locate the absolute extrema of the function
on the closed interval

The true statement for all invertible matrices A and B are

1. (In−A)(In+A)=In−A².

2. (AB)⁻¹=A⁻¹B⁻¹

4. A⁷ is invertible.

The given statement is true for all invertible matrices A. To prove this statement, we can expand the left-hand side of the equation as follows:

(In−A)(In+A) = In(In) + In(A) − A(In) − A(A)

= In² + InA − AIn − A²

= In + InA − AIn − A²

= In − A²

Therefore, we have shown that (In−A)(In+A)=In−A2 is true for all invertible matrices A.

The statement is true for all invertible matrices A and B. To prove this statement, we can use the definition of the inverse of a matrix. The inverse of a matrix A is a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I, where I is the identity matrix. Using this definition, we can show that:

(AB)(A⁻¹B⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = AA⁻¹ = I

(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹IB = BB⁻¹ = I

Therefore, we have shown that (AB)⁻¹ = A⁻¹B⁻¹ is true for all invertible matrices A and B.

The statement is false in general. For instance, consider the matrices A = [1 0] and B = [−1 0]. Both A and B are invertible matrices, but A + B = [0 0] which is not invertible as it is not a full rank matrix.

The statement is true for all invertible matrices A. To prove this statement, we can use the fact that the product of invertible matrices is also invertible. Since A is invertible, we can write:

A⁷ = AAAA...A

= A⁶A

= (A⁻¹)⁻¹A⁶A

= (A⁻¹A)⁻¹A⁶A

= IA⁶A

= A⁶

We can repeat this process until we get A⁷ = (A⁻¹)⁻¹. Thus, A⁷ is invertible for all invertible matrices A.

The statement is false in general. To show this, we can use a counterexample. Let A = [1 0] and B = [0 −1]. Then,

(A + B)² = [1 −1][1 −1]

= [0 0]

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The probability of choosing a letter from n to q would be = 2/5

Probability is defined as the expression that is used to represent the possibility of an outcome of an event which can be solved with the formula = chosen events/ total outcomes.

The number of cards in the box = 10

The various cards are labelled as follows= j, k, l, m, n, o, p, q, r, and s.

The number of cards from n to q = 4

Therefore the probability that a number from n to q will be chosen = 4/10 = 2/5.

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Where the above Terms exists, the amount to be paid in each of the above invoices are given as follows;

Note that the terms are defined as follows;

Terms A: 2/10, n/60 (2% discount if paid within 10 days, net due in 60 days)

Terms B: 1/15, EOM (1% discount if paid within 15 days, end of month terms)

Terms C: 1/10, n/30 (1% discount if paid within 10 days, net due in 30 days)

Terms D: 3/15, n/45 (3% discount if paid within 15 days, net due in 45 days.

So to compute the amount to be paid for each of the four separate invoices assuming that all invoices are paid within the discount period, we need to calculate the amount of the discount and subtract it from the gross merchandise amount.

For terms 2/10, n/60:

Discount = 2% of $8,000 = $160

Amount to be paid = $8,000 - $160
= $7,840

For terms 1/15, EOM:

Discount = 1% of $24,500 = $245

Amount to be paid = $24,500 - $245
= $24,255

For terms 1/10, n/30:

Discount = 1% of $81,000 = $810

Amount to be paid = $81,000 - $810
= $80,190

For terms 3/15, n/45:

Discount = 3% of $17,500 = $525

Amount to be paid = $17,500 - $525
= $16,975

Therefore, the amount to be paid for each of the four separate invoices assuming that all invoices are paid within the discount period are:

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The common ratio (r) of this geometric sequence is found by dividing any term by its preceding term, such as:

r = a2/a1 = 6/10 = 0.6

We can use this common ratio to find any term in the sequence using the recursive formula:

a(n) = r * a(n-1)

where a(1) is the first term in the sequence, a(n) is the nth term, and a(n-1) is the (n-1)th term

Using this formula, we can find any term in the sequence. For example:

a(2) = r * a(1) = 0.6 * 10 = 6

a(3) = r * a(2) = 0.6 * 6 = 3.6

a(4) = r * a(3) = 0.6 * 3.6 = 2.16

and so on

Therefore, the complete recursive formula for this geometric sequence is:

a(n) = 0.6 * a(n-1), where a(1) = 10 and a(n) = a(n-1) for all n > 1

Answer:

A. The volume of the triangular prism can be calculated using the formula V = (1/2)bh × h, where b is the base of the triangular face and h is the height of the prism. Thus, V = (1/2)(7 cm)(10 cm) × 20 cm = 700 cubic centimeters.

B. If the height of the prism is tripled to 60 cm, then the new volume would be V' = (1/2)(7 cm)(10 cm) × 60 cm = 2100 cubic centimeters. Thus, the volume is tripled.

C. If the base of the triangular face is tripled to 21 cm, then the new volume would be V' = (1/2)(21 cm)(10 cm) × 20 cm = 2100 cubic centimeters. Thus, the volume is tripled.

D. If the height of the triangular face is tripled to 30 cm, then the new volume would be V' = (1/2)(7 cm)(30 cm) × 20 cm = 2100 cubic centimeters. Thus, the volume is tripled.

E. If all three dimensions (base, height of triangular face, and height of prism) are tripled, then the new volume would be V' = (1/2)(21 cm)(30 cm) × 60 cm = 18900 cubic centimeters. Thus, the volume is multiplied by a factor of 27.

By answering the presented question, we may conclude that Therefore, 1  expressions cm on the map corresponds to a real distance of 3.2 km. 

In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.

Scale 1:

320000 means that 1 unit on the map represents his 320000 units in the real world.

To find the actual distance represented by 1 cm on the map, you need to convert the units to the same scale.

1 kilometer = 100000 cm

So,

1 unit on the map = 320000 units in the real world

1 cm on the map = (1/100000) km in the real world

Multiplying both sides by 1 cm gives:

1 cm on the map = (1/100000) km * 320000

A simplification of this expression:

1 cm on the map = 3.2 km

Therefore, 1 cm on the map corresponds to a real distance of 3.2 km. 

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Answer:

24 356 ÷ 5 using long division.

Step-by-step explanation:

See the image

Answer:

x = 17

Step-by-step explanation:

Subtract 6x from both sides:

2x - 18 = 16

Add 18 on both sides to isolate the variable:

2x = 18 + 16

2x = 34

Divide by 2: x = 17

The margin of error in a confidence interval for the population mean includes the standard error of the sample mean and the z or t value associated with a certain level of confidence, usually 95% or 99%. Therefore, the correct answer is (C).

Sampling bias and nonresponse bias are potential sources of error in survey or study design and data collection, but they are not directly related to the construction of a confidence interval.

The desired level of confidence is a key input for determining the z or t value used in the calculation of the margin of error, but it is not included in the margin of error itself. The correct answer is  z or t value associated with a 95% confidence level.

So, the correct option is (C).

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Answer:

10,000

Step-by-step explanation:

The first step is to divide 500 hats by 2 hours (500 ÷ 2 = 250)

Second, multiply 250 hats by 40 hours (250 x 40 = 10,000)

Conversely, as the confidence level decreases, the margin of error becomes smaller, and the confidence interval becomes narrower.

In statistics, a confidence interval is a range of values that is likely to contain the true value of a population parameter (such as a mean or a proportion), based on a sample from that population. The confidence interval is typically expressed as an interval around a sample statistic, such as a mean or a proportion, and is calculated using a specified level of confidence, typically 90%, 95%, or 99%.

Here,

To develop a confidence interval, we need to use the following formula:

Confidence Interval = sample mean ± margin of error

where the margin of error is calculated as:

Margin of Error = z* (sample standard deviation/ √n)

where z* is the critical value from the standard normal distribution table based on the chosen confidence level.

a. For a 90% confidence interval, the critical value (z*) is 1.645. Thus, the margin of error is:

Margin of Error = 1.645 * (4 / √25) = 1.317

So, the 90% confidence interval for the population mean is:

30 ± 1.317, or (28.683, 31.317)

b. For a 95% confidence interval, the critical value (z*) is 1.96. Thus, the margin of error is:

Margin of Error = 1.96 * (4 / √25) = 1.568

So, the 95% confidence interval for the population mean is:

30 ± 1.568, or (28.432, 31.568)

c. For a 99% confidence interval, the critical value (z*) is 2.576. Thus, the margin of error is:

Margin of Error = 2.576 * (4 / √25) = 2.0656

So, the 99% confidence interval for the population mean is:

30 ± 2.0656, or (27.9344, 32.0656)

d. As the confidence level increases, the margin of error also increases, because we need to be more certain that our interval includes the true population mean. This means that the confidence interval becomes wider as the confidence level increases.

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Answer:

0.54545454545

Step-by-step explanation:

Answer:

-8: (-4, -3]

-7: (-3, -1]

(-5, -1]: (-5, -1]

(-4, -1]: (-4, -1]

[-3, 1]: [-3, 1]

(2, 1): (1, 2]

The equation for a linear regression model is represented as: y = B0 + B1x + error.

What is Slope ?

Slope refers to the measure of steepness of a line. It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. It is also called the gradient of the line.

In statistics, a linear regression model is used to describe the relationship between a dependent variable (usually denoted as "y") and one or more independent variables (usually denoted as "x").

The equation for a linear regression model is represented as: y = B0 + B1x + error

where B0 is the y-intercept, B1 is the slope of the line relating y and x, and error is the random error term. The population parameters B0 and B1 are the true values of the y-intercept and slope that would be obtained if the entire population were studied, rather than just a sample.

These parameters are estimated using the sample data, and the resulting estimates are used to construct the regression equation. The estimates of the population parameters are based on the assumption that the sample is representative of the population, and that the errors are normally distributed with constant variance.

Therefore, The equation for a linear regression model is represented as: y = B0 + B1x + error

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Answer:

3mn^4 - 6m^2n^4

a) The expression for the stable fixed point is P* = (1 + sqrt(1 - 4h)) / 2

b) There exists a fixed point for all values of h less than or equal to 1/4.

The fixed points of the model are the values of P at which dP/dt = 0. Therefore, we need to solve the equation

P(1-P) - h = 0

Expanding the left-hand side, we get

P - P^2 - h = 0

Rearranging, we get a quadratic equation

P^2 - P + h = 0

Using the quadratic formula, the two solutions for P are

P = (1 ± sqrt(1 - 4h)) / 2

a) The larger root is the stable fixed point, as it corresponds to a minimum of the fish population growth function. Therefore, the expression for the stable fixed point is

P* = (1 + sqrt(1 - 4h)) / 2

b) For the model to have a fixed point, the quadratic equation must have real roots. This occurs when the discriminant (the expression inside the square root) is non-negative

1 - 4h ≥ 0

Solving for h, we get

h ≤ 1/4

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The given question is incomplete, the complete question is:

This problem explores some questions regarding the fishery model

dP/dt =P(1−P)−h

If you have not yet run the Jupyter notebook please do so now. Find analytical expressions for the two fixed points of the model, in terms of h

a) Give an expression for the stable fixed point. You may assume that the larger fixed point is the stable one. b) For what values of h does there exist a fixed point?

Answer:

[tex]\sqrt {45}[/tex]

Step-by-step explanation:

Distance between two points, (x1, y1) and (x2, y2)  in a 2D cartesian coordinate is given by
[tex]d = \sqrt {(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]

here the points are (6, - 2) and (3, 4)

We get

[tex]d = \sqrt {(3 - 6)^2 + (4 - (-2))^2}\\\\d = \sqrt {(-3)^2 + (6)^2}\\\\d = \sqrt {{9} + {36}}\\\\d = \sqrt {45}[/tex]

The cardinality of set A, n(A) = 29

The cardinality of a set is the total number of elements in the set

Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.

Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9

= 29

So, n(A) = 29

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Answer:

497/√302 = 49.7

z-score = (44-0)/49.7 = 0.88

Probability = 0.8133

We can represent the number of elements in a set using the symbol |A| or n(A). So, we can say |A| = 5 or n(A) = 5. We can make generalizations about the equality of sets based on the definition above.

In mathematics, a set is a collection of distinct objects, which can be anything such as numbers, letters, or other mathematical objects. These objects are called the elements or members of the set. Sets can be defined using various methods such as listing the elements, describing properties of the elements, or using set-builder notation. For example, we can define the set of even numbers using set-builder notation as {x | x is an integer and x is divisible by 2}. Sets play a fundamental role in various branches of mathematics, including algebra, geometry, and analysis. They are used to define mathematical structures such as groups, rings, and fields, and to study various mathematical concepts such as functions, relations, and cardinality.

Here,

11.1.4 To determine the number of elements of a set, you need to count the number of distinct elements in the set. For example, if you have a set A = {1, 2, 3, 4, 5}, then the number of elements in set A is 5.

11.1.5 Two sets are equal if they have exactly the same elements. For example, if we have set A = {1, 2, 3} and set B = {3, 2, 1}, then A and B are equal sets because they contain the same elements, even though the order of the elements is different. We can write this as A = B. If two sets are not equal, then we use the symbol ≠ to denote inequality. For example, if set A = {1, 2, 3} and set B = {4, 5, 6}, then A ≠ B.

Some of these generalizations are:

Two sets are equal if and only if they have the same elements.

The order of the elements in a set does not matter for equality.

If two sets have different elements, then they are not equal.

If two sets have the same elements, but with different multiplicities, then they are not equal.

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Answer:

47 the answer is simply 47

Step-by-step explanation:

To find the width of the strip of uncovered floor, we need to subtract the area of the covered floor from the total area of the room, and then divide by the width of the strip.

The total area of the room is:

9 ft x 12 ft = 108 ft^2

Let's assume the width of the strip is x.

Then the dimensions of the covered floor are:

Length = 9 - 2x Width = 12 - 2x

The area of the covered floor is:

(9 - 2x) x (12 - 2x) = 108 - 30x + 4x^2

We know that the area of the uncovered floor is 270 ft^2, so we can set up the equation:

108 - 30x + 4x^2 = 270

Simplifying and rearranging:

4x^2 - 30x - 162 = 0

Dividing by 2:

2x^2 - 15x - 81 = 0

Using the quadratic formula:

x = [15 ± sqrt(15^2 + 4(2)(81))]/4

x = [15 ± sqrt(1089)]/4

x = [15 ± 33]/4

x = 12 or x = -3/2

Since the width of the strip cannot be negative, we can discard the negative root, and the width of the strip is:

x = 12/2 = 6 ft

Therefore, the strip of uncovered floor is 6 ft wide.

The appropriate 0.5 adjusted formula for normal approximation is option (d) p(x > 8.5)

The appropriate 0.5 adjusted formula for normal approximation to approximate binomial probabilities when n is large is

P(Z > (x + 0.5 - np) / sqrt(np(1-p)))

where Z is the standard normal variable, x is the number of successes, n is the number of trials, and p is the probability of success in each trial.

To approximate binomial probability p(x > 8) when n is large, we need to use the continuity correction and find the appropriate 0.5 adjusted formula for normal approximation. Here, x = 8, n is large, and p is unknown. We first need to find the value of p.

Assuming a binomial distribution, the mean is np and the variance is np(1-p). Since n is large, we can use the following approximation

np = mean = 8, and

np(1-p) = variance = npq

8q = npq

q = 0.875

p = 1 - q = 0.125

Now, using the continuity correction, we adjust the inequality to p(x > 8) = p(x > 8.5 - 0.5)

P(Z > (8.5 - 0.5 - 8∙0.125) / sqrt(8∙0.125∙0.875))

= P(Z > 0.5 / 0.666)

= P(Z > 0.75)

Therefore, the correct option is (d) p(x > 8.5)

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The given question is incomplete, the complete question is:

To approximate binomial probability p(x > 8) when n is large, identify the appropriate 0.5 adjusted formula for normal approximation. a) p(x > 7.5) b)  p(x >= 9) c) p(x > 9) d) p(x > 8.5)

Step-by-step explanation:

To find the maximum/minimum value of the quadratic function q² + 22q = y - 85, we can complete the square as follows:

q² + 22q = y - 85

q² + 22q + 121 = y - 85 + 121 (adding (22/2)² = 121 to both sides)

(q + 11)² = y + 36

Now, we have a square of a binomial on the left side, which means the minimum value of the quadratic function is y + 36, and it occurs when (q + 11) = 0. Thus, the minimum value is:

y + 36 = 36 - 85 = -49

Similarly, the maximum value of the quadratic function occurs when (q + 11) = 0, but this time the value of y will be as large as possible. The largest possible value of y occurs when STU is a permutation of the digits 9, 8, and 7 (since PQR must be 4, 0, and 3 in some order). Thus, the maximum value is:

y + 36 = 9 + 8 + 7 - 85 + 36 = -25

Therefore, the options are incorrect and the correct answers are:

Minimum value: -49

Maximum value: -25

Answer:

$18.03

Step-by-step explanation:

10.30 x 1.7 = 18.025

round answer to 18.3

Answer: $18.03

Step-by-step explanation:

1.75 x 10.3 = 18.025

18.025 ≈ 18.03

Var(y) = E(y2) – [E(y)]2

E(y2) = 3² * 0.2 + 5² * 0.3 + 6² * 0.4 + 9² * 0.1 = 68.6

E(y) = 3 * 0.2 + 5 * 0.3 + 6 * 0.4 + 9 * 0.1 = 6.2

Var(y) = 68.6 – 6.22 = 37.44

σ = √Var(y) = √37.44 = 6.11

a) D. No. A person can be both male and bet on professional sports at the same time

b) If the events A and B are independent, P(A&B) = P(A) x P(B)

P(male and also bets on professional sports) = 0.484x0.13 = 0.0629

c) P(male or bets in professional sports) = P(male) + P(bets in professional sports) - P(male and also bets on professional sports)

= 0.484 + 0.13 - 0.0629

= 0.5511

d) A. The assumption was incorrect and the events are not independent

(if the were independent, the percentage would have been 6.29)

e) A. P(male or bets on professional sports = 0.484 + 0.13 - 0.081

= 0.5330

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Using graphs, we can see that the point (4,2) can be a coordinate where y will represent x.

The graph is simply a structured representation of the data. The numerical information gathered through observation is referred to as data.

If there is just one value of y (output) for every value of x, the relationship between x and y is said to be a function (input).

In other words, there can only be one value of y for each value of x.

Determine each plotted point's coordinates first:

(-4,4)

(-2,3)

(0,1)

(2, -1)

(3,0)

The following point cannot have any of the x-coordinates of the displayed points, which are -4, -2, 0, 2, and 3.

Options include:

A (0,1) →The relationship cannot be regarded as a function at this stage as the x-coordinate zero already has a corresponding value of y.

B (2,2) →Although there is already a value of y for the location x=2, the relationship cannot be regarded as a function at this point.

C (3,4) →Although there is already a value of y for the location x=3, the relationship cannot be regarded as a function at this point.

D (4,2) → The relationship will still be regarded as a function even though there are no points on the graph with the coordinates x=4 displayed.

Therefore, option D (4,2) is the point where y will represent x.

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The complete question:

Please see attached question