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The real part of the particular solution to the differential equation is [tex](1/30)Re(e^(3it))(sin(3t) - cos(3t))[/tex]

The real part of the particular solution to the differential equation:

[tex]\frac{d^2y}{dt^2} +3\frac{dy}{dt} +7y = e^(3it)[/tex]

First, we assume a particular solution of the form:

[tex]y(t) = Bcos(3t) + Csin(3t)[/tex]

where B and C are real fractions.

Taking the first and second derivatives of y(t), we get:

[tex]\frac{dy}{dt} = -3Bsin(3t) + 3Ccos(3t)[/tex]

[tex]\frac{d^2y}{dt2} = -9Bcos(3t) - 9Csin(3t)[/tex]

Substituting these into the differential equation, we get:

[tex](-9Bcos(3t) - 9Csin(3t)) + 3(-3Bsin(3t) + 3Ccos(3t)) + 7(Bcos(3t) + Csin(3t)) = e^(3it)[/tex]

Simplifying and collecting terms, we get:

[tex](-9B + 21C)*cos(3t) + (-9C - 9B)*sin(3t) = e^(3it)[/tex]

Comparing the coefficients of cos(3t) and sin(3t), we get:

[tex]-9B + 21C = Re(e^(3it))[/tex]

[tex]-9C - 9B = 0[/tex]

Solving for B and C, we get:

[tex]B = -C[/tex]

[tex]C = (1/30)*Re(e^(3it))[/tex]

Therefore, the particular solution is:

[tex]y(t) = -Ccos(3t) + Csin(3t) = (1/30)Re(e^(3it))(sin(3t) - cos(3t))[/tex]

A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).

Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.

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Therefore, when the equation  x = 72 - 3x, the value of 2 - 3x is -52.

An equation is a mathematical statement that shows the equality of two expressions. It typically contains one or more variables, which are symbols that can represent any number or value. The expressions on both sides of the equal sign are called the left-hand side (LHS) and the right-hand side (RHS) of the equation. Equations are used to describe relationships between quantities or to solve problems. They can be represented in various forms, including linear equations, quadratic equations, exponential equations, and trigonometric equations. Equations can be solved by performing operations on both sides of the equation to isolate the variable or variables.

Here,

When we are given that x = 72 - 3x, we can solve for x by first adding 3x to both sides of the equation:

x + 3x = 72

Combining like terms, we get:

4x = 72

Dividing both sides by 4, we get:

x = 18

Now that we know x = 18, we can substitute this value into the expression 2 - 3x:

2 - 3x = 2 - 3(18)

2 - 3x = 2 - 54

2 - 3x = -52

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Answer:

Step-by-step explanation:

2(2x-5) = 18 (FOIL)

4x-10=18

4x = 28

x= 7

Answer:

x = -2

Step-by-step explanation:

2(2x - 5) = -18

Divide both sides by 2.

2x - 5 = -9

Add 5 to both sides.

2x = -4

Divide both sides by 2.

x = -2

The probability of the given event is 11/13.

A card is drawn randomly from a deck of 52 cards.

The probability that the card drawn is neither an ace nor a king is a question that can be answered with probability.

We know that there are 4 aces and 4 kings in a deck of 52 cards.

The probability of drawing an ace or a king is P(Ace or King) = 4/52 + 4/52 = 8/52 = 2/13

There are four of each card rank in a deck of 52 cards, and there are a total of 52 cards. A player must choose one of the 52 cards at random.

The probability of drawing an ace or a king is P(Ace or King) = 4/52 + 4/52 = 8/52 = 2/13

The probability of at drawing neither an ace nor a king is:

P(Not Ace and Not King) = 1 - P(Ace or King) = 1 - 2/13 = 11/13

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The total cost of compensated absences for Tamarisk Company for the years 2019 and 2020 was $348 + $1,399 = $1,747.

To calculate the cost of compensated absences for Tamarisk Company, we need to calculate the number of vacation days and sick days earned by the employees in 2019 and 2020, and then calculate the cost of the days earned but not taken.

Each employee earns 9 vacation days per year. As they can be taken after January 15th of the year following the year in which they are earned, the vacation days earned by the employees in 2019 can be taken in 2020. Therefore, in 2019, no vacation days were taken by any employee.

In 2020, the employees took a total of 8 vacation days. As there are 9 employees, the total vacation days taken in 2020 were 9 x 8 = 72.

Sick Days:

Each employee earns 7 sick days per year, and unused sick days accumulate. In 2019, the employees used a total of 5 sick days. Therefore, the unused sick days at the end of 2019 were 9 x 7 - 5 = 58.

In 2020, the employees used a total of 6 sick days, and the unused sick days at the end of 2020 were 58 + 9 x 7 - 6 = 109.

To find the cost of compensated absences. The unused sick days and vacation days must be multiplied to get the hourly wage rate in effect in a year.

In 2019, the cost of compensated absences was 58 x $6 = $348.

In 2020, the cost of compensated absences was (72 + 109) x $7 = $1,399.

Therefore, the total cost of compensated absences for Tamarisk Company for the years 2019 and 2020 was $348 + $1,399 = $1,747.

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To determine the distance traveled by Katie at different marks, we can use the formula Distance = Rate × Time.

To determine how far Katie has traveled at each mark, we can use the formula:
Distance = Rate × Time

So, the elements in the range of the function that would be used to determine how far Katie has traveled at each mark are 384 miles, 416 miles, and 448 miles.

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Step-by-step explanation:

x²-6x+9=0

using the almighty formula where a=1 , b=-6 , c=9

Completing the proof that if A has strictly positive diagonal entries, then all of its eigenvalues have a positive real part.

For any square matrix A, a diagonal element is any element that is on the main diagonal, i.e., (i,i) for 1<=i<=n. If A has strictly diagonal dominance, then for each row i of A, the absolute value of the diagonal element of that row is more than the sum of the absolute values of the non-diagonal elements of that row.If A is invertible, then det(A) is nonzero. If all the diagonal entries are positive, then det(A) is also positive, which implies that all the eigenvalues of A have the same sign (either all are positive or all are negative).As a result, suppose all of A's diagonal entries are positive. Then all of A's eigenvalues are positive since they all have the same sign as det(A).

And if all of A's eigenvalues are positive, then all of their real parts are positive as well, hence completing the proof that if A has strictly positive diagonal entries, then all of its eigenvalues have a positive real part.

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The smallest number of batteries in the package for the probability to exceed 1% is a) 17. This can be calculated using the binomial distribution with parameters n=17 and b)p=0.2927 and number of batteries is c)4. (Where p is the probability from part a).

a) The probability that a battery does not reach the significant milestone after 8 hours of usage is 0.2927.

This can be calculated using the cumulative normal distribution function. The parameters are μ=7.36, σ=0.29, and x=8.

b) The probability that at least 10 batteries will last more than 7.5 hours is 0.7012.

This can be calculated using the binomial distribution with parameters n=12 and p=0.2927 (where p is the probability from part a).

c) The number of batteries should be in package is μ*4.2/7.5 = 4.

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The new position of Rhombus ABCD after the translation can be described as follows: point A is now at (-1,-8), point B is at (2,-3), point C is at (7,0), and point D is at (4,-5).

To translate Rhombus ABCD using the rule (x, y) = (x+2, y-6), we add 2 to the x-coordinate and subtract 6 from the y-coordinate for each vertex.

Thus, the new vertices for the translated rhombus are:

A' = (-3+2, -2-6) = (-1, -8)

B' = (0+2, 3-6) = (2, -3)

C' = (5+2, 6-6) = (7, 0)

D' = (2+2, 1-6) = (4, -5)

Therefore, the coordinates for the translated Rhombus ABCD are A'(-1,-8), B'(2,-3), C'(7,0), and D'(4,-5).

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Step-by-step explanation:

1/(x - 5) < 3

1 < 3(x - 5)

1/3 < x - 5

1/3 + 5 < x

5 1/3 = 16/3 = 5.33333... < x

or

x > 5 1/3

As per the confidence interval, the true proportion of people in the population who prefer Candidate A lies somewhere between 0.622 and 0.698, based on the results of the survey of 1050 voters.

To calculate the margin of error, we will use the following formula:

Margin of error = z x standard error

Where z is the critical value from the standard normal distribution corresponding to our desired confidence level (95%), and the standard error is given by:

Standard error = √[(sample proportion x (1 - sample proportion)) / sample size]

Using a z-table, we find that the critical value for a 95% confidence interval is 1.96

Plugging in the values we have, we get:

Standard error = √[(0.66 x 0.34) / 1050] = 0.0193

Margin of error = 1.96 x 0.0193 = 0.0378

Therefore, the 95% confidence interval for the proportion of people who prefer Candidate A is:

0.66 ± 0.0378

Rounded to the nearest thousandth, the lower bound of the confidence interval is 0.622 and the upper bound is 0.698.

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Answer:

2

Step-by-step explanation:

literally just 2 it's quite simple

2/3 rd or 0.667th part of an hour passes from 1:35 P.M to 2:15 P.M

To determine the fraction of an hour that passes from 1:35 P.M. to 2:15 P.M., we need to calculate the elapsed time between the two times and express it as a fraction of an hour.

The elapsed time between 1:35 P.M and 2:15 P.M is 40 minutes. To convert minutes to fractions of an hour, we can divide the number of minutes by 60.

40 minutes / 60 minutes per hour = 2/3 hours = 0.67 hours

Therefore, the fraction of an hour that passes from 1:35 P.M. to 2:15 P.M. is 0.67. This means that approximately two-thirds of an hour, or 67% of an hour, has passed between these two times.

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The two options are as follows: a. 20, 24, 21, 19, 18.

This sample is a possible bootstrap sample.

A bootstrap sample is created by taking samples from the original sample with replacement. Therefore, we can include the original data multiple times.

Here, 20, 24, 19, 23, 18 is the original sample.

Sample a includes 20, 24, 19, 23, and 18 which are part of the original data, and 21 which is not part of the original data. Since we can include the original data multiple times in bootstrap samples, a is possible.b. 20, 20, 20, 20, 20This sample is not a possible bootstrap sample.

In the original sample, we have five unique data points: 20, 24, 19, 23, and 18. Sample b includes only one unique data point, 20, repeated five times. We cannot create a bootstrap sample with the same data point repeated multiple times. Hence A is correct.

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The number of different selections possible if at least 3 balls must be blue is 35


1. Calculate the total number of possible selections (6 red + 7 blue = 13 total): 13C4 = 715
2. Calculate the number of possible selections that have fewer than 3 blue balls: 6C4 = 15
3. Subtract the number of possible selections with fewer than 3 blue balls from the total number of possible selections: 715 - 15 = 700
4. Divide the answer by the number of selections with exactly 3 blue balls: 700/7 = 100
5. Multiply the result by the number of selections with exactly 4 blue balls: 100 * 7 = 700
6. Finally, subtract the number of selections with 4 blue balls from the total number of possible selections: 715 - 700 = 35

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Answer:

n > 10

Step-by-step explanation:

2n > 20

2n/2 > 20/2

n > 10

Answer:

The perimeter of a rectangle is given by:

P = 2(L + W)

where P is the perimeter, L is the length, and W is the width.

In this case, we know that P = 270 cm and L = 90 cm, so we can solve for W as follows:

270 = 2(90 + w)

Divide both sides by 2:

135 = 90 + w

Subtract 90 from both sides:

w = 45

Therefore, the width of the map is 45 cm.

Step-by-step explanation:

Therefore, the smallest positive integer with at least 8 odd factors and at least 16 even factors is N = 1800.

In mathematics, combination is a way to count the number of possible selections of k objects from a set of n distinct objects, without regard to the order in which they are selected.

The number of combinations of k objects from a set of n objects is denoted by [tex]nCk[/tex] or [tex]C(n,k),[/tex] and is given by the formula:

[tex]nCk = n! / (k! *(n-k)!)[/tex]

where n! denotes the factorial of n, i.e., the product of all positive integers up to n.

by the question.

Now, let's consider the parity (evenness or oddness) of the factors of N. A factor of N is odd if and only if it has an odd number of factors of each odd prime factor of N. Similarly, a factor of N is even if and only if it has an even number of factors of each odd prime factor of N. Therefore, the condition that N has at least 8 odd factors and at least 16 even factors can be expressed as:

[tex](a_{1} +1) * (a_{2} +1) * ... * (an+1) = 8 * 2^{16}[/tex]

Let's consider the factor 2 separately. Since N has at least 16 even factors, it must have at least 16 factors of 2. Therefore, we have a_i >= 4 for at least one prime factor p_i=2. Let's assume without loss of generality that p[tex]1=2[/tex] and [tex]a1 > =4.[/tex]

Now, let's consider the remaining prime factors of N. Since N has at least 8 odd factors, it must have at least 8 factors that are not divisible by 2. Therefore, the product (a2+1) * ... * (an+1) must be at least 8. Let's assume without loss of generality that n>=2 (i.e., N has at least three distinct prime factors).

Since a_i >= 4 for i=1, we have:

[tex]N > = 2^4 * p2 * p3 > = 2^4 * 3 * 5 = 240[/tex]

Let's now try to find the smallest such N. To minimize N, we want to make the product (a2+1) * ... * (an+1) as small as possible. Since 8 = 2 * 2 * 2, we can try to distribute the factors 2, 2, 2 among the factors (a2+1), (a3+1), (a4+1) in such a way that their product is minimized. The only possibility is:

[tex](a2+1) = 2^2, (a3+1) = 2^1, (a4+1) = 2^1[/tex]

This gives us:

[tex]N = 2^4 * 3^2 * 5^2 = 1800[/tex]

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The triangle DEF is twice the size of the triangle ABC and the triangles are similar triangles

Given the triangles ABC and DEF

From the figure, we have

AB = 1

DE = 2

This means that the triangle DEF is twice the size of the triangle ABC

Yes, the triangles ABC and DEF are similar triangles

This is because the corresponding sides of  DEF is twice the corresponding sides of triangle ABC

Multipliying the coordinates of A by 2 gives coordinates of D

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False, the probability that A or B will occur is P(A or B) = P(A) + P(B) - P(A and B).

Probability refers to the measure of the likelihood or chance of a particular event occurring. It is expressed as a number between 0 and 1, where 0 indicates an impossible event, and 1 indicates a certain event.

This formula is known as the Addition Rule for Probability and states that to calculate the probability of either event A or event B occurring (or both), we add the probability of A happening to the probability of B happening, but then we need to subtract the probability of both A and B happening at the same time to avoid double counting.

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Answer:

Step-by-step explanation:

You want to make 5.4 in three different ways.

You can add, subtract, multiply, or divide numbers to obtain a result of 5.4:

  2.9 +2.5 = 10.0 -4.6 = 2.0×2.7 = 10.8÷2 = 5.4

  √29.16 = ∛157.464 = 5.4

  [tex]\displaystyle\sum_{n=1}^\infty{10.8(3^{-n})}=5.4[/tex]

The values in the interval [0, 2π) for which the two points would intersect as required is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.

Recall from the task content; the given functions are;

f (x) = 2cos2θ and g(x) = −1 − 4cos θ − 2cos2θ

Therefore, for intersection; f (θ) and g(θ):

2 cos²θ = −1 − 4cos θ − 2cos²θ

4cos²θ + 4cosθ + 1 = 0

let cos θ = y;

4y² + 4y + 1 = 0

y = -1/2

Therefore; -1/2 = cos θ

θ = cos-¹ (-1/2)

θ = 2π/3, 4π/3.

Ultimately, the correct answer choice is; Choice C; theta equals 2 times pi over 3 comma 4 times pi over 3.

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The given functions form a fundamental set of solutions of the given system.

The solution of the given system of differential equations is shown below.a) To prove that the given functions X(1) and x(2) are the solutions of the given system of differential equations, we must substitute these functions into the given system to show that they satisfy the equations.In the given system, we have the following equations:

X_1' (t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4X_1 (t) - 2X_2 (t)

Now, let's substitute the given vector-valued functions X(1) and x(2) into the above equations and check if they satisfy these equations.

a. For X(1) = [1, 2]e^2t
Substituting X(1) into the given system, we get:

X_1' (t) = [1, 2] * 2e^2t = 2X_1 (t) - X_2 (t)
X_2' (t) = [1, 2] * 4e^2t = 4X_1 (t) - 2X_2 (t)

Therefore, the given function X(1) is a solution to the given system of differential equations.

b. To prove that X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2, we need to X into the given system of equations and check if it satisfies the equations.

So, we have:

X = C_1[1, 2]e^2t + C_2[1, -1]e^-t
X_1 = C_1e^2t + C_2e^-t
X_2 = 2C_1e^2t - C_2e^-t

Differentiating X_1 and X_2 with respect to t, we get:

X_1' = 2C_1e^2t - C_2e^-t
X_2' = 4C_1e^2t + C_2e^-t

Substituting X_1 and X_2 into the given system, we get:

X_1' (t) = 2(C_1e^2t - C_2e^-t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4(C_1e^2t + C_2e^-t) = 4X_1 (t) - 2X_2 (t)

Therefore, the given function X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2.

c. To show that the given functions form a fundamental set of solutions of the given system, we need to prove that they are linearly independent and that their Wronskian is non-zero.

We know that the vectors [1, 2] and [1, -1] are linearly independent, therefore the functions x(1) and x(2) are also linearly independent.

Also, the Wronskian of x(1) and x(2) is given by:

W(x1, x2) = | x1  x2 |
         | x1' x2' |

Substituting x(1) and x(2) into the above equation, we get:

W(x1, x2) = | e^2t  e^-t  |
          | 2e^2t -e^-t |

Simplifying the above equation, we get:

W(x1, x2) = 3e^(3t) ≠ 0

Therefore, the given functions form a fundamental set of solutions of the given system.

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Answer:

35 passengers produce the maximum revenue for the bus company.

Step-by-step explanation:

Define the variables:

If there are 30 or fewer passengers, each passenger will be charged $80. Therefore, the equation for the total revenue for 30 or fewer passengers is:

[tex]y = 80x,\quad x \leq 30[/tex]

If there are more than 30 passengers, each passenger will be charged $80 minus $2 for every passenger over 30. Therefore, the equation for the total revenue for more than 30 passengers is:

[tex]y = [80 - 2(x - 30)]x, \quad x > 30[/tex]

This simplifies to:

[tex]y = [80 - 2x + 60]x, \quad x > 30[/tex]

[tex]y = [140 - 2x]x, \quad x > 30[/tex]

[tex]y = 140x - 2x^2, \quad x > 30[/tex]

To maximize revenue, we need to find the value of x that maximizes the above equation.

Since this is a quadratic function, the maximum value occurs at the vertex of the parabola.

The x-value of the vertex of a parabola in the form y = ax² + bx + c is when x = - b/2a. Therefore, the x-value of the vertex is:

[tex]\implies x_{\sf vertex}=\dfrac{-140}{2(-2)}=\dfrac{-140}{-4}=35[/tex]

Therefore, the number of passengers that produce the maximum revenue for the bus company is 35.

The line that best fits the data has a value of around y=-1.98x+7.97 is False.

A line is defined as a straight, one-dimensional figure that extends infinitely in both directions. It is a basic geometrical object that has no thickness or width, and can be defined as the set of all points that are equidistant from two fixed points, called the endpoints. It is represented by y=mx+c.

given y=-1.98x+7.97

Number of passenger=75

Number of Suitcases=159

75=-1.98×159+7.97

75=-305.26

Hence, the line does not satisfy the data.

The line that best fits the data has a value of around y=-1.98x+7.97 is False.

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Answer:

Step-by-step explanation:

When we divide two exponential expressions with the same base, we can subtract their exponents. Using this property, we can simplify the given expression as:

2^(x+2) /2^(x-1) = 2^x * 2^2 / 2^x * 2^(-1)

= 2^(x-1+2)

= 2^(x+1)

Therefore, the simplified expression is 2^(x+1).

Answer:

Cos2a - cot2a/sin2a - tan2a

Step-by-step explanation:

Step-by-step explanation:

X = third side

  due to the triangle side length rule: sum of any two sides must be greater than the third side

so

4 + 6 > x       so x <10

x + 4 > 6       so  x > 2

x + 6 > 4     and  any value of x will work here

so  

2 < x < 10    <=====use this to answer your question as you didn't list the choices in your post