(L5) Given: ΔABC with AC>AB;BD¯ is drawn so that AD¯≅AB¯Prove: m∠ABC>m∠C

This gives us a dendrogram with three levels, where the first level has two clusters {{7,10},{20,28}} and {35}, the second level has two clusters {{7,10,20,28},35}, and the third level has only one cluster {{7,10,20,28,35}}.

What is a sequence?

A sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms).

To perform hierarchical clustering using single linkage, we start by treating each point as its own cluster, and then iteratively merge the two closest clusters until only one cluster remains. We use the single linkage method, which defines the distance between two clusters as the minimum distance between any two points in the clusters.

First, we calculate the pairwise distances between each point:

  7    10   20   28   35

7   -    3    13   21   28

10  3    -    10   18   25

20  13   10   -    8    15

28  21   18   8    -    7

35  28   25   15   7    -

Next, we find the two closest points/clusters and merge them:

  7,10  20   28   35

7,10  -    10   18   25

20    10   -    8    15

28    18   8    -    7

35    25   15   7    -

The closest points/clusters are 7 and 10, so we merge them to form a new cluster {7,10}.

  7,10  20,28  35

7,10  -    18     25

20,28 18   -      7

35    25   7      -

The closest points/clusters are now {20,28} and 35, so we merge them to form a new cluster {{20,28},35}.

 7,10  {20,28,35}

7,10  -    7

{20,28,35} 7    -

The closest points/clusters are now {7,10} and {{20,28},35}, so we merge them to form a new cluster {{{7,10},{20,28}},35}.

Hence, This gives us a dendrogram with three levels, where the first level has two clusters {{7,10},{20,28}} and {35}, the second level has two clusters {{7,10,20,28},35}, and the third level has only one cluster {{7,10,20,28,35}}.

The dendrogram can be visualized as in the attached image.

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The estimated number of whole cookies of the remaining 60 is 50.  

Estimating is the process of making an approximate calculation or judgment based on incomplete or uncertain information.

It involves using reasoning and previous experience to make an educated guess about a quantity, value, or outcome.

Estimating is often used when exact calculations are not feasible or practical, or when only a rough estimate is needed.

Here we have

Tara made 6 dozen cookies and brought them to the school bake sale in a big container.

She has taken 12 cookies out of the container to give to customers. Of the cookies she has taken out, 2 were broken and 10 were whole.

The total number of cookies Tara made = 6 × 12 = 72 cookies.

She took 12 cookies out of the container i.e 1 dozen

In which broken cookies = 2 i.e 2/12 = 1/6

The whole cookies = 10/12 = 5/6

Hereafter taking 12 cookies, remaining cookies = 72 - 12 = 60

Hence, the estimated number of whole cookies calculated as

the estimated number of whole cookies = (5/6)(60) = 5(10) = 50

Therefore,

The estimated number of whole cookies of the remaining 60 is 50.  

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The 95% confidence interval for the proportion of the population who prefer brand named items is:

CI = (0.102, 0.232)

What is confidence interval?

A confidence interval is a statistical tool used to estimate the range of possible values in which a population parameter, such as the mean or proportion, is expected to lie with a certain level of confidence based on the observed sample data.

To find the 95% confidence interval for the population proportion, we use the formula:

CI = p′ ± z*[tex]\sqrt{(p'q'/n)[/tex]

where:

CI: confidence interval

p′: sample proportion

q′: 1 - p′

z: z-score from the standard normal distribution for the desired confidence level (95% in this case)

n: sample size

Substituting the given values, we get:

CI = 0.167 ± 1.96[tex]\sqrt{((0.1670.833)/180)[/tex]

Simplifying, we get:

CI = 0.167 ± 0.065

Therefore, the 95% confidence interval for the proportion of the population who prefer brand named items is:

CI = (0.102, 0.232)

This means that we can be 95% confident that the true population proportion of people who prefer brand named items falls within this range.

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The ordered pair of one of the zeros for the following function is (x +8)

We have that the function is a quadratic function written as;

f(x)=x2+7x−8

Using the factorization method of solving quadratic functions;

Multiply the coefficient of  x squared by the constant in the expression, we have;

1(-8) = -8

Now, find the pair factors of the product that sum up to give 7, we have;

8x and -x

Substitute the values

x² + 8x - x - 8

group in pairs

(x² + 8x) - (x - 8)

factorize

x(x + 8) - 1(x + 8)

Then, we have;

x = 1

x = -8

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The least number of units that should be produced in order for the average total cost per unit to be $11 or less is 200.

The average total cost per unit is the total cost divided by the number of units produced. Let's denote the total cost as C and the number of units as x. Then the average total cost per unit is:

ATC = C/x

The total cost is the sum of the initial cost and the cost per unit of the product times the number of units produced:

C = 800 + 7x

Substituting this expression for C into the formula for ATC, we get:

ATC = (800 + 7x)/x

We want to find the least number of units, x, that should be produced in order for the average total cost per unit to be $11 or less. This can be written as an inequality:

ATC ≤ 11

(800 + 7x)/x ≤ 11

Multiplying both sides by x, we get:

800 + 7x ≤ 11x

4x ≤ 800

x ≤ 200

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Regression equation are calculations used to predict a person's to score on one variable when that person's score on another variable is already known. So, option(C) is right one.

Statistical study is used to collect and analyze data and is useful in census. The collected data is used to interpret economic activities. Statistics can be qualitative or quantitative in nature. The regression analysis is used to determine the line of best fit for the dependent variable and independent variables. The equation form of regression line is written as, Y= a + bX, where

It is an analysis to measure the relationship between a dependent variable and two or more. independent variables. So the correct choice is the regression equation.

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Complete question:

____ ____ are calculations used to predict a person's to score on one variable when that person's score on another variable is already known.

A. Pearson product-moment correlation coefficient

B. Coefficient of determination

C. Regression analysis

D. Point-biserial correlation coefficient

The new price of the playground slide after a 45 1/8% markup is $476.25.

To find the new price after a 45 1/8% markup, we need to first calculate the markup amount and then add it to the original price.

Markup amount = original price x markup rate

Markup rate = 45 1/8% = 45.125%

We need to convert the percentage to a decimal by dividing by 100

Markup rate = 45.125% ÷ 100 = 0.45125

Now we can calculate the markup amount

Markup amount = $328 x 0.45125 = $148.25

To find the new price, we add the markup amount to the original price

New price = original price + markup amount

New price = $328 + $148.25 = $476.25

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Answer:

Step-by-step explanation:

3 -2

Using the following pattern and inductive reasoning, predicted answer to 9 x 7,654,321 - 1 is 73,888,889.

The pattern:

When you subtract 1 from a number that ends with a sequence of n consecutive digits (all equal to d), the result is a number that ends with the same n digits followed by ([tex]10^{n}[/tex] - 1) -d.

For example:

If you subtract 1 from a number that ends with three 7's, the result is a number that ends with three 6's, i.e., (777-1=776).

If you subtract 1 from a number that ends with four 2's, the result is a number that ends with four 1's, i.e., (2222-1=2221).

Applying this pattern to 9 x 7,654,321 - 1:

The number ends with one 9, so n=1 and d=9.

Therefore, the result will end with one 8 (one less than 9), followed by ([tex]10^{1}[/tex] - 1) - 9 = 0, i.e., it will end with 8.

So, the predicted answer is 73,888,889.

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i got you, buddy!

To rewrite the equation in standard form, we need to complete the square for both x and y terms.

Starting with the x terms: x^2 + 4x + y^2 - 10y = 7 (x^2 + 4x) + y^2 - 10y = 7 (x^2 + 4x + 4) + y^2 - 10y = 7 + 4 (adding and subtracting 4 to complete the square for x) (x + 2)^2 + y^2 - 10y = 11

Now completing the square for y terms: (x + 2)^2 + y^2 - 10y = 11 (x + 2)^2 + (y^2 - 10y + 25) = 11 + 25 (adding and subtracting 25 to complete the square for y) (x + 2)^2 + (y - 5)^2 = 36

Therefore, the equation in standard form is: (x + 2)^2 + (y - 5)^2 = 36

a)  The equation for the exponential distribution of waiting times is given by [tex]f(x) = \lambda e^{-\lambda x}[/tex]

b) The probability of waiting less than 2 minutes to check out is 0.427

c) The probability of waiting between 4 and 6 minutes is 0.242

d) The probability of waiting more than 5 minutes to check out is 0.082

a. The equation for the exponential distribution of waiting times is given by:

[tex]f(x) = \lambda e^{-\lambda x}[/tex]

where λ is the rate parameter of the distribution, and e is the natural logarithmic constant (approximately equal to 2.71828). The graph of the exponential distribution is a decreasing curve that starts at λ and approaches zero as x approaches infinity. The mean waiting time, denoted by E(X), is equal to 1/λ.

b. To find the probability that a customer waits less than 2 minutes to check out, we need to calculate the area under the exponential distribution curve between zero and 2 minutes. This can be expressed mathematically as:

P(X < 2) = [tex]\int_0^2 \lambda e^{-\lambda x} dx[/tex]

Solving this integral yields:

P(X < 2) = 1 - [tex]e^{(-2\lambda)}[/tex]

Substituting the given average waiting time of 2.5 minutes into the formula for the mean waiting time, we can calculate λ as:

E(X) = 1/λ

2.5 = 1/λ

λ = 0.4

Therefore, the probability of waiting less than 2 minutes to check out is:

P(X < 2) = 1 - [tex]e^{-2*0.4}[/tex]

P(X < 2) ≈ 0.427

c. To find the probability of waiting between 2 and 4 minutes, we need to calculate the area under the exponential distribution curve between 2 and 4 minutes. This can be expressed mathematically as:

P(2 < X < 4) =[tex]\int_2^4 \lambda e^{(-\lambda x)} dx[/tex]

Solving this integral yields:

P(2 < X < 4) = [tex]e^{(-2\lambda)} - e^{(-4\lambda)}[/tex]

Substituting the value of λ obtained in part (b), we get:

P(2 < X < 4) = [tex]e^{(-20.4)} - e^{(-40.4)}[/tex]

P(2 < X < 4) ≈ 0.242

d. To find the probability of waiting more than 5 minutes to check out, we need to calculate the area under the exponential distribution curve to the right of 5 minutes. This can be expressed mathematically as:

P(X > 5) = [tex]\int_5^{ \infty} \lambda e^{(-\lambda x)} dx[/tex]

Solving this integral yields:

P(X > 5) = [tex]e^{(-5\lambda)}[/tex]

Substituting the value of λ obtained in part (b), we get:

P(X > 5) = [tex]e^{(-5*0.4)}[/tex]

P(X > 5) ≈ 0.082

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Answer:

10x^2 - 40x = 10x(x - 40)

H.A. y = 1/2

V.A. x = 0, x = 40

The mean is higher than the median because the data is skewed to the right. The median is more resistant to the skew in the data.

To calculate the standard deviation of the number of lines in use that this support center expects to have at noon, we would need to have a dataset of the number of lines in use at different times.

If we have this dataset, we can use the following formula to calculate the standard deviation:

Standard deviation = √(sum((x - mean)²) / n)

Where:

x is the number of lines in use at a given time

mean is the mean of the number of lines in use across all times

n is the total number of times in the dataset

We can calculate the mean of the number of lines in use by adding up all the values and dividing by the total number of times. Once we have the mean, we can calculate the standard deviation using the formula above. However, without access to the dataset, it is not possible to provide a specific answer.

Therefore, The mean is higher than the median because the data is skewed to the right. The median is more resistant to the skew in the data.

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Complete question:

Let the random variable X represent the number of telephone lines in use by the technical support center of a software manufacturer at noon each day. The probability distribution of X is shown in the table below.

In a sentence of two, comment on the relationship between the mean and the median relative to the shape of this distribution.

The fraction of days the man does not wear a hat is 0.

To solve this problem, let's analyze the possible scenarios:

The man chooses a drawer with hats in the morning and returns the hat to the same drawer in the evening.

The man chooses a drawer with hats in the morning and returns the hat to the other drawer in the evening.

The man chooses an empty drawer in the morning and does not wear a hat.

Let's calculate the probabilities for each scenario:

Probability of choosing a drawer with hats in the morning: There are two drawers, so the probability is 2/2 = 1.

Probability of returning the hat to the same drawer in the evening: There is a 1/2 chance of choosing the same drawer, so the probability is 1/2.

Probability of choosing a drawer with hats in the morning: 2/2 = 1.

Probability of returning the hat to the other drawer in the evening: There is a 1/2 chance of choosing the other drawer, so the probability is 1/2.

Probability of choosing an empty drawer in the morning: There is a 0/2 chance of choosing a drawer with hats, so the probability is 0.

Now, let's calculate the fraction of days the man does not wear a hat:

Fraction of days without a hat = (Probability of scenario 3) = 0.

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The area of the rhombus is 108 meters²

A rhombus is a type of quadrilateral with four sides of equal length. It is also known as a diamond or a lozenge.

The diagonals of a rhombus bisect each other at right angles, and they also bisect the angles of the rhombus. The area of a rhombus can be found by multiplying the lengths of its diagonals and dividing by 2.  

Hence, the Area of the rhombus (A) = d₁d₂/2  

Where d₁ and d₂ are lengths of diagonals

Here we have

A drawing of a rhombus with both diagonals bisecting each other at right angles. The vertical diagonal is divided into two lengths of 6 meters each and the horizontal diagonal is divided into two lengths of 9 meters each

From the given data,

Length of vertical diagonal (d₁) = 2 × 6 = 12 meters

Length of horizontal diagonal (d₂) = 2 × 9 = 18 meters  

Using the formula, Area of the rhombus (A) = d₁d₂/2

= (12)(18)/2 = 6 (18) = 108 meters²  

Therefore,

The area of the rhombus is 108 meters²  

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Answer:

the boxplot tells us that the median height of males in the data set is approximately 174.7 cm. The middle 50% of the males in the data set have heights between approximately 153 cm (25th percentile) and 194 cm (75th percentile). There are no outliers in the data set.

Step-by-step explanation:

The boxplot provides a visual representation of the distribution of the heights of males in the data set. The box represents the middle 50% of the data, with the bottom of the box indicating the 25th percentile and the top indicating the 75th percentile. The line within the box represents the median height, which is the middle value of the data set. The whiskers represent the range of the data, with the bottom whisker extending from the bottom of the box to the smallest observation within 1.5 times the interquartile range (IQR) below the bottom of the box, and the top whisker extending from the top of the box to the largest observation within 1.5 times the IQR above the top of the box. Any outliers beyond the whiskers are indicated as individual points.

The a survey of 1,150 people for choose their favorite laundry detergent from brands. If percent value of people who choose brand A is 3, then the number of people who chose A as their favorite brand are 30 , 35 , 40.

We have a survey results of total 1150 people. That is sample size = 1150

The survey is based on their favorite laundry detergent from brands A, B, and C. The percent of people who chose A as their favorite brand = x %

If x = 3, we have to determine the number of people who chose A as their favorite brand. So, we use percentage formula, for each option and see for which number of people percent is equals to 3. Now, percent formula is [tex] \frac{value}{total \: value}×100%.[/tex]

a) [tex] (\frac{20}{1150 }) 100 \% = 1.73 \%[/tex] ≈ 2%

b) [tex] (\frac{25}{1150 }) 100 \% = 2.17 \%[/tex] ≈ 2%

c) [tex] (\frac{30}{1150 }) 100 \% = 2.60\%[/tex] ≈ 3 %

d) [tex] (\frac{35}{1150 }) 100 \% = 3.04\%[/tex]≈ 3 %

e)[tex] (\frac{40}{1150 }) 100\% = 3.48 \%[/tex] ≈ 3%

f)[tex] (\frac{45}{1150 }) 100 \% = 3.91 \%[/tex]≈ 4%

g)[tex] (\frac{50}{1150 }) 100 \% =4.35 \%[/tex] ≈ 4%

Hence, all such numbers are 30 , 35 , 40.

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Complete question:

A survey asked 1,150 people to choose their favorite laundry detergent from brands A, B, and C. Of the people surveyed, x percent chose A as their favorite brand. If x is rounded to the nearest integer, the result is 3. Which of the following could be the number of people who chose A as their favorite brand? Indicate all such numbers.

a)20

b) 25

c)30

d) 35

e) 40

f) 45

g) 50

If x percent of people chose A as their favorite brand, then the number of people who chose A can be found by multiplying x percent by the total number of people surveyed:

number of people who chose A = (x/100) * 1150

Since x is rounded to the nearest integer and equals 3, we have:

number of people who chose A = (3/100) * 1150 = 34.5, which is closest to 35.

If x percent chose brand A as their favorite and x is rounded to the nearest integer, it means that x lies between x-0.5 and x+0.5. In other words, x-0.5 <= actual percentage of people who chose A <= x+0.5.

From the given information, we know that x rounded to the nearest integer is 3. Therefore, we have:

2.5 <= x <= 3.5

Since x represents the percentage of people who chose brand A, we can find the number of people who chose A as their favorite by multiplying x with the total number of people surveyed (1150). Therefore, the number of people who chose brand A lies between:

2.5% of 1150 <= number of people who chose A <= 3.5% of 1150

28.75 <= number of people who chose A <= 40.25

Since the number of people who chose A must be a whole number, the only possible value for the number of people who chose is 29.

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The researcher interested in analyzing the frequency of some combination of possible responses to multiple categorical variables should construct a contingency table.

A contingency table is a two-way table that displays the frequency of observations or counts for two or more categorical variables. The table is constructed by tabulating the counts or percentages of the variables in rows and columns. The contingency table can be used to identify relationships between variables and can be helpful in analyzing data and developing hypotheses.

In a contingency table, the frequency of observations is summarized in rows and columns, as well as in the margins. The marginal totals represent the total counts or percentages for each variable, and they are typically displayed at the bottom or the right side of the table.

The marginal totals provide an overview of the overall distribution of the variables and can be helpful in identifying patterns or trends. A researcher interested in analyzing multiple categorical variables may need to construct a multiple contingency table. This type of table displays the frequency of observations for more than two categorical variables.

The table is constructed by tabulating the counts or percentages of the variables in rows and columns, as well as in the margins. In summary, a researcher interested in analyzing the frequency of some combination of possible responses to multiple categorical variables should construct a contingency table.

The contingency table summarizes the frequency of observations in rows, columns, and margins, providing a comprehensive overview of the distribution of the variables. The table can be helpful in identifying relationships between variables and developing hypotheses.

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The test statistic value is -2.03.

What is the mean and standard deviation?

In statistics, the measurement of variability known as the standard deviation (SD) is frequently utilised. It displays the degree of variance from the mean (average). While a high SD shows that the data are dispersed throughout a wide range of values, a low SD suggests that the data points tend to be close to the mean.

We can use a one-sample t-test to test whether the mean endurance time of the tires is significantly different from the claimed value of 40000 km. The test statistic is given by:

[tex]t = (x - \mu) / (s / \sqrt{(n)})[/tex]

where x is the sample mean (39750 km), μ is the claimed mean (40000 km), s is the sample standard deviation (387 km), and n is the sample size (25).

Substituting the values, we get:

[tex]t = (39750 - 40000) / (387 / \sqrt{25})[/tex]

t = -250 / (387/5)

t = -2.03

Therefore, the test statistic value is -2.03.

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Answer:

B. no, the survey represented both groups in the population of students

Step-by-step explanation:

It says, "surveyed all students"

The expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high is 60.

To determine the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high, we need to use the information provided in the table.

Here we need to find the total number of respondents in each group For those living less than 25 miles from the waterfall,

The total is 150.

For those living more than 25 miles from the waterfall,

the total is 100.

Again,we need to find the proportion of respondents in each group who rated the cultural importance as high.

For those living less than 25 miles from the waterfall,

the proportion is 60/150 = 0.4.

For those living more than 25 miles from the waterfall,

the proportion is 40/100 = 0.4.

Now, we can find the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high by multiplying the total number of respondents in that group (150) by the proportion who rated the cultural importance as high (0.4). Expected count = 150 x 0.4 = 60

Therefore, the expected count of members living less than 25 miles from the waterfall who rated the cultural importance as high is 60.

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In the given equation, find sales = 0.22 + 1.8 (degrees over 32 Fahrenheit), the "y-intercept" is the value of "find sales" when the "degrees over 32 Fahrenheit" is equal to 0.

To find the y-intercept, substitute 0 for "degrees over 32 Fahrenheit":

Find sales = 0.22 + 1.8(0)
Find sales = 0.22

So, the y-intercept is 0.22.

In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. As such, these points satisfy x = 0.

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The sample that could constitute a discrete random variable is the number of near collisions of aircraft in a year.

This is because it is a countable, finite number and not a continuous measurement like the height of the ocean's tide. The total number of points scored in a football game could also be considered a discrete random variable because it is a countable, finite number. However, the height of the ocean's tide is a continuous measurement and cannot be counted as a discrete random variable.
The sample that could constitute a discrete random variable is: i. total number of points scored in a football game.
A discrete random variable represents a countable number of distinct values or outcomes. In this case, the total number of points scored in a football game can be counted and listed, making it a discrete random variable.
On the other hand, the height of the ocean's tide at a given location (ii) is a continuous random variable, as it can take any value within a given range, and the number of near collisions of aircraft in a year (iii) could also be considered as a discrete random variable, but it's not one of the options in your question.

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Part A) the p-value is 0.0025.

Part B) the positive critical value is:zα/2 = |1.645| = 1.645 (rounded to three decimal places).

what is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.

Part A) For a left-tailed test with a test statistic of z = -2.816, the p-value can be calculated using a standard normal distribution table or a calculator. The p-value is the area to the left of the test statistic in the standard normal distribution.

Using a standard normal distribution table, the area to the left of z = -2.816 is 0.0025. Therefore, the p-value is 0.0025.

Alternatively, using a calculator such as the TI-84, the p-value can be found by entering the command "normalcdf(-9999,-2.816)" which gives a result of 0.0025, accurate to 4 decimal places.

Therefore, the p-value is 0.0025.

Part B) For a two-tailed test at a significance level of α = 0.10, the critical values can be found using a standard normal distribution table or a calculator. The critical values are the z-scores that leave α/2 in each tail.

Using a standard normal distribution table, the critical value for α/2 = 0.05

Therefore, the positive critical value is:zα/2 = |1.645| = 1.645 (rounded to three decimal places)

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The absolute maximum value of f(x,y) subject to the constraint [tex]x^2 + y^2 = 49[/tex] is 30√2, which occurs at the point (5√2/2, 3√2/2), and the absolute minimum value is -30√2, which occurs at the point (-5√2/2, -3√2/2).

Lagrange Multiplier is a  method used to find the extreme values of a function subject to one or more constraints. The method involves introducing a new variable, called a Lagrange multiplier, for each constraint in the problem.

We can use the method of Lagrange multipliers to find the absolute extrema of the function f(x,y) = 5x + 9y subject to the constraint [tex]x^2 + y^2 = 49.[/tex] We start by defining the Lagrangian function L(x,y,λ) as:
L(x,y,λ) = f(x,y) - λg(x,y)

where g(x,y) = [tex]x^2 + y^2 - 49[/tex] is the constraint function and λ is the Lagrange multiplier.
Taking partial derivatives of L with respect to x, y, and λ, we get:
∂L/∂x = 5 - 2λx = 0

∂L/∂y = 9 - 2λy = 0

∂L/∂λ = x² + y² - 49 = 0

∂L/∂λ [tex]= x^2 + y^2 - 49 = 0[/tex]

Solving these equations simultaneously, we get:
x = ±5√2/2, y = ±3√2/2, λ = 5/7
These are the critical points of f(x,y) subject to the constraint [tex]x^2 + y^2 = 49.[/tex]
To determine which of these critical points are absolute maxima and minima, we need to evaluate the function f(x,y) at these points and compare the values. We have:
f(5√2/2, 3√2/2) = 5(5√2/2) + 9(3√2/2) = 30√2

f(-5√2/2, -3√2/2) = 5(-5√2/2) + 9(-3√2/2) = -30√2

So, the absolute maximum value of f(x,y) subject to the constraint [tex]x^2 + y^2 = 49[/tex] is 30√2, which occurs at the point (5√2/2, 3√2/2), and the absolute minimum value is -30√2, which occurs at the point (-5√2/2, -3√2/2).

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The values of the trig ratios is Sin X = a/c, Cos X = b/c and Tan X = a/b.

The value of each of the trig ratio is determined by applying a short formula known as SOH CAH TOA  as shown below;

SOH CAH TOA

SOH = sin θ = opposite /hypothenuse side

TOA = tan θ = opposite side / adjacent side

CAH = cos θ = adjacent side / hypothenuse side

Let the opposite side of angle X = a

Let the adjacent side of angle X = b

Let the hypothenuse side of angle X =  c

The values of the trig ratios is calculated as follows;

Sin X = a/c

Cos X = b/c

Tan X = a/b

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The favorable outcome is the number of all-women groups (126) and the total possible outcomes are all possible groups (3003). Therefore, P(all-women) = 126/3003 ≈ 0.0419 (rounded to four decimal places).

(a) To determine the number of different groups of applicants that can be selected for the positions, we use the combination formula: C(n, k) = n! / (k!(n-k)!) where n is the total number of applicants (9 women + 6 men = 15) and k is the number of positions (5). So, C(15, 5) = 15! / (5!(15-5)!) = 3003.

(b) To find the number of different groups of trainees consisting entirely of women, we use the same formula but with only the 9 women as applicants: C(9, 5) = 9! / (5!(9-5)!) = 126.

(c) To calculate the probability that the trainee class will consist entirely of women, we can use the formula P(event) = Number of favorable outcomes / Total possible outcomes. In this case, the favorable outcome is the number of all-women groups (126) and the total possible outcomes are all possible groups (3003). Therefore, P(all-women) = 126/3003 ≈ 0.0419 (rounded to four decimal places).

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The probability distribution of random variable x is

a) 1.78 is the anticipated value of x;

b) The variance of x is 0.6484.

(a) The expected value of x can be found using the formula:

[tex]E(x) = Σ[x * f(x)][/tex]

where x's potential values are all added up.

Using the given probability distribution, we have:

[tex]E(x) = (0 * 0.27) + (1 * 0.35) + (2 * 0.05) + (3 * 0.25) + (4 * 0.08)[/tex]

[tex]E(x) = 1.78[/tex]

As a result, 1.78 is the expected value of x.

(b) The variance of x can be found using the formula:

[tex]Var(x) = E(x^2) - [E(x)]^2[/tex]

where E(x) represents the anticipated value of x and E(x2) represents the expected value of x2.

To find E(x^2), we can use the formula:

[tex]E(x^2) = Σ[x^2 * f(x)][/tex]

Using the given probability distribution, we have:

[tex]E(x^2) = (0^2 * 0.27) + (1^2 * 0.35) + (2^2 * 0.05) + (3^2 * 0.25) + (4^2 * 0.08)[/tex]

[tex]E(x^2) = 3.33[/tex]

Consequently, the variation of x is:

[tex]Var(x) = E(x^2) - [E(x)]^2[/tex]

[tex]var(x) = 3.33 - (1.78)^2[/tex]

[tex]var(x) = 0.6484[/tex] (rounded to four decimal places)

x's variance is 0.6484

As a result, x's variance is 0.6484.

1.78 is the anticipated value of x

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The ordered pairs of the linear expression 3x + 3y = 9 is (0, 3)

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

The linear expression 3x+3y=9

To determine the ordered pairs of the linear expression, we set x to any value say x = 0 0 and then calculate the value of y

Using the above as a guide, we have the following:

3(0) + 3y = 9

Evauate

3y = 9

Divide both sides by 3

y = 3

This means that the value of y is equal to 3

So, we have (0, 3)

Hence, the ordered pairs of the linear expression is (0, 3)

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(i) T-interval

(ii) F. Because I don't have a large number of professors and I don't know the standard deviation of all FSU professors' heights.

What is standard deviation?

Standard deviation is a measure of the amount of variation or dispersion of a set of data values from the mean value.

When constructing a confidence interval for the mean of a population using a sample, we use either a Z-interval or a T-interval based on the sample size and whether we know the population standard deviation.

If the sample size is large (usually taken to be greater than or equal to 30) and/or we know the population standard deviation, then we can use a Z-interval.

However, if the sample size is small (usually less than 30) and/or we don't know the population standard deviation, we should use a T-interval.

In this case, we don't have a large sample size (just the professors the student has this term), and we don't know the standard deviation of all FSU professors' heights, so we would use a T-interval.

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