a quadrilateral must be a parallelogram if one pair of opposite sides is:

Answer:

Options A, C and D are true.

- The average daily revenue for days when the Red Sox do not play away is $1,768.32.

- The average daily revenue for days when the Red Sox play away is $2,264.57.

- On average, the bar’s revenue is $496.25 higher on days when the Red Sox play away than on days when they do not.

Step-by-step explanation:

The complete Question is presented in the attached image to this solution.

Analyzing the options at a time

A) The average daily revenue for days when the Red Sox do not play away is $1,768.32.

This option is true as 1768.32 is the intercept which is the average daily revenue when the Red Sox=0, that is, 0=no, when red sox do not play away.

B) The average daily revenue for days when the Red Sox play away is $1,768.32.

This is false because when the Red Sox play away, the value is 1 and the average revenue = 1768.32 + 496.25 = $2,264.57

C) The average daily revenue for days when the Red Sox play away is $2,264.57.

This is true. I just gave the explanation under option B.

D) The average daily revenue for days when the Red Sox do not play away is $1,272.07.

This is false. The explanation is under option A.

E) On average, the bar’s revenue is $496.25 higher on days when the Red Sox play away than on days when they do not.

This is true. It is evident from the table that the 0 and 1 coefficient is 496.25. This expresses the difference in average daily revenue when the Red Sox games are played away and when they are not.

Hope this Helps!!!

Answer:

65%

Step-by-step explanation:

Nadine mixes a juice solution that is made from 3 gallons of an 80% juice solution and 1 gallon of a 20% juice solution. What is the percent concentration of the final solution?

3 gallons of 80% juice solution contains this amount of juice:

80% * 3 gal = 0.8 * 3 gal = 2.4 gal

1 gallon of 20% juice solution contains this amount of juice:

20% * 1 gal = 0.2 * 1 gal = 0.2 gal

The total amount of juice in the final juice solution is

2.4 gal + 0.2 gal = 2.6 gal

The total amount of juice solution made is 3 gal + 1 gal = 4 gal

The 4 gal juice solution contains 2.6 gallons of juice.

2.6 gallons is what percent of 4 gallons?

2.6/4 * 100% = 0.65 * 100% = 65%

Answer: 65%

Answer:

65% i got the answer right on the question

Step-by-step explanation:

Answer:

4.11% probability that he has lung disease given that he does not smoke

Step-by-step explanation:

We use the conditional probability formula to solve this question. It is

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Does not smoke

Event B: Lung disease

Lung Disease/Nonsmoker 0.03

This means that [tex]P(A \cap B) = 0.03[/tex]

Lung Disease/Nonsmoker 0.03

No Lung Disease/Nonsmoker 0.7

This means that [tex]P(A) = 0.03 + 0.7 = 0.73[/tex]

What is the probability of the following event: He has lung disease given that he does not smoke?

[tex]P(B|A) = \frac{0.03}{0.73} = 0.0411[/tex]

4.11% probability that he has lung disease given that he does not smoke

Probabilities are used to determine the chances of an event.

The  probability that he has lung disease given that he does not smoke is 0.231

The required probability is calculated as:

[tex]\mathbf{P = \frac{P(Lung\ Disease\ and\ Non\ Smoker)}{P(Lung\ Disease)}}[/tex]

From the question, we have:

[tex]\mathbf{P(Lung\ Disease\ and\ Non\ Smoker) = 0.03}[/tex]

[tex]\mathbf{P(Lung\ Disease) = P(Has Lung Disease/smoker) + P(Lung Disease/Nonsmoker)}[/tex]

[tex]\mathbf{P(Lung\ Disease) = 0.1 + 0.03}[/tex]

[tex]\mathbf{P(Lung\ Disease) = 0.13}[/tex]

So, we have:

[tex]\mathbf{P = \frac{P(Lung\ Disease\ and\ Non\ Smoker)}{P(Lung\ Disease)}}[/tex]

[tex]\mathbf{P = \frac{0.03}{0.13}}[/tex]

[tex]\mathbf{P = 0.231}[/tex]

Hence, the  probability that he has lung disease given that he does not smoke is 0.231

Read more about probabilities at:

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Answer:

308.67 cm ^ 3 / week

Step-by-step explanation:

A cantaloupe is approximately a sphere, therefore its approximate volume would be:

V = (4/3) * pi * (r ^ 3)

They tell us that dr / dt 0.5 cm / week and the radius is 6.4 cm

if we derive the formula from the volume we are left with:

dV / dt = (4/3) * pi * d / dr [(r ^ 3)]

dV / dt = (4/3) * pi * 3 * (r ^ 2) * dr / dt

dV / dt = 4 * pi * (r ^ 2) * dr / dt

we replace all the values and we are left with:

dV / dt = 4 * 3.14 * (6.4 ^ 2) * 0.6

dV / dt = 308.67

Therefore the volume is changing at a rate of 308.67 cm ^ 3 / week

Answer:

0.1587

Step-by-step explanation:

According to the situation, the solution and the data provided is as follows

mean = 12.45 ounces

Standard deviation = 0.30 ounces

maximum = 12.75 ounces

More than ounces of soda = 12.75

Based on the above information, the probability is

[tex]Z=\frac{X-\mu }{\sigma } \\\\Z=\frac{12.75-12.45 }{0.30 } \\\\\Z=\frac{0.30 }{0.30 } \\\\Z= 1 \\\\P(X> 12.75)=1-P(X< 12.75) \\\\\P(X> 12.75)=1-P(Z< 1) \\\\[/tex]

As we know that

P(Z<1) = 0.8413

So,

P (X > 12.75) = 1 - 0.8413

= 0.1587

Answer:

x= 9 inches

Step-by-step explanation:

Hello

I can help you with this.

in this case, we have two similar triangles, let's see

Step 1

identify the rigth triangles.

1) the first triangle has these dimensions

hypotenuse( remember, the longest side)= unknown=H

adjacent side(the horizontal)=6 +x

opposite side(the vertical)=10

2) the second triangle has these dimensions

hypotenuse( remember, the longest side)= unknown=h

adjacent side(the horizontal)=6

opposite side(the vertical)=4.

As these triangles keep the same proportion and in both cases we know the length of the legs, we can establish a relationship

Step 2

establish a relationship

let's compare the opposite side and the adjacent side

triangle 1 (the bigger)

[tex]proportion= \frac{opposite\ side}{adjacent\ side}\\proportion= \frac{10}{6+x}[/tex]

Triangle 2

[tex]proportion= \frac{opposite\ side}{adjacent\ side}\\proportion= \frac{4}{6}\\proportion=\frac{2}{3}[/tex]

were the proportions are equal, so

[tex]\frac{10}{6+x}=\frac{2}{3}[/tex]

at this point, just isolate x to find its value

Step 3

isolate x

[tex]\frac{10}{6+x}=\frac{2}{3}\\multiply\ both\ sides\ by\ 3\\\frac{10*3}{6+x}=\frac{2*3}{3}\\\frac{30}{6+x} =2\\\\Multiply\ both\ sides\ by (6+x)\\\frac{30(6+X)}{6+x} =2(6+x)\\30=12+2x\\30-12=2x\\18=2x\\so\\x=\frac{18}{2} \\x=9[/tex]

remember the units of measure ( Inches)

x= 9 inches

I really hope it helps, have a nice day.

Answer:

42

Step-by-step explanation:

Number of students whose favorite class is Math:

60*0.85=51

Number of students whose favorite class is Science:

15% is equal to 0.15.

60*0.15=9

Subtract number of students who like science from number of students who like math.

51-9=42

42 more students in the survey prefer math over science.

Answer:

42

Step-by-step explanation:

85% math

15% science

Subtract

85-15 = 70

The difference is 70 %

70% of 60 students

.70 * 60 = 42

There is a 42 student difference

Answer:

-18 < -x-4 < -8

Step-by-step explanation:

We start with the initial range as:

4 < x < 14

we multiplicate the inequation by -1, as:

-4 > -x > -14

if we multiply by a negative number, we need to change the symbols < to >.

Then, we sum the number -4, as:

-4-4> -x-4 > -14-4

-8 > -x-4 > -18

Finally, the range for -x-4 is:

-18 < -x-4 < -8

Answer:

B

Step-by-step explanation:

It can be figured out by using graph transformations.

When when subtracting directly next to x, it shifts the graph to the left while doing the opposite when adding. Since the graph is to the left, we know it has to be A or B since those are subtracting by 5

Outside of the absolute value, when subtracting, it makes the graph move down. That means we are looking for a -4 which is found in  B

Answer:

See below for the table.

Step-by-step explanation:

The results for the thirty cases are listed below:

U U U I F O O U I F F O U I I F I I O U I F F U U I I O F U

The table summarizing the frequency distribution of the primary causes leading to student dropout is:

[tex]\left|\begin{array}{c|c}$Cause&$Frequency\\----------&----\\\\$Unexcused absences (U)&9\\$Illness (I)&9\\$Family problems (F)&7\\$Other causes (O)&5\\-----------&---\\$Total&30\end{array}\right|[/tex]

Answer:

The margin of error for the true mean number of hours a teenager spends on their phone is of 0.4 hours a day.

Step-by-step explanation:

We have the standard deviation of the saple, so we use the t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 50 - 1 = 49

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 49 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.95}{2} = 0.975[/tex]. So we have T = 2

The margin of error is:

[tex]M = T\frac{s}{\sqrt{n}} = 2\frac{1.3}{\sqrt{50}} = 0.4[/tex]

In which s is the standard deviation of the sample and n is the size of the sample.

The margin of error for the true mean number of hours a teenager spends on their phone is of 0.4 hours a day.

Answer:

The work will be "1909212.015 J". The further explanation is given below.

Step-by-step explanation:

The given values are:

Liquid's density

= 760 kg/m³

Height

= 3 meters

Gravity

g = 3.8 m/s²

Value of y is:

y = 5 log (x-2)

y = 0

y = 4

As we know,

⇒  [tex]\Delta V=\pi r^2 \Delta y[/tex]

⇒  [tex]y =5log(x-2)[/tex]

⇒  [tex]\frac{y}{5} =log (x-2)[/tex]

⇒  [tex]e^{\frac{y}{5}}=(x-2)[/tex]

⇒  [tex]x=e^{\frac{y}{5}}+2[/tex]

Now,

[tex]\Delta F=ma[/tex]

      [tex]=760 \pi (e^{\frac{y}{5}}+2)^2(9.8)\Delta y[/tex]

So that,

⇒  [tex]\Delta W = \Delta F.distance[/tex]

            [tex]=\Delta F(4-y)[/tex]

The required work will be:

⇒  [tex]W=760\times 9.8 \pi \int_{3}^{0}(e^{\frac{y}{5}}+2)^2 (\Delta-y)dy[/tex]

         [tex]=760\times 9.8 \pi[{-20(y-9)^{e^{\frac{y}{5}}}-2(y-8)y}][/tex]

         [tex]=760\times 9.8 \pi[81.455][/tex]

         [tex]=1909212.015 \ J[/tex]

Answer: (5, 42)

Step-by-step explanation:

22 + 4x= 42

if we test the options we will see this is the only one that works

42 - 22 = 20

4x = 20

x= 5

which is equal to X the number of weeks they have gotten the allowance.

Answer:

Step-by-step explanation:

a) If we assume the annual withdrawals are at the beginning of the year, we can use the formula for an annuity due to compute the necessary savings.

The principal P that must be invested at rate r for n annual withdrawals of amount A is ...

  P = A(1+r)(1 -(1 +r)^-n)/r

  P = $60,000(1.08)(1 -1.08^-20)/0.08 = $636,215.95

__

b) 20 withdrawals of $60,000 each total ...

  20×$60,000 = $1,200,000

__

c) The excess over the amount deposited is interest:

  $1,200,000 -636,215.95 = $563,784.05

Answer:

June Financial Reports

a) Cost of goods available for sale = $5,250

b) Moving-Average unit cost for:

i) June 1:  = $5

ii)        12:  = $4.75

iii)       15: = $4.75

iv)      23:  = $5.75

v)       27:  = $5.25

Step-by-step explanation:

a) Calculations:

Date     Explanation   Units     Unit Cost    Total Cost   Moving Average Cost

June 1 Inventory          150        $4                $600         $4.000

      12 Purchase         450          5               2,250            4.750

      15 Sale                 500          7                      3,500     4.750

     23 Purchase         400          6               2,400            5.750

     27 Sale                 420          8                      3,360     5.250

     30 Inventory           80

Cost of goods available for sale = Cost of Beginning Inventory + Cost of Purchases = $5,250 + ($600 + 2,250 + 2,400)

b) Moving-Average unit cost for:

i) June 1: Cost of goods available/Units of goods available = $5 ($600/150)

ii)        12: Cost  of goods available/Units of goods available = $4.75 ($600 + 2,250/600)

iii)       15: Cost  of goods available/Units of goods available = $4.75 ($475/100)

iv)      23: Cost of goods available/Units of goods available = $5.75 ($475 + 2,400)/500

v)       27: Cost of goods available/Units of goods available = $5.25 ($420/80)

Answer:

Step-by-step explanation:

(4+1)/(8-2)= 5/6

y + 1 = 5/6(x - 2)

y + 1 = 5/6x - 5/3

y + 3/3 = 5/6x - 5/3

y = 5/6x - 8/3

6(y = 5/6x - 8/3)

6y = 5x - 16

-5x + 6y = -16

Answer:

x= -14

Step-by-step explanation:

Please see attached picture for full solution.

Answer:

The Pythagorean Triplet that has 5 is 3-4-5

Step-by-step explanation:

We can prove this using Pythagorean Theorem: a² + b² = c²

3² + 4² = 5²

9 + 16 = 25

25 = 25

Answer:

n^2+3

Step-by-step explanation:

As we can see in the diagram

1st pattern consists from 1 square 1x1 +3 squares 1x1 each

2nd pattern consists from 1 square 2x2 +3 squares 1x1 each

3-rd pattern consists from 1 square 3x3 +3 squares 1x1 each

4-th pattern consists from 1 square 4x4 + 3 squares 1x1  each

We can to continue :

5-th pattern consists from 1 square 5x5+3 squares 1x1 each

So the nth    pattern consists from 1 square nxn+3 squares 1x1 each

Or total amount of 1x1 squares in nth pattern N= n^2+3

The expression for the numbers of squares in the nth pattern of the sequence is  [tex]n^{2} +3[/tex].

"The nth term of a sequence is a formula that enables us to find any term in the sequence. We can make a sequence using the nth term by substituting different values for the term number(n) into it."

From the given diagram

We can see that every term is made up with a square which side is n and three small square side is 1.

So,

1st term is 1 × 1 + 3 = 4

2nd term is 2 × 2 + 3 = 4

3rd term is  3 × 3 + 3 = 12

4th term is 4 × 4 + 3 = 19

So, nth term is [tex]n^{2} +3[/tex]

Hence, The expression for the numbers of squares in the nth pattern of the sequence is  [tex]n^{2} +3[/tex].

Learn more about nth term of a sequence here

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Answer:

AB=4

Step-by-step explanation:

The transitive property states if A=B and B+C than A+C  Next substitute

AB=x and x=4 so AB=4

Hope this helps, if it did, please give me brainliest, it helps me a lot. :)

Have a good day!

see if other people has already answered this question

Answer:

-1

1

1/2(1±i√3)

Step-by-step explanation:

1. x+1=0 ⇒ x= -1

2. x-1= 0 ⇒ x= 1

3. x^2-x+1=0

Answer:

the probability that the number of heads is between 40 and 60 is 0.9535

the probability that the number of heads is between 50 and 55 is 0.3557

Step-by-step explanation:

From the given information:

A fair coin is tossed 100 times.

Let consider n to be the number of time the coin is tossed, So n = 100 times

In a fair toss of a coin; the probability of getting a head P(Head) = 1/2 = 0.5

If we assume X to be the random variable which follows a binomial distribution of n and p; therefore , the mean and the standard deviation can be  calculated as follows:

Mean μ = n × p

Mean μ = 100 × 1/2

Mean μ = 100 × 0.5

Mean μ =  50

Standard deviation σ =  [tex]\sqrt{n \times p \times (1-p)}[/tex]

Standard deviation σ =  [tex]\sqrt{100 \times 0.5 \times (1-0.5)}[/tex]

Standard deviation σ = [tex]\sqrt{50 \times (0.5)}[/tex]

Standard deviation σ = [tex]\sqrt{25}[/tex]

Standard deviation σ = 5

Now, we've made it easier now to estimate  the probability that the number of heads is between 40 and 60 and the probability that the number is between 50 and 55.

To start with the probability that the number of heads is between 40 and 60 ; we have:

P(40 < X < 60) = P(X < 60)- P(X < 40)

Applying  the central limit theorem , for X is 40 which lies around 39.5 and 40.5  and X is 60 which is around 59.5 and 60.5 but the inequality signifies less than sign ;

Then

P(40 < X < 60) = P(X < 59.5) - P(X < 39.5)

[tex]P(40 < X < 60) = P( \dfrac{X - \mu}{\sigma}< \dfrac{59.5 - 50 }{5}) - P( \dfrac{X - \mu}{\sigma}< \dfrac{39.5 - 50 }{5})[/tex]

[tex]P(40 < X < 60) = P( Z < \dfrac{9.5 }{5}) - P( Z< \dfrac{-10.5 }{5})[/tex]

[tex]P(40 < X < 60) = P( Z <1.9}) - P( Z< -2.1)[/tex]

[tex]P(40 < X < 60) =0.9713 -0.0178[/tex]

[tex]P(40 < X < 60) =0.9535[/tex]

Therefore; the probability that the number of heads is between 40 and 60 is 0.9535

To estimate the probability that the number is between 50 and 55.

P(50 < X < 55) = P(X < 55)- P(X < 50)

Applying  the central limit theorem , for X is 50 which lies around 49.5 and 50.5  and X is 55 which is around 54.5 and 55.5 but the inequality signifies less than sign ;

Then

P(50 < X < 55) = P(X < 54.5) - P(X < 49.5)

[tex]P(50 < X < 55) = P( \dfrac{X - \mu}{\sigma}< \dfrac{54.5 - 50 }{5}) - P( \dfrac{X - \mu}{\sigma}< \dfrac{49.5 - 50 }{5})[/tex]

[tex]P(50 < X < 55) = P( Z < \dfrac{4.5 }{5}) - P( Z< \dfrac{-0.5 }{5})[/tex]

[tex]P(50 < X < 55) = P( Z <0.9}) - P( Z< -0.1)[/tex]

[tex]P(50 < X < 55) =0.8159 -0.4602[/tex]

[tex]P(50 < X < 55) =0.3557[/tex]

Therefore; the probability that the number of heads is between 50 and 55 is 0.3557

Answer:

Area: 7 units²

Perimeter: 14 units

Step-by-step explanation:

Area of each square:

1 unit × 1 unit = 1 unit²

There are 7 squares:

1 unit² × 7 (squares) = 7 units²

The area of the shape is 7 units².

The perimeter of the shape is the length of the outer sides.

1 + 1 + 1 + 1/2 + 1/2 + 1 + 1 + 1/2 + 1 + 1/2 + 1 + 1 + 1 + 1 + 1 + 1 =  14 units

Answer:

[tex]x=-1[/tex]

Step-by-step explanation:

Cross multiply.

10(4x + 1) = 15(2x)

Expand brackets.

40x + 10 = 30x

Add -30x and 10 on both sides.

40x - 30x = -10

10x = -10

Divide both sides by 10.

10/10x = -10/10

x = -1

Answer:

x = 465 km

y = 735 km

Step-by-step explanation:

Step 1: Write out equations

x + y = 1200

y = x + 270

Step 2: Find x using substitution

x + (x + 270) = 1200

2x + 270 = 1200

2x = 930

x = 465

Step 3: Plug in x to find y

y = 465 + 270

y = 735

Answer:

They traveled 780

Step-by-step explanation:

Got it right on the test

Answer:

The average revenue per passenger is about $13.85

μ = $13.85

The corresponding standard deviation is $14.51

σ = $14.51

The airline should expect revenue of $1,662 with a standard deviation of $14.51 for a flight of 120 passengers.

Expected revenue = $1,662 ± 14.51

Step-by-step explanation:

An airline charges the following baggage fees:

$25 for the first bag and $35 for the second

Suppose 51% of passengers have no checked luggage,

P(0) = 0.51

33% have one piece of checked luggage and 16% have two pieces.

P(1) = 0.33

P(2) = 0.16

a. Build a probability model, compute the average revenue per passenger, and compute the corresponding standard deviation.

The average revenue per passenger is given by

μ = 0×P(0) + 25×P(1) + 35×P(2)

μ = 0×0.51 + 25×0.33 + 35×0.16

μ = 0 + 8.25 + 5.6

μ = $13.85

Therefore, the average revenue per passenger is about $13.85

The corresponding standard deviation is given by

σ = √σ²

Where σ² is the variance and is given by

σ² = (0 - 13.85)²×0.51 + (25 - 13.85)²×0.33 + (35 - 13.85)²×0.16

σ² = 97.83 + 41.03 + 71.57

σ² = 210.43

So,

σ = √210.43

σ = $14.51

Therefore, the corresponding standard deviation is $14.51

b. About how much revenue should the airline expect for a flight of 120 passengers? With what standard deviation?

For 120 passengers,

Expected revenue = 120×$13.85

Expected revenue = $1,662 ± 14.51

Therefore, the airline should expect revenue of $1,662 with a standard deviation of $14.51 for a flight of 120 passengers.

Answer:

19.49% probability that no more than 16 of them are strikes

Step-by-step explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].

In this problem, we have that:

[tex]n = 30, p = 0.626[/tex]

So

[tex]\mu = E(X) = np = 30*0.626 = 18.78[/tex]

[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{30*0.626*(1-0.626)} = 2.65[/tex]

What is the probability that no more than 16 of them are strikes

Using continuity correction, this is [tex]P(X \leq 16 + 0.5) = P(X \leq 16.5)[/tex], which is the pvalue of Z when X = 16.5. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{16.5 - 18.78}{2.65}[/tex]

[tex]Z = -0.86[/tex]

[tex]Z = -0.86[/tex] has a pvalue of 0.1949

19.49% probability that no more than 16 of them are strikes

The question is incomplete. Here is the complete question.

In a certain online dating service, participants are given a 4-statement survey to determine their compatibility with other participants. Based on the questionnaire, each particpant is notified if they are compatible with another participant. Each question is multiple choice with the possible responses of "Agree" or "Disagree", and these are assigned the numbers 1 or -1, respectively. pArticipnat's responses to the survey are encoded as a vector in R4, where coordinates coreespond to their answers to each question. Here are the questions:

Question #1: I prefer outdoor activities, rather than indoor activities.

Question #2: I prefer going out to eat in restaurants, rahter than cooking at home.

Question #3: I prefer texting, rather than talking on the phone.

Question #4: I prefer living in a small town, rather than in a big city.

Here are the results for the questionaire, with a group of 5 participants:

                        Question1     Question2   Question3       Question4

participant A           1                      1                   -1                      -1

participant B           -1                     1                    1                       1

participant C           -1                    -1                    1                       1

participant D           1                     -1                   -1                      -1

participant E            1                    -1                    1                       1

Two participants are considered to be "compatible" with each other if the angle between their compatibility vectors is 60° or less. Participants are considered to be "incompatible" if the angle between their compatibility vectors is 120° or larger. For angles between 60° or 120°, pairs of participants are warned that they "may or may not be compatible".

(a) Which pairs of paricipants are compatible?

(b) Which pairs of participants are incompatible?

(c) How would this method of testing compatibility change if the questionnaire also allowed the answer "Neutral", which would correspond to the number zero in a participant's vector? Would this be better than only

allowing  "Agree" or "Disagree"? Could anything go wrong if we allowed "Neutral" as an answer?

Answer: (a) Participants A and D; B and C; C and E.

(b) Participants A and B; A and C; A and E; B and D; C and D;

Step-by-step explanation: Vectors in R4 are vectors in a 4 dimensional space and are determined by 4 numbers.

Vectors form angles between themselves and can be found by the following formula:

cos α = [tex]\frac{A.B}{||A||.||B||}[/tex]

which means that the cosine of the angle between two vectors is equal the dot product of these vectors divided by the product of their magnitude.

For the compatibility test, find the angle between vectors:

1) The vectors magnitude:

Magnitude of a vector is given by:

||x|| = [tex]\sqrt{x_{i}^{2} + x_{j}^{2}}[/tex]

Since all the vectors have value 1, they have the same magnitude:

||A|| = [tex]\sqrt{1^{2} + 1^{2} + (-1)^{2} + (-1)^{2}}[/tex] = 2

||A|| = ||B|| = ||C|| = ||D|| = ||E|| = 2

2) The dot product of vectors:

A·B = 1(-1) + 1(1) + (-1)1 + (-1)1 = -2

cos [tex]\alpha_{1}[/tex] = [tex]\frac{-2}{4}[/tex] = [tex]\frac{-1}{2}[/tex]

The angle that has cosine equal -1/2 is 120°, so incompatible

A·C = 1(-1) + 1(-1) + (-1)1 + (-1)1 = -4

cos [tex]\alpha _{2}[/tex] = -1

Angle = 180° --------> incompatible

A·D = 1(1) + 1(-1) + (-1)(-1) + (-1)(-1) = 2

cos [tex]\alpha _{3}[/tex] = 1/2

Angle = 60° ---------> COMPATIBLE

A·E = 1.1 + 1(-1) + (-1)1 + (-1)1 = -2

cos [tex]\alpha_{4}[/tex] = -1/2

Angle = 120° --------> incompatible

B·C = (-1)(-1) + 1(-1) + 1.1 + 1.1 = 2

cos [tex]\alpha _{5}[/tex] = 1/2

Angle = 60° -------------> COMPATIBLE

B·D = (-1)1 + 1(-1) + 1(-1) + 1(-1) = -4

cos[tex]\alpha_{6}[/tex] = -1

Angle = 180° -----------> incompatible

B·E = (-1)1 + 1(-1) + 1.1 + 1.1 = 0

cos[tex]\alpha _{7}[/tex] = 0

Angle = 90° -------------> may or may not

C·D = (-1)1 + (-1)(-1) + 1(-1) + 1(-1) = -2

cos[tex]\alpha_{8} =[/tex] -1/2

Angle = 120° ---------------> Incompatible

C·E = (-1)1 + (-1)(-1) + 1.1 + 1.1 = 2

cos [tex]\alpha_{9}[/tex] = 1/2

Angle = 60° ---------------> COMPATIBLE

D·E = 1.1 + (-1)(-1) + (-1)1 + (-1)1 = 0

cos [tex]\alpha_{10}[/tex] = 0

Angle = 90° -----------------> may or may not

(c) Adding zero (0) as a component of the vectors would have to change the method of compatibility because, to determine the angle, it is necessary to calculate the magnitude of a vector and if it is a zero vector, the magnitude is zero and there is no division by zero. So, unless the service change the method, adding zero is not a good option.

Answer:

slope = 2

Step-by-step explanation:

All four points lie on the same line.

Taking the first and fourth points, the slope can be found by the formula

slope, m = (y2-y1)/(x2-x1) = (3- -3) / (1- -2) = 6/3 =2

See attached diagram.

Answer:

2

Step-by-step explanation:

edge 2020