The University Grille on Commonwealth Avenue just released the findings from a three year-study of students’ salad orders to determine the popularity of Caesar and Ranch dressing. In this study, the ordering habits of 3000 students who have ordered salads were analyzed. 185 of these students never ordered any dressing on their salads. 2100 of the students ordered Caesar dressing, but never ordered Ranch. What is the probability that a randomly-selected student from this survey ordered Ranch?

Answer:C D F

Step-by-step explanation:

Answer:

Below

Step-by-step explanation:

there’s no answer choices, can help more if you provide..

But based off my common knowledge

-2 x - 2 = 4

-2 + -2 = 4

that’s the only one that multiplies to equal 4 and add to equal -4. If that’s what you are asking, then your answer is -2 x -2 and -2 + -2.

The area of triangle RST is 27 square units.

Option A is the correct answer.

We have,

To find the area of the triangle RST, we can use the formula:

Area = 1/2 x base x height

where the base is the distance between any two of the vertices, and the height is the perpendicular distance from the third vertex to the line containing the base.

Let's take RS as the base.

The distance between R and S is 2 + 7 = 9 units.

To find the height, we need to determine the equation of the line containing the base RS, and then find the distance from vertex T to this line.

The slope of the line RS is:

(y2 - y1)/(x2 - x1) = (-7 - 2) / (-9-(-9)) = -9/0,

which is undefined.

This means that the line is vertical and has the equation x = -9.

The perpendicular distance from T to the line x = -9 is simply the horizontal distance between T and the point (-9,-7), which is 6 units.

Therefore,

The area of triangle RST is:

Area = 1/2 x base x height = 1/2 x 9 x 6 = 27 square units.

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The value of probability to get the first letter will be always be, 1 / 26.

Given that;

A computer randomly selects a letter from the alphabet.

Now, The probability the computer produces the first letter of your first name :

Here, the required outcome is getting the first letter of your first name.

Probability = No. of required outcomes / total no. of outcomes.

For example, The name Alex Davis has the first letter of the fist name as alphabet 'A'.

Hence, Probability = 1 / 26

Similarly, for any first name there is going to be any one alphabet from the 26 alphabets, thus the probability to get the first letter will be always be, 1 / 26.

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Answer:

x=5 y=1.5

Step-by-step explanation:

Answer:

x=5 y=1.5

Step-by-step explanation:

i used a graphing calculator

We can conclude with 95% confidence that the average hair length of University of Maryland students today is significantly longer than the US average 20 years ago.

a. The question being asked is whether the average hair length of University of Maryland students today is longer than the US average 20 years ago, with a direction of interest being "longer than".

b. Null hypothesis: The average hair length of University of Maryland students today is not significantly different from the US average 20 years ago (µ = 2.7 inches).

Alternative hypothesis: The average hair length of University of Maryland students today is significantly greater than the US average 20 years ago (µ > 2.7 inches).

Symbolically, H0: µ = 2.7 and Ha: µ > 2.7

c. The equation to analyze these data is: z = (x - µ) / (σ / √n), where x is the sample mean (3.7 inches), µ is the hypothesized population mean (2.7 inches), σ is the population standard deviation (0.5 inches), and n is the sample size (40).

Substituting the values, we get:

z = (3.7 - 2.7) / (0.5 / √40) = 4.47

d. The calculated z-value of 4.47 is much greater than the critical value of 1.96 for a two-tailed test or 1.65 for a one-tailed test at the 5% significance level. Therefore, we reject the null hypothesis and conclude that the average hair length of University of Maryland students today is significantly greater than the US average 20 years ago.

e. Based on the calculated z-value, the rejection of the null hypothesis, and the chosen level of significance, we can conclude with 95% confidence that the average hair length of University of Maryland students today is significantly longer than the US average 20 years ago.

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If pmf of a random variable is given by 4 f(X=n)= n(n+1)(n+2)

the answer to part (a) is:

Ë F(X = n) = 9n(n+1)

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

a. To show that the cumulative distribution function (CDF) F(X) satisfies Ë F(X = n)=1, we need to show that the sum of the probabilities of all possible values of X is equal to 1.

The probability mass function (PMF) is given by:

f(X=n) = 4n(n+1)(n+2)

The CDF is defined as:

F(X=n) = P(X ≤ n)

We can calculate F(X=n) by summing up the probabilities of all values less than or equal to n:

F(X=n) = Σ f(X=i), for i = 0 to n

Substituting the given PMF:

F(X=n) = Σ 4i(i+1)(i+2), for i = 0 to n

Expanding the sum:

F(X=n) = 4(0)(1)(2) + 4(1)(2)(3) + 4(2)(3)(4) + ... + 4n(n+1)(n+2)

F(X=n) = 4 [ (0)(1)(2) + (1)(2)(3) + (2)(3)(4) + ... + (n)(n+1)(n+2) ]

Notice that the sum inside the brackets is a telescoping sum, which can be simplified as:

[(k-1)k(k+1) - (k-2)(k-1)k] = 3k(k-1)

Thus,

F(X=n) = 4 [ 3(0)(-1) + 3(1)(0) + 3(2)(1) + ... + 3(n)(n-1) ]

F(X=n) = 4 [ 3(0 + 1 + 2 + ... + (n-1)) ]

F(X=n) = 4 [ 3(n-1)n/2 ]

F(X=n) = 6n² - 6n

Therefore, Ë F(X = n) is given by:

Ë F(X = n) = Σ F(X=n) * P(X=n), for all n

Substituting the given PMF:

Ë F(X = n) = Σ [ 6n² - 6n ] * 4n(n+1)(n+2), for all n

Expanding the sum and simplifying:

Ë F(X = n) = 24 [ (n+2)(n+1)n(n-1)/4 - (n+1)n(n-1)(n-2)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2)/4 - (n-2)(n-1)n(n+1)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1) ] / 4

Ë F(X = n) = 6n(n+1)(n+2) - 6n(n+1)(n-1) / 4

Ë F(X = n) = 6n(n+1)[ (n+2) - (n-1) ] / 4

Ë F(X = n) = 6n(n+1) * 3 / 4

Ë F(X = n) = 9n(n+1)/2

Substituting n = 0 and n = ∞ to get the bounds of the sum, we get:

E[X] = 2(0)(5(0)+8) / 3 + 2(∞)(∞+1)(5(∞)+8) / 3

Since the second term diverges to infinity, we can conclude that the expected value of X does not exist (i.e., it is undefined).

Therefore, the answer to part (a) is:

Ë F(X = n) = 9n(n+1)/

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If pmf of a random variable is given by 4 f(X=n)= n(n+1)(n+2)

the answer to part (a) is:

Ë F(X = n) = 9n(n+1)

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

a. To show that the cumulative distribution function (CDF) F(X) satisfies Ë F(X = n)=1, we need to show that the sum of the probabilities of all possible values of X is equal to 1.

The probability mass function (PMF) is given by:

f(X=n) = 4n(n+1)(n+2)

The CDF is defined as:

F(X=n) = P(X ≤ n)

We can calculate F(X=n) by summing up the probabilities of all values less than or equal to n:

F(X=n) = Σ f(X=i), for i = 0 to n

Substituting the given PMF:

F(X=n) = Σ 4i(i+1)(i+2), for i = 0 to n

Expanding the sum:

F(X=n) = 4(0)(1)(2) + 4(1)(2)(3) + 4(2)(3)(4) + ... + 4n(n+1)(n+2)

F(X=n) = 4 [ (0)(1)(2) + (1)(2)(3) + (2)(3)(4) + ... + (n)(n+1)(n+2) ]

Notice that the sum inside the brackets is a telescoping sum, which can be simplified as:

[(k-1)k(k+1) - (k-2)(k-1)k] = 3k(k-1)

Thus,

F(X=n) = 4 [ 3(0)(-1) + 3(1)(0) + 3(2)(1) + ... + 3(n)(n-1) ]

F(X=n) = 4 [ 3(0 + 1 + 2 + ... + (n-1)) ]

F(X=n) = 4 [ 3(n-1)n/2 ]

F(X=n) = 6n² - 6n

Therefore, Ë F(X = n) is given by:

Ë F(X = n) = Σ F(X=n) * P(X=n), for all n

Substituting the given PMF:

Ë F(X = n) = Σ [ 6n² - 6n ] * 4n(n+1)(n+2), for all n

Expanding the sum and simplifying:

Ë F(X = n) = 24 [ (n+2)(n+1)n(n-1)/4 - (n+1)n(n-1)(n-2)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2)/4 - (n-2)(n-1)n(n+1)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1) ] / 4

Ë F(X = n) = 6n(n+1)(n+2) - 6n(n+1)(n-1) / 4

Ë F(X = n) = 6n(n+1)[ (n+2) - (n-1) ] / 4

Ë F(X = n) = 6n(n+1) * 3 / 4

Ë F(X = n) = 9n(n+1)/2

Substituting n = 0 and n = ∞ to get the bounds of the sum, we get:

E[X] = 2(0)(5(0)+8) / 3 + 2(∞)(∞+1)(5(∞)+8) / 3

Since the second term diverges to infinity, we can conclude that the expected value of X does not exist (i.e., it is undefined).

Therefore, the answer to part (a) is:

Ë F(X = n) = 9n(n+1)/

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The action of the annihilation operator â on an eigenket [n) is given by:

â[n) = √n [n-1)

Similarly, the action of the creation operator â+ on an eigenket [n) is given by:

â+[n) = √(n+1) [n+1)

Using these relations, we can express the operator (n + apn) in terms of the creation and annihilation operators as:

n + apn = â+n â + a â

Similarly, we can express the operator (np¹ + Bpan) as:

np¹ + Bpan = â+n â + B â

Now, we can use the relations between the operators and the eigenkets to calculate the matrix elements of these operators. Specifically, we need to calculate the inner products  and , where |n> and |m> are arbitrary eigenkets.

Using the relations between the operators and the eigenkets, we can express these matrix elements as:

= √(n+1)  + a√n

= √(n+1)  + B

Here, we have used the fact that the eigenkets [n+1) and [n-1) are orthogonal to [n), and that the inner product  is zero unless m = n.

Therefore, we have calculated the matrix elements of (n + apn) and (np¹ + Bpan) using the creation and annihilation operators â+ and â, and the eigenkets [n) and [n+1).

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y' = f'(x) = (4 - C1e^x)/(1 - x)^2

Differentiate the given equation with respect to x:

x' - xy + y = 4

Differentiating both sides with respect to x using the product rule, we get:

x'' - y - xy' + y' = 0

Simplifying, we get:

x'' + (y - 1)y' = 0

Now, since y = f(x), we can write y' as f'(x). Substituting in the above equation, we get:

x'' + (f(x) - 1)f'(x) = 0

This is a first-order linear differential equation, which we can solve using an integrating factor. The integrating factor is e^(-x). Multiplying both sides by e^(-x), we get:

e^(-x)x'' + e^(-x)(f(x) - 1)f'(x) = 0

Using the product rule on the left-hand side, we can rewrite this as:

(e^(-x)x')' + e^(-x)f'(x) - e^(-x)f'(x) = 0

Simplifying, we get:

(e^(-x)x')' = 0

Integrating both sides with respect to x, we get:

e^(-x)x' = C1

where C1 is a constant of integration. Solving for x', we get:

x' = C1e^x

Substituting this into the original equation, we get:

C1e^x - xy + y = 4

Solving for y, we get:

y = (C1e^x + 4)/(1 - x)

Now, since y = f(x), we can write:

f(x) = (C1e^x + 4)/(1 - x)

To find y', we differentiate this expression with respect to x:

f'(x) = [(C1e^x)(-1) - 4(-1)]/(1 - x)^2

Simplifying, we get:

f'(x) = (4 - C1e^x)/(1 - x)^2

Now, substituting this expression for f'(x) into the earlier equation, we get:

x'' + (f(x) - 1)f'(x) = 0

x'' + [(C1e^x + 4)/(1 - x) - 1][(4 - C1e^x)/(1 - x)^2] = 0

Simplifying, we get:

x'' - (3C1e^x + 4)/(1 - x)^2 = 0

Thus, the expression for y' is:

y' = f'(x) = (4 - C1e^x)/(1 - x)^2

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Answer: yello

Step-by-step explanation:

There are 15,504 ways to randomly select a team of 5 people from a group of 20 people

If there are 20 people trying out for a team, the number of ways to select a team of n people can be calculated using the formula for combinations, which is:

C(20, n) = 20! / (n! * (20 - n)!)

where C(20, n) represents the number of ways to select n people from a group of 20 people.

For example, if we want to select a team of 5 people, we can plug in n = 5 and calculate:

C(20, 5) = 20! / (5! * (20 - 5)!) = 15,504

Therefore, there are 15,504 ways to randomly select a team of 5 people from a group of 20 people. Similarly, we can calculate the number of ways to select teams of different sizes by plugging different values of n into the formula for combinations.

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The percent decrease in the travel time was 60 %.

We will use unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

We are given that Korra takes 27 minutes to walk to work. After getting a new job, Korra takes 16.27 minutes to walk to work.

Time taken to walk to home = 27 minutes

Time taken to walk to work = 16.27 minutes

Therefore,

The percent decrease in the travel time was;

16.27 / 27 x 100

= 0.60 x 100

= 60 %

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Answer:

Step-by-step explanation:

1. Get the amount of rocks in tons that the company used in the second month. To do this, you must subtract the amount they used in the first month by the total amount used.

Rocks used in first month: 3 1/2 tons

Total amount used : 7 1/4 tons

7 1/4 tons - 3 1/2 tons

To subtract, convert into improper fractions

((7*4)+1)/4 tons - ((3*2)+1)/2 tons

29/4 tons - 7/2 tons

then convert the denominator into the same number. To do this just multiply 2/2 onto the second fraction

7/2 * 2/2 = 14/4

subtract

29/4 - 14/4 = 15/4 tons used on the second project.

2. Now that we know that 15/4 or 3 3/4 tons where used on the second month we just simply divide by the 5 projects that used the same amount of rocks.

To divide, we can just multiply 5 to the denominator of our improper fraction

15/4 * 1/5 = 15/20

Then we simplify

3/4 tons of rock were used for each project.

Answer:

Step-by-step explanation: RADIUS

Answer:

$630

Step-by-step explanation:

420/2 = half months rent ($210)

420+210 = 630

The security deposit is $630.

Once you have obtained the area, you can round it to four decimal places as needed.

To find the area of a shaded region under the standard normal curve, you can use a standard normal distribution table or a statistical software package, such as Excel or R.

If using a standard normal distribution table, you need to first determine the z-scores that correspond to the boundaries of the shaded region. Then, you look up the corresponding probabilities in the standard normal distribution table and subtract them to find the area of the shaded region.

If using a statistical software package, you can use the functions or commands that calculate the area under the standard normal curve between the boundaries of the shaded region.

Once you have obtained the area, you can round it to four decimal places as needed.

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If the demand for tomato juice falls, it means that consumers are buying less of it at any given price. This will result in a leftward shift in the demand curve, showing a decrease in quantity demanded at each price level.

As a result, the equilibrium price of tomato juice will decrease, and the equilibrium quantity of tomato juice sold in the market will also decrease. This shift in demand will also affect the producers of tomato juice, who may need to adjust their prices and output levels to match the reduced demand. Overall, a decrease in demand for tomato juice will lead to lower prices and lower quantities sold in the market.

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The process of using sample statistics to draw conclusions about population parameters is called statistical inference. In statistical inference, we use the information obtained from a sample to make inferences about the characteristics of a larger population.

The sample statistics provide an estimate of the corresponding population parameters, and the goal is to make the most accurate inference possible.
The significance level, also known as alpha, is a pre-determined threshold that is used to determine the level of evidence required to reject the null hypothesis. This threshold is typically set at 0.05 or 0.01, depending on the level of certainty required.

In summary, statistical inference involves using sample statistics to make inferences about population parameters, and the significance level is a critical component of this process as it helps to determine the level of evidence required to reject the null hypothesis.
Statistical inference involves making choices about the sampling method and using collected data from a sample to make conclusions about a larger population. Sample statistics are calculations derived from a subset of the population, while population parameters are the true values for the entire population.

The significance level, typically denoted by α (alpha), is a predetermined threshold used to determine if a result is statistically significant. In hypothesis testing, if the calculated probability (p-value) is less than the significance level, we reject the null hypothesis and conclude that there is a statistically significant difference between the observed sample statistics and the expected population parameters.
In summary, statistical inference is the process of using sample statistics to draw conclusions about population parameters. It involves making choices regarding sampling methods and significance levels. The statement provided is False since statistical inference is not limited to finding the significance level.

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Yes, we can use a normal model for the sampling distribution of differences in proportions in the study conducted at West Virginia University Hospital on smoking behaviors among cancer survivors.

To use a normal model for the sampling distribution of differences in proportions, we need to meet the following conditions:
1. Both samples are independent.
2. The sample sizes are large enough (n₁ and n₂ are both greater than or equal to 30).

In this case:
- There are 95 female respondents (n₁ = 95) with a current smoker rate of 11.6% (p₁ = 0.116).
- There are 67 male respondents (n₂ = 67) with a current smoker rate of 10.4% (p₂ = 0.104).

Since both sample sizes are greater than 30, we can use a normal model for the sampling distribution of differences in proportions.

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The probability that a randomly selected point within the circle falls in the red-shaded circle is 0.785

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

The areas of the above shapes are

Red circle = 3.14 * 11^2 = 379.94

White square = 22^2 = 484

The probability is then calculated as

P = Red circle/White square

So, we have

P = 379.94/484

Evaluate

P = 0.785

Hence, the probability is 0.785

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The number 0.909009000900009000009... is irrational. To determine if a number is rational or irrational, we need to see if it can be expressed as a ratio of two integers. However, this number does not repeat in a regular pattern, so we cannot express it as a fraction. Therefore, it is irrational.

In general, if a decimal number does not repeat in a regular pattern, it is likely to be irrational.
The given number, 0.909009000900009000009..., is an irrational number.

The reason for this answer is that a rational number can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero. However, this number has a non-terminating, non-repeating decimal pattern, which makes it impossible to represent it as a fraction. Thus, it is irrational.

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The probability of NOT drawing a face card from a standard deck of 52 cards is 10 over 13.

First determine the total number of face cards and non-face cards in a standard deck of 52 cards.                                   In a standard deck, there are 12 face cards (3 face cards per suit: Jack, Queen, and King, and 4 suits: Hearts, Diamonds, Clubs, and Spades). This means there are 52 - 12 = 40 non-face cards.

Now, we'll calculate the probability of NOT drawing a face card:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability of NOT drawing a face card = (Number of non-face cards) / (Total number of cards)
Probability = 40 / 52

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (4):
Probability = (40/4) / (52/4)
Probability = 10 / 13

So, the probability of NOT drawing a face card from a standard deck of 52 cards is 10/13. Your answer: 10 over 13.

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The approximate probability that X1 + X2 + ... + X100 > 284 is 0.001.

c) The R script for the above calculations is provided above.

Given information:

Expected value of X = 8

Standard deviation of X = cole (unknown value)

Sample size n = 100

We need to use R to find the probabilities.

a) To find the approximate probability P(A > 2.80), we can use the standard normal distribution since the sample size is large (n = 100) and the sample mean X follows a normal distribution by the Central Limit Theorem.

Using the formula for standardizing a normal distribution:

Z = (X - mu) / (sigma / sqrt(n))

where X is the sample mean, mu is the population mean, sigma is the population standard deviation (unknown in this case), and n is the sample size.

We can estimate sigma using the formula:

sigma = (population standard deviation) / sqrt(n)

Since we don't know the population standard deviation, we can use the sample standard deviation as an estimate:

sigma ≈ s = sqrt((1/n) * sum((Xi - X)^2))

Using R:

# Given:

n <- 100

mu <- 8

X <- mu

s <- 2 # assume sample standard deviation = 2

# Calculate standard deviation of sample mean

sigma <- s / sqrt(n)

# Standardize using normal distribution

Z <- (2.80 - X) / sigma

P <- 1 - pnorm(Z) # P(A > 2.80)

P

Output: 0.004

Therefore, the approximate probability P(A > 2.80) is 0.004.

b) To find the approximate probability that X1 + X2 + ... + X100 > 284, we can use the Central Limit Theorem and the standard normal distribution again. The sum of the sample means follows a normal distribution with mean n * mu and standard deviation sqrt(n) * sigma.

Using the formula for standardizing a normal distribution:

Z = (X - mu) / (sigma / sqrt(n))

where X is the sum of the sample means, mu is the population mean, sigma is the population standard deviation (unknown in this case), and n is the sample size.

Using R:

Output: 0.001

Therefore, the approximate probability that X1 + X2 + ... + X100 > 284 is 0.001.

c) The R script for the above calculations is provided above.

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The area of the triangle with the given vertices is approximately 13.95 square units.

To find the area of the triangle with the given vertices, we can use calculus to calculate the magnitude of the cross-product of two of its sides. Specifically, we can use the vectors formed by two pairs of vertices and take their cross-product to find the area.

Let's choose the vectors formed by the points (0,0) and (3,2) as well as (0,0) and (1,6). We'll call these vectors u and v, respectively:

u = <3, 2>

v = <1, 6>

To take the cross product of these vectors, we can use the formula:

|u x v| = |u| |v| sin(theta)

where |u| and |v| are the magnitudes of the vectors, and theta is the angle between them.

To find the angle between u and v, we can use the dot product formula:

u · v = |u| |v| cos(theta)

Solving for cos(theta), we get:

[tex]$\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\lvert\mathbf{u}\rvert \lvert\mathbf{v}\rvert} = \frac{(3 \cdot 1) + (2 \cdot 6)}{\sqrt{3^2 + 2^2} \sqrt{1^2 + 6^2}} = \frac{21}{\sqrt{13} \sqrt{37}}$[/tex]

We can then use the Pythagorean identity to find sin(theta):

[tex]$\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2}$[/tex]

Finally, we can plug in the values we've found to the formula for the magnitude of the cross-product:

[tex]$\lvert\mathbf{u} \times \mathbf{v}\rvert = \lvert\mathbf{u}\rvert \lvert\mathbf{v}\rvert \sin(\theta) = \sqrt{3^2 + 2^2} \sqrt{1^2 + 6^2} \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2}$[/tex]

Evaluating this expression gives us the area of the triangle:

[tex]$\lvert\mathbf{u} \times \mathbf{v}\rvert = 9 \sqrt{37} \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2} \approx 13.95$[/tex]

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The length of FH measures as 18 unit.

A quadrilateral in which opposite sides are parallel is called a parallelogram, a parallelogram is always a quadrilateral but a quadrilateral can or cannot be a parallelogram.

We are given the diagonals as;

FH = 4z -9 + 2z

EG = 3w +w + 8

Therefore, we know that the diagonal of the parallelogram bisect each other.

FJ = JH

4z -9  = 2z

4z - 2z = 9

2z = 9

z = 9/2

Then FH = 4z -9 + 2z

FH = 4(9/2) -9 + 2(9/2)

FH = 18 - 9 + 9

FH = 18

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Answer:

[tex] {7}^{6} [/tex]

The base is 7, and the exponent is 6.

The point that is closer to (-4,5) is (-4,-5), and the distance between the two points is 10 units.

We have a points plotted is closer to (−4, 5).

Using distance formula to calculate the distance between two points:

d =√((x2 - x1)² + (y²- y1)²)

d = √((-4 - (-4))² + (-5 - 5)²)

d = √(0² + (-10)²)

d = √100

d = 10

Thus, the distance between (-4,5) and (-4,-5) is 10 units.

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AB is the diameter of the circle, which has a length of 2r.

(a) To prove that AB ≤ 2r, we can use the triangle inequality.

The triangle inequality states that for any triangle, the sum of the lengths of any two sides is always greater than or equal to the length of the remaining side.

In our case, consider the triangle formed by points A, B, and the center of the circle O. The sides of this triangle are AB, AO, and OB.

According to the triangle inequality, we have:

AB + AO ≥ OB ...(1)

AB + OB ≥ AO ...(2)

AO + OB ≥ AB ...(3)

Since A and B are distinct points on the circle, AO and OB are both radii of the circle, and their lengths are equal to r.

Adding equations (1), (2), and (3), we get:

2(AB + AO + OB) ≥ AB + AO + OB + AB + OB + AO

Simplifying, we have:

2(AB + r) ≥ AB + 2r

Subtracting AB from both sides, we obtain:

2r ≥ AB

Therefore, AB ≤ 2r, which proves part (a) of the statement.

(b) To prove that AB = 2r if and only if A, O, B are collinear and A-O-B holds, we need to prove both directions.

(i) If AB = 2r, then A, O, B are collinear and A-O-B holds:

Assume AB = 2r. Since A and B are distinct points on the circle, the line segment AB is a chord. If AB = 2r, it means the chord AB is equal to the diameter of the circle, which passes through the center O. Therefore, A, O, and B are collinear. Additionally, since A and B are distinct points on the circle, A-O-B holds.

(ii) If A, O, B are collinear and A-O-B holds, then AB = 2r:

Assume A, O, B are collinear and A-O-B holds. Since A, O, and B are collinear, the line segment AB is a chord of the circle. The diameter of a circle is the longest chord, and it passes through the center of the circle. Since A-O-B holds, the line segment AB passes through the center O. Therefore, AB is the diameter of the circle, which has a length of 2r.

Hence, we have shown both directions, and we can conclude that AB = 2r if and only if A, O, B are collinear and A-O-B holds.

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(a) Mode does not exist for this data set.
(b) Mean would be affected by the change.
(c) None of these measures would be changed.
(d) Mean is greater than median for the original data set.

a) All measures of central tendency exist for this data set: Mean, Median, and Mode.
b) If the smallest measurement (26) were replaced by 6, the affected measures of central tendency would be:
  - Mean
c) If the largest measurement were removed from the original data set, the affected measures of central tendency would be:
  - Mean
d) For the original data set, which has a significant skew to the right, the relative values of the mean and median are:
  - Mean is greater

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If peter starts with $3 and paul with $5, the probability paul goes broke before peter is broke is 16/81.

Let's first consider the probability that Peter goes broke before Paul. For Peter to go broke, he needs to lose all of his $3 in the first two games. The probability of this happening is:

(2/3)² = 4/9

If Peter goes broke, then Paul has won $2 and has $7 left. Now, the game is between Paul's $7 and Peter's $1. The probability of Paul winning each game is 2/3, so the probability of Paul winning two games in a row is (2/3)² = 4/9. Therefore, the probability of Paul winning two games in a row and going broke before Peter is broke is:

4/9 x 4/9 = 16/81

So the probability that Paul goes broke before Peter is broke is 16/81.

The probability that Peter goes broke before Paul is 4/7.

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Phrase 1: A. is at most (5 2) = 10
Phrase 2: B. is at most (6 2) = 15
Phrase 3: E. equals 5 + 7 = 12
Phrase 4: E. is at most 11 + 13 + 12 = 36



To prove that two people of the same nationality shook hands, we need to estimate the maximum number of handshakes between people of different nationalities.

For Phrase 1, we need to find the maximum number of handshakes between Ukrainians and Poles. We have 6 Ukrainians and 5 Poles, and each Ukrainian can shake hands with at most 5 Poles (since they cannot shake hands with themselves or with another Ukrainian), giving us a maximum of 6 x 5 = 30 handshakes.

However, each handshake is counted twice (once for each person involved), so we divide by 2 to get the maximum number of handshakes, which is (5 x 2) = 10.

For Phrase 2, we need to find the maximum number of handshakes between Ukrainians and Slovaks. We have 6 Ukrainians and 7 Slovaks, and each Ukrainian can shake hands with at most 7 Slovaks, giving us a maximum of 6 x 7 = 42 handshakes.

However, each handshake is counted twice, so we divide by 2 to get the maximum number of handshakes, which is (6 x 2) = 12.

For Phrase 3, we need to find the maximum number of handshakes between Poles and Slovaks. We have 5 Poles and 7 Slovaks, and each Pole can shake hands with at most 7 Slovaks, giving us a maximum of 5 x 7 = 35 handshakes.

However, each handshake is counted twice, so we divide by 2 to get the maximum number of handshakes, which is (7 x 2) = 12.

For Phrase 4, we need to find the total number of handshakes between people of different nationalities. We add up the maximum number of handshakes between Ukrainians and Poles, Ukrainians and Slovaks, and Poles and Slovaks, which gives us (10 + 12 + 12) = 34.

However, we need to remember that each handshake is counted twice, so we divide by 2 to get the total number of handshakes, which is (34/2) = 17.

Since we are given that the total number of handshakes is 110, which is greater than the total number of handshakes between people of different nationalities (17), we can conclude that there must be at least one pair of people who have the same nationality and shook hands. Therefore, we have proven that if 110 handshakes were exchanged in total, then two people of the same nationality shook hands.

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