/The volume of a square based pyramid is 120cm³. If its height is 10 cm, what is the length of its base?​

The size of the sample has to be 119 since you do not want to  deviate from the population

We have to assume that the estimated proportion is given as 0.5

From the standard normal table, we have to solve for the z critical value

a=0.05, Z(0.025) =1.96

The formula for n can be gotten through  n=(Z/E)^2*p*(1-p)

When we put in the values we will have

=(1.96/0.09)^2*0.5*0.5

Thus n

= 118.5679

This is approximated as n = 119

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The probability of rolling an even number in at least one of the throws is 31/32 or approximately 0.969.

The probability of not rolling an even number in a single throw is 1/2, since there are three odd numbers and three even numbers on the die. Therefore, the probability of not rolling an even number in five independent throws is (1/2)^5 = 1/32.

The probability of rolling an even number in at least one of the throws is the complement of the probability of not rolling an even number in any of the five throws. So:

P(rolling an even number in at least one of the throws) = 1 - P(not rolling an even number in any of the five throws)

= 1 - (1/32)

= 31/32

Therefore, the probability of rolling an even number in at least one of the throws is 31/32 or approximately 0.969.

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Therefore,  Selecting a five from a deck of 52 playing cards is not a simple event. This is due to the multiple outcomes (one for each suit).

whether selecting a five from a standard deck of 52 playing cards is a simple event, and you would like the main answer in the last two lines.
To determine if this event is a simple event, we must first define a simple event. A simple event is an event that consists of only one outcome or one possible result. In a standard deck of 52 playing cards, there are 4 fives (one in each suit: hearts, diamonds, clubs, and spades).
Since there are 4 fives in the deck, selecting a five is not considered a simple event because there are multiple outcomes (one for each suit).

Therefore,  Selecting a five from a deck of 52 playing cards is not a simple event. This is due to the multiple outcomes (one for each suit).

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The relationship between chocolate sales and student happiness can be tested using the mean difference.

Specifically, a statistical test such as a correlation analysis or regression analysis can be used to examine the relationship between the amount of chocolate sales and the level of student happiness.

The mean difference refers to the difference in the average level of student happiness between two groups, such as those who consume more chocolate and those who consume less chocolate. This can help determine if there is a significant association between chocolate consumption and student happiness.

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Answer: the answer is 2 I believe

Step-by-step explanation:as you look at “stem” and go down to the number 4. You count below that. So 8, and 6. That’s 2 numbers :) hope this helps!

The stem and leaf plot shows the ages of people at the library. 2 people are under the age of 40.

The duration of a being's or thing's existence; the length of its existence or life up until the moment being discussed or alluded to age. A time frame for humans that begins at birth and is measured in years.

This time frame is typically characterised by a specific stage or level of physical or mental development as well as the potential for legal responsibility. The stem and leaf plot shows the ages of people at the library. 2 people are under the age of 40.

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From the perimeter of rectangle painting with frame, the width of the frame, i.e., x is equals to the three inches.

We have a rectangular painting measures 13 inches by 14 inches. Also, it contains a frame of uniform width around the four edges. Let the uniform width of frame be 'x inches'. See the above figure of painting, it is a rectangle.

Length of rectangular painting = 14 inches

Width of rectangle= 13 inches

The perimeter of the rectangle formed by the painting and its frame = 70 inches

We have to determine width of the frame.

Length of rectangle painting with frame = (14+ 2x) in

Width of rectangle painting with frame = (13+ 2x) in

As we know perimeter is sum of boundary lengths of a shape or geometry. So, the perimeter of rectangle= 2 ( l + w)

Substitute values of length, width and perimeter for painting with frame, 70 = 2( 14 + 2x + 13 + 2x )

Simplify the expression, 70 = 28 + 4x + 26 + 4x

=> 70 = 8x + 54

=> 8x = 70 - 54 = 24

=> x = 3 inches

Hence, required value is 3 inches.

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Complete question:

A rectangular painting measures 13 inches by 14 inches and contains a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 70 inches. Determine the width of the frame What is the width of the frame? inches

The limitation of cross-sectional research among the given choices is: Differences between age groups may be due to different experiences.

Explanation:

The limitation of cross-sectional research is that differences between age groups may be due to different experiences. Cross-sectional research is a type of observational study design that collects data at a single point in time from a sample of individuals of different ages, groups, or populations. This type of research is used to study differences between groups, such as age, gender, or socioeconomic status.

However, one of the main limitations of cross-sectional research is that the differences observed between groups may not necessarily be due to age or any other factor of interest, but rather due to different experiences, histories, or social contexts. For instance, if researchers find that older adults perform worse on a cognitive test than younger adults, it is unclear whether this difference is due to age-related decline, or to factors such as education, occupation, or health.

Other limitations of cross-sectional research include the inability to establish causal relationships between variables, the possibility of selection bias or confounding, and the inability to track changes over time. Additionally, the other options presented in the multiple-choice question, such as test-wise effects, drop-out rates, and questionnaire completion by children, are not specific limitations of cross-sectional research, but rather potential issues that may affect any type of research design.

Therefore, the limitation of cross-sectional research is that differences between age groups may be due to different experiences.

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The correct option is b. judgmental sampling; convenience sampling.  

Judgmental sampling was used in Mexico to ensure a representative sample of target respondents, while convenience sampling with the snowball procedure was employed in Saudi Arabia due to the absence of sampling frames and social customs prohibiting personal interviews.

The question is about the different sampling techniques used for the same consumer research study in Mexico and Saudi Arabia. In Mexico, judgmental sampling was used by having experts identify neighborhoods where the target respondents lived; homes were then randomly selected for interviews.

In Saudi Arabia, convenience sampling employing the snowball procedure was used because there were no lists from which sampling frames could be drawn and social customs prohibited spontaneous personal interviews.

Judgmental sampling, also known as expert sampling or purposive sampling, is a non-probability sampling technique where the researcher selects participants based on their expertise or knowledge about the population being studied.

In the case of Mexico, experts identified neighborhoods where target respondents lived, and homes were randomly selected for interviews. This technique helped ensure that the sample was representative of the target population.

On the other hand, convenience sampling is another non-probability sampling technique where participants are selected based on their availability and ease of access.

In Saudi Arabia, due to the absence of lists for sampling frames and social customs prohibiting spontaneous personal interviews, the snowball procedure was used. Snowball sampling is a type of convenience sampling where existing participants recruit additional participants from their network. This process continues until the desired sample size is achieved.

Therefore, the correct answer is b. judgmental sampling; convenience sampling.

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To find the time required for more than 200 cars to have entered the parking lot with a probability of 0.9, we need to use the properties of the Poisson distribution and its cumulative distribution function (CDF).

Given:

- Poisson distribution rate (λ) = 100 cars per hour

- Desired probability (P(X > 200)) = 0.9

The Poisson distribution probability mass function (PMF) is given by:

P(X = k) = (e^(-λ) * λ^k) / k!

To find the time required, we need to determine the value of λt (λ multiplied by time) that corresponds to the desired probability.

Using the CDF of the Poisson distribution, we can express the desired probability as:

P(X > 200) = 1 - P(X ≤ 200)

Since the CDF is a cumulative probability, we want to find the time (t) at which P(X ≤ 200) is less than or equal to 0.1. We can incrementally increase t until we reach the desired probability.

Step 1: Calculate the rate parameter for the desired time:

λt = λ * t

100 * t = λt

Step 2: Calculate the cumulative probability P(X ≤ 200) using the Poisson CDF:

P(X ≤ 200) = ∑[k=0 to 200] (e^(-λt) * (λt)^k) / k!

Step 3: Incrementally increase t until P(X ≤ 200) ≤ 0.1:

Start with t = 0, and increase it until P(X ≤ 200) ≤ 0.1 is achieved. This can be done using numerical methods or a Poisson distribution calculator.

The specific value of time required can vary based on the precise probability calculation and numerical approximation used. To obtain an accurate and precise result, it is recommended to use specialized software, statistical calculators, or programming languages with built-in functions for Poisson distributions.

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Answer:

2 days

Step-by-step explanation:

The probability of choosing 6 people at random from the group and having all of them be left-handed is approximately 0.00002599 or 0.0026%.

We can approach this problem using the binomial distribution, which models the probability of a certain number of successes (in this case, choosing left-handed people) in a fixed number of trials (choosing 6 people).

Let p be the probability of choosing a left-handed person, which is given as 15% or 0.15. Then, the probability of choosing all 6 people to be left-handed can be calculated as:

P(6 left-handed) = (0.15[tex])^6[/tex] * (1 - 0.15)[tex]^(6 - 6) * C(6, 6)[/tex]

where C(6, 6) represents the number of ways to choose 6 items from a set of 6, which is equal to 1.

Plugging in the values, we get:

P(6 left-handed) = (0.15[tex])^6[/tex] * (0.85)^0 * 1

= 0.00002599

Therefore, the probability of choosing 6 people at random from the group and having all of them be left-handed is approximately 0.00002599 or 0.0026%.

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The effect size associated with a statistical test indicates the magnitude or strength of the relationship between variables being tested, regardless of statistical significance.

The p-value associated with a statistical test indicates the probability of observing the given data, or more extreme data, under the null hypothesis.

A smaller p-value suggests that the observed data is less likely to have occurred by chance alone, and it's often used to determine statistical significance.

Common thresholds for significance are p < 0.05 or p < 0.01.
On the other hand, the associated effect size indicates the magnitude or strength of the relationship between the variables being studies.

Effect sizes can be measured in different ways, such as Cohen's d, correlation coefficient, or odds ratio, depending on the type of data and analysis.

Effect size is important because it helps to determine the practical significance of the findings, beyond just the statistical significance indicated by the p-value.

In summary, the p-value assesses statistical significance, while the effect size indicates the magnitude of the relationship between the variables.

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The total prize money in a tournament with n players is proportional to n log n.

To find the total prize money in the tournament, we need to consider the number of players and their winnings. Let T(n) be the total prize money in a tournament with n players.

If a player wins in a subtournament of size k, then their winnings will be k * 100 dollars. We can divide the tournament into two subtournaments of size n/2 and calculate the winnings for each half separately. Let's consider the player with the highest subtournament size in each half. They will win in their subtournament and the rest of the players in their half will have subtournament sizes less than or equal to k/2. Therefore, the total winnings for each half will be:

T(n/2) = (n/2) * 100 + T(n/2)

The first term in the equation represents the winnings of the player with the highest subtournament size in that half, and the second term represents the total prize money for the rest of the players in that half.

Using the above equation, we can write the recurrence relation for T(n) as:

T(n) = n * 100 + 2T(n/2)

This recurrence relation represents the total prize money in a tournament with n players, where n is a power of 2. We can solve this recurrence relation using the Master Theorem, which gives us:

T(n) = O(n log n)

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We need a sample size of at least 876 adults to estimate the percentage of adults.

To find the sample size needed for estimating the percentage of adults who have consulted fortune-tellers with a 0.04 margin of error and a 99% confidence level, we can use the following formula:

n = (Z² * p * (1 - p)) / E²

where:

n = sample size

Z = Z-score associated with the confidence level (in this case, 2.58 for a 99% confidence level)

p = the expected proportion of adults who have consulted fortune-tellers (in this case, 0.18 based on the prior reliable poll)

E = the margin of error (in this case, 0.04)

Plugging in the values, we get:

n = (2.58² * 0.18 * (1 - 0.18)) / 0.04²

n ≈ 875.85

Rounding up to the nearest whole number, we need a sample size of at least 876 adults to estimate the percentage of adults who have consulted fortune-tellers with a 0.04 margin of error and a 99% confidence level, assuming the prior reliable poll suggested that 18% of adults have consulted fortune-tellers.

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The width of a confidence interval for the population mean depends on the sample size, standard deviation, and the desired level of confidence. Generally, for a greater confidence level, the width of the confidence interval will be wider.

This is because a higher confidence level means that we want to be more certain that the true population mean falls within the interval. To achieve this higher level of certainty, we need to widen the interval to include a larger range of possible values for the population mean.

However, it's important to note that the increase in width may not be proportional to the increase in confidence level. In fact, the width of the confidence interval increases at a decreasing rate as the confidence level increases.

This means that the increase in width from a 90% confidence interval to a 95% confidence interval will be less than the increase in width from an 80% confidence interval to a 85% confidence interval, even though both involve a 5% increase in the level of confidence.

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To determine the probabilities, we need to use the standard normal distribution table or a calculator that has this function. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

(a) P(−0.79 ≤ z ≤ −0.57) = We need to find the area under the standard normal distribution curve between z = -0.79 and z = -0.57. Using a standard normal distribution table or calculator, we find this probability to be 0.0767.
(b) P(z > 1) = We need to find the area under the standard normal distribution curve to the right of z = 1. Using a standard normal distribution table or calculator, we find this probability to be 0.1587.

(c) P(z ≥ −3.36) = We need to find the area under the standard normal distribution curve to the right of z = -3.36 (or the area to the left of z = 3.36, which is the same thing since the standard normal distribution is symmetrical). Using a standard normal distribution table or calculator, we find this probability to be 1.

(d) P(z < 4.96) = We need to find the area under the standard normal distribution curve to the left of z = 4.96. Since this value is greater than any z-score on the standard normal distribution table, we can say that this probability is extremely close to 1 (or practically 1).


To find the probabilities for a standard normal distribution, you can use a z-table or an online calculator based on the given intervals, rounding to four decimal places:

(a) P(-0.79 ≤ z ≤ -0.57) = P(z ≤ -0.57) - P(z ≤ -0.79) = 0.2843 - 0.2148 = 0.0695
(b) P(z > 1) = 1 - P(z ≤ 1) = 1 - 0.8413 = 0.1587
(c) P(z ≥ -3.36) = 1 - P(z < -3.36) = 1 - 0.0004 = 0.9996
(d) P(z < 4.96) = 0.9999 (since the probability of z > 4.96 is extremely small)

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The mean wait time for callers during the tax filing season was greater than 15 minutes.

What is the mean and standard deviation?

The standard deviation is a summary measure of the differences of each observation from the mean. If the differences themselves were added up, the positive would exactly balance the negative and so their sum would be zero. Consequently, the squares of the differences are added.

To test whether there is enough evidence to reject the IRS claim that the mean wait time for callers is at most 15 minutes,

we can use a one-sample t-test with a significance level of α = 0.05.

The null hypothesis is that the true mean wait time μ is equal to 15 minutes,

and the alternative hypothesis is that μ is greater than 15 minutes. Mathematically, this can be expressed as:

H₀: μ ≤ 15

Ha: μ > 15

We can calculate the test statistic t using the formula:

t = (x - μ) / (s / √(n))

where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

Substituting the given values, we get:

t = (16.7 - 15) / (2.7 / √(40))

t = 4.07

Using a t-table or calculator with 39 degrees of freedom (n-1), we find that the p-value associated with this test statistic is less than 0.0001.

This means that if the true mean wait time is really 15 minutes, the probability of obtaining a sample mean of 16.7 minutes or greater is less than 0.0001.

Hence, this p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is enough evidence to suggest that the true mean wait time for callers during the tax filing season was greater than 15 minutes.

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Option (A) is the correct option and on simplifying the expression by combining like terms we get,

24 - 3p.

Here, we have,

Given, the expression 6(4 - 2p) + 9p

On simplifying, we get

24 - 12p + 9p

24 - 3p

So, on simplifying the given expression, we get

24 - 3p

Hence, Option (A) is the correct option and on simplifying the expression by combining like terms we get,

24 - 3p

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Therefore, there were 75 downloads of the standard version with total size downloaded is 75 MB + 3(25 MB) = 150 MB.

To solve this problem, we can use a system of equations. Let x be the number of downloads for the standard version and y be the number of downloads for the high-quality version.

From the problem, we know that:
- The size of the standard version is x MB
- The size of the high-quality version is 3y MB
- The total size downloaded for the two versions was x + 3y MB

Putting these equations together, we get:

x + 3y = 150

We also know that the size of the standard version is smaller than the high-quality version, so we can assume that the standard version was downloaded more times than the high-quality version:

x > y

Now we have two equations and two unknowns. We can solve for x and y by substitution or elimination.

Let's use substitution:

From the second equation, we can isolate y:

y = x/3

Substitute this into the first equation:

x + 3(x/3) = 150

Simplify:

x + x = 150

2x = 150

x = 75

So there were 75 downloads of the standard version.

To check our answer, we can find the number of downloads for the high-quality version:

y = x/3 = 75/3 = 25

And verify that the total size downloaded is 75 MB + 3(25 MB) = 150 MB.

Therefore, there were 75 downloads of the standard version.

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Each team member should run 9073.6 miles in the race on Course B.

Let's break down the problem step by step:

1. First, we need to find the length of Course B. We know that Course B is 212 times as long as Course A, and Course A is 214 miles. So, to find the length of Course B, we multiply:
  Course B length = 212 × 214 miles

2. Calculate the length of Course B:
  Course B length = 45368 miles

3. Now, we need to determine the distance each team member should run. There are 5 members on the team, and they need to run an equal distance on Course B. To find the distance for each member, we divide the total length of Course B by the number of team members:
  Distance per member = Course B length / number of members

4. Calculate the distance per team member:
  Distance per member = 45368 miles / 5

5. Finally, we find the distance each team member should run:
  Distance per member = 9073.6 miles

So, each team member should run 9073.6 miles in the race on Course B.

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The educational researcher will need a sample size of at least 385 community college students to estimate the mean score on the math part of the SAT with a margin of error of 10 and 95% confidence.

To calculate the sample size required to estimate the mean score on the math part of the SAT of all community college students in the state, we can use the following formula:

n = (Zα/2 × σ / E) ^ 2

where:

n = sample size

Zα/2 = the z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to a z-score of 1.96)

σ = the standard deviation of the population (given as 100)

E = the desired margin of error (given as 10)

Substituting the given values into the formula, we get:

n = (1.96 × 100 / 10) ^ 2

n = 384.16

Rounding up to the nearest integer, we get a sample size of 385. Therefore, the educational researcher will need a sample size of at least 385 community college students to estimate the mean score on the math part of the SAT with a margin of error of 10 and 95% confidence.

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The  Halting Problem is undecidable, it follows that PCP is undecidable over the binary alphabet Σ=[0,1].

The Post Correspondence Problem (PCP) is a decision problem that asks whether there exists a sequence of pairs of strings from a given finite set of pairs which, when concatenated in order, yield the same result for the first and second components of the pairs.

To show that PCP is undecidable over the binary alphabet Σ=[0,1], we can reduce the Halting Problem, which is known to be undecidable, to PCP.

Given a Turing machine M and an input w, we can construct a finite set of pairs of strings S such that there is a sequence of pairs in S that corresponds to an accepting computation of M on w if and only if M halts on w. Specifically, we can construct S such that the first component of each pair encodes a configuration of M on w, and the second component of each pair encodes the next configuration of M on w according to the transition function of M. If M halts on w, there exists a sequence of pairs in S that concatenate to the same string in the first and second components, corresponding to an accepting computation of M. Otherwise, there is no such sequence of pairs.

Since the Halting Problem is undecidable, it follows that PCP is undecidable over the binary alphabet Σ=[0,1].

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1 gallon contains 231 cubic inches. A rectangular container has dimensions of 32 in. by 16 in. by 28 in. A comparable container is one-quarter the size. Larger container holds 61.1 more gallons.

To solve this question, we have to find the volume of both the containers. First, we will find the volume of the container whose sides are 32 in. by 16 in. by 28 in.

Volume = multiplication of sides

= 32 × 16 × 28 in

= 14,336 inches cube

Now, we will find the sides of the smaller container which is scale factor of 1/4

Side 1 of small container = 1/4 × 32 = 8 in

Side 2 of small container = 1/4 × 16 = 4 in

Side 3 of small container = 1/4 × 28 = 7 in

Volume of container which is small = 8 × 4 × 7 = 224 cu in

Now, we will calculate the difference of volume to find the difference in capacity of containers.

Larger container holds more = 14,336 cu in - 224 cu in

= 14,112 cu in

Now, we will convert it into galloons.

231 cu in = 1 galloon

14,112 cu in = 14,112 / 231 galloons

= 61.1 galloons

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Strengths Finder 2.0 is a valuable tool that can help individuals identify and develop their unique strengths. By leveraging these strengths, individuals can achieve greater success in their personal and professional lives, and organizations can create a more positive and productive work environment.

Strengths Finder 2.0 is a popular survey tool that helps individuals identify their unique strengths and talents. The survey has been completed by 10 million people to date and has been adopted by many universities and organizations to assist individuals in identifying their strengths.

The survey consists of a series of questions designed to assess an individual's natural talents and abilities. The results are then used to provide personalized feedback and coaching on how to leverage those strengths in their personal and professional lives.

The use of Strengths Finder 2.0 has become increasingly popular in recent years as individuals and organizations recognize the importance of focusing on strengths rather than weaknesses. By identifying and developing their strengths, individuals can improve their performance, increase their engagement and job satisfaction, and achieve greater success in their careers.

Organizations that use Strengths Finder 2.0 have reported increased productivity, employee engagement, and retention rates. By providing employees with the tools and resources to develop their strengths, organizations can create a more positive and fulfilling work environment.

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The  determinant of this matrix is -67/16.

To find the determinant of a matrix by row reduction to echelon form, we perform elementary row operations until the matrix is in echelon form, and then take the product of the diagonal entries. The determinant is the product of the pivots, with a factor of -1 for each row exchange.

Let's start with Exercise 5:

-1 -3  1
1  4  9
-1  0  5

We can start by adding the first row to the third row to eliminate the -1 in the (3,1) position:

-1 -3  1
1  4  9
0 -3  6

Next, we can multiply the third row by -1/3 to get a pivot in the (3,3) position:

-1 -3  1
1  4  9
0  1 -2

Then we add 3 times the second row to the third row to eliminate the 3 in the (3,2) position:

-1 -3  1
1  4  9
0  0  3

We now have an upper triangular matrix, which is in echelon form. The product of the diagonal entries is -1*4*3 = -12. Since we did not perform any row exchanges, there is no factor of -1. Therefore, the determinant of this matrix is -12.

Now let's move on to Exercise 6:

-1  0  5
3 -32  3
3  3  -1

We can start by adding 3 times the first row to the second row to eliminate the 3 in the (2,1) position:

-1  0   5
0 -32  18
3  3  -1

Then we can add the first row to the third row to eliminate the 3 in the (3,1) position:

-1  0   5
0 -32  18
0  3  4

Next, we can multiply the second row by -1/32 to get a pivot in the (2,2) position:

-1  0   5
0  1  -9/16
0  3   4

Then we add 9/16 times the second row to the third row to eliminate the -9/16 in the (3,2) position:

-1  0  5
0  1  -9/16
0  0  67/16

We now have an upper triangular matrix, which is in echelon form. The product of the diagonal entries is -1*1*67/16 = -67/16. Since we performed two row exchanges, there is a factor of (-1)^2 = 1. Therefore, the determinant of this matrix is -67/16.

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Answer:

N′(−9, −2), M′(−6, −1), O′(−9, −5)

Step-by-step explanation:

it would be N′(−9, −2), M′(−6, −1), O′(−9, −5) because you would subtract 5 from each of the x values

Thus,  there are 720 different Loco Mocos that can be ordered if we consider all six protein options, but this number can vary depending on individual preferences and availability.

A Loco Moco is a popular Hawaiian dish that usually consists of a base of white rice topped with a hamburger patty, a fried egg, and brown gravy.

However, it can be customized by adding different types of proteins such as Portuguese sausage, bacon, turkey, hot dog, salmon, or mahi. So, how many different Loco Mocos can be ordered?

If we consider all six proteins listed in the question, we have six options for the first protein, and then five options left for the second protein (since one has already been used), four options for the third protein, three for the fourth, two for the fifth, and one for the sixth.

Using the multiplication principle, we can calculate the total number of different Loco Mocos as follows:

6 x 5 x 4 x 3 x 2 x 1 = 720

Therefore, there are 720 different Loco Mocos that can be ordered if we consider all six proteins. However, this number can vary depending on the number of protein options available at a particular restaurant or if a customer chooses to only add one or two proteins to their Loco Moco.

In conclusion,  there are 720 different Loco Mocos that can be ordered if we consider all six protein options, but this number can vary depending on individual preferences and availability.

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Thus, working together, Marie and Gloria would take 56/15 hours, or approximately 3.73 hours, to paint the living room.

Let's analyze the situation using the terms "work rate" and "combined work rate" to determine how long it would take Marie and Gloria to paint the living room together.

Marie's work rate is 1/7, as she can complete the job in 7 hours.

Gloria's work rate is 1/8, as she can finish the task in 8 hours. To find their combined work rate, we simply add their individual work rates: (1/7) + (1/8).

To add these fractions, we need a common denominator, which in this case is 56.

So, we can rewrite the fractions as (8/56) + (7/56). Adding them together, we get a combined work rate of 15/56.

Now that we have their combined work rate, we can find out how long it would take them to complete the job together. To do this, we need to find the reciprocal of their combined work rate, which is 56/15.

Therefore, working together, Marie and Gloria would take 56/15 hours, or approximately 3.73 hours, to paint the living room.

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The  integer coordinates will be strictly inside the rectangular region is 80.

We can find the dimensions of the rectangle by taking the absolute value of the difference between the x-coordinates and y-coordinates of two adjacent vertices. In this case, the dimensions are |5 - (-5)| = 10 and |4 - (-4)| = 8.

To find the number of integer coordinates strictly inside the rectangle, we can count the number of lattice points (points with integer coordinates) inside the rectangle using Pick's theorem. Pick's theorem states that the area of a lattice polygon (a polygon whose vertices have integer coordinates) can be found using the formula A = I + {B}\{2} - 1, where A is the area of the polygon, I is the number of lattice points strictly inside the polygon, and B is the number of lattice points on the boundary of the polygon.

In this case, the area of the rectangle is 10 \times 8 = 80. The boundary of the rectangle consists of 10 lattice points on the top and bottom sides and 8 lattice points on the left and right sides, for a total of 36 lattice points on the boundary.

Using Pick's theorem, we can solve for I:

80 = I + {36}÷{2} - 1

80 = I + 18 - 1

I = 63

Therefore, there are 63 lattice points strictly inside the rectangle.

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The displacement and distance traveled depend on the given time interval.

To find the displacement, we need to integrate the velocity function from 0 to 5 or 0 to 5π, depending on the given time interval.

If the time interval is [0,5], we have:

Displacement = ∫[0,5] v(t) dt
Displacement = ∫[0,5] 6cos(t) dt
Displacement = [6sin(t)] from 0 to 5
Displacement = 6sin(5) - 6sin(0)
Displacement ≈ 2.785 meters

If the time interval is [0,5π], we have:

Displacement = ∫[0,5π] v(t) dt
Displacement = ∫[0,5π] 6cos(t) dt
Displacement = [6sin(t)] from 0 to 5π
Displacement = 0 - 6sin(0)
Displacement = 0 meters

To find the distance traveled, we need to integrate the absolute value of the velocity function over the given time interval.

If the time interval is [0,5], we have:

Distance = ∫[0,5] |v(t)| dt
Distance = ∫[0,5] |6cos(t)| dt
Distance = ∫[0,π/2] 6cos(t) dt + ∫[π/2,5] -6cos(t) dt
Distance = [6sin(t)] from 0 to π/2 + [-6sin(t)] from π/2 to 5
Distance = 6 - 6sin(5) ≈ 2.215 meters

If the time interval is [0,5π], we have:

Distance = ∫[0,5π] |v(t)| dt
Distance = ∫[0,5π] |6cos(t)| dt
Distance = ∫[0,π] 6cos(t) dt + ∫[π,2π] -6cos(t) dt + ∫[2π,3π] 6cos(t) dt + ∫[3π,4π] -6cos(t) dt + ∫[4π,5π] 6cos(t) dt
Distance = [6sin(t)] from 0 to π + [-6sin(t)] from π to 2π + [6sin(t)] from 2π to 3π + [-6sin(t)] from 3π to 4π + [6sin(t)] from 4π to 5π
Distance = 0 meters

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