a group of students measure the length and width of a random sample of beans. they are interested in investigating the relationship between the length and width. their summary statistics are displayed in the table below. all units, if applicable, are millimeters. mean width 7.55 standard deviation of width 0.88 mean height 14.737 standard deviation of height 1.845 correlation coefficient 0.916 round your answers to three decimal places. the students are interested in using the width of the beans to predict the height. calculate the slope of the regression equation. write the equation of the line of best fit that can be used to predict bean heights. use to represent width and to represent height. what fraction of the variability in bean heights can be explained by the linear model of bean height vs. width? express your answer as a decimal. if, instead, the students are interested in using the height of the beans to predict the width, calculate the slope of this new regression equation. write the equation of the line of best fit that can be used to predict bean widths. use to represent height and to represent width.

To find positive numbers x and y that satisfy the given condition, we need to minimize the sum x + y while keeping the product xy constant. We can use the Arithmetic Mean-Geometric Mean (AM-GM) Inequality for this problem. The AM-GM inequality states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean of the same numbers.

In this case, we have two numbers, x and y. The arithmetic mean of x and y is (x + y)/2, and the geometric mean is √(xy). According to the AM-GM inequality, we have:

(x + y)/2 ≥ √(xy)

Multiplying both sides by 2, we get:

x + y ≥ 2√(xy)

Now, we want to minimize the sum x + y, which means we need to find the minimum value for the right-hand side of the inequality. The minimum value occurs when the inequality becomes an equality:

x + y = 2√(xy)

To achieve this equality, x must be equal to y (x = y). This is because the arithmetic mean and geometric mean are equal only when all the numbers in the set are equal. Therefore, x = y and the product xy will have the minimum sum. The exact values of x and y will depend on the given constraint for the product xy.

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All the points (1,1), (1,2), (1,3) and (1,4) lie on a same line, because the pair of points have the same slope.

We use the "slope-concept" to determine if the points (1,1), (1,2), (1,3), and (1,4) lie on a straight line.

We know that, slope of line passing through two points (x₁, y₁) and (x₂, y₂) is given by : slope = (y₂ - y₁)/(x₂ - x₁);

If the slope is the same for all pairs of points, then the points lie on the same straight line.

For first two points, (1,1) and (1,2):

⇒ slope = (2 - 1)/(1 - 1) = 1/0;

The slope is undefined and so line passing through (1,1) and (1,2) is vertical.

For second pair of points, (1,2) and (1,3):

⇒ slope = (3 - 2)/(1 - 1) = 1/0;

The slope is undefined and line passing through (1,2) and (1,3) is vertical and the same as the line passing through (1,1) and (1,2).

For third pair of points, (1,3) and (1,4):

⇒ slope = (4 - 3)/(1 - 1) = 1/0;

Once again, the slope is undefined and the line passing through (1,3) and (1,4) is vertical and the same as the previous two lines.

Therefore, we see that all pairs of points have the same "x-coordinate" of 1 and same undefined slope which means that points (1,1), (1,2), (1,3), and (1,4) all lie on same vertical-line.

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The given question is incomplete, the complete question is

Do the points (1,1), (1,2), (1,3) and (1,4) lie on a same line?

We have shown that E(Z) = E(X) + E(Y) for nonnegative random variables X and Y.

The alphabetic character that expresses a numerical value or a number is known as a variable in mathematics. A variable is used to represent an unknown quantity in algebraic equations.

To show that E(Z) = E(X) + E(Y), we need to use the definition of the expected value of a random variable and some properties of max function.

The expected value of a random variable X is defined as E(X) = ∑x P(X = x), where the sum is taken over all possible values of X.

Now, let's consider the random variable Z = max(X, Y). The probability that Z is less than or equal to some number z is the same as the probability that both X and Y are less than or equal to z. In other words, P(Z ≤ z) = P(X ≤ z and Y ≤ z).

Using the fact that X and Y are nonnegative, we can write:

P(Z ≤ z) = P(max(X,Y) ≤ z) = P(X ≤ z and Y ≤ z)

Now, we can apply the distributive property of probability:

P(Z ≤ z) = P(X ≤ z)P(Y ≤ z)

Differentiating both sides of the above equation with respect to z yields:

d/dz P(Z ≤ z) = d/dz [P(X ≤ z)P(Y ≤ z)]

P(Z = z) = P(X ≤ z) d/dz P(Y ≤ z) + P(Y ≤ z) d/dz P(X ≤ z)

Since X and Y are nonnegative, we have d/dz P(X ≤ z) = P(X = z) and d/dz P(Y ≤ z) = P(Y = z). Therefore, we can simplify the above expression as:

P(Z = z) = P(X = z) P(Y ≤ z) + P(Y = z) P(X ≤ z)

Now, we can calculate the expected value of Z as:

E(Z) = ∑z z P(Z = z)

    = ∑z z [P(X = z) P(Y ≤ z) + P(Y = z) P(X ≤ z)]

    = ∑z z P(X = z) P(Y ≤ z) + ∑z z P(Y = z) P(X ≤ z)

Since X and Y are nonnegative, we have:

∑z z P(X = z) P(Y ≤ z) = E(X) P(Y ≤ Z) and

∑z z P(Y = z) P(X ≤ z) = E(Y) P(X ≤ Z)

Substituting these values in the expression for E(Z) above, we get:

E(Z) = E(X) P(Y ≤ Z) + E(Y) P(X ≤ Z)

Finally, we note that P(Y ≤ Z) = P(X ≤ Z) = 1, since Z is defined as the maximum of X and Y. Therefore, we can simplify the above expression as:

E(Z) = E(X) + E(Y)

Thus, we have shown that E(Z) = E(X) + E(Y) for nonnegative random variables X and Y.

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The study aims to investigate whether the presence of popular cartoon characters on food packages affects the food choices of young children aged four to six. The study could be designed as an observational study or an experimental study.

In an observational study, the researchers could observe the children's food choices when presented with food packages with and without popular cartoon characters. The study could be conducted in a naturalistic setting, such as a school cafeteria or a grocery store. However, in an observational study, it may be difficult to control for other factors that could influence children's food choices, such as their previous exposure to the food and their preferences.

In an experimental study, the researchers could randomly assign children to two groups. One group would be presented with food packages with popular cartoon characters, while the other group would be presented with food packages without popular cartoon characters. The researchers could then measure the children's food choices and compare them between the two groups. By randomly assigning children to the groups, the researchers can control for other factors that could influence children's food choices.

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The standard deviation of a sample mean is a measure of how much the sample mean varies from the true population mean.

As the sample size increases, the standard deviation of the sample mean decreases. This is because larger samples are more representative of the population and are less likely to contain extreme values that can skew the sample mean.

The decrease in standard deviation is due to the central limit theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the true population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

This means that the standard deviation of the sample mean decreases as the square root of the sample size increases. Therefore, the correct answer to the question is b) decreases as the sample size increases. This is an important concept to understand in statistical analysis, as it affects the precision and accuracy of estimates made from sample data.

A larger sample size generally leads to more reliable and accurate estimates of population parameters. The standard deviation of a sample mean decreases as the sample size increases. As the sample size gets larger, the sample mean becomes a more accurate estimate of the population mean,

And the spread of the sample means around the population mean becomes smaller. This decrease in variability is captured by the standard deviation of the sample mean, which is calculated as the population standard deviation divided by the square root of the sample size. So, the correct answer is option b.

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Answer:

Use the information I provided below and your own knowledge to finish.  I'm not sure were to go from here, but maybe it has something to do with the 90 degree angle and the other two I just figured out??  I know it has to do with the provided leg, 34, but don't know how to calculate the other to find the hypotenuse, x.

Step-by-step explanation:

All angles add up to 180 degrees.

180 - (90 + 27) = J

180 - 117 = J

63 = J

btw, do you do VLACS too!?

The volume of the composite solid, can be found to be C. 22. 53mm ³.

Seeing as this is a composite solid, we can find the volume by finding the volumes of the constituent shapes which are a triangular prism and a cuboid.

Volume of triangular prism :

= 1 / 2 x base x h x l

= 1 / 2 x 2. 1 x 2. 4 x 2. 9

= 7. 308 mm ³

The volume of the cuboid would be:

= Length x height x width

= 2. 1  x 2. 5 x 2. 9

= 15. 225 mm ³

The volume is therefore:

= 15. 225 + 7. 308

= 22. 53 mm ³

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The volume of given sphere is 3054 cm³

Given that a sphere with diameter of 18 cm we need to find its volume,

Volume of a sphere = 4/3 × π × radius³

So,

Volume = 4/3 × 3.14 × 9³

= 9156.24 / 3

= 3053.08

≈ 3054 cm³

Hence the volume of given sphere is 3054 cm³

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Using a system of equations, the number of hours the artisan worked overtime, y, is 25 hours.

A system of equations is two or more equations solved concurrently.

A system of equations is called simultaneous equations because they are solved at the same time.

The normal hourly rate = sh. 30

The overtime hourly rate = sh. 50

The total hours worked  for a week = 65 hours

The total remuneration for the week = sh. 2,450

Let the normal hours = x

Let the overtime hours = y

x + y = 65 Equation 1

30x + 50y = 2,450 Equation 2

Multiply Equation 1 by 30:

30x + 30y = 1,950 Equation 3

Subtract Equation 3 from Equation 2:

30x + 50y = 2,450

-

30x + 30y = 1,950

20y = 500

y = 25 hours

x = 40 hours

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The probability that exactly 5 students out of 24 randomly chosen students are repeating the module is approximately 25.83%.

We are given that n = 24, k = 5, and p = 0.24. We can plug these values into the formula to calculate the probability:

[tex]P(X = 5) = C(24, 5) * (0.24)^5 * (0.76)^(^2^4^-^5^)[/tex]

First, calculate the binomial coefficient C(24, 5):

[tex]C(24, 5)=\frac{24!}{5!(24-5)!}[/tex]

[tex]C(24, 5) = \frac{24!}{5!19!}[/tex]

C(24, 5) = 42,504

Now, plug the values into the formula:

[tex]P(X = 5) = 42,504 * (0.24)^5 * (0.76)^1^9[/tex]

P(X = 5)

= 0.2583

So, the probability that exactly 5 students out of 24 randomly chosen students are repeating the module is approximately 25.83%.

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The calculated value of the temperature at 2 hours is -6 °C

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

This means that we have

Initial value = 0

Rate of change = 3

The equation of the fuction is represented as

f(x) = Initial value - Rate of change * x

Where

x = hours

substitute the known values in the above equation, so, we have the following representation

f(2) = 0 - 3 * 2

Evaluate

f(2) = -6

Hence, the temperature at 2 hours is -6 °C

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The formula to calculate the standard error of the mean is: Standard error of the mean = σ/√n a) σ=20, n=25
Standard error of the mean = 20/√25 = 4


The formula to calculate the standard error of the mean (SEM) is:
SEM = σ / √n

where σ is the population standard deviation and n is the sample size.

a) σ = 20, n = 25
SEM = 20 / √25
SEM = 20 / 5
SEM = 4.00

b) σ = 20, n = 100
SEM = 20 / √100
SEM = 20 / 10
SEM = 2.00

c) σ = 20, n = 400
SEM = 20 / √400
SEM = 20 / 20
SEM = 1.00

So, the standard error of the mean for each situation is:
a) 4.00
b) 2.00
c) 1.00

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let's divide the whole of 320 by (3 + 5) and then distribute accordingly to each portion

[tex]3~~ : ~~5\implies 3\cdot \frac{320}{3+5}~~ : ~~5\cdot \frac{320}{3+5}\implies 3\cdot 40~~ : ~~5\cdot 40\implies 120~~ : ~~\text{\LARGE 200}[/tex]

A z-score of 2.14 indicates that Katie's score on the test is quite high and unusual, and places her in the top 2% of the scores in the population.

Katie scored a 93 on a test and her calculated z score was 2.14, that means that her score is 2.14 standard deviations above the mean of the test scores.

A z score represents the number of standard deviations a data point is from the mean of the data set.

A positive z score means that the data point is above the mean, while a negative z score means that the data point is below the mean.

The mean of the test scores was, 80 with a standard deviation of 5, then Katie's z score would be calculated as:

z = (x - μ) / σ

= (93 - 80) / 5

= 2.6

Z scores are useful for comparing data points from different data sets or for comparing data points within the same data set that are measured on different scales.

Katie's score is 2.6 standard deviations above the mean.

A z score of 2.14 would mean that Katie's score is slightly below this value, but still significantly above the mean.

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The 95% confidence interval for the population mean is (16.426, 26.574

To find the 95% confidence interval, we can use the formula:

Confidence interval = sample mean ± (z-score)*(population standard deviation/√n)

where the z-score for a 95% confidence level is 1.96.

Plugging in the values given in the question, we get:

Confidence interval = 21.5 ± (1.96)*(19.4/√49)

Simplifying this expression, we get:

Confidence interval = 21.5 ± 5.074

Therefore, the 95% confidence interval for the population mean is (16.426, 26.574) as an open-interval.

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There are six possible arrangements of the letters in the word "mat": "mat", "mta", "am t", "atm", "tam", and "tma". The tree diagram is attached.

To create a tree diagram of all possible arrangements of the letters in the word "MAT," we start with the first letter, which can be either "M," "A," or "T." Then, for each of these possibilities, we add the next letter, and so on until we have included all three letters. The resulting tree diagram shows all possible combinations of the letters in the word "MAT."

There are a total of 6 possible arrangements of the letters in the word "MAT," as we can see from the tree diagram. Out of these 6 arrangements, only one has the letters in the order "M-A-T," which is "MAT." Therefore, the fractional probability of writing the letters in that order is 1/6 or approximately 0.1667.

This means that if we randomly order the letters of "MAT" many times, we can expect to get the letters in the order "M-A-T" about 16.67% of the time.

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The median of the triangle is a segment that runs from the vertex of a triangle to the midpoint of the opposing side.

A median is a line segment that extends from a vertex of a triangle to the midpoint of the opposite side. The centroid has several interesting properties. It is always located inside the triangle, and it divides each median into two segments in a 2:1 ratio.  Additionally, the centroid is the center of mass of the triangle, meaning that if the triangle were a physical object with uniform density, it would balance perfectly on the centroid. The centroid is also equidistant from the three vertices of the triangle.

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It is true that, Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.

Hypothesis testing is a statistical method for making inferences about a population based on sample data.

It involves formulating a null hypothesis, which is a statement about the population that we want to test, and an alternative hypothesis, which is a statement that contradicts the null hypothesis.

Here we have a statement

Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.

The above statement is true

Z-scores are a measure of the distance between a sample mean and the population mean in standard deviation units.

By using z-scores, we can compare a group mean to a population mean and determine whether the difference is statistically significant.

To do this, we calculate the z-score using the formula:

z = (x - μ) / (σ / sqrt(n))

If the resulting z-score falls in the rejection region of the standard normal distribution, we can reject the null hypothesis that the sample mean is not significantly different from the population mean.

In summary, comparing a group mean to a population mean using z-scores is a common statistical technique to determine whether there is a significant difference between the two means.

Therefore,

It is true that, Statistically, using z-scores, a group mean can be compared to a population mean to ascertain whether or not the group mean is different from the population mean.

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We can approach this problem using a recursive strategy. We can't choose 0 or a multiple of 3 for the first digit, and we can't choose a multiple of 3 for the second digit if the first digit is not a multiple of 3)

Let's define a function f(n) as the number of valid strings of length n, where no multiples of 3 are beside one another. We want to find f(10), the number of valid strings of length 10.

To compute f(n) for a given n, we can use the following recursive formula:

f(n) = (8n-1) f(n-1) - g(n-1)

where (8n-1) represents the number of possible choices for the first digit in a string of length n (since the first digit can't be 0, and we can't have any multiples of 3 beside it), f(n-1) represents the number of valid strings of length n-1, and g(n-1) represents the number of invalid strings of length n-1 that end in a multiple of 3.

To compute g(n-1), we need to consider all possible strings of length n-2 that end in a digit that is a multiple of 3, and multiply by the number of valid choices for the next digit. Let h(n-2) be the number of valid strings of length n-2 that end in a digit that is a multiple of 3. Then we can compute g(n-1) as:

g(n-1) = h(n-2) × 2n-1

where the factor of 2n-1 represents the number of choices for the final digit in a string of length n-1 (since we can't choose a multiple of 3).

To compute h(n-2), we can use another recursive formula. Let h(k) be the number of valid strings of length k that end in a digit that is a multiple of 3. Then we can compute h(k) as:

h(k) = f(k-1) - h(k-1)

where f(k-1) represents the total number of valid strings of length k-1 (since any valid string of length k-1 can be extended with a multiple of 3 to form a valid string of length k), and h(k-1) represents the number of invalid strings of length k-1 that end in a multiple of 3.

Using these formulas, we can compute f(10) as follows:

h(1) = 0 (there are no valid strings of length 1 that end in a multiple of 3)

f(1) = 9 (there are 9 possible choices for the first digit)

h(2) = 2 (there are 2 invalid strings of length 2 that end in a multiple of 3: 30 and 60)

f(2) = (8×2-1) f(1) - h(1) = 143 (there are 143 possible choices for the first two digits,

since we can't choose 0 or a multiple of 3 for the first digit, and we can't choose a multiple of 3 for the second or third digit if the first and second digits are not multiples of 3) h(4).

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Ben used wrong the Pythagorean's theorem, the solution that he should have got is x = 2√5

The Pythagorean's theorem says that for a right triangle, the sum of the squares of the cathetus is equal to the square of the hypotenus.

Then the rule that Ben should have written is:

x² + 4² = 6²

So he started wrong.

Solving that we will get:

x = √(6² - 4²) = √20 = 2√5

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Step-by-step explanation:

Let's start by setting up the problem. We know that one side of the rectangle is 11 cm, and we'll call the other side x (which is increasing at a rate of 3 cm per minute). The formula for the area of a rectangle is A = l x w, where l is the length and w is the width. In this case, the length is 11 cm and the width is x. So:

A = 11x

To find how fast the area is changing, we need to take the derivative with respect to time (t):

dA/dt = 11(dx/dt)

Now we can plug in the values we know:

x = 26 cm

dx/dt = 3 cm/min

So:

dA/dt = 11(3) = 33 cm^2/min

Therefore, the area of the rectangle is changing at a rate of 33 cm^2/min at the instant when the other side is 26 cm and increasing at a rate of 3 cm per minute.

The measure of ∠POX  = 160°  The length of XOA = 16.95 mm ,We know that PM¯, PN¯, and PO¯ are angle bisectors of triangle MNO, so they divide the opposite sides in two equal parts. Let x = MY, y = NY, and z = OY. Then, we have:

MX / NO = MY / NY (by the angle bisector theorem)

MX / (MX + XO) = x / (x + y)

MX(x + y) = x(MX + XO)

MXy = XOx

NO / OX = NY / OY (by the angle bisector theorem)

(OX + XO) / OX = y / z

1 + XO/OX = y/z

XO/OX = (z - y)/y

Now, we can use these equations to solve the problem:

To find PX¯, we need to find MX. Using the angle sum property of triangles, we have:

m∠M = 180 - m∠MON = 140°

m∠PMX = m∠M/2 = 70°

m∠PMO = m∠MON/2 = 20°

m∠XMO = m∠PMX + m∠PMO = 70° + 20° = 90°

Therefore, PX¯ is the altitude from M to XO¯, so we have:

tan(30°) = PX / MX

MX = PX / tan(30°)

= 23 / √(3)

= 13.31 mm

To find m∠PMX, we can use the fact that PM¯ is an angle bisector:

m∠PMX = m∠M + m∠PMO

= 140° + 20°

= 160°

To find m∠POX, we can use the fact that PO¯ is an angle bisector:

m∠POX = m∠O + m∠PNO

= 180° - m∠MON + m∠PNO

= 180° - 40° + 20°

= 160°

To find XO¯, we need to find y and z. Using the fact that PX¯ is an angle bisector, we have:

PY / OY = PM / OM

23 / z = 52 / (x + y + z)

y + z = 52z / 23

z = 23y / (52 - 23)

Using the equation XO/OX = (z - y)/y, we have:

XOA / 52 = (23y / (52 - 23) - y) / x

XOA= 52 * 23y / ((52 - 23) * x - 23y)

Substituting MX = 23/√(3) - PX = 23/√(3) - 13.31, we get:

y = NO * PY / (PM + PN + PO) = 56.17 mm

z = OY + PY = 79.17 mm

XOA = 16.95 mm

Therefore, the answers are:

Length of PX¯: 13.31 mm

Measure of ∠PMX: 160°

Measure of ∠PO

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In a cookie dunk experiment, the dependent variable could be the amount of time it takes for the cookie to become fully saturated and break apart.

What is independent variable?

In scientific experiments, an independent variable is the variable that is intentionally changed or manipulated by the experimenter. It is the variable that is being studied to determine its effect on the dependent variable. n summary, the independent variable is the variable that is being changed in the experiment to determine its effect on the dependent variable.

In a cookie dunk experiment, the controlled variable is the variable that is kept constant throughout the experiment. This could be the type of cookie being used, the temperature of the milk, the amount of milk in each glass, or the time each cookie is dunked.

The independent variable is the variable that is intentionally changed by the experimenter. In this case, it could be the type of liquid being used for dunking, such as milk, coffee, or tea.

The dependent variable is the variable that is being measured and observed as a result of changing the independent variable.

Hence, In a cookie dunk experiment, the dependent variable could be the amount of time it takes for the cookie to become fully saturated and break apart.

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The function f(x) = ab^x is an exponential function.

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

f(x) = ab^x

As a general rule, and exponential functtion is represented as

f(x) = ab^x

Where

Hence, the function is an exponential function

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Yes, it is true that if matrix B is formed by adding to one row of matrix A a linear combination of the other rows, then the determinant of B is equal to the determinant of A.

Proven using elementary row operations are operations that can be performed on the rows of a matrix without changing its determinant.

One such operation is adding a multiple of one row to another row.

A linear combination of the other rows to a particular row of A to form B, we can express this operation as a matrix multiplication:

B = P × A

P is a matrix that represents the elementary row operation.

P is an identity matrix with the (i, j)-th entry equal to 1 if i = j or equal to a constant c if i is the row being modified and j is the row being added to, and 0 otherwise.

Since P is an elementary matrix, it has determinant equal to 1 or -1, depending on whether an odd or even number of row swaps were performed.

Therefore,

det(B) = det(P × A) = det(P) × det(A) = ±det(A)

det(P) is either 1 or -1.

P is formed by adding a linear combination of the other rows to a particular row, we can easily see that P is a triangular matrix with 1's on the diagonal, and thus its determinant is equal to 1.

Therefore, we have

det(B) = det(A)

as claimed.

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Answer:

A = (1/2)(4)(5) + 7(7) + 3(2) = 65 mm^2

Yes,

It is true that if A is invertible, then det(A⁽⁻¹⁾) = det(A).

The determinant of a matrix and its inverse are closely related.

The determinant of a matrix is nonzero, then the matrix is invertible, and its inverse has the same determinant as the original matrix.

The following properties of determinants:

det(AB) = det(A)det(B) for any square matrices A and B

If A is invertible, then det(A⁽⁻¹⁾) = 1/det(A)

Using these properties, we can write:

det(A⁽⁻¹⁾) = det(A⁽⁻¹⁾)det(AI) = det(A⁽⁻¹⁾A) = det(I) = 1

And:

det(A) = det(AA⁽⁻¹⁾) = det(A)det(A⁽⁻¹⁾)

Multiplying both sides by det(A⁽⁻¹⁾), we get:

det(A)det(A⁽⁻¹⁾) = det(A⁽⁻¹⁾)det(A⁽⁻¹⁾) = det(A⁽⁻¹⁾)det(A)

det(A⁽⁻¹⁾) = det(A).

The determinant of a matrix and its inverse are closely related.

The determinant of a matrix is nonzero, then the matrix is invertible, and its inverse has the same determinant as the original matrix.

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The maximum value of f (x, y, z) =  [tex](xyz)^{1/3}[/tex] given that x, y, and z are non- negative numbers and x + y + z = 4 is 4/3

Using arithmetic mean and geometric mean relation

Arithmetic mean is dividing the sum of the values of a set by the number of values in the set

Geometric mean is the nth root of the product of the n values

AM ≥ GM

AM = [tex]\frac{x+y+z}{3}[/tex]

GM = [tex](xyz)^{1/3}[/tex]

[tex]\frac{x+y+z}{3}[/tex] ≥ [tex](xyz)^{1/3}[/tex]

x + y + z = 4

[tex]\frac{4}{3}[/tex] ≥ [tex](xyz)^{1/3}[/tex]

maximum possible value is 4/3

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The original weight of the sample is determined as 3.24×10⁻⁶ grams.

If each batch weighs 3.6×10⁻⁷ grams, then the total weight of all 9 batches can be found by multiplying the weight of one batch by the number of batches.

Total weight = 9 × 3.6×10⁻⁷ grams

= 32.4×10⁻⁷ grams

To express this number in scientific notation, we can move the decimal point one place to the left and adjust the exponent accordingly:

Total weight = 3.24×10⁻⁶ grams

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