A committee of size 5 is selected from a group of 8 math majors and 14 computer science majors. (a) How many different committees are possible? (b) How many committees are possible if exactly 2 math majors must be on the committee? (c) How many committees are possible if the computer science majors must outnumber the math majors on the committee?

It will take Mthwetew and Andy 1.875 days to build the block wall if they work together.

Hello! I understand that you'd like to know how long it will take Mthwetew and Andy to build a block wall if they work together. To answer this question, we will use the concept of work rates.
Mthwetew can build a wall in 3 days. This means Mthwetew's work rate is 1/3 (wall/day).
Andy can build the same wall in 5 days. This means Andy's work rate is 1/5 (wall/day).
To find out how long it will take them to build the wall together, we will add their work rates and solve for the time it would take them to complete the task.
Combined work rate: (1/3) + (1/5)
To add these fractions, find a common denominator, which in this case is 15.
(1/3) * (5/5) = 5/15
(1/5) * (3/3) = 3/15
Now, add the numerators:
5/15 + 3/15 = 8/15 (wall/day)
Their combined work rate is 8/15 (wall/day). To find out how long it takes them to build the wall together, we need to calculate the reciprocal of their combined work rate:
15/8 = 1.875 days
So, it will take Mthwetew and Andy 1.875 days to build the block wall if they work together.

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The equation of the ellipse is: [tex](x^2/27) + (y^2/36) = 1[/tex]

What is equation?

An equation is a statement in mathematics asserting that two expressions have the same value. It consists of different mathematical operations like addition, subtraction, multiplication, and division, as well as constants and variables.

The equation for an ellipse with vertical major axis and center at the origin is given by:

[tex](x^2/b^2) + (y^2/a^2) = 1[/tex]

where a is the length of the semi-major axis and b is the length of the semi-minor axis.

For an ellipse with eccentricity e, we have:

[tex]e = sqrt(1 - (b^2/a^2))[/tex]

In this case, the foci are at (0, +3) and (0, -3), which means that the distance between the foci is:

2c = 6

And since the eccentricity is 1/2, we have:

e = 1/2 = c/a

Solving for c, we get:

c = a/2

Substituting this into the equation for the distance between the foci, we get:

2c = 6

2(a/2) = 6

a = 6

Now we can find b using the equation for eccentricity:

[tex]e = \sqrt{(1 - (b^2/a^2))}\\\\1/2 = \sqrt{(1 - (b^2/36))}\\\\1/4 = 1 - (b^2/36)\\\\b^2/36 = 3/4\\\\b^2 = 27\\\\b = \sqrt{(27)}[/tex]

Therefore, the equation of the ellipse is:

[tex](x^2/27) + (y^2/36) = 1[/tex]

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An arrow diagram and give the range of function of f: {0, 1}² → {0, 1}³ is illustrated below.

An arrow diagram is a visual representation of a function that helps us understand how the elements of its domain map to the elements of its codomain. In this explanation, we will use arrow diagrams to represent two different functions and determine their ranges.

Consider the function f: {0, 1}² → {0, 1}³.

This means that f takes inputs from the set {0, 1}² (which contains four elements: 00, 01, 10, and 11) and outputs elements from the set {0, 1}³ (which contains eight elements: 000, 001, 010, 011, 100, 101, 110, and 111).

To create an arrow diagram for this function, we draw a box representing the domain on the left and a box representing the codomain on the right.

Then, we draw arrows connecting each element in the domain to its corresponding output in the codomain. In this case, the function f is defined as f(x) = x0, where x0 denotes the first bit of x. Therefore, the arrow diagram for f would look like the following.

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Answer:

The statement 5.5 + 0 = 5.5 is an example of the additive identity property. This property states that when you add zero to any number, the sum is that same number. In other words, zero is the "identity" element for addition because it does not change the value of the other number.

In this case, adding zero to 5.5 results in 5.5 because 0 doesn't add or subtract anything from 5.5. So, the sum is simply 5.5, which is the original number.

The associative property of addition states that you can change the grouping of the numbers being added without changing the result. The commutative property of addition states that you can change the order of the numbers being added without changing the result. However, neither of these properties apply to the expression 5.5 + 0 because there is only one number being added and no grouping or order to change.

therefore the correct answer is Additive Identity.

The predicted mean franchise value for a sports team that generates $200 million of annual revenue is $450 million (rounded to the nearest integer).

To predict the mean franchise value (in millions of dollars) of a sports team that generates $200 million of annual revenue, we can use the linear regression equation:

franchise value = a + b * annual revenue

where a is the intercept and b is the slope of the regression line.

From the information given in the problem, we know that:

When the annual revenue is zero, the mean franchise value is $50 million, so the intercept is a = 50.

When the annual revenue increases by $1 million, the franchise value is expected to increase by $2 million, so the slope is b = 2.

Substituting these values into the regression equation, we get:

franchise value = 50 + 2 * annual revenue

To predict the mean franchise value for an annual revenue of $200 million, we plug in this value into the equation:

franchise value = 50 + 2 * 200 = 450

Therefore, the predicted mean franchise value for a sports team that generates $200 million of annual revenue is $450 million (rounded to the nearest integer).

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Cannot conclude mean pH < 7.0, p-value = 0.088.95% CI for mean pH: 6.44 to 6.92.

In view of the given data, we can play out a speculation test to decide if we can reason that the mean pH is under 7.0. Utilizing a one-followed t-test with an importance level of 0.05 and 5 levels of opportunity, we compute a t-measurement of - 1.475 and a p-worth of 0.088. Since the p-esteem is more prominent than the importance level, we neglect to dismiss the invalid speculation and reason that there isn't sufficient proof to help the case that the mean pH is under 7.0.

To find a 95% certainty span for the mean pH, we can utilize the recipe x ± tα/2 * (s/√n), where x is the example mean, s is the example standard deviation, n is the example size, and tα/2 is the t-an incentive for a two-followed test with a 95% certainty level and n-1 levels of opportunity. Connecting the qualities, we view the certainty stretch as 6.44 to 6.92. This implies we can be 95% certain that the genuine mean pH of the slop buildups falls inside this reach.

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Two children are both 26.25 inches tall, but one has a head circumference of 17.2 inches and the other has a head circumference of 17.4 inches. The circumference is the distance around the edge of a circular object. It is the measurement of the perimeter of a circle.

The formula for calculating the circumference of a circle is:

C = 2πr

where C is the circumference, π (pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle (the distance from the center of the circle to any point on the edge).

This can be because every individual, including children, have unique body proportions. Factors such as genetics, nutrition, and health can influence the differences in head circumferences, even if the children have the same height. So, it is quite normal for two children with the same height to have different head circumferences.

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For a function f(x, y, z) = x²y + y²z + z²x and the point P = (1, 2, 3).

a) The gradient of function at P is equal to the 20.92.

b) Rate of change in the direction in vector v = (2, 2, -1) at P is equal to the 14.

c) The maximum rate of change of f at P is equal to the 20.92.

We have a function, f(x, y, z) = x²y + y²z + z²x and the point P = (1, 2, 3). Check. the function and separate all three vector parts of it that is fₓ = x²y + z²x

fᵧ = y²z + x²y

fz = y²z + z²x

So,[tex] f(x, y, z) = (x²y + z²x)\hat i + ( y²z + x²y)\hat y + (y²z + z²x) \hat z \\ [/tex]a) The gradient of any function say f, denoted as ∇ f and it is the collection of all its partial derivatives into a vector. The gradient of function is [tex]∇ f = ( \frac{∂ }{∂x }\hat i + \frac{∂ }{∂ y }\hat j + \frac{∂}{∂ z }\hat k) f(x,y,z) \\ [/tex]

[tex]= ( \frac{∂( x²y + z²x ) }{∂x}, \frac{∂(x²y + y²z)}{∂y}, \frac{∂ (z²x + y²z)})\\ [/tex]

( since dot product of i.j = j.k = k.i = 1)

= (2xy + z² , x² + 2yz ,2zx + y²)

At point P = (1,2,3) = (2× 1× 2 + 4 , 1 + 2×2×3 , 4 + 2×3×1 ) = ( 13, 13,10)

b) As we know, gradient is work as slope ( The rate of change of dependent variable with respect to independent). Here, v = (2, 2, -1), and

fₓ = x²y + z²x fᵧ = y²z + x²y ; fz = y²z + z²x.

unit vector is written as [tex] \vec u = (\frac{2}{3},\frac{2}{3}, \frac{1}{3} )[/tex]

So, direction derivative, [tex]D_u = ( \frac{2}{3})f_x + \frac{2}{3}f_y + \frac{1}{3}f_z [/tex]

= [tex] ( \frac{2}{3})(x²y + z²x) + \frac{2}{3}( y²z + x²y) + \frac{1}{3}(y²z + z²x) \\ [/tex]

At point P, [tex]D_4 f(1,2,3,) = ( \frac{2}{3})(13) + \frac{2}{3}( 13) + \frac{1}{3}(10) \\ [/tex] = 14

c) the maximum rate of change of f at P is same as 20.92. Hence required value is 20.92.

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For the first question, the statement is true. Since the probability of any single value for a continuous random variable is infinitesimally small, we say that the probability of X taking any specific value x is equal to 0.

For the second question, the statement is true. The area under the PDF curve must be equal to 1 since the total probability of all possible values of X must add up to 1.
For the third question, the statement is false. For continuous random variables, P(X ≤ x) is equal to P(X < x) since the probability of X taking any specific value is infinitesimally small.
For the fourth question, the statement is false. The antiderivative of the PDF is actually the CDF, since the CDF is the integral of the PDF. So, the antiderivative of the PDF will always be equal to the CDF.

1. For continuous distributions, P(X = x) = 0 for all x in the support of X. This statement is TRUE. In continuous distributions, the probability of any single point is always 0 because the possible values for X are infinitely many.
2. For f(x) to be a valid PDF, integrating f(x) dx over the support of X must be equal to 1. This statement is TRUE. A probability density function (PDF) must integrate to 1 over the support of the random variable, as this represents the total probability of all possible outcomes.
3. For continuous random variables, P( X ≤ x) doesn't equal P( X < x), similar to how probabilities work with discrete random variables. This statement is FALSE. For continuous random variables, P( X ≤ x) = P( X < x) because the probability of any single point is 0, so including or excluding it does not change the probability.
4. The antiderivative of the PDF will not always equal the CDF. This statement is FALSE. The cumulative distribution function (CDF) is the antiderivative of the PDF, as the CDF represents the probability that the random variable is less than or equal to a certain value, and this is obtained by integrating the PDF up to that point.

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By the Integral Test, the series ∑ (n=2 to infinity) 1/[n*(ln n)^(3/2)] converges.

To determine whether the series ∑ (n=2 to infinity) 1/[n*(ln n)^(3/2)] converges or diverges, we can use the Integral Test.

Step 1: Check if the function is positive, continuous, and decreasing for n ≥ 2.
The function f(n) = 1/[n*(ln n)^(3/2)] is positive, continuous, and decreasing for n ≥ 2.

Step 2: Evaluate the improper integral.
Consider the integral ∫ (from 2 to infinity) 1/[x*(ln x)^(3/2)] dx.

Step 3: Apply substitution.
Let u = ln x, so du = (1/x) dx. When x = 2, u = ln 2, and as x approaches infinity, u also approaches infinity. Now, the integral becomes:

∫ (from ln 2 to infinity) 1/(u^(3/2)) du.

Step 4: Evaluate the integral.
∫ 1/(u^(3/2)) du = ∫ u^(-3/2) du = [2/(-1/2)] * u^(-1/2) evaluated from ln 2 to infinity.

Step 5: Determine if the integral converges or diverges.
As u approaches infinity, the term u^(-1/2) approaches 0, so the integral converges.

Since the integral converges, by the Integral Test, the series ∑ (n=2 to infinity) 1/[n*(ln n)^(3/2)] also converges.

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The given statement "The apparent horizon is always visible in a "low oblique" photo" is false because the visible horizon may be below the apparent horizon due to the curvature of the Earth, which means that the apparent horizon is not visible.

The apparent horizon is the theoretical line that separates the visible sky and the hidden sky, which is the part of the sky that is blocked by the Earth's curvature. It is always located at eye level, regardless of the observer's altitude.

Therefore, in a "low oblique" photo, if the visible horizon is below eye level, the apparent horizon will not be visible in the photo. Conversely, in a "high oblique" photo taken from a high altitude and at an angle, the apparent horizon may be visible because the visible horizon is above eye level.

Therefore, the given statement is false.

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Random sampling errors may arise because different random samples can produce different results due to the different people being surveyed.

Based on the news article, the poll may have practical difficulties that cause errors in addition to the 63 percentage point margin of error. These difficulties include undercoverage, response bias, nonresponse, and random sampling errors. Undercoverage refers to the possibility of systematically leaving out certain groups of people who should have been included in the sample. Response bias may occur when people answer questions in a way that they think is socially acceptable, rather than their true opinion. Nonresponse may occur when some people are not available or refuse to answer the poll. Finally, random sampling errors may arise because different random samples can produce different results due to the different people being surveyed.

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Answer:

1125

Step-by-step explanation:

First do  (5)^3 = 125

then 3 x 125= 375

after that you do 3 x 375 = 1125

Answer:

The answer to your problem is, D. [2,3,8]

Step-by-step explanation:

In our question, we would need to remember that domain is set of first coordinates of it.

Thus it would be: [2,3,8]

( Always find the domain first then answer )

Answer of differential functions are;

1. dy/dt = (-14/9y)

2. dx/dt = (-1/x⁶)

1. Using implicit differentiation, we can find:
9x⁷y = 18

Taking the derivative on both sides:
63x⁶(dx/dt)(y) + 9x⁷(dy/dt) = 0

Simplifying and solving for dy/dt:
dy/dt = (-7/9)(dx/dt)(x⁶/y)

Substituting dx/dt = 2 and x = 1, we get:
dy/dt = (-7/9)(2)(1⁶/y)
dy/dt = (-14/9y)

2. Using the same implicit differentiation approach:
9x⁷y = 18

Taking the derivative on both sides:
63x⁶(dx/dt)(y) + 9x⁷(dy/dt) = 0

Simplifying and solving for dx/dt:
dx/dt = (-1/6)(dy/dt)(y/x⁶)Substituting dy/dt = 3 and y = 2, we get:
dx/dt = (-1/6)(3)(2/x⁶)
dx/dt = (-1/x⁶)

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To find the percentage rate of change of f(x) at x=9, we first need to calculate the derivative of f(x). Using the power rule of differentiation, we get: f'(x) = 32.

This means that the rate of change of f(x) is constant and equal to 32. To find the percentage rate of change at x=9, we can calculate the ratio of the change in f(x) to the original value of f(x), and then multiply by 100 to get the percentage.
So, the change in f(x) from x=9 to x=9+Δx is: f(9+Δx) - f(9) = 32Δx. And the original value of f(x) at x=9 is: f(9) = 126 + 32(9) = 414.

Therefore, the percentage rate of change of f(x) at x=9 is: [(32Δx)/414] x 100, Note that the value of Δx is not given, so we cannot calculate the exact percentage rate of change without more information. However, we know that the rate of change is constant and equal to 32, which means that the percentage rate of change will also be constant.

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Answer:

≈ 35,53 m

Step-by-step explanation:

Given:

∠GCM = 76° (central angle, which is equal to the arc on which it rests on)

arc FG = 7,5 m

Find: C (circumference) - ?

The whole circle forms an angle of 360°

Since we don't know the length of the radius, we can make a proportion to find C:

76° - 7,5 m

360° - C m

Cross-multiply to find C:

[tex]c = \frac{360° \times 7.5}{76°} ≈35.53 \: m[/tex]

This means that a ⊕ b = (a − b) ∪ (b − a), as desired.

Why will be show that a ⊕ b = (a − b) ∪ (b − a)?

we need to demonstrate that any element in either set is also in the other set.

First, let's consider (a − b) ∪ (b − a). This set includes any element that is in a but not in b, or in b but not in a.

Now, let's look at a ⊕ b. This set includes any element that is in a or b, but not in both.

To show that these sets are equal, we need to show that any element that is in (a − b) ∪ (b − a) is also in a ⊕ b, and vice versa.

First, let's consider an element x that is in (a − b). This means that x is in a but not in b. Therefore, x is also in a ⊕ b, since it is in a but not in both a and b.

Next, let's consider an element y that is in (b − a). This means that y is in b but not in a. Again, y is also in a ⊕ b, since it is in b but not in both a and b.

Therefore, we have shown that any element in (a − b) ∪ (b − a) is also in a ⊕ b, and vice versa. This means that a ⊕ b = (a − b) ∪ (b − a), as desired.

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Y_1, Y_2, ..., Y_n are a random sample from a Laplace distribution with density function f(y|θ) = (1/2θ)e^(-|y|/θ) for -∞ < y < ∞, where θ > 0. The first two moments of the distribution are E(Y) = 0 and E(Y^2) = 2θ^2. The likelihood function of the sample is L(θ|y) = (1/2^nθ^n)e^(-∑|y_i|/θ).

a) The likelihood function is the product of the individual probability density functions for each observation. So, for a sample of size n, the likelihood function can be expressed as L(θ|y) = ∏(1/2θ)e^(-|y_i|/θ), i=1 to n. Simplifying this expression, we get L(θ|y) = (1/2^nθ^n)e^(-∑|y_i|/θ).

b) The sum of absolute values of the observations, ∑|y_i|, is a sufficient statistic for θ.

c) To find the maximum likelihood estimator (MLE) of θ, we differentiate the likelihood function with respect to θ and set it equal to zero. Solving for θ, we get the MLE as θ = ∑|y_i|/n.

d) The standard deviation of the Laplace distribution is given by σ = √(2)θ. Therefore, the MLE of the standard deviation is √(2)(∑|y_i|/n).

e) The method of moments estimator of θ is obtained by equating the sample mean absolute deviation to the population mean absolute deviation, which gives θ = ∑|y_i|/n.

f) To show that the MLE of θ is a minimum variance unbiased estimator, we can use the Fisher information. The Fisher information for θ is given by I(θ) = n/θ^2. The variance of the MLE is then given by Var(θ) = 1/I(θ) = θ^2/n. Therefore, the MLE is unbiased and has minimum variance. The variance of Y is Var(Y) = 4θ^2, so Var(X) = Var(Y)/n = 4θ^2/n.

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Answer: I believe C. 18a5 is the correct answer

Business Requirement Document (BRD)

Project Title: Loan Origination Process Automation for AER Bank

Project Objective: To automate the loan origination process of AER Bank to improve efficiency and reduce processing time.

I. Introduction

AER Bank is a financial institution that provides various loan products to its customers. The current loan origination process involves manual processing of loan applications, which is time-consuming and prone to errors. To improve the process efficiency, AER Bank intends to automate the loan origination process.

II. Scope

The loan origination process includes the following steps:

III. User Stories

IV. Acceptance Criteria

V. Priority

The priority of each user story is as follows:

VI. Constraints

VII. Queries

In case of any doubts, mail your queries to the trainer.

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The essential elements of the job are:

- The vertices are (-1, 0)

- From x = -1 to infinity, the function is diminishing.

- The function's domain includes all sets of real numbers.

- The set of all real numbers less than or equal to 0 is the function's range.

It is specified as follows:

f(x) = -(x + 1)²

The answer to the question is:

Vertex = (-1, 0)

-(x + 1)² is a negative function because the function has a negative leading term, or a negative leading sign.

This indicates that the function is inherently negative.

Additionally, because the function is exponential, its domain includes all sets of real numbers.

The function is always negative and the vertex's y value is 0.

The range, therefore, includes all real values that are less than or equal to 0.

The function is always negative and the vertex's x value is -1. Thus, from x = -1 to infinity, the function is diminishing.

Therefore, the essential elements of the job are:

- The vertices are (-1, 0)

- From x = -1 to infinity, the function is diminishing.

- The function's domain includes all sets of real numbers.

- The set of all real numbers less than or equal to 0 is the function's range.

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True, simple exponential smoothing is a type of forecasting for which the forecasted values tend to lag behind the actual values.

Simple exponential smoothing (SES) is a time series forecasting method that aims to predict future data points based on the historical data available. The method involves assigning exponentially decreasing weights to past observations, with the most recent observations given more importance than older ones. This is done using a smoothing constant, alpha (α), which ranges from 0 to 1.
The main idea behind SES is that recent data points are more likely to be representative of future values than older data points, hence the exponential decay in weights. However, one of the limitations of SES is that it tends to lag behind actual values, especially when dealing with data that exhibits a trend or seasonality.
This lag occurs because the model is heavily reliant on the past data and does not account for any trend or seasonality components. In cases where data exhibits trends or seasonal patterns, more advanced forecasting methods such as Holt's linear trend model or Holt-Winters seasonal method may provide better predictions.
To summarize, simple exponential smoothing is a forecasting method that places more weight on recent observations in predicting future values. While this method can be useful for certain types of data, it tends to lag behind actual values, especially when dealing with data that exhibits a trend or seasonality.

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In order to determine how many defective samples can be acceptable, we need to consider the concept of Acceptable Quality Level (AQL). AQL is the maximum percentage or number of defective units in a batch or lot that can be considered acceptable by the consumer.

AQL is determined based on the criticality of the product and the level of risk that the defects pose to the end user.
The AQL is usually determined through statistical sampling methods. In general, the higher the AQL, the higher the risk that the product may contain defective units. For example, if the AQL is 1%, it means that for every 100 units, one defective unit is acceptable.
In your case, you have not specified the criticality of the product or the level of risk associated with the defects. Therefore, it is difficult to determine the appropriate AQL for your situation. However, assuming a standard AQL of 2.5%, which is commonly used in the industry, it means that out of 100 samples provided by the manufacturer, at most 2.5 units can be defective for you to agree to use the new product.
It is important to note that the AQL is not a guarantee that the product is defect-free. Rather, it is a measure of the acceptable level of defects in a batch or lot. Therefore, it is important to establish appropriate quality control measures to ensure that the product meets the required standards and specifications.

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Katy's house is about 8.1 miles from Camilla's house, measured in a straight line.

we apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Using the Pythagorean theorem, we have:

H^2 = west distance^2 + north distance^2

H^2 = 4.6^2 + 6.7^2

H^2 = 21.16 + 44.89

He^2 = 66.05

We take the square root of both sides, we get:

hypotenuse = 8.13

We then can say that  Katy's house is about 8.1 miles from Camilla's house, measured in a straight line.

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Expanding along the first column, the determinant of the given matrix is -cos(0) * sin(0).

The given matrix is:

\left[\begin{array}{ccc}tan(0)&sin(0)&cos(8)\\cos(0)&0&-sin(0) \\sin(0)&cos(8) &0\end{array}\right] \\

Expanding along the first column, we get:

det = tan(0) *

(0 * cos(8) - sin(0) * cos(0))- cos(0) *(sin(0) * cos(8) - cos(0) * 0)+ sin(0) *  (sin(0) * 0 - cos(0) * cos(8))

Simplifying, we get:

det = 0 - cos(0) * sin(0) + 0

det = -cos(0) * sin(0)

Therefore, the determinant of the given matrix is -cos(0) * sin(0).

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Note that the domain of the graph is:

-7 ≤ x < -4 or -4 < x < 3.5 or 3.5 < x ≤ 4

The domain of the graph is the set of all possible values of x for which the function is defined.

Since the graph starts at (-7,-9) and ends at (4,3), we know that the domain of the function includes all values of x between -7 and 4.

Also, we know that the graph has two x-intercepts at x=-4 and x=3.5. Therefore, the domain of the function must exclude the values -4 and 3.5, since the function is undefined at those points (the function has vertical asymptotes at those points).

So the domain of the graph is:

-7 ≤ x < -4 or -4 < x < 3.5 or 3.5 < x ≤ 4

(Note that we use < instead of ≤ at the endpoints where the function is undefined.)

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The power series expansion of the function 6/(7-x) with center c=0 is

[tex]\frac{6}{(7-x)} = (6/7) \sum_{n=0}^{\infty} (x^n/7^n)[/tex].

To expand the function 6/(7-x) in a power series with center c=0 using the equation [tex]1/(1-x) = \sum_{n=0}^{\infty} x^n[/tex] for |x|<1, proceed as follows

Firstly, rewrite the given function.

Divide both the numerator and denominator by 7 to get the function [tex]\frac{(6/7)}{(1-(x/7))}[/tex].

Apply the given equation: [tex]1/(1-x) = \sum_{n=0}^{\infty} x^n[/tex] for |x|<1.

Replace x with x/7 in the sum:

[tex](6/7) \sum_{n=0}^{\infty} (x/7)^n[/tex].

Simplify the expression to get:

[tex](6/7) \sum_{n=0}^{\infty} (x^n/7^n)[/tex].

So, the power series expansion of the function 6/(7-x) with center c=0 is [tex]\frac{6}{(7-x)} = (6/7) \sum_{n=0}^{\infty} (x^n/7^n)[/tex].

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Both AB and BA exists and AB = [30] and BA= [tex]\left[\begin{array}{cccc}1&2&3&4\\2&4&6&8\\3&6&9&12\\4&8&12&16\end{array}\right][/tex]

To multiply two matrices we need to check if the number of columns of first matrix is equal to number of rows in second matrix. Then the multiplication exists.

Hence the order of the resulting matrix formed will be equal to number of rows of first matrix and number of columns of second matrix.

Thus AB exists since, Number of columns in matrix A = Number of rows in matrix B.

Therefore, AB = [ 1 2 3 4 ] [tex]\left[\begin{array}{ccc}1\\2\\3\\4\end{array}\right][/tex]

⇒AB = [ (1)(1) +(2)(2) +(3)(3) + (4)(4)]

⇒ AB = [ 1+ 4+ 9+ 16]

⇒AB = [30]

Thus BA exists since, Number of columns in matrix B = Number of rows in matrix A.

Therefore, BA =[tex]\left[\begin{array}{ccc}1\\2\\3\\4\end{array}\right][/tex] [ 1 2 3 4 ]

⇒BA = [tex]\left[\begin{array}{cccc}(1)(1)&(1)(2)&(1)(3)&(1)(4)\\(2)(1)&(2)(2)&(2)(3)&(2)(4)\\(3)(1)&(3)(2)&(3)(3)&(3)(4)\\(4)(1)&(4)(2)&(4)(3)&(4)(4)\end{array}\right][/tex][tex]{4*4}[/tex]

⇒BA= [tex]\left[\begin{array}{cccc}1&2&3&4\\2&4&6&8\\3&6&9&12\\4&8&12&16\end{array}\right][/tex]

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Step-by-step explanation:

There are 2pi radians in a full circle ....pi radians is 1/2 of the circle

   so ENTIRE arc length (circumference)  of the circle is 18 pi

      ENTIRE circumference = 18pi = pi * d

                                                   d = 18  units then    r = 9 units