Let the sample space be S={1,2,3,4,5,6,7,8,9,10}. Suppose the outcomes are equally likely. Compute the probability of the event E={1,2,4}. P(E)= (Type an integer or a decimal. Do not round.)

The correct choice to prove that ΔBAC ≅ ΔDCA based on the given information is ASA (Angle-Side-Angle).

Let's break down the given information and the steps of the ASA congruence:

Given:

∠BCA ≅ ∠DAC (Angle-Angle)

∠BAC ≅ ∠DCA (Angle-Angle)

ASA Congruence:

Angle-Angle (AA): Two triangles are congruent if they have two pairs of corresponding angles that are congruent.

Side-Side-Angle (SSA): The SSA condition is not a valid congruence criterion.

Proof using ASA Congruence:

∠BCA ≅ ∠DAC (Given)

∠BAC ≅ ∠DCA (Given)

BC ≅ DA (Given)

ΔBAC ≅ ΔDAC (ASA Congruence)

Explanation:

In the given information, we have two pairs of corresponding angles that are congruent (∠BCA ≅ ∠DAC and ∠BAC ≅ ∠DCA). This satisfies the Angle-Angle (AA) condition for congruence.

Additionally, we are given that BC is congruent to DA, which provides the side (included between the two congruent angles) for the congruence.

By the ASA congruence criterion, when two pairs of corresponding angles are congruent, and the included sides are congruent, the two triangles ΔBAC and ΔDAC are congruent.

Therefore, the correct choice is:

ASA.

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The solution to yh is obtained by solving the homogeneous differential equation and for dy/dt = 4y, the characteristic equation is r = 4.Then, the general solution to the homogeneous equation is given by;

Thus, the general solution for the given differential equation will be Where, c1 is the constant of integration and yp is the particular solution.b.) Given that, f(t) = cos(2t) The guess yp = acos(2t) will not work as it is already present in the homogeneous solution.

Therefore, it is necessary to multiply by t such that we obtainyp = t * acos(2t). To show that this guess works, differentiate the guess to get the first derivative of the guessed particular solution as follows;yp' = acos(2t) - 2t * asin(2t)The second derivative of the guessed particular solution is;yp'' = -4acos(2t) - 4t * asin(2t)

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The volume of the frustum of the square pyramid is 1/3 [√(2s² - 2b√((s - b)² + (s - b)² + l²) - b²/4 - s²/4 + bs/2)] [b² + bs + s²].

Given, The frustum of the pyramid has a square base of side b and a square top of side s.

Now, let us find the formula for the volume of the frustum of the pyramid.

Let, V be the volume of the frustum of the pyramid.

The formula for the volume of the frustum of a pyramid is given by:

V = 1/3 h (A + √Aa + a²)

Where,h = height of the frustum

A = area of the base of the frustum

a = area of the top of the frustum.

Now, the given frustum of a pyramid is a square pyramid.

So, A = b² and a = s² and the height h can be obtained by considering a right-angled triangle as shown below.

Now, using the Pythagorean theorem, we get:    

h² = l² - (b/2 - s/2)²

= l² - (b²/4 - bs/2 + s²/4)    (as (a-b)² = a² - 2ab + b²)  

= l² - b²/4 - s²/4 + bs/2

Now, we have to find the value of l.

Now, turn the frustum and arrange it along the x-axis as shown below.

Now, the co-ordinates of A, B, C and D are:(0,0,0), (b, 0, 0), (b, b, l) and (s, s, l) respectively.

Using the distance formula, we get:

AB = bBC = bCD = √((s - b)² + (s - b)² + l²)DA

= √(s² + s² + l²)

We know that AB + CD = AD

Therefore, b + √((s - b)² + (s - b)² + l²) = √(s² + s² + l²)

∴ b² + 2b√((s - b)² + (s - b)² + l²) + (s - b)² + l² = 2s²

∴ b² + (s - b)² + l² = 2s² - 2b√((s - b)² + (s - b)² + l²)

Now, using the equation (1), we can get the value of l as:

l² = 2s² - 2b√((s - b)² + (s - b)² + l²) - b²/4 - s²/4 + bs/2

Therefore, the volume of the frustum of the square pyramid is given by:

V = 1/3 h (A + √Aa + a²)

= 1/3 [√(2s² - 2b√((s - b)² + (s - b)² + l²) - b²/4 - s²/4 + bs/2)] [b² + bs + s²] {as A = b², a = s² and h = √(l² - (b²/4 - bs/2 + s²/4)}

= √[2s² - 2b√((s - b)² + (s - b)² + l²) - b²/4 - s²/4 + bs/2])

Therefore, the volume of the frustum of the square pyramid is 1/3 [√(2s² - 2b√((s - b)² + (s - b)² + l²) - b²/4 - s²/4 + bs/2)] [b² + bs + s²].

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For annual demand up to 28,750 units, purchasing from the vendor at $10 per unit would be the best option. For annual demand exceeding 28,750 units, producing in-house would be more cost-effective.

To determine the range of annual demand for each alternative, we need to compare the costs of purchasing and producing for different demand levels.

1. Purchasing from the vendor:

The cost of purchasing is $10 per unit, regardless of the annual demand. Hence, the total cost for purchasing is simply the cost per unit multiplied by the annual demand.

2. Producing in-house:

The in-house process has a fixed cost of $230,000 per year, which remains constant regardless of the demand. The variable cost per unit is $8. Therefore, the total cost of producing in-house consists of the fixed cost plus the variable cost per unit multiplied by the annual demand.

By comparing the total costs for each alternative, we can determine the range of demand for which each option is optimal.

Let's denote the annual demand as "D":

- For purchasing from the vendor, the total cost is 10D.

- For producing in-house, the total cost is 230,000 + 8D.

To find the range of demand, we set the two costs equal to each other:

10D = 230,000 + 8D.

Simplifying the equation:

2D = 230,000,

D = 115,000.

Therefore, for annual demand up to 28,750 units (115,000/4), purchasing from the vendor at $10 per unit is the best option. For annual demand exceeding this threshold, producing in-house would be more cost-effective.

Based on the analysis, the manager should choose to purchase the appliance from the vendor when the annual demand is up to 28,750 units. Beyond that demand level, it would be more economical to produce the appliances in-house due to the fixed cost being spread over a larger number of units, resulting in a lower cost per unit.

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The common difference is 4. The 0th term is 9. The nth term is 9 + 4n. The 100th term is 409. Yes, 273 is a term in the sequence, specifically the 66th term.

To determine the properties of the given sequence (9, 13, 17, 21, 25, 29, ...), we can analyze the pattern and answer each question:

1. What is the common difference?

The common difference is the constant value added to each term to obtain the next term. In this sequence, the common difference is 4. Each term is obtained by adding 4 to the previous term.

2. What is the 0th term?

To find the 0th term, we need to identify the term that corresponds to n = 0. In this case, the 0th term is the first term of the sequence, which is 9.

3. What is the nth term?

The nth term refers to the general formula or expression that describes any term in the sequence. In this case, the sequence follows the pattern of adding 4 to the previous term. Thus, the nth term can be expressed as 9 + 4n, where n represents the position of the term in the sequence.

4. What is the 100th term?

To find the 100th term, we substitute n = 100 into the formula for the nth term:

Term(100) = 9 + 4 * 100 = 9 + 400 = 409

Therefore, the 100th term of the sequence is 409.

5. Is 273 a term in the sequence? (yes or no) If yes, which term is it?

To determine if 273 is a term in the sequence, we need to check if there is an n value that satisfies the equation 9 + 4n = 273. By solving this equation, we can determine if 273 is a valid term in the sequence.

9 + 4n = 273

4n = 273 - 9

4n = 264

n = 264 / 4

n = 66

Since n = 66 satisfies the equation, 273 is indeed a term in the sequence. It is the 66th term.

In summary:

- The common difference is 4.

- The 0th term is 9.

- The nth term is 9 + 4n.

- The 100th term is 409.

- Yes, 273 is a term in the sequence, specifically the 66th term.

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The daily log return of a security follows the model Tt = 0.02 + 0.5r-2 + € where {e} is a Gaussian white noise series with mean zero and variance 0.02.The mean and variance of the return series r can be calculated as follows:

μr = E(r) = E(Tt - 0.02 - 0.5r-2) = -0.01σ2r = Var(r) = Var(Tt - 0.02 - 0.5r-2) = 0.02 + 0.25Var(r-2)

The lag-1 and lag-2 autocorrelations of rt can be calculated as:

ρ1 = Cov(r100, r99) / Var(r99)ρ2 = Cov(r100, r98) / Var(r98)

The 1- and 2-step-ahead forecasts of the return series at the forecast origin t = 100 can be calculated as:

rt+1 = E(rt+1| rt, rt-1) = E(0.02 + 0.5rt-1 + €t+1| rt, rt-1) = 0.02 + 0.5rt-1rt+2 = E(rt+2| rt, rt-1) = E(0.02 + 0.5rt+1 + €t+2| rt, rt-1) = 0.02 + 0.5E(rt+1| rt, rt-1)

The associated standard deviation of the forecast errors can be calculated as follows:

σ(1) = sqrt(Var(rt+1 - E(rt+1| rt, rt-1)))σ(2) = sqrt(Var(rt+2 - E(rt+2| rt, rt-1)))

The final answers can be given in values by using the given values in the equation of μr = E(r) and σ2r = Var(r).

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To avoid bias in the survey, the human resources director creates a list of all the employees and randomly selects 150 of them to complete the survey.

In this scenario, we have a company that employs 650 people and wishes to survey a sample of its employees about the company culture.  Let's match each term with its corresponding description:

1. Population: The population refers to the entire group of individuals that the survey aims to represent. In this case, the population is the total number of employees in the company, which is 650.

2. Sample: A sample is a subset of the population that is selected for data collection and analysis. It represents a smaller portion of the population. In this scenario, the sample consists of the 150 employees randomly selected by the human resources director.

3. Random Selection: Random selection is the process of choosing individuals from the population in a way that ensures each member has an equal chance of being included in the sample. By randomly selecting the 150 employees, the human resources director avoids bias and increases the likelihood that the sample represents the entire population.

4. Survey: A survey is a data collection method used to gather information from individuals within the sample. In this case, the selected employees will be asked to complete a survey about the company culture.

By randomly selecting 150 employees from the total population of 650, the company aims to create a sample that is representative of the entire workforce. This helps to avoid bias and increase the generalizability of the survey findings. The survey responses from the selected employees will provide insights into the company culture, which can then be used to make informed decisions or improvements. It's important to note that the quality of the survey and the representativeness of the sample can impact the validity and reliability of the survey results. Therefore, careful consideration should be given to the sampling method and survey design to ensure accurate and meaningful findings.

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The provided digraph:

```

       ----> 1 ----> 2 <----

       |  ^  |       |     |

       |  |  |       |     |

       |  |  v       v     |

       ----  4 <---- 3 ----

```

Adjacency Matrix:

```

   | 1 2 3 4

----|--------

 1 | 0 1 0 0

 2 | 1 0 0 1

 3 | 0 0 0 1

 4 | 1 0 1 0

```

Sparse Matrix:

```

(1, 2)

(1, 4)

(2, 1)

(2, 4)

(3, 4)

(4, 1)

(4, 3)

```

Adjacency List:

```

1 -> 2 -> 4

2 -> 1 -> 4

3 -> 4

4 -> 1 -> 3

```

Incidence Matrix:

```

   | 1 2 3 4

----|--------

 1 | 1 0 0 1

 2 | 1 1 0 0

 3 | 0 0 0 1

 4 | 0 1 1 0

```

The adjacency matrix is a square matrix that represents the connections between vertices in a graph or digraph. In the adjacency matrix, rows represent the starting vertices, and columns represent the ending vertices. A value of 1 in the matrix indicates an edge or connection between the corresponding vertices. In this case, the adjacency matrix is a 4x4 matrix representing the connections between vertices 1, 2, 3, and 4.

The sparse matrix representation lists only the non-zero elements of the adjacency matrix along with their corresponding row and column indices. This representation is useful when the graph has a large number of vertices but relatively few edges. The sparse matrix for this digraph includes tuples indicating the non-zero elements and their indices.

The adjacency list representation consists of a list of vertices, where each vertex is associated with a list of its adjacent vertices. In this case, we have the adjacency list representation of the digraph with vertex 1 connected to vertices 2 and 4, vertex 2 connected to vertices 1 and 4, vertex 3 connected to vertex 4, and vertex 4 connected to vertices 1 and 3.

The incidence matrix representation is used for directed graphs and bipartite graphs. It represents the relationship between vertices and edges. In this case, the incidence matrix is a 4x4 matrix representing the connections between vertices 1, 2, 3, and 4. Each row represents a vertex, and each column represents an edge. A value of 1 indicates that the vertex is the starting point of the corresponding edge, -1 indicates the endpoint, and 0 indicates no connection between the vertex and edge.

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The partial derivative of z with respect to r (∂z/∂r) is 1, and the partial derivative of z with respect to θ (∂z/∂θ) is r. These results are obtained using the chain rule and evaluating at r = 1 and θ = 0.

z = yexy, x = r cos and y = r sin e

We are to find  ∂z/∂r and ∂z/∂θ using Chain Rule, when r = 1 and θ = 0

We know that, Chain Rule states that if z = f(y) and y = g(x) , then [tex]$$\frac{dz}{dx}=\frac{dz}{dy}\cdot \frac{dy}{dx}$$[/tex]

Now,

Let [tex]$$z=y\cdot e^{xy}$$[/tex]

Using product rule, we have  ∂z/∂r as,  ∂z/∂r = ∂z/∂y * ∂y/∂r + ∂z/∂x * ∂x/∂r ………… (1)

Now,  ∂z/∂y = [tex]e^{(xy)} + y*x*e^{(xy)[/tex]  [Using product rule and chain rule]

∂y/∂r = sinθ∂x/∂r = cosθ

We get, ∂z/∂r = [tex](e^{xy} + y*x*e^{xy})[/tex] * sinθ + cosθ

Now,

Let's calculate  ∂z/∂θ.

Using product rule, we have  ∂z/∂θ as,  ∂z/∂θ = ∂z/∂y * ∂y/∂θ + ∂z/∂x * ∂x/∂θ ………… (2)

Now, ∂z/∂y = [tex]e^{xy} + y*x*e^{xy}[/tex]  [Using product rule and chain rule]

∂y/∂θ = r cosθ

∂x/∂θ = -r sinθ

Now, substituting the given values of r and θ in equations (1) and (2), we have

∂z/∂r = [tex]e^{xy} + y*x*e^{xy}[/tex] * sin(0) + cos(0)  = e⁰  = 1

∂z/∂θ = [tex]e^{xy} + y*x*e^{xy}[/tex] * r cos(0) + (-r sin(0)) = e⁰ * r = r  

The required answers are as follows:

∂z/∂r = 1

∂z/∂θ = r.

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The only solution is (x, y) = (2, 3) in positive integer. If we allow one of the either x or y to be negative, are there is no solution.

Given the equation x³ + y³ = 35, we have to find the following solutions in positive integers. Using factorization to determine solutions For x = 2, we have 2³ + y³ = 35, then y³ = 27. Taking the cube root of both sides, y = 3.  Hence, (x, y) = (2, 3) is one of the solutions.

For x = 3, we have 3³ + y³ = 35, then y³ = 26. As there is no whole number y that satisfies this equation, there is no solution when x = 3.For x = 4, we have 4³ + y³ = 35, then y³ = -27. As y is a positive integer, there is no solution when x = 4.

Thus, the only solution in positive integers is (x, y) = (2, 3). Now, we have to find whether there are any other solutions by allowing either x or y to be negative. By letting x = -2, we get (-2)³ + y³ = 35, then y³ = 43. As there is no whole number y that satisfies this equation, there is no other solution when x is negative.

By letting y = -3, we get x³ + (-3)³ = 35, then x³ = 62. As there is no whole number x that satisfies this equation, there is no other solution when y is negative. Therefore, the only solution is (x, y) = (2, 3).

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complete question

Given the equation x^3 + y^3 = 35, answer the following:

a) Find all solutions in positive integers. While it is easy to guess the solution, you must use factorization to determine the solutions.

b) If you allow one of either x or y to be negative, are there any other solutions?

In statistics, a confidence interval is a range of values, derived from a data sample,

That is used to estimate an unknown Population  Parameter.

The interval has an associated confidence level that quantifies the level of confidence that the parameter lies in the interval

The formula for calculating the confidence interval for the difference between two means is given below: CI = (X1 - X2) ± t(α/2, n1 + n2 - 2) × s√(1/n1 + 1/n2)

Where CI is the confidence interval, X1 and X2 are the sample means,

s is the pooled standard deviation, n1 and n2 are the sample sizes, t(α/2, n1 + n2 - 2) is the critical value from the t-distribution with α/2 level of significance and n1 + n2 - 2 degrees of freedom.

We can use this formula to find the 95% confidence interval for the difference between the means of the two populations:

First, we need to calculate the pooled standard deviation:

s = sqrt(((n1 - 1) × s1^2 + (n2 - 1) × s2^2) ÷ (n1 + n2 - 2))s = sqrt(((11 - 1) × 1.3^2 + (15 - 1) × 1.16^2) ÷ (11 + 15 - 2))s = sqrt(169.46 ÷ 24)s = 1.87

Next, we need to calculate the critical value from the t- distribution: t(0.025, 24) = 2.064

Finally, we can calculate the confidence interval: CI = (X1 - X2) ± t(α/2, n1 + n2 - 2) × s√(1/n1 + 1/n2)CI = (3.27 - 2.53) ± 2.064 × 1.87 √(1/11 + 1/15)CI = 0.74 ± 0.963CI = (−0.223, 1.703)

Therefore, the 95% confidence interval for the difference between the means of the two populations is (−0.223, 1.703).

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Height from Bottom=-137.14 feet

To determine the height of a person from the bottom of the London Eye after traveling 5/12 of the way around, we can use the following calculations:

Radius of the London Eye (r) = 221 feet

Circumference of the London Eye (C) = 2πr = 2 * 3.14159 * 221 = 1387.92 feet

Arc Length for 5/12 of the circumference = (5/12) * C = (5/12) * 1387.92 = 579.14 feet

Total Height of the London Eye = 2 * r = 2 * 221 = 442 feet

Height from Bottom = Total Height - Arc Length = 442 - 579.14 = -137.14 feet

The negative value indicates that the person is below the starting point or at a height below the ground level of the London Eye.

Please note that the height calculated is relative to the bottom of the London Eye, and a negative value suggests that the person has gone below the initial starting point.

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The identities among the given options are:

B. (sin x + cos x)² = 1 + sin 2x

C. sin 6x = 2 sin 3x cos 3x

Therefore, options B and C are the identities.

Among the given options, the identities are as follows:

B. (sin x + cos x)² = 1 + sin 2x

C. sin 6x = 2 sin 3x cos 3x

Let's examine each option:

A. This equation is not an identity since it does not hold true for all values of x.

B. This equation is an identity.

It is known as the Pythagorean Identity, which states that the square of the sum of sine and cosine is equal to 1 plus the sine of twice the angle.

C. This equation is also an identity. It is derived from the double angle formula for sine, which states that sin(2x) = 2sin(x)cos(x).

By substituting 3x for x, we get sin(6x) = 2sin(3x)cos(3x), which is the given equation.

D. The equation given here, "4 cos x sec x = tan x," is not an identity since it does not hold true for all values of x.

To summarize, the identities among the given options are B. (sin x + cos x)² = 1 + sin 2x and C. sin 6x = 2 sin 3x cos 3x.

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(a) The expression [tex]\(\log(2-3i)\)[/tex] in Cartesian coordinates is [tex]\(\log(\sqrt{13}) + i \tan^{-1}\left(\frac{-3}{2}\right)\).[/tex]

(b) The expression [tex]\((4i)^i\)[/tex] in Cartesian coordinates is [tex]\(e^{-\frac{\pi}{4}} \cdot e^{i \log(4) \frac{\pi}{2}}\).[/tex]

(a) To express [tex]\(\log(2-3i)\)[/tex] in Cartesian coordinates, we can use the properties of logarithms and complex numbers.

Let's write [tex]\(2-3i\)[/tex]  in polar form.

The modulus r is given by [tex]\(r = \sqrt{2^2 + (-3)^2} = \sqrt{13}\)[/tex], and the argument θ can be found using the inverse tangent function:

[tex]\(\theta = \tan^{-1}\left(\frac{-3}{2}\right)\)[/tex].

Therefore, [tex]\(2-3i\)[/tex] in polar form is [tex]\(r \cdot \text{cis}(\theta) = \sqrt{13} \cdot \text{cis}\left(\tan^{-1}\left(\frac{-3}{2}\right)\right)\).[/tex]

Applying the logarithmic property [tex]\(\log(ab) = \log(a) + \log(b)\)[/tex] and the logarithm of a complex number in polar form

[tex]\(\log(r \cdot \text{cis}(\theta)) = \log(r) + i \theta\)[/tex], we have:

[tex]\(\log(2-3i) \\= \log(\sqrt{13} \cdot \text{cis}\left(\tan^{-1}\left(\frac{-3}{2}\right)\right)) \\= \log(\sqrt{13}) + i \tan^{-1}\left(\frac{-3}{2}\right)\).[/tex]

Therefore,[tex]\(\log(2-3i)\)[/tex] in Cartesian coordinates is[tex]\(\log(\sqrt{13}) + i \tan^{-1}\left(\frac{-3}{2}\right)\).[/tex]

(b) To express [tex]\((4i)^i\)[/tex] in Cartesian coordinates, we'll use the properties of complex numbers and exponential functions.

First, we'll write 4i in polar form.

The modulus r is given by [tex]\(r = \sqrt{0^2 + 4^2} = 4\)[/tex], and the argument [tex]\(\theta\)[/tex] can be found using the inverse tangent function: [tex]\(\theta = \tan^{-1}\left(\frac{4}{0}\right) = \frac{\pi}{2}\).[/tex]

Therefore, 4i in polar form is [tex]\(4 \cdot \text{cis}\left(\frac{\pi}{2}\right)\).[/tex]

Using the Euler's formula [tex]\(e^{ix} = \text{cis}(x)\)[/tex], we can express[tex]\((4i)^i\)[/tex] as [tex]\(e^{i\frac{\pi}{2}}\).[/tex]

Next, we'll use the exponential property [tex]\(a^b = e^{b \log(a)}\)[/tex] to compute [tex]\((4i)^i\)[/tex]:

[tex]\((4i)^i = e^{i\frac{\pi}{2} \log(4i)}\).[/tex]

Now, let's compute [tex]\(\log(4i)\)[/tex] in Cartesian coordinates.

Writing 4i in polar form, we have [tex]\(4i = 4 \cdot \text{cis}\left(\frac{\pi}{2}\right)\)[/tex].

The logarithm of a complex number in polar form is [tex]\(\log(r \cdot \text{cis}(\theta)) = \log(r) + i \theta\)[/tex].

Therefore, [tex]\(\log(4i) = \log(4) + i \frac{\pi}{2}\).[/tex]

Substituting this back into [tex]\((4i)^i\)[/tex], we get:

[tex]\((4i)^i \\\\= e^{i\frac{\pi}{2} \log(4i)} \\\\= e^{i\frac{\pi}{2} (\log(4) + i \frac{\pi}{2})}\).[/tex]

Expanding this expression, we have:

[tex]\((4i)^i = e^{-\frac{\pi}{2}^2} \cdot e^{i \log(4) \frac{\pi}{2}}\).[/tex]

Simplifying further, we get:

[tex]\((4i)^i = e^{-\frac{\pi}{4}} \cdot e^{i \log(4) \frac{\pi}{2}}\).[/tex]

Therefore, [tex]\((4i)^i\)[/tex] in Cartesian coordinates is [tex]\(e^{-\frac{\pi}{4}} \cdot e^{i \log(4) \frac{\pi}{2}}\).[/tex]

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The molar flow of each component in the product stream will be: C₂H₄ = 50 kmol, O₂ = 0 kmol, and C₂H₄O = 50 kmol.

To determine the limiting reactant, we need to compare the number of moles of each reactant to the stoichiometric ratio in the balanced equation.

From the balanced equation: C₂H₄ + 1/2O → C₂H₄O

1 mole of C₂H₄ reacts with 1/2 mole of O₂ to produce 1 mole of C₂H₄O.

Number of moles of ethene (C₂H₄) = 100 kmol

Number of moles of oxygen (O₂) = 100 kmol

Since the stoichiometric ratio between ethene and oxygen is 1:1/2, we can see that 1 mole of ethene requires 1/2 mole of oxygen.

Considering the number of moles available for both reactants, we find that 1 mole of ethene requires 1/2 mole of oxygen, but we have equal moles of each. Therefore, the limiting reactant is oxygen (O₂).

The maximum extent of reaction is determined by the limiting reactant, which is oxygen. Thus, the maximum extent of reaction is 100/2 = 50 kmol.

To calculate the percentage excess of the excess reactant (ethene in this case), we can compare the number of moles actually used with the number of moles initially available.

Number of moles of ethene used = 50 kmol (since oxygen is limiting)

Percentage excess of ethene = [(100 kmol - 50 kmol) / 100 kmol] * 100% = 50%

If the reaction proceeds to completion, all the limiting reactant (oxygen) will be consumed, and the molar flow of each component in the product stream will be as follows:

Molar flow of C₂H₄ = 100 kmol - 50 kmol = 50 kmol

Molar flow of O₂ = 0 kmol (all consumed)

Molar flow of C₂H₄O = 50 kmol (produced)

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a) Higher risk factors for material misstatements: significant increase in revenue and margins, significant increase in inventory and trade receivables. b) Briefing items to discuss with the engagement team: factors for assessing the reliability of audit evidence. c) The sale of sawmill machines as a subsequent event may require adjustment or disclosure in the financial statements, with potential impact on asset values, related accounts, and appropriate disclosures to ensure fair presentation.

(a) Issues or conditions indicating a higher risk of material misstatements in the potential engagement with Orchid Limited may include:

Significant increase in revenue and margins: The substantial increase in revenue and margins may indicate a higher risk of aggressive financial reporting or potential manipulation of financial results to meet targets or investor expectations.

Significant increase in inventory and trade receivables: The significant increase in inventory and trade receivables suggests a higher risk of overstatement of assets or potential difficulties in collecting receivables, which may require further assessment for potential impairment or obsolescence.

Introduction of online ordering system: The implementation of a new online ordering system and its growth in terms of sales volume may increase the risk of data integrity issues, such as unauthorized access, errors in processing orders, or potential fraud related to online transactions.

CEO's intention to sell shareholding: The CEO's intention to sell his 100% shareholding introduces the risk of management bias or conflict of interest, which may impact the accuracy and completeness of financial reporting. It requires assessment to ensure fair presentation of the financial statements and proper disclosure of related party transactions.

(b) Briefing items to discuss with the engagement team:

i) Factors to be considered in assessing the reliability of audit evidence:

Source and nature of the evidence: Assess the credibility and competence of the provider, considering factors such as independence, qualifications, and reputation.

Objectivity and consistency: Evaluate the consistency and compatibility of the evidence with other available information.

Timeliness: Consider the relevance and freshness of the evidence in relation to the audit objectives and the financial statement date.

ii) Why an audit can only provide reasonable assurance:

Inherent limitations of financial reporting: Financial statements are prepared based on management's judgments and estimates, making them inherently subjective and prone to error or misstatement.

Sampling and testing: Auditors use sampling techniques to select items for testing, and there is a risk that material misstatements may not be detected due to the nature and extent of testing.

iii) Key activities of audit planning:

Understanding the entity and its environment: Gather knowledge of Orchid's business, industry, and regulatory environment to identify risks and assess the impact on financial statements.

Establishing materiality levels: Determine materiality thresholds to set the scope and nature of audit procedures.

Identifying significant risks: Assess risks of material misstatement, including fraud risks, and develop appropriate responses.

Developing an audit strategy: Determine the overall approach, timing, and resources required for the audit.

Assembling the audit team: Assign appropriate team members with the necessary skills, knowledge, and experience to perform the engagement effectively.

(c) Significance and impact of the sale of sawmill machines:

The sale of sawmill machines after the year-end but before the issuance of the auditor's report is a subsequent event that may require adjustment or disclosure in the financial statements as at December 31, 2021. The event is significant as the value of the machines sold represents 18% of total assets, indicating a material impact.

The financial statements as at December 31, 2021, need to be evaluated to determine if the sale should be recognized in the period-end financial statements. Depending on the specific circumstances, the impact may include:

Adjusting the carrying amount of the machines sold: The sale would involve derecognizing the machines from the balance sheet and recognizing the proceeds or gain/loss on the sale.

Assessing the impact on other financial statement items: The sale may affect related accounts, such as depreciation expense, accumulated depreciation, and profit or loss.

Disclosures: Proper disclosure is required to provide information about the sale, its financial impact, and any subsequent events that may materially affect the financial statements.

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A) The sample proportion of teenagers is 0.165.

B) The test statistic is -2.42 rounded to 2 decimal places.

C)  The p-value is  0.0076.

D) The conclusion regarding the null hypothesis is :The evidence suggests that the proportion of teenagers  is less than 0.2.

E)  The appropriate concluding statement is: The data supports the claim that less than 200 of all teenagers use tobacco.

(a) Sample proportion of teenagers who use tobacco is given by:

P = 132/800P = 0.165

(b) The null hypothesis states that 200 or more of all teenagers use tobacco and the alternative hypothesis is that less than 200 of all teenagers use tobacco.

The sample proportion is given by 0.165 and population proportion is 0.200.z-test statistic is given by, z = (P - p) / sqrt(pq/n)

Here, p = 0.200q = 1 - p = 0.800n = 800z = (0.165 - 0.200) / sqrt(0.200 * 0.800 / 800)z = -2.42z = -2.42

(c) The p-value of the test statistic can be found using the standard normal distribution table.

p-value for z = -2.42 is 0.0076.

Therefore, the p-value of the test statistic is 0.0076.

(d) The hypothesis is tested at the 0.01 significance level. Since the p-value of the test statistic (0.0076) is less than the level of significance (0.01), we reject the null hypothesis.

(e) The appropriate concluding statement is: The data supports the claim that less than 200 of all teenagers use tobacco. Therefore, the correct option is: The data supports the claim that less than 200 of all teenagers use tobacco.

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(a) The values of a and b are 0 and π/2, respectively. (b) The values of c and d are 0 and k, respectively.

(c) In polar coordinates, θ is represented by t. Therefore, g(r, t) remains similar to the authentic characteristic: g(r, t) = [tex]2 - r^2 * e^-(r^2)[/tex]. (d) The price of I is equal to [tex]\int\limit{($0 to \pi/2)}[/tex] [tex]\int\limits {$(0 to k)}[/tex][tex](2 - r^2 * e^-(r^2))[/tex] * [tex]rdrdt[/tex].

(e) As k processes infinity, the upper restriction of the second critical (∫(0 to k)) turns endless. Thus, the restrict of I as k procedures infinity diverges or is undefined. (f) None of the integrals corresponding to lim(k→∞) I for the reason that restricts of I is undefined.

(a) The values of a and b inside the polar coordinate shape of the vital are 0 and π/2, respectively. This is due to the fact the limits of integration for θ are from 0 to π/2, representing 1 / 4 of the overall circle.

(b) The values of c and d within the polar coordinate shape of the indispensable are 0 and k, respectively. This represents the variety of integration for the radial variable r.

(c) Using t in the area of θ, g(r,t) is identical to [tex]r * e^(-r^2).[/tex] This is received by substituting the expression for g(r,t) = [tex]e^(-x^2-y^2)[/tex] in phrases of polar coordinates.

(d) The fee of I is π[tex](1 - e^(-k^2)).[/tex] This can be observed by way of comparing the crucial usage of the given limits of integration and the expression for g(r,t).

(e) The restrict as k tactics infinity of I is π. This is due to the fact as okay turns into infinitely large, the term[tex]e^(-k^2)[/tex] tactics 0, resulting in π(1 - 0) = π.

(f) The critical that corresponds to lim k→∞ I is (C) [tex]\int\limits {0 [infinity] }[/tex][tex]\int\limits {0 [infinity] e^(-(x^2 + y^2)) } \, dxdy[/tex]. This may be inferred from the unique definition of I in Cartesian coordinates and thinking about the boundaries of integration for x and y.

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The given system of differential equations is [y₁ = 3y₂ + 2y₂ |y₂ = 2y₁ + 3y₂]To find the general solution of the given system of differential equations, we need to write the corresponding characteristic equation, find eigenvalues, eigenvectors, and then the general solution for the given system.

Step 1: Write the characteristic equationThe characteristic equation is obtained by replacing each y in the differential equation with (λ - A), where A is the matrix [3 2 | 2 3].Hence, the characteristic equation becomes:|[λ - 3 2 | 2 λ - 3]| = (λ - 3)(λ - 3) - 4|2| = λ² - 6λ + 1This is the characteristic equation of the given system of differential equations.

Step 2: Find the eigenvalues The eigenvalues are found by setting the characteristic equation equal to zero and solving for λ.λ² - 6λ + 1 = 0Solving this quadratic equation using the quadratic formula, we get:λ = (6 ± √32)/2 = 3 ± √8, the eigenvalues of the given system of differential equations are λ1 = 3 + √8 and λ2 = 3 - √8.

Step 3: Find the eigenvectorsWe now find the eigenvectors corresponding to each eigenvalue λ1 and λ2.To find the eigenvector corresponding to λ1 = 3 + √8, we solve the equation:[A - λ1I]X = 0where I is the identity matrix of order 2. [3 2 | 2 3] - [(3 + √8) 1 | 0 (3 + √8)] =|√8 2 | 2 √8| 2 √8 |√8 2|So, the augmented matrix is:|√8 2 | 2 √8| 2 √8 |√8 2|≡ |1 √8/2 | 1| 0 |0 - √8/2 | 0|, the corresponding eigenvector is:|X1| |√8/2| |X2| = |-1/2|

Thus, the eigenvector corresponding to λ1 is X1 = (√8/2, -1/2)To find the eigenvector corresponding to λ2 = 3 - √8, we solve the equation:[A - λ2I]X = 0where I is the identity matrix of order 2. [3 2 | 2 3] - [(3 - √8) 1 | 0 (3 - √8)] =| -√8 2 | 2 -√8| -√8 | -√8 2|So, the augmented matrix is:| -√8 2 | -√8| -√8 | -√8 2|≡ |1 -√8/2 | 1| 0 |0 √8/2 | 0|, the corresponding eigenvector is:|X1| |-√8/2| |X2| = | 1/2|

Thus, the eigenvector corresponding to λ2 is X2 = (-√8/2, 1/2)

Step 4: Find the general solutionThe general solution of the given system of differential equations is given by:y(t) = c1 e^(λ1 t) X1 + c2 e^(λ2 t) X2where c1 and c2 are constants determined by the initial conditions on the system.Therefore, the general solution of the given system of differential equations is:y(t) = c1 e^[(3 + √8) t] (√8/2, -1/2) + c2 e^[(3 - √8) t] (-√8/2, 1/2) where c1 and c2 are arbitrary constants. This is the required solution.

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Answer:

x = 6

MP = 30

Step-by-step explanation:

5x = 3x + 12

2x = 12

x = 6

MP = 5x = 5 × 6

MP = 30

Therefore, the matrix A representing the distribution of trucks after 1 week is:

A = [341 259]

[316 308]

To find the matrix A that represents the distribution of trucks after 1 week, we can use the information provided in the problem statement.

We know that about 40 percent of the trucks in Tucson move to Memphis each week, and about 55 percent of the trucks in Memphis move to Tucson each week. Let's represent these percentages as decimals: 0.40 and 0.55, respectively.

The matrix A will have the form:

A = [a b]

[c d]

To determine the values of a, b, c, and d, we can use the given percentages and the total number of trucks.

Since Tucson initially has 220 trucks and 40% of those trucks move to Memphis, the number of trucks moving from Tucson to Memphis is 0.40 * 220 = 88.

Similarly, since Memphis initially has (600 - 220) = 380 trucks and 55% of those trucks move to Tucson, the number of trucks moving from Memphis to Tucson is 0.55 * 380 = 209.

Now, let's determine the values of a, b, c, and d:

a (number of trucks in Tucson after 1 week) = (1 - 0.40) * 220 (trucks remaining in Tucson) + 0.55 * 380 (trucks moving from Memphis to Tucson)

a = (1 - 0.40) * 220 + 0.55 * 380

= 132 + 209

= 341

b (number of trucks in Memphis after 1 week) = 0.40 * 220 (trucks moving from Tucson to Memphis) + (1 - 0.55) * 380 (trucks remaining in Memphis)

b = 0.40 * 220 + (1 - 0.55) * 380

= 88 + 171

= 259

c (number of trucks in Tucson after 1 week) = 0.40 * 220 (trucks moving from Tucson to Memphis) + (1 - 0.40) * 380 (trucks remaining in Memphis)

c = 0.40 * 220 + (1 - 0.40) * 380

= 88 + 228

= 316

d (number of trucks in Memphis after 1 week) = (1 - 0.55) * 220 (trucks remaining in Tucson) + 0.55 * 380 (trucks moving from Memphis to Tucson)

d = (1 - 0.55) * 220 + 0.55 * 380

= 99 + 209

= 308

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The amount of work required to pump the water out of the pool when it is full is 1036800P Joules.

The formula that will be used to solve the problem is

W = ∫ a b QgA(y)D(y)dy,

where Q =  volume flow rate of water,

g =  acceleration due to gravity,

A(y) =  cross-sectional area of water, and

D(y) =  depth of the water at height y.

The cross-sectional area of water in the pool is given by

A(y) = 30m x 12m

= 360m².

Height of water in the pool is 4m, hence

D(y) = 4m.

Substituting these values in the formula, we get

W = ∫ 0 4 QgA(y)D(y)dy.

Since we don't have the value of Q, we will use the formula,

F = ∫ 0 a Pg(a - y)w(y)dy, where

P =  density of water,

w(y) =  width of water at height y, and

a =  4m.

Substituting the values given,

F = ∫ 0 4 P(12)(30)(4 - y)dy

= 259200P.

Work is required to pump the water out of the pool is equal to potential energy of the water when it is full.

The potential energy of the water is given by W = Fh,

where h is the height of the water in the pool when it is full.

Substituting the values,

W = 259200P(4)

= 1036800P Joules.

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Answer:

-6 divided by 5

Step-by-step explanation:

The most suitable parameter to use when designing a wastewater treatment plant is BOD5/BOD.

BOD5/BOD stands for Biochemical Oxygen Demand ratio. This parameter is a measure of the amount of organic matter present in wastewater and the amount of oxygen needed to break down that organic matter. It helps in determining the level of pollutants in the wastewater and the efficiency of the treatment process.

By using BOD5/BOD, engineers can assess the strength of the wastewater and design treatment processes accordingly. It provides valuable information about the organic load and the oxygen requirements for biological treatment.

Other parameters like COD (Chemical Oxygen Demand) and TDS (Total Dissolved Solids) are also important in wastewater treatment plant design, but BOD5/BOD specifically focuses on the organic content, which is a crucial aspect to consider for efficient treatment.

In summary, the BOD5/BOD ratio is the most suitable parameter to use when designing a wastewater treatment plant as it provides valuable information about the organic load and oxygen requirements for effective treatment.

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The function y(x) is given by [tex]y(x) = 17 + 9x + 25e^{(5x).[/tex]

To find the function y(x) given the conditions, we need to solve the given third-order linear homogeneous differential equation:

y(4) - 10y'''+ 25y'' = 0

We also have the initial conditions:

y(0) = 17

y'(0) = 9

y''(0) = 25

y'''(0) = 0

The characteristic equation associated with the differential equation is:

[tex]r^4 - 10r^3 + 25r^2 = 0[/tex]

Factoring out an r^2 term:

[tex]r^2(r^2 - 10r + 25) = 0[/tex]

The roots of this equation are r = 0 (with multiplicity 2) and r = 5 (with multiplicity 2).

Therefore, the general solution for y(x) is:

[tex]y(x) = (c1 + c2x) + (c3 + c4x)e^{(5x)[/tex]

To find the particular solution, we can use the initial conditions.

Using the initial condition y(0) = 17:

17 = c1

Using the initial condition y'(0) = 9:

9 = c2 + 5c4

Using the initial condition y''(0) = 25:

25 = c3 + 25c4

Using the initial condition y'''(0) = 0:

0 = 10c4

From the last equation, we find that c4 = 0.

Substituting the values of c1 and c4 into the equations for c2 and c3, we get:

9 = c2

25 = c3

Therefore, the particular solution for y(x) is:

[tex]y(x) = 17 + 9x + 25e^{(5x)[/tex]

Thus, the function y(x) is given by:

[tex]y(x) = 17 + 9x + 25e^{(5x)[/tex]

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Therefore, the area of the upper portion of the figure bounded by the equations y² = 4x and x = 4 is 8 square units.

Given the equations are y² = 4x and x = 4, we need to find the area of the upper portion of the figure that is bounded by the given equations.

Here, y² = 4x is a parabolic equation that opens towards the right.

This curve intersects the line x = 4 at (4, ± 4).

So, the points of intersection are (4, 4) and (4, -4).

Let's find the equation of the curve by squaring both sides:

y² = 4x  ⇒ x = y²/4

Now we know that the curve and the line intersect at x = 4.

So, let's put x = 4 in the curve equation and solve for y.

4 = y²/4⇒ y² = 16 ⇒ y = ±4

Therefore, the coordinates of the points of intersection are (4, 4) and (4, -4).

Therefore, the area of the upper portion of the figure bounded by the equations y² = 4x and x = 4 is shown below:

Now, the required area of the shaded region can be calculated as follows:

Area = Total area - Area below the curve and above the line= 16 - 8 = 8 sq. units

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The solution to the simultaneous differential equations is:

x(t) = 0

y(t) = t

Let's solve the first system of differential equations using Laplace transform.

Taking the Laplace transform of both sides of the equations, we get:

sX(s) - x(0) + 8Y(s) = 0

sY(s) - y(0) + 4X(s) + e^(-3s)Y(s) = 0

Substituting the initial conditions, we have:

sX(s) + 8Y(s) = 1

sY(s) + 4X(s) + e^(-3s)Y(s) = 0

Solving for X(s) and Y(s), we have:

X(s) = (1/((s^2)+32)) * s

Y(s) = (-4/(((s^2)+32)*(s+e^(-3s))))

Using partial fraction decomposition, we can rewrite Y(s) as:

Y(s) = (-1/(s+e^(-3s))) + ((s-3)/((s^2)+32))

Taking the inverse Laplace transform of X(s) and Y(s), we obtain:

x(t) = cos(4t) / 4

y(t) = -(1/4)e^(-3t) + (1/4)sin(4t) - (3/4)cos(4t)

Therefore, the solution to the simultaneous differential equations is:

x(t) = cos(4t) / 4

y(t) = -(1/4)e^(-3t) + (1/4)sin(4t) - (3/4)cos(4t)

Now let's solve the second system of differential equations using Laplace transform.

Taking the Laplace transform of both sides of the equations, we get:

sX(s) - x(0) = 0

sY(s) - y(0) + sX(s) = 0

Substituting the initial conditions, we have:

sX(s) = 0

sY(s) + sX(s) = 1

Solving for X(s) and Y(s), we have:

X(s) = 0

Y(s) = 1/(s^2)

Taking the inverse Laplace transform of X(s) and Y(s), we obtain:

x(t) = 0

y(t) = t

Therefore, the solution to the simultaneous differential equations is:

x(t) = 0

y(t) = t

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To determine the width of the river, we can use trigonometry and the information provided. Let's denote the width of the river as "x." From the surveyor's measurements, we have formed a right triangle with the baseline along the riverbank.

the width of the river as the opposite side, and the line of sight to the tree as the hypotenuse.  The angle between the baseline and the line of sight is 39 degrees. Using the trigonometric function tangent (tan), we can set up the following equation:

tan(39 degrees) = opposite/adjacent

tan(39 degrees) = x/110

To find the width of the river (x), we can rearrange the equation:

x = 110 * tan(39 degrees) Evaluating this expression gives us the width of the river: x ≈ 77.96 meters Therefore, the width of the river is approximately 77.96 meters.

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The average temperature difference in a heat exchanger can be calculated by subtracting the outlet temperature of the hot fluid (crude oil in this case) from the inlet temperature of the hot fluid, and then subtracting the outlet temperature of the cold fluid (reactant in this case) from the inlet temperature of the cold fluid.

For the co-current flow, the average temperature difference is:

Inlet temperature difference = 250°C - 100°C = 150°C
Outlet temperature difference = 180°C - 160°C = 20°C

Average temperature difference for co-current flow = Inlet temperature difference - Outlet temperature difference = 150°C - 20°C = 130°C

For the counter-current flow, the average temperature difference is:

Inlet temperature difference = 250°C - 100°C = 150°C
Outlet temperature difference = 180°C - 160°C = 20°C

Average temperature difference for counter-current flow = Inlet temperature difference + Outlet temperature difference = 150°C + 20°C = 170°C

So, the average temperature difference for co-current flow is 130°C and for counter-current flow is 170°C.

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Answer:

AB = 5.4

Step-by-step explanation:

In a ΔPQR,

Law of sines : [tex]\frac{sin(P)}{QR} = \frac{sin(Q)}{PR}= \frac{sin(R)}{PQ}[/tex]

Law of cosines:

[tex]cos(P) = \frac{PQ^2 + PR^2 - QR^2}{2(PQ)(PR)}\\\\cos(Q) = \frac{PQ^2 + QR^2 - PR^2}{2(PQ)(QR)}\\\\cos(R) = \frac{PR^2 + QR^2 - PQ^2}{2(PR)(QR)}[/tex]

In ΔCDE, by cosine law,

[tex]cos(DCE) = \frac{DC^2 + CE^2 - DE^2}{2(DC)(CE)}\\\\= \frac{10^2 + 8^2 - 9^2}{2(10)(8)}\\\\= \frac{100 + 64- 81}{160}\\\\= \frac{83}{160}\\\\cos(DCE) = \frac{83}{160} \\\\\implies \angle DCE = cos^{-1}(\frac{83}{160})\\\\\implies \angle DCE = 58.75[/tex]

In ΔABC, sine law,

[tex]\frac{sin(BAC)}{BC} =\frac{sin(ABC)}{AC}=\frac{sin(ACB)}{AB}\\\\\implies \frac{sin(BAC)}{BC} =\frac{sin(72)}{6}=\frac{sin(ACB)}{AB}\\\\\implies \frac{sin(72)}{6}=\frac{sin(ACB)}{AB}\\\\\implies AB =\frac{6sin(ACB)}{sin(72)}\\\\[/tex]

∠DCE = ∠ACB (vertically opposite angles)

[tex]\implies AB = \frac{6sin(DCE)}{sin(72)} \\\\= \frac{6*sin(58.75)}{sin(72)} \\\\= \frac{6*0.85}{0.95} \\\\= 5.4[/tex]

Answer:

5.39 m

Step-by-step explanation:

As line segments AE and BD intersect at point C, m∠ACB ≅ m∠ECD according to the vertical angles theorem.

As we have been given the lengths of all three sides of triangle DCE, we can use the Law of Cosines to find the measure of angle ECD, and thus the measure of angle ACB.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Cosine Rule} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

Given:

Therefore:

[tex]9^2=10^2+8^2-2(10)(8) \cos ECD[/tex]

[tex]81=100+64-160 \cos ECD[/tex]

[tex]81=164-160 \cos ECD[/tex]

[tex]160 \cos ECD=164-81[/tex]

[tex]160 \cos ECD=83[/tex]

[tex]\cos ECD=\dfrac{83}{160}[/tex]

[tex]m \angle ECD= \cos^{-1}\left(\dfrac{83}{160}\right)[/tex]

According to the vertical angles theorem, m∠ACB ≅ m∠ECD. Therefore:

[tex]m \angle ACB= \cos^{-1}\left(\dfrac{83}{160}\right)[/tex]

We now have two internal angles and one side length of triangle ACB:

    [tex]\bullet \quad m \angle ACB= \cos^{-1}\left(\dfrac{83}{160}\right)[/tex]

    [tex]\bullet \quad m \angle ABC=72^{\circ}[/tex]

    [tex]\bullet \quad AC=6\; \sf m[/tex]

The distance between points A and B is the length of line segment AB.

To find this, we can use the Law of Sines.

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]

Substitute the values of AC, ∠ACB and ∠ABC into the formula and solve for AB:

[tex]\dfrac{AB}{\sin \angle ACB}=\dfrac{AC}{\sin \angle ABC}[/tex]

[tex]\dfrac{AB}{\sin \left(\cos^{-1}\left(\dfrac{83}{160}\right)\right)}=\dfrac{6}{\sin 72^{\circ}}[/tex]

[tex]\dfrac{AB}{\left(\dfrac{9\sqrt{231}}{160}\right)}=\dfrac{6}{\sin 72^{\circ}}[/tex]

[tex]AB=\dfrac{6}{\sin 72^{\circ}}\cdot \left(\dfrac{9\sqrt{231}}{160}\right)[/tex]

[tex]AB=5.39353425...[/tex]

[tex]AB=5.39\; \sf m\;(nearest\;hundredth)[/tex]

Therefore, posts A and B are 5.39 meters apart (rounded to the nearest hundredth of a meter).