Consider the function f(x)=7x+28/x​ on the interval [0.01,4]. (a) f′(x)=−28/x2​+7 (b) f(x) has an absolute minimum equal to which occurs at x=__

The estimated distance the security guard needs to patrol is **11,660 feet, the runways at an airport are arranged to intersect and are bordered by fencing.

The security guard needs to patrol the outside fence of the runways once per shift. The shape of the runways is a right triangle, with the two legs being the lengths of the two runways.

The hypotenuse of the triangle is the length of the outside fence that the security guard needs to patrol.

Let's say that the lengths of the two runways are $x$ feet and $y$ feet. Then, the length of the hypotenuse is $\sqrt{x^2+y^2}$ feet.

We can estimate the distance the security guard needs to patrol by assuming that the two runways are equal in length. In this case, the length of the hypotenuse is $\sqrt{2x^2} = 2x\sqrt{2}$ feet.

If the lengths of the two runways are each 1000 feet, then the estimated distance the security guard needs to patrol is $2 \cdot 1000 \sqrt{2} = \boxed{11,660}$ feet.

The shape of the runways:

The runways at an airport are arranged to intersect and are bordered by fencing. This creates a right triangle, with the two legs being the lengths of the two runways. The hypotenuse of the triangle is the length of the outside fence that the security guard needs to patrol.

We can estimate the distance the security guard needs to patrol by assuming that the two runways are equal in length. In this case, the length of the hypotenuse is $\sqrt{2x^2} = 2x\sqrt{2}$ feet.

If the lengths of the two runways are each 1000 feet, then the estimated distance the security guard needs to patrol is $2 \cdot 1000 \sqrt{2} = \boxed{11,660}$ feet.

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It is not possible to determine the level of water in the tank using only the given information. To determine the level of water in the tank, we need to know either the height of the water column or the total pressure at the bottom of the tank, which includes the pressure due to the water column and the pressure due to the atmosphere.

Therefore, we can't fill the blank with any value since the problem does not provide any information regarding it. In order to find the level of water in the tank, we need to know either the height of the water column or the total pressure at the bottom of the tank, which includes the pressure due to the water column and the pressure due to the atmosphere.

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The required probabilities are as follows:
(a) P(X ≤ 6, Y = 6) = 33

(b) P(X > 6, Y ≤ 6) = 25

(c) P(X > Y) = 66

(d) P(X + Y = 13) = 13

To find the probabilities, we need to calculate the sum of the joint probability values for the given events.

(a) P(X ≤ 6, Y = 6):

We need to sum the joint probability values for X ≤ 6 and Y = 6.

P(X ≤ 6, Y = 6) = f(4, 6) + f(5, 6) + f(6, 6)

= (4 + 6) + (5 + 6) + (6 + 6)

= 10 + 11 + 12

= 33

Therefore, P(X ≤ 6, Y = 6) = 33.

(b) P(X > 6, Y ≤ 6):

We need to sum the joint probability values for X > 6 and Y ≤ 6.

P(X > 6, Y ≤ 6) = f(7, 5) + f(7, 6)

= (7 + 5) + (7 + 6)

= 12 + 13

= 25

Therefore, P(X > 6, Y ≤ 6) = 25.

(c) P(X > Y):

We need to sum the joint probability values for X > Y.

P(X > Y) = f(5, 4) + f(6, 4) + f(6, 5) + f(7, 4) + f(7, 5) + f(7, 6)

= (5 + 4) + (6 + 4) + (6 + 5) + (7 + 4) + (7 + 5) + (7 + 6)

= 9 + 10 + 11 + 11 + 12 + 13

= 66

Therefore, P(X > Y) = 66.

(d) P(X + Y = 13):

We need to find the joint probability value for X + Y = 13.

P(X + Y = 13) = f(6, 7)

P(X + Y = 13) = 6 + 7

= 13

Therefore, P(X + Y = 13) = 13.

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the antiderivative of x √(1 − x²) dx is −√(1 − x²) + C Where C is the constant of integration and The value of  ∫ln(x)dx is

x (ln(x) − 1) + C

a) Using substitution, find the antiderivative of x √(1 − x²) dx The integral can be evaluated using the substitution u = 1 − x², so that du/dx = −2x. Then the integral becomes

∫x √(1 − x²) dx

= −∫√(1 − x²) d(1 − x²)

= −(1/2) ∫u^(-1/2) du

= −(1/2) 2u^(1/2) + C

= −√(1 − x²) + C Where C is the constant of integration.

b) Using integration by parts, find the antiderivative of ln(x) dx The integral can be evaluated using integration by parts with u = ln(x) and dv/dx = 1, so that du/dx = 1/x and v = x. Then the integral becomes

∫ln(x) dx = x ln(x) − ∫x (1/x) dx

= x ln(x) − x + C

= x (ln(x) − 1) + C

Where C is the constant of integration. This is the required antiderivative of ln(x) dx.

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Marginal profit function, f'(x) = 0.08x f'(9500) = 0.08(9500) = 760Thus, p(9500) = 760.

Given: $G(n)=150t+12,000$ and $A(n)=−0.04x^2+000x$

The profit function, f(x) is given by subtracting the cost function, C(x) from the revenue function, R(x)

So, f(x) = R(x) - C(x)Where, R(x) = G(n) = 150t + 12,000 and C(x) = A(n) = −0.04x² + 000x

On substituting the values, we get,

                                    f(x) = 150t + 12,000 - (-0.04x² + 000x) = 150t + 0.04x² - 000x + 12,000

Thus, the profit function, f(x) = 150t + 0.04x² - 000x + 12,000.

Marginal profit function is the derivative of profit function with respect to x.

It gives the rate of change of profit function with respect to x.So, to find marginal profit, we need to differentiate profit function w.r.t x.

                                         f(x) = 150t + 0.04x² - 000x + 12,000

Differentiating w.r.t x, we getf'(x) = d/dx (150t) + d/dx (0.04x²) - d/dx (000x) + d/dx (12,000)

                                                 = 0 + 0.08x - 000 + 0 = 0.08x

Thus, the marginal profit function is given by f'(x) = 0.08x.(e)To find f(9200), we need to substitute x = 9200 in profit function,

                                 f(x) = 150t + 0.04x² - 000x + 12,000 f(9200) = 150t + 0.04(9200)² - 000(9200) + 12,000

                                     = 150t + 338400 - 0 + 12,000 = 150t + 350,400

Thus, f(9200) = 150t + 350,400

To find p(9500), we need to substitute x = 9500 in marginal profit function,

f'(x) = 0.08x f'(9500) = 0.08(9500) = 760Thus, p(9500) = 760.

Hence, the required value is 760.

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Therefore, the linear convolution of the two sequences is \( y(n) = \{0, 1, 3, 8\} \). Therefore, the periodic convolution of the two sequences is \( y_p(n) = \{0, 1, 3, 0\} \).

To determine the linear convolution of two sequences, we convolve the two sequences by taking the sum of the products of corresponding elements. For the given sequences \( x_1(n) = \{0, 1, 2, 3\} \) and \( x_2(n) = \{1, 1, 2, 2\} \), the linear convolution can be calculated as follows:

\( y(n) = x_1(n) * x_2(n) \)

\( y(0) = 0 \cdot 1 = 0 \)

\( y(1) = (0 \cdot 1) + (1 \cdot 1) = 1 \)

\( y(2) = (0 \cdot 2) + (1 \cdot 1) + (2 \cdot 1) = 3 \)

\( y(3) = (0 \cdot 2) + (1 \cdot 2) + (2 \cdot 1) + (3 \cdot 1) = 8 \)

To determine the periodic convolution, we need to consider the periodicity of the sequences. Since both sequences have a length of 4, their periods are also 4. We calculate the periodic convolution by performing the linear convolution modulo 4.

\( y_p(n) = (x_1(n) * x_2(n)) \mod 4 \)

\( y_p(0) = 0 \)

\( y_p(1) = 1 \)

\( y_p(2) = 3 \)

\( y_p(3) = 0 \)

The output sequence depends on the specific application or context in which the convolution is used. The linear convolution and periodic convolution represent the relationships between the input sequences, but the output sequence may have different interpretations based on the system being analyzed.

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(-4 + 7i)² = 9 + 56i ; Where x + iy is complex form.

To find the square of (-4 + 7i), we can use the formula for squaring a complex number, which states that (a + bi)² = a² + 2abi - b².

In this case, a = -4 and b = 7. Applying the formula, we have:

(-4 + 7i)² = (-4)² + 2(-4)(7i) - (7i)²

= 16 - 56i - 49i²

Since i² is equal to -1, we can substitute -1 for i²:

(-4 + 7i)² = 16 - 56i - 49(-1)

= 16 - 56i + 49

= 65 - 56i

So, (-4 + 7i)² simplifies to 65 - 56i.

If we multiply the result by 4, we get:

4(-4 + 7i)² = 4(65 - 56i)

= 260 - 224i

Therefore, 4(-4 + 7i)² is equal to 260 - 224i.

The square of (-4 + 7i) is 65 - 56i. Multiplying that result by 4 gives us 260 - 224i.

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The functions represented on the graph are (b)

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

The graph

On the graph, we have the following intervals:

When the intervals are represented as inequalities, we have the following:

This means that the intervals of the graphs are x ≤ 2 and x > 2

From the list of options, we have the graph to be option (b

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Based on the provided sample output, it seems that you have a 4x4 matrix, and you want to calculate the sum of each row. Here's an example implementation in Python:

python

Copy code

def calculate_row_sums(matrix):

   row_sums = []

   for row in matrix:

       row_sum = sum(row)

       row_sums.append(row_sum)

   return row_sums

# Get the size of the matrix from the user

size = int(input("Enter the size of the matrix: "))

# Get the matrix elements from the user

matrix = []

print("Enter the matrix:")

for _ in range(size):

   row = list(map(int, input().split()))

   matrix.append(row)

# Calculate the row sums

row_sums = calculate_row_sums(matrix)

# Print the row sums

for i, row_sum in enumerate(row_sums):

   print("Sum of the", i, "row is =", row_sum)

Sample Input:

mathematica

Copy code

Enter the size of the matrix: 4

Enter the matrix:

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

Output:

csharp

Copy code

Sum of the 0 row is = 4

Sum of the 1 row is = 4

Sum of the 2 row is = 4

Sum of the 3 row is = 4

This implementation prompts the user to enter the size of the matrix and its elements.

It then calculates the sum of each row using the calculate_row_sums() function and prints the results.

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The answer is: 1 local maximum at x = 24/7, which is the only local maximum of the function.

Given a function h(x) = x7 - 4x6 + 10

We have to find the x-coordinates of all local maxima, using the second derivative test.

Second Derivative Test

If the second derivative of the function at a point is positive, the function has a relative minimum at that point.

If the second derivative of the function at a point is negative, the function has a relative maximum at that point.

If the second derivative of the function at a point is zero, the test is inconclusive.

x-coordinates of all local maxima:

The first derivative of the given function is

h'(x) = 7x6 - 24x5

The second derivative of the given function is

h''(x) = 42x4 - 120x3h''(x) = 6x3(7x - 20)

The critical values are found by setting the first derivative to zero.

h'(x) = 7x6 - 24x5 = 0x5

(7x - 24) = 0

x = 0 and x = 24/7, which are the critical values.

We use the second derivative test to classify each critical point as a relative minimum, a relative maximum, or neither.

If the second derivative is positive at a critical point, the point is a relative minimum.

If the second derivative is negative at a critical point, the point is a relative maximum.

If the second derivative is zero at a critical point, the test is inconclusive.

The critical point must be tested by another method.

Using the second derivative test,

h''(0) = 6(0) (7(0) - 20) = 0

h''(24/7) = 6(247)

(7(247) - 20) > 0

The second derivative is positive at x = 24/7.

Therefore, the function h(x) has a local maximum at x = 24/7.

The answer is: 1 local maximum at x = 24/7, which is the only local maximum of the function.

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The area of the resulting surface is approximately 22π square units.

Therefore, the correct option is option D.

The given curve is rotated about the x-axis.

We are supposed to find the area of the resulting surface.

Let us first obtain the differential element of the given curve.

We know that the area of a surface obtained by rotating a curve around the x-axis is given by:

S=2π∫abf(x)√(1+(dy/dx)²)dx

where f(x) is the function of the curve which is being rotated and dy/dx is its differential element obtained as:

dy/dx=−2x

Let us now substitute the values into the formula:

S=2π∫−325−x2​(1+(−2x)²)dx

=2π∫−324(1+4x²)dx

=2π[1x+4x3/3]−324

=2π(11/3)

≈22π

The area of the resulting surface is approximately 22π square units.

Therefore, the correct option is option D.

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If a 4x1 MUX is not available, we can also implement the expression F(A, B, C) using a 2x1 MUX. In this case, we would need to use multiple 2x1 MUXes and combine their outputs to achieve the desired function. However, the 4x1 MUX is more straightforward and efficient for this particular expression.

To implement the Boolean expression F(A, B, C) = (A + B + C)(A' + C')(B + C') using a 4x1 multiplexer (MUX), we can consider the inputs A, B, and C as the select lines of the MUX, while the complement of A (A'), the complement of C (C'), and the expression (B + C') can be used as the data inputs. The output of the MUX will represent the function F.

The inputs A, B, and C are used to select the appropriate data input. We can set up the MUX as follows:

• Connect A' to one of the data inputs of the MUX.

• Connect C' to the other data input.

• Connect B + C' to the MUX's single-bit output.

By setting up the MUX in this way, we effectively implement the expression (A' + C')(B + C'), which is equivalent to the expression F(A, B, C).

If a 4x1 MUX is not available, we can also implement the expression F(A, B, C) using a 2x1 MUX. In this case, we would need to use multiple 2x1 MUXes and combine their outputs to achieve the desired function. However, the 4x1 MUX is more straightforward and efficient for this particular expression.

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The area of the composite figure is  40 in²

The formula for the area of a rectangle is expressed as;

A = length ×width

Substitute the value, we get;

Area = 7(3)

Multiply the value, we have;

Area = 21 in²

Also, we have that;

Area of the second rectangle = 2(7) = 14 in²

Then, area of the triangle is expressed as;

Area = 1/2bh

Area = 1/2 × 5 × 2

Area = 5 in²

Total area = 40 in²

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Answer: Approximately $759.96.

Step-by-step explanation:

To calculate the monthly payment for Bill's loan, we can use the formula for calculating the monthly payment of a loan:

Monthly Payment = P * r * (1 + r)^n / ((1 + r)^n - 1)

Where:

P = Principal amount (loan amount)

r = Monthly interest rate

n = Total number of monthly payments

Let's calculate the monthly payment using the given information:

Principal amount (P) = $40,000

Annual interest rate = 6%

Monthly interest rate (r) = Annual interest rate / 12 = 6% / 12 = 0.06 / 12 = 0.005

Total number of monthly payments (n) = 5 years * 12 months/year = 60 months

Plugging these values into the formula, we get:

Monthly Payment = 40,000 * 0.005 * (1 + 0.005)^60 / ((1 + 0.005)^60 - 1)

Calculating this expression gives us the monthly payment Bill needs to make to pay off the loan.

The derivative of h(x) at x = 1 is 12.

For a function y=f(u) and u=g(x), the derivative of y with respect to x is [tex]dy/dx=dy/du * du/dx[/tex]. Here, [tex]u = g(x)[/tex] and [tex]y = h(x)[/tex], so [tex]dy/dx=dh/du * du/dx.[/tex]

Given that [tex]h(x)=f(g(x))[/tex] => [tex]u = g(x)[/tex] and [tex]y = f(u)[/tex]. Then, [tex]h'(1) = f'(g(1)) * g'(1)h'(1) = f'(2) * 4[/tex]. Hence, [tex]h'(1) = 3 * 4 = 12[/tex]. So, the derivative of h(x) at x = 1 is 12. Therefore, the correct option is (D) 12.

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A linear differential operator that annihilates the given function e^(-x) + 6xe^x - x^2e^x is (D^3 - 3D^2 + 4D - 2)where D denotes the differential operator d/dx and '^' is the exponentiation operator.

An explanation for this answer is given below.Differential Operator:In calculus, a differential operator is a mathematical operator defined on a function to obtain the function's derivative. Differential operators can also be used to describe the solution space for specific differential equations. These operators are linear; in other words, if they are applied to a sum of functions, the result is the sum of the functions that have been individually operated on.The given function:  e^(-x) + 6xe^x - x^2e^x

The first derivative of the given function with respect to x is:-e^(-x) + 6e^x + 6xe^x - 2xe^x

The second derivative of the given function with respect to x is:e^(-x) + 12xe^x - 4xe^xThe third derivative of the given function with respect to x is:

-e^(-x) + 12e^x + 24xe^x - 4e^x + 4xe^x

The differential operator (D^3 - 3D^2 + 4D - 2) when applied to the given function, yields:

(D^3 - 3D^2 + 4D - 2)(e^(-x) + 6xe^x - x^2e^x)

= -e^(-x) + 12e^x + 24xe^x - 4e^x + 4xe^x - 3[-e^(-x) + 6e^x + 6xe^x - 2xe^x]+ 4[-e^(-x) + 6e^x + 6xe^x - 2xe^x] - 2[e^(-x) + 6xe^x - x^2e^x]

= 0

This implies that the differential operator (D^3 - 3D^2 + 4D - 2) annihilates the given function.

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The value of the sum [tex]$$\sum_{n=1}^{50} n^{2}=42925$$[/tex]and the value of the sum [tex]$$\sum_{n=1}^{20} n^{3}=44100$$[/tex]

Given :

[tex]$$\sum_{n=1}^{50} n^{2}=1^{2}+2^{2}+3^{2}+\cdots 50^{2}$$[/tex]

We know that,

[tex]$$\sum_{n=1}^{n} n^{2} = \frac{n(n+1)(2n+1)}{6}$$[/tex]

Putting n=50, we get,

[tex]$$\sum_{n=1}^{50} n^{2}= \frac{50*51*101}{6} = 42925 $$[/tex]

Given,

[tex]$$\sum_{n=1}^{20} n^{3}=1^{3}+2^{3}+3^{3}+\cdots 20^{3}$$[/tex]

We know that

[tex],$$\sum_{n=1}^{n} n^{3} = \frac{n^{2}(n+1)^{2}}{4}$$[/tex]

Putting n=20, we get,

[tex]$$\sum_{n=1}^{20} n^{3} = \frac{20^{2}*21^{2}}{4} = 44100$$[/tex]

Hence, the value of the sum [tex]$$\sum_{n=1}^{50} n^{2}=42925$$[/tex]

and the value of the sum [tex]$$\sum_{n=1}^{20} n^{3}=44100$$[/tex]

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The 11th term of the arithmetic sequence is 34. Hence, the correct option is C.

To find the 11th term of an arithmetic sequence, you can use the formula:

nth term = first term + (n - 1) * difference

Given that the first term is -6 and the difference is 4, we can substitute these values into the formula:

11th term = -6 + (11 - 1) * 4
         = -6 + 10 * 4
         = -6 + 40
         = 34

Therefore, the 11th term of the arithmetic sequence is 34. Hence, the correct option is C.

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Prefix notation, infix notation, and postfix notation are three different ways to represent mathematical expressions.

They differ in the placement of operators and operands within the expression.

1. Prefix Notation (also known as Polish Notation):

In prefix notation, the operator is placed before its operands. It does not require the use of parentheses to indicate the order of operations. Here's an example:

Expression: + 5 3

Explanation: In prefix notation, the addition operator '+' is placed before its operands '5' and '3'. The expression evaluates to 8.

2. Infix Notation:

In infix notation, the operator is placed between its operands. It is the most commonly used notation in mathematics and is familiar to most people. Parentheses are used to indicate the order of operations. Here's an example:

Expression: 5 + 3

Explanation: In infix notation, the addition operator '+' is placed between the operands '5' and '3'. The expression evaluates to 8.

3. Postfix Notation (also known as Reverse Polish Notation):

In postfix notation, the operator is placed after its operands. Similar to prefix notation, postfix notation does not require the use of parentheses to indicate the order of operations. Here's an example:

Expression: 5 3 +

Explanation: In postfix notation, the addition operator '+' is placed after the operands '5' and '3'. The expression evaluates to 8.

To evaluate expressions in prefix, infix, or postfix notation, different algorithms or parsing techniques are used. For example, to evaluate postfix expressions, a stack-based algorithm known as the postfix evaluation algorithm can be applied.

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Selective Perception and Action Path can help in making a decision on whether to continue or modify the policy by considering biases in perception and developing a clear plan of action based on gathered information and stakeholder input.

In the given scenario, several of the mentioned best practices can be useful for making a decision on how to proceed with the office policy. Let's explore some of them:

1. Deviation from Standard: This best practice suggests considering alternative approaches to the existing policy. You can analyze whether the current policy of bringing people back into the office is still effective and explore other possibilities, such as a hybrid model or flexible work arrangements.

This allows you to deviate from the standard approach and adapt to the current situation.

2. Six Honest Servingmen: This principle encourages asking critical questions to gather relevant information. You can apply this by gathering feedback from employees to understand their perspective on productivity, job satisfaction, and the impact of working in the office versus remotely.

By considering the opinions and experiences of your team members, you can make a more informed decision.

3. So What? What if?: This approach involves considering the potential consequences and exploring different scenarios. You can ask questions such as "What if we continue with the current policy?" and "What if we modify the policy to accommodate remote work?"

By evaluating the potential outcomes and weighing the pros and cons of each option, you can make a decision based on informed reasoning.

4. Meaningful Experience: This principle emphasizes the importance of drawing insights from past experiences. In this case, you can review the productivity data from the two years of remote work and compare it to the three months since the return to the office.

If there is a noticeable decrease in productivity, you can take this into account when deciding whether to continue with the current policy or make adjustments.

5. Action Path: This best practice involves developing a clear plan of action. Once you have considered the various factors and options, you can create an action plan that outlines the steps to be taken.

This could involve conducting surveys, seeking input from team members, analyzing data, and consulting with relevant stakeholders. Having a well-defined action path can help you make an informed decision and communicate it effectively to your team.

By applying these best practices, you can gather information, analyze the situation, consider different perspectives, and develop a well-thought-out plan for how to proceed with the office policy.

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Using the cosine rule, the distance the plane has flown during Jack's phone call can be calculated by taking the square root of the sum of the squares of the initial and final distances, minus twice their product, multiplied by the cosine of the angle difference.

To determine the distance the plane has flown during Jack's phone call, we can use the cosine rule in trigonometry.

The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles.

Let's denote the initial distance from Jack to the plane as d1 and the final distance as d2.

We know that the altitude of the plane remains constant at 2400 meters.

According to the cosine rule:

[tex]d^2 = a^2 + b^2 - 2ab \times cos(C)[/tex]

Where d is the side opposite to the angle C, and a and b are the other two sides of the triangle.

For the initial angle of elevation (70°), we have the equation:

[tex]d1^2 = (2400)^2 + a^2 - 2 \times 2400 \times a \timescos(70)[/tex]

Similarly, for the final angle of elevation (50°), we have:

[tex]d2^2 = (2400)^2 + a^2 - 2 \times 2400 \times a \times cos(50)[/tex]

To find the distance the plane has flown, we subtract the two equations:

[tex]d2^2 - d1^2 = 2 \times 2400 \times a \times (cos(70) - cos(50))[/tex]

Now we can solve this equation to find the value of a, which represents the distance the plane has flown.

Finally, we calculate the square root of [tex]a^2[/tex] to find the distance in meters.

It's important to note that the angle of elevation assumes a straight-line path for the plane's movement and does not account for any changes in altitude or course adjustments that might occur during the phone call.

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The solution of the differential equation using Laplace transform (Heaviside Expansion Theorem only) is;

y(t) = [3 sin t + 2 cos t - 2 e^(-t) + (6/5) e^(2t)] u(t) - (3/5) t sin t u(t)

Given differential equation is y" - y = 4e^(-x) + 3e^(2x); y(0) = 0, y'(0) = -1

Now, taking Laplace transform of both sides of the differential equation, we get;

[s² Y(s) - s y(0) - y'(0)] - Y(s) = [4 / (s + 1)] + [3 / (s - 2)]

On substituting y(0) = 0 and y'(0) = -1, we get;

s² Y(s) + Y(s) = [4 / (s + 1)] + [3 / (s - 2)] + s …(1)

We know that Heaviside Expansion Theorem states that if f(s) is a rational function of s of degree less than N, then:

f(s) = [(ak s + bk-1 s^{k-1} + ....+ b1 s + b0)] / [A(s - p1)^q1 (s - p2)^q2 ......(s - pr)^qr]

where (s - pi) are distinct linear factors. Here, k < N, and q1, q2, ..., qr are positive integers such that q1 + q2 + ...+ qr = N - kAlso, a coefficient ak should be nonzero.

Hence, using Heaviside Expansion Theorem in equation (1), we get;

Y(s) = [As + B] / [s² + 1] + [C / (s + 1)] + [D / (s - 2)] + E(s) ... (2)

Differentiating both sides of equation (2) with respect to s, we get:

Y'(s) = [A(s² + 1) - 2Bs] / (s² + 1)² - [C / (s + 1)²] - [D / (s - 2)²] + E'(s) ... (3)

We are also given y(0) = 0 and y'(0) = -1 which gives Y(0) = 0 and Y'(0) = -1

Substituting these values in equation (2) and equation (3) and then solving for A, B, C, D and E(s), we get;

A = 3/5, B = 2/5, C = -2, D = 6/5 and E(s) = s / (s² + 1)²

On applying inverse Laplace transform on Y(s), we get;

y(t) = [3 sin t + 2 cos t - 2 e^(-t) + (6/5) e^(2t)] u(t) - (3/5) t sin t u(t) where u(t) is the unit step function.

Hence, the solution of the differential equation using Laplace transform (Heaviside Expansion Theorem only) is;

y(t) = [3 sin t + 2 cos t - 2 e^(-t) + (6/5) e^(2t)] u(t) - (3/5) t sin t u(t)

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Answer: -7.87 × 108

Step-by-step explanation: Hope this helps:)

The interval(s) where the function is increasing are (-5, 0) and (0, 5), and the interval(s) where it is decreasing are (-, -5) and (5, ).

We have the function given as g(x) = x⁴ - 50x² + 5. Now, we have to determine the interval(s) where the function is increasing and the interval(s) where it is decreasing. To determine where a function is increasing or decreasing, we need to find its first derivative and check the sign of the first derivative. If the sign of the first derivative is positive, the function is increasing in that interval. If the sign of the first derivative is negative, the function is decreasing in that interval.

Let's differentiate g(x) with respect to x to find its first derivative as follows: g'(x) = 4x³ - 100xWe can factorize g'(x) as shown below:g'(x) = 4x(x² - 25) = 4x(x - 5)(x + 5)Now we can create a sign chart for g'(x) as shown below :x -5 0 +5 x-5(-) (-) (+)x (-) 0 (+)x +5 (+) (+)From the above sign chart, we can see that g'(x) is negative for x < -5 and x > 5, and positive for -5 < x < 0 and 0 < x < 5.

Therefore, the function g(x) is decreasing on the intervals (-∞, -5) and (5, ∞), and it is increasing on the intervals (-5, 0) and (0, 5).

Thus, we can say that the interval(s) where the function is increasing is (-5, 0) and (0, 5), and the interval(s) where the function is decreasing is (-∞, -5) and (5, ∞).

The interval(s) where the function is increasing is (-5, 0) and (0, 5), and the interval(s) where the function is decreasing is (-∞, -5) and (5, ∞).

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MATLAB is a programming environment that is commonly used for numerical analysis, signal processing, data analysis, and graphics visualization. In MATLAB, the symbolic expression of Fourier transforms of the given functions, x1(t) and x2(t), can be generated using the syms and fourier functions. The commands for generating the symbolic expression of Fourier transforms of the given functions are shown below:

To find the symbolic expression of Fourier transform of \( x_{1}(t)=e^{-|t|} \),

use the following command: syms t;
fourier(e^(-abs(t)))The symbolic expression of the Fourier transform of x1(t) is as follows:
\( \frac{2}{\pi \left(\omega^{2}+1\right)} \)

To find the symbolic expression of Fourier transform of \( x_{2}(t)=t e^{-t^{2}} \),

use the following command: syms t;
fourier(t*e^(-t^2))

The symbolic expression of the Fourier transform of x2(t) is as follows:

\( \frac{i}{2} \sqrt{\frac{\pi}{2}} e^{-\frac{\omega^{2}}{4}} \)

Given the function \( x(t)=e^{-2 t} \cos (t) t u(t) \),

we can find its Fourier transform using the following command: syms t;
syms w;
fourier(t*exp(-2*t)*cos(t)*heaviside(t))

The symbolic expression of the Fourier transform of x(t) is as follows:
\( \frac{\frac{w+2}{w^{2}+9}}{2i} \)

Hence, the symbolic expression of the Fourier transforms of the given functions, x1(t), x2(t), and x(t), using the syms and fourier functions in MATLAB are provided in this solution.

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The given differential equation is, x²y′′ − 4xy’ + 6y = 0Now, we have verified that x² and x³ are linearly independent solutions of the above second-order, homogeneous differential equation on the interval (0, ∞).

Therefore, the general solution of the given differential equation is given by the linear combination of the two fundamental solutions, y₁ and y₂ as follows, y = c₁y₁ + c₂y₂, where c₁ and c₂ are arbitrary constants. To find the values of the constants c₁ and c₂, we substitute the fundamental solutions, y₁ = x² and y₂ = x³ in the general solution, y = c₁y₁ + c₂y₂, and their respective derivatives in the differential equation, x²y′′ − 4xy’ + 6y = 0. Now, solving this system of two equations in two unknowns yields the values of c₁ and c₂. So, the general solution of the given differential equation is given by y = c₁x² + c₂x³.

Let, y = xᵐ Now, differentiate both sides of this equation w.r.t. x, we get; y' = mx^(m-1)Differentiating both sides of this equation again w.r.t. x, we get; y'' = m(m-1)x^(m-2) Now, substitute y, y' and y'' in the given differential equation x²y′′ − 4xy’ + 6y = 0,

we get;x²y′′ − 4xy’ + 6y = x²(m(m-1)x^(m-2)) - 4x(mx^(m-1)) + 6xᵐ

= xᵐ(x²m(m-1)x^(m-2)) - xᵐ(4mx^(m-1)) + xᵐ(6)

= xᵐ(m(m-1)x^(m)) - xᵐ(4mx^m) + xᵐ(6)

= xᵐ(x^2m(m-1) - 4mx + 6)Since xᵐ ≠ 0, cancelling xᵐ on both sides,

we get;x^2m(m-1) - 4mx + 6 = 0

=> x^2(m^2 - m) - 4mx + 6 = 0

By substituting the given fundamental solution y₁ = x² in the differential equation,

we get;x²y′′ − 4xy’ + 6y = 0x²y'' − 4xy' + 6y

= x²(2) − 4x(2x) + 6(x²)

= 2x² − 8x³ + 6x²

= 8x² − 8x³

Therefore, the solution is not zero if x ≠ 0. Thus, x² is a non-trivial solution of the given differential equation. Similarly, we can show that x³ is also a non-trivial solution of the given differential equation. Thus, x² and x³ form a fundamental set of solutions of the given differential equation.

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The graph that consists of equations, intersecting at x = -1 and y = 8, is graph A, because it represents the solution of the two equations.

The solution of the two system of equations is calculated by applying the following formula as follows;

The given system of equations are;

-3y - 3x = - 21  ----- (1)

0 = y - x - 9   ------- (2)

From equation (2), make y the subject of the formula;

y = x + 9

Substitute the value of y into equation (1);

-3y - 3x = - 21

-3(x + 9) - 3x = -21

-3x - 27 - 3x = -21

-6x = 6

x = -1

y = x + 9

y = -1 + 9

y = 8

The solution of the equations = (-1, 8)

The graph that consists of equations, intersecting at x = -1 and y = 8, is graph A, so graph A is the solution of the two equations.

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The value of A is the difference of this integral evaluated at x = -2 and x = 2 found as: A = 20/3.

The region described is the region between y = 5/3 and y = 1/√(4 − x²).

To find the area of this region, integrate the difference between the two functions with respect to x between x = -2 and x = 2

(since the denominator of the second function is sqrt(4-x^2),

the region exists only between x = -2 and x = 2).

Hence,

Area of the region bounded by y=5/3​ and y=1/√(4−x2)​ is given by:

A=∫dx∫(5/3 − 1/√(4−x2))dy

=∫[5/3 − 1/√(4−x2)]dx

Area A is given by

∫(5/3 − 1/√(4−x2))dx

= [5/3]x − arcsin(x/2) + C

Where C is the constant of integration.

The value of A is the difference of this integral evaluated at x = -2 and x = 2.

Hence,

A = [5/3](2) − arcsin(1) − [5/3](-2) + arcsin(-1)

= [10/3] + [π/6] + [10/3] − [π/6]

= 20/3.

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The system is absolutely summable and hence the given DT-LTI system is stable.

The given system has impulse response as:\[h[n] = \left( {\frac{{ - 1}}{2}} \right)^{ - n}u[ - n]\]

Let's check whether the given system is stable or not.

The DT-LTI system is said to be stable, if and only if its impulse response is absolutely summable. i.e., if the system impulse response, h[n] satisfies the condition of the absolute summability, then the system is said to be stable.

Thus,\[\mathop \sum \limits_{n =  - \infty }^\infty \left| {h[n]} \right| = \mathop \sum \limits_{n =  - \infty }^\infty \left| {\left( {\frac{{ - 1}}{2}} \right)^{ - n}u[ - n]} \right| = \mathop \sum \limits_{n = 0}^\infty {\left( {\frac{1}{2}} \right)^n} \le \infty \]

Thus, the system is absolutely summable and hence the given DT-LTI system is stable.

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The event with the highest probability is selecting a bus on Route R3, with a probability of 0.42.

The data given is a bus schedule for three bus routes, and we are to select the event with the highest probability of occurring when a bus is chosen at random.

The events are each bus route represented by R1, R2, and R3.

Total Number of Buses = 15 + 20 + 25

                                        = 60

The probability of each event occurring is calculated by dividing the number of buses on each route by the total number of buses.

P(R1) = 15/60 = 0.25

P(R2) = 20/60 = 0.33

P(R3) = 25/60 = 0.42

Therefore, the event with the highest probability is selecting a bus on Route R3, which has a probability of 0.42. This means that if you select a bus randomly, the probability that you would select a bus on Route R3 is the highest.

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