(b) does not equal sin(a). To get the sine of an angle, you would do opposite over hypotenuse. For angle B, that would be 3/5, while for angle A, that would be 4/5.

As stated above, sin(B) is 3/5. Now, if you did cos(90 - B), it would be the same thing as cos(A). This is because the triangle is a right triangle. Since a triangle has 180 degrees, and one angle is a right triangle, the other two angles will add up to be 90 degrees. So, 90 - B = A. cos(A) is the same thing as adjacent over hypotenuse, which is 3/5. So, sin(B) = cos(90 - B) must be true.

Let's just check the others to make sure they are false.

cos(B) = 4/5.

sin(180 - B) is basically the same thing as sin(A + C), which is definitely NOT 4/5.

cos(B) = 4/5, which is NOT the same as A.

So, your answer is sin(B) = cos(90 - B).

Hope this helps!

Answer:

When both the conditions hold true, F is prime.

Step-by-step explanation:

AB, CD, and EF are two-digit numbers, where A, B, C, D, E and F represent distinct digits from 1 to 9.

  AB

+ CD

--------

  EF

1st condition, B and D are consecutive.

Adding B and D gives us F.

Possible values can be (F being the unit value after adding not considering the carry over):

B + D = F

1+2=3

2+3=5

3+4=7

4+5=9

5+6=1

6+7=3

7+8=5

8+9=7

Here F is not prime (because 9 is not prime).

Now, let us consider the 2nd condition as well.

i.e. C = 8

For the following

  AB

+ CD

--------

  EF

C is 8 then A must be 1 because any value other than 1 for A will make the sum of A and C greater than 9 and there will be a carry which is not the case here.

So, E = 8 + 1 = 9

Now, B  and D are consecutive and can not be 1, 8 or 9.

So, possible values are:

B + D = F

2 + 3 = 5

3 + 4 = 7

Here F is prime.

So, when both the conditions hold true, F is prime.

The answer in positive form using the law of exponents is 3/16.

The given expression is\(2^_(-4)\)\(- 2^_(-3)\)\(+ 2^_(-2).\)

To find the answer in positive form using the law of exponents, we need to take the reciprocal of each term and then simplify.

Let's begin:

\(2^_(-4)\ - 2^_(-3)\ + 2^_(-2)\)

= \((1/2^4) - (1/2^3) + (1/2^2)\)(using the law of exponents)

= (1/16) - (1/8) + (1/4) (taking the reciprocal of each term)

Now, we can simplify this expression by taking the common denominator. The common denominator is 16.

So, the given expression can be written as:

(1/16) - (2/16) + (4/16) = 3/16

Hence, the answer in positive form using the law of exponents is 3/16.

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Answer:

\(y>\frac{19}{3}\)

Step-by-step explanation:

\(3y-28>-9\\\\3y-28+28>-9+28\\\\3y>19\\\\\frac{3y>19}{3}\\\\ \boxed{y>\frac{19}{3}}\)

Hope this helps.

Answer:

angle b = 21 degrees

Step-by-step explanation:

102+57+ x =180 all triangles equal 180 degrees so set equal to 180

159+ x =180

x= 21

Answer:

The answer is 21 degrees.

Step-by-step explanation:

First of all, you should know that all the triangle sides add up to 180 degrees. Add up all the degrees, 102 + 57 + x.

102 + 57 + x = 180

159 + x = 180

x = 21

Answer:

Anna's wand can cast 70 spells

Step-by-step explanation:

Elsa's wand can cast 53 spells

her wand casts 17 fewer cells than her sister Anna's

Amount of spells cast by Anna's wand = ?

We write the question down in the form of an equation

\(x - 17 = 53\)

where \(x\) is the amount of spells Anna's wand can cast.

we then proceed to solve by collecting like terms to different sides of the equation. We'll have

\(x = 53 + 17\)

which leaves us with

\(x = 70\)

This means that Anna's wand can cast 70 spells.

Answer:

x=-(7 \(\pm i\sqrt(7)\))

Step-by-step explanation:

Answer:

-2

Step-by-step explanation:

the rate of change or slope (M) is -2

Hope this helps:P

Answer

iStep-by-step explanation:

hi.......

To find the z-score on a TI-84 calculator with the mean and standard deviation, use the "invNorm" function.

The z-score measures how many standard deviations a data point is away from the mean. On a TI-84 calculator, you can find the z-score using the "invNorm" function, which calculates the inverse of the cumulative distribution function (CDF) for the standard normal distribution.

To find the z-score, follow these steps:

1. Press the "2nd" button, followed by the "Vars" button to access the DISTR menu.

2. Scroll down or press "3" to select "invNorm(" and press "Enter".

3. Enter the desired area under the normal curve (probability), followed by a comma.

4. Enter the mean (μ) followed by another comma.

5. Enter the standard deviation (σ) and close the parentheses by pressing ")".

6. Press "Enter" to calculate the z-score.

The result displayed on the calculator will be the z-score corresponding to the given area, mean, and standard deviation. The z-score helps in understanding the relative position of a data point within a normal distribution by indicating the number of standard deviations it is above or below the mean.

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The correct answer is D) A critical aspect of designing a survey is ensuring that a sample which typifies the population is drawn.

Sample surveys are used to characterise an entire population. This is done by determining the characteristics of some portion of the population.

In order to achieve this, the survey needs to be carefully designed, taking into account the characteristics of the population, such as age, gender, education level, ethnicity, etc. This will help ensure that the sample group is representative of the entire population. Furthermore, it is important to draw a large enough sample size to ensure the results are accurate and generalizable.

Finally, the results should be compared to other independently derived estimates in order to ensure the survey results are accurate.

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we conclude the national average is greater than the alaska average  if the population standard deviation in each case is 3

Nationwide scores as  n1=1000, ` = 21.4 ,σ1 = 3

Carolina scores as n2=500,  ` = 21.2 ,σ2 = 3

So from here we can say

µ represents population mean.

represent population mean for nationwide scores.

represents population mean for Carolina scores.

is  =  and for   (claim)

Given Question is incomplete, Complete Question here,

a survey of 1000 students nationwide showed a mean act score of 21.4. a survey of 500 south Carolina scores showed a mean of 21.2. if the population standard deviation in each case is 3, can we conclude the national average is greater than the south Carolina average?

use α=0.01 and  for the nationwide mean act score.

State the hypotheses and identify the claim.

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Answer:

12.79/4.89 is $2.62 per gallon

2.62x<=21.44

Step-by-step explanation:

You could buy 8 because 21.44 divided by 2.62 is 8.1832...

When testing the claim that σ21≠σ22, the null hypothesis states that the variances of the two populations are equal, while the alternative hypothesis states that the variances are not equal. To test this claim, we use an F-test, which involves calculating the ratio of the variances of the two samples.

If the test statistic is f=1, this means that the ratio of the variances is equal to 1. This indicates that there is no significant difference between the variances of the two populations. In other words, we cannot reject the null hypothesis that the variances are equal.
However, it is important to note that a test statistic of f=1 does not necessarily mean that the two samples are identical. It is possible for two samples to have slightly different variances that still result in a test statistic of f=1. Additionally, a sample size that is too small or too large can affect the accuracy of the F-test.
Overall, if the test statistic is f=1 when testing the claim that σ21≠σ22, we can conclude that there is not enough evidence to support the alternative hypothesis that the variances are different.

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To solve a word problem using a one-step linear inequality, follow these steps: identify the given information, translate it into an inequality, isolate the variable, and write the solution. For example, if a store sells T-shirts for $15 each and you have at most $100 to spend, the number of T-shirts you can buy is represented by the inequality x ≤ 6, which means you can buy at most 6 T-shirts.

To solve a word problem using a one-step linear inequality, follow these steps:

For example, let's say you have the word problem: 'A store sells T-shirts for $15 each. You have at most $100 to spend. Write an inequality to represent the number of T-shirts you can buy.'

Step 1: Identify the given information. The store sells T-shirts for $15 each and you have at most $100 to spend.
Step 2: Translate the given information into an inequality. Let x represent the number of T-shirts. The inequality is 15x ≤ 100, since the total cost of the T-shirts should be at most $100.
Step 3: Isolate the variable. Divide both sides of the inequality by 15 to get x ≤ 6.67. Since you can't buy a fraction of a T-shirt, round down to the nearest whole number. The solution is x ≤ 6.
Step 4: Write the solution. The number of T-shirts you can buy is represented by the inequality x ≤ 6, which means you can buy at most 6 T-shirts.

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Answer:

1 = 175ft cubed

2 = 58m cubed

3 = 140m cubed

4 = 180in cubed

Step-by-step explanation:

You have to split these complex shapes into parts and find the volume of each part.  Then, add the parts' volumes together, which will be the total volume of these.  The equation is V=bhw.  V means volume, b means base, h means height, and w means width.  Hope this helps!

AB - DE, BC - EF, AC – DF, and 2A - 2D, and C - ZF lists the other corresponding sides and angles.

The two triangles below are similar because MZA = m2E and m2B = m_F.

Option that lists the other corresponding sides and angles is AB - DE, BC - EF, AC – DF, and 2A - 2D, and C - ZF. To justify why two triangles are similar, we have to state that they have the same shape, but not necessarily the same size. It is important to remember that corresponding angles are equal and that corresponding sides are in proportion.

Explanation:The two triangles below are similar because of the following reasons:MZA = m2E: These are corresponding angles.m2B = m_F:

These are corresponding angles. Therefore, the two triangles are similar. Corresponding sides and angles are: AB - DE: These are corresponding sides. BC - EF:

These are corresponding sides.AC – DF: These are corresponding sides.2A - 2D: These are corresponding angles. C - ZF: These are corresponding sides.

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To find the exact value of P(X ≤ 3) for a binomial distribution with n = 20 and p = 0.1, we can use the table of binomial probabilities. Answer : P(X ≤ 3) using the table of binomial probabilities.

The probability mass function (PMF) for a binomial distribution is given by the formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where C(n, k) represents the binomial coefficient.

To find P(X ≤ 3), we need to calculate the probabilities for X = 0, 1, 2, and 3 and sum them up.

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Using the binomial probability formula, we can calculate each term:

P(X = 0) = C(20, 0) * (0.1)^0 * (1 - 0.1)^(20 - 0)

        = 1 * 1 * 0.9^20

P(X = 1) = C(20, 1) * (0.1)^1 * (1 - 0.1)^(20 - 1)

        = 20 * 0.1 * 0.9^19

P(X = 2) = C(20, 2) * (0.1)^2 * (1 - 0.1)^(20 - 2)

        = 190 * 0.01 * 0.9^18

P(X = 3) = C(20, 3) * (0.1)^3 * (1 - 0.1)^(20 - 3)

        = 1140 * 0.001 * 0.9^17

Now, we can calculate the sum:

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

        = 0.9^20 + 20 * 0.1 * 0.9^19 + 190 * 0.01 * 0.9^18 + 1140 * 0.001 * 0.9^17

Evaluating this expression will give you the exact value of P(X ≤ 3) using the table of binomial probabilities.

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The probability that the sample mean will be within plus minus 2 of the population mean if the sample size is n = 100 between z-scores of 0 and 2.5 using a z-table.

The standard deviation of the sample distribution, commonly known as the standard error, can be computed using the formula given that the population mean is 100 and the standard deviation is 16:

Standard Error = Standard Deviation / sqrt(sample size)

Let's determine the likelihoods for sample sizes of n = 100 and n = 400:

For n = 100:

Standard Error = 16 / sqrt(100) = 16 / 10 = 1.6

We can determine the z-scores for the upper and lower boundaries to establish the likelihood that the sample mean will be within plus or minus 2 of the population mean:

Lower Bound z-score = (Sample Mean - Population Mean) / Standard Error

Lower Bound z-score = (100 - 100) / 1.6

Lower Bound z-score = 0

Upper Bound z-score = (Sample Mean - Population Mean) / Standard Error

Upper Bound z-score = (104 - 100) / 1.6

Upper Bound z-score = 4 / 1.6

Upper Bound z-score = 2.5

We can calculate the region under the normal distribution curve between z-scores of 0 and 2.5 using a z-table or statistical software. This shows the likelihood that the sample mean will be within +/- 2 standard deviations of the population mean.

For n = 400:

Standard Error = 16/√400

Standard Error = 16/20

Standard Error = 0.8

We determine the z-scores by following the same procedure as above:

Lower Bound z-score = (Sample Mean - Population Mean) / Standard Error

Lower Bound z-score = (100 - 100) / 0.8

Lower Bound z-score = 0

Upper Bound z-score = (Sample Mean - Population Mean) / Standard Error

Upper Bound z-score = (104 - 100) / 0.8

Upper Bound z-score = 4 / 0.8

Upper Bound z-score = 5

Once more, we may determine the region under the normal distribution curve between z-scores of 0 and 5 using a z-table or statistical software.

A larger sample size, like n = 400, has the benefit of a lower standard error. The sampling distribution of the sample mean will be more constrained and more closely resemble the population mean if the standard error is less.

As a result, there is a larger likelihood that the sample mean will be within +/- 2 of the population mean. In other words, the estimate of the population mean gets more accurate and dependable as the sample size grows.

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Answer:

False

Step-by-step explanation:

\(2\sin x \cos x =\sin 2x \implies 4 \sin x \cos x=2\sin 2x \\ \\ \therefore \int 4\sin x \cos x dx=2\int \sin 2x dx \\ \\ 2\int \sin 2x dx=\frac{2}{2}(-\cos 2x)+C=-\cos 2x+C\)

A and B have the same elements, the measure of AUB will be equal to the measure of B.

To show that if m*(A) = 0, then m*(AUB) = m*(B), we need to prove the following:
1. If m*(A) = 0, then A is a null set.
2. If A is a null set, then AUB = B.
3. If AUB = B, then m*(AUB) = m*(B).
Let's break down each step:
1. If m*(A) = 0, then A is a null set:
  - By definition, a null set has a measure of 0.
  - Since m*(A) = 0, it implies that A has no elements or its measure is 0.
  - Therefore, A is a null set.
2. If A is a null set, then AUB = B:
  - Since A is a null set, it means that it has no elements or its measure is 0.
  - In set theory, the union of a null set (A) with any set (B) results in B.
  - Therefore, AUB = B.
3. If AUB = B, then m*(AUB) = m*(B):
  - Since AUB = B, it implies that both sets have the same elements.
  - The measure of a set is defined as the sum of the measures of its individual elements.
  - Since A and B have the same elements, the measure of AUB will be equal to the measure of B.
  - Therefore, m*(AUB) = m*(B).
By proving these three steps, we have shown that if m*(A) = 0, then m*(AUB) = m*(B).

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Answer:

the correct answer would be d.

Step-by-step explanation:

These lines should be straight line, since they have no slope.

Answer:

D.

Step-by-step explanation:

Since we see that any x-value for the piecewise function will either be always y-value 4 or 2, we know that we are looking for a graph with all horizontal lines. Therefore, our answer is D.

Answer:

1,578

Step-by-step explanation:

I took the test

Since the provided equation is inconsistent, it cannot intersect.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign. A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.

Here,

we can see that the second set of equation,

4x+2y=12

20x+10y=30

4/20=2/10≠12/30

1/5=1/5≠2/5

The given equation is inconsistent so it does not intersect.

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a) The length of the projection of vector B onto vector A is sqrt(7/2).

b) The length of the projection of vector A onto vector B is sqrt(455/361).

c) The scalar projection of vector B onto vector A is (7 / sqrt(14)).

d) The vector projection of vector A onto vector B is (35/38, -21/38, 7/19).

e) The projection of vector B onto vector A is (3/2, 1, -1/2).

a. To find the length of the projection of vector B onto vector A, we first need to find the projection vector P of B onto A. The projection vector P is given by:

P = (B dot A / \(||A||^{2}\) ) * A

where "dot" represents the dot product of two vectors and ||A|| is the magnitude of vector A.

The dot product of vectors A and B is given by:

A dot B = (3 * 5) + (2 * -3) + (-1 * 2) = 15 - 6 - 2 = 7

The magnitude of vector A is:

||A|| = \(\sqrt{3^{2}+2^{2}+(-1)^{2} }\) = \(\sqrt{14}\)

Substituting these values into the formula for the projection vector P, we get:

P = (7 / 14) * (3, 2, -1) = (3/2, 1, -1/2)

The length of the projection of vector B onto vector A is simply the magnitude of the projection vector P. That is:

||P|| = \(\sqrt{(3/2)^{2}+1^{2}+(-1/2)^{2} }\) = \(\sqrt{7/2}\)

b. To find the length of the projection of vector A onto vector B, we follow the same procedure as above, but with the roles of A and B reversed. That is, we need to find the projection vector Q of A onto B, which is given by:

Q = (A dot B / \(||B||^{2}\)) * B

The dot product of vectors A and B is the same as above, which is 7. The magnitude of vector B is:

||B|| = \(\sqrt{5^{2}+(-3)^{2}+2^{2} }\) = \(\sqrt{38}\)

Substituting these values into the formula for the projection vector Q, we get:

Q = (7 / 38) * (5, -3, 2) = (35/38, -21/38, 7/19)

The length of the projection of vector A onto vector B is the magnitude of the projection vector Q, which is:

||Q|| = \(\sqrt{(35/38)^{2}+(-21/38)^{2}+(7/19)^{2} }\) = \(\sqrt{455/361}\)

c. The scalar projection of vector B onto vector A is given by:

B scalar projection A = (B dot A) / ||A||

where "dot" represents the dot product of two vectors and ||A|| is the magnitude of vector A.

The dot product of vectors A and B is given by:

A dot B = (3 * 5) + (2 * -3) + (-1 * 2) = 15 - 6 - 2 = 7

The magnitude of vector A is:

||A|| = \(\sqrt{3^{2}+2^{2}+(-1)^{2} }\)=    \(\sqrt{14}\)

Substituting these values into the formula for the scalar projection, we get:

B scalar projection A = (7 /   )

d. The vector projection of vector A onto vector B is given by:

A vector projection B = (A dot B /  \(||B||^{2}\)) * B

The dot product of vectors A and B is given by:

A dot B = (3 * 5) + (2 * -3) + (-1 * 2) = 15 - 6 - 2 = 7

The magnitude of vector B is:

||B|| =  \(\sqrt{5^{2}+(-3)^{2}+2^{2} }\) = \(\sqrt{38}\)

Substituting these values into the formula for the vector projection, we get:

A vector projection B = (7 / 38) * (5, -3, 2) = (35/38, -21/38, 7/19)

e. The projection of vector B onto vector A is given by:

B projection A = (B dot A / \(||A||^{2}\) ) * A

The dot product of vectors A and B is given by:

A dot B = (3 * 5) + (2 * -3) + (-1 * 2) = 15 - 6 - 2 = 7

The magnitude of vector A is:

||A|| =  \(\sqrt{3^{2}+2^{2}+(-1)^{2} }\)=    \(\sqrt{14}\)

Substituting these values into the formula for the projection, we get:

B projection A = (7 / 14) * (3, 2, -1) = (3/2, 1, -1/2)

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