rovide a proof to show that the box-muller method produces standard normal random variates

The Cartesian equation of the curve is  Y = 1/X.

To the parameter and find a Cartesian equation of the curve using the given terms. Consider the following:

X = sin(t), Y = csc(t), and 0 ≤ t ≤ 2π

Step 1: Rewrite Y in terms of sin(t)
Since Y = csc(t), we know that csc(t) = 1/sin(t). Therefore, Y = 1/sin(t).

Step 2: Eliminate the parameter t
We already have X = sin(t), so we can substitute this into the equation for Y:
Y = 1/X

Step 3: Write the Cartesian equation of the curve
Now that we have eliminated the parameter t, the Cartesian equation of the curve is simply:

Y = 1/X

Therefore, the Cartesian equation of the curve is Y = 1/X.

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Answer:

$546

Step-by-step explanation:

It is given that the variables:

a = adults

c = students.

There are 6 adults, and 38 students. Plug in 6 for a, and 38 for c in the given expression:

[tex]15a + 12c\\15(6) + 12(38)[/tex]

Simplify. Remember to follow PEMDAS. PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, multiply 15 & 6, and 12 & 38 together:

[tex](15 * 6) + (12 * 38)\\(90) + (456)[/tex]

Next, simplify by adding:

[tex]90 + 456 = 546[/tex]

$546 is your total cost.

~

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Using the concept of proportion, we have that:

1) Approximate number of students who prefer R & B is: 176 students

2) Approximate number of students who prefer R & B is:  464 students

Normally in table of values, we can easily tell if a table shows a proportional relationship by calculating the ratio of each pair of values. If those ratios are all the same, the table shows a proportional relationship.

We are told that there are a total of 1200 total students on the campus.

1) Total number of students = 1200

Total number surveyed = 150

Total who prefer R & B music = 22

Thus:

Approximate number of students who prefer R & B = (22/150) * 1200

= 176 students

2) Total number of students = 1200

Total number surveyed = 150

Total who prefer Pop or Rock music = 30 + 28 = 58

Thus:

Approximate number of students who prefer R & B = (58/150) * 1200

= 464 students

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The number of different fruit salads that can be made using up to three  different fruits is 41

The number of ways of selecting r objects from n unlike objects is given by combinations

ⁿCₓ = n! / ( ( n - x )! x! )

where

n = total number of objects

x = number of choosing objects from the set

Given data ,

For each possible number of fruits (1, 2, or 3), we can count the number of ways to choose that many fruits from the available options and then sum up the results

To make a fruit salad with one fruit, we can choose any of the six available fruits. There are 6 ways to do this

To make a fruit salad with two fruits, we can choose any two fruits from the six available options. This can be done using the combination formula

C(6,2) = 6! / (2! x (6-2)!) = 15

So, there are 15 ways to make a fruit salad with two fruits

And , To make a fruit salad with three fruits, we can choose any three fruits from the six available options. This can be done using the combination formula:

C(6,3) = 6! / (3! x (6-3)!) = 20

So , the total number of ways A = 6 + 15 + 20

A = 41 ways

Hence , the different fruit salads is A = 41

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The expression for the nth term an, for the infinite series s=∑n=1[infinity]an is ∑n=4¹⁶an = 468

We know that the sum of the first n terms of the series is given by sn. Therefore, we can find an expression for the nth term an by taking the difference between successive values of sn:

sn - sn-1 = an

(4-4n²) - (4-4(n-1)²) = an

Simplifying this expression, we get:

an = 8n - 4

Now we can use this expression to find the values of ∑n=1¹⁰an and ∑n=4¹⁶an:

∑n=1¹⁰an = a1 + a2 + ... + a10

= (81 - 4) + (82 - 4) + ... + (8*10 - 4)

= 76

Therefore, ∑n=1¹⁰an = 76.

Similarly,

∑n=4¹⁶an = a4 + a5 + ... + a16

= (84 - 4) + (85 - 4) + ... + (8*16 - 4)

= 468

Therefore, ∑n=4¹⁶an = 468.

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Approximately 25% of the tomatoes weigh less than, given the standard deviation and mean, A. 93 g.

25 % of tomatoes is the value that we are looking for. On the z - distribution table, the closest to this amount is 0.2486, and this has a z - score of - 0. 674.

With this z - score, we can use the z - score formula to find the amount that the 25 % of tomatoes weigh:

z = (x - μ) / σ

-0. 674 = (x - 95 ) / 3.6

x - 95 = -0. 674 x 3. 6

x = 95 - 2. 4264

x = 92. 5736

x = 93 grams

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a. We have shown that α + 1 = β.

b. We have shown that α² = β.

A substance that cannot be broken down into another substance is referred to as an element. Because each element is made up of its own type of atom, each element is distinct and distinct from the others.

Since K is a field with 4 elements, the nonzero elements of K form a cyclic group of order 3 under multiplication. Let us call this group G. Since G is cyclic, it has a unique generator, say g. Then we can write G = {1, g, g²}.

Now, since α and β are both in K but not in Z2, they must be elements of G. Moreover, since K contains Z2 as a sub-field, α and β must be roots of the polynomial x² + x + 1 over Z2. Therefore, we have the following possibilities for α and β:

α = g

β = g²

or

α = g²

β = g

a. Let us first consider the case where α = g and β = g². Then we have:

α + 1 = g + 1

= g² + g + 1 (since g³ = 1)

= β + α + 1 (since β = g² and α = g)

= β

Therefore, we have shown that α + 1 = β.

b. Next, we will show that α² = β. Using the same assumption that α = g and β = g², we have:

α² = g²

= β

Therefore, we have shown that α² = β.

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β is an angle with cos(β) = -5/6 and β is not in the third quadrant, we need to compute the exact value of csc(β), without rationalizing the denominator.

Step 1: Determine the quadrant of angle β.
Since cos(β) = -5/6 and β is not in the third quadrant, then β must be in the second quadrant, as cosine values are negative in the second quadrant.

Step 2: Compute the sine of angle β.
We know that sin^{2}(β) + cos^{2}(β) = 1 (Pythagorean identity). So,
sin^{2}(β) + (-5/6)^{2} = 1
sin^{2}(β) + 25/36 = 1
sin^{2}(β) = 11/36
sin(β) = √(11/36) (since sin is positive in the second quadrant)

Step 3: Compute the exact value of csc(β).
Since csc(β) is the reciprocal of sin(β), then csc(β) = 1/sin(β).
csc(β) = 1/(√(11/36))

Therefore, the exact value of csc(β) is 1/(√(11/36)).

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β is an angle with cos(β) = -5/6 and β is not in the third quadrant, we need to compute the exact value of csc(β), without rationalizing the denominator.

Step 1: Determine the quadrant of angle β.
Since cos(β) = -5/6 and β is not in the third quadrant, then β must be in the second quadrant, as cosine values are negative in the second quadrant.

Step 2: Compute the sine of angle β.
We know that sin^{2}(β) + cos^{2}(β) = 1 (Pythagorean identity). So,
sin^{2}(β) + (-5/6)^{2} = 1
sin^{2}(β) + 25/36 = 1
sin^{2}(β) = 11/36
sin(β) = √(11/36) (since sin is positive in the second quadrant)

Step 3: Compute the exact value of csc(β).
Since csc(β) is the reciprocal of sin(β), then csc(β) = 1/sin(β).
csc(β) = 1/(√(11/36))

Therefore, the exact value of csc(β) is 1/(√(11/36)).

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The second derivative of y = 3sin(x) + [tex]x^{2 cosx}[/tex] is [tex]d^{2y}[/tex]/dx^2 = -3sin(x) - x²cos(x) + 2cos(x) - 2xsin(x).

To find the second derivative of y = 3 sin(x) + x² cos(x), we need to take the derivative of the first derivative of y with respect to x.

First, let's find the first derivative of y:

dy/dx = 3cos(x) - [tex]x^{2sin(x)}[/tex] + 2xcos(x)

Now, let's take the derivative of this expression with respect to x to find the second derivative:

[tex]d^{2y}[/tex]/dx² = -3sin(x) - x²cos(x) + 2cos(x) - 2xsin(x)

Therefore, the second derivative of y = 3sin(x) + [tex]x^{2 cosx}[/tex] is [tex]d^{2y}[/tex]/dx^2 = -3sin(x) - x²cos(x) + 2cos(x) - 2xsin(x).

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She will need to purchase 3 boxes of gloves.

He exceeded his goal by 10 devices.

There are 28 phalange bones in each foot.

There will be 2 smoothies left in stock after 4 days from one case.

Frank needs to consume no more than 15 grams of fat for the rest of the day.

Since there are 100 gloves in each box, she will need 222/100 = 2.22 boxes of gloves. Since she cannot purchase a partial box, she will need to purchase 3 boxes of gloves.

The medical sales rep sold devices for a total of 17 x 30 = 510 devices in November. Since his goal was to sell 500 devices, he exceeded his goal by 510 - 500 = 10 devices.

Since there are 56 phalange bones in the body and 14 phalange bones in each hand, there are 56 - (14 x 2) = <<56-(14*2)= 28 phalange bones in each foot.

Frank needs to consume no more than 56 - 41 = 15 grams of fat for the rest of the day.

The rec center sells 12 smoothies per day for 4 days, for a total of 12 x 4 = 48 smoothies. Therefore, there will be 50 - 48 = 2 smoothies left in stock after 4 days from one case.

Since Ashton drank a 24 oz bottle of water, he still needs to drink 64 - 24 = 40 ounces of water for the rest of the day.

Kathy walks a total of 3 x 2.5 =7.5 miles with her friend each week. Therefore, she still needs to walk 10 - 7.5 = 2.5 more miles to meet her goal for the week.

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Answer:

78.274 (Steps shown below)

Step-by-step explanation:

Rectangle Area

6 x 12 =50

Circle (put the 2 half circles together)

Circle area = π * r² = π * 9 [inch²] ≈ 28.274 [in²]

π ≈ 3.14159265 ≈ 3.14

d = r * 2 = 3 [inch] * 2 = 6 [inch]

Area of Circle= 28.274

To determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we need to calculate the Wronskian W(y1, y2, ..., yk) and check whether it is zero for some x in (-[infinity], [infinity]). If it is never zero, then the set of solutions is linearly independent. If it is zero for some x, then the set of solutions is linearly dependent.

To determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we can use the Wronskian determinant. The Wronskian of a set of k functions is defined as:

W(y1, y2, ..., yk) = det [y1, y2, ..., yk; y1', y2', ..., yk'; ..., ..., ..., ...; y1^(k-1), y2^(k-1), ..., yk^(k-1)]

where y1', y2', ..., yk' are the first derivatives of y1, y2, ..., yk, respectively, and y1^(k-1), y2^(k-1), ..., yk^(k-1) are their (k-1)th derivatives.

If the Wronskian is nonzero for all x in (-[infinity], [infinity]), then the set of solutions is linearly independent. If the Wronskian is zero for some x in (-[infinity], [infinity]), then the set of solutions is linearly dependent.

Therefore, to determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we need to calculate the Wronskian W(y1, y2, ..., yk) and check whether it is zero for some x in (-[infinity], [infinity]). If it is never zero, then the set of solutions is linearly independent. If it is zero for some x, then the set of solutions is linearly dependent.

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A) y = x + 3 represents a straight line with a slope of 1 passing through the point (0,3).

B) y = -x + 3 represents a straight line with a slope of -1 passing through the point (0,3).

C) y = |x| + 3 represents two lines; one with a slope of 1 passing through the point (-3,0) and another with a slope of -1 passing through the point (3,0).

The points (x,y) that satisfy all three equations must lie on the lines with slope 1 and -1 passing through the point (0,3).

These lines intersect at the point (1,4) and the set of points that satisfy all three equations is the single point (1,4).

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To find the linear approximation of f(x) at x = 3, we need to calculate the derivative f'(x) and then use the formula for the linear approximation: L(x) = f(a) + f'(a)(x-a).

Step 1: Calculate the derivative f'(x) of the given function f(x).
As the function is not provided, I'll assume it's a general rational function, f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. To find the derivative, use the quotient rule: f'(x) = (P'(x)Q(x) - P(x)Q'(x))/Q(x)^2.

Step 2: Evaluate f(3) and f'(3).
Once you find f'(x), plug in x=3 to get f(3) and f'(3).

Step 3: Use the linear approximation formula.
L(x) = f(3) + f'(3)(x-3).

Now, estimate L(2.9):
L(2.9) = f(3) + f'(3)(2.9-3) = f(3) - 0.1f'(3).

To provide an exact answer in fractional form, compute the numerical values of f(3) and f'(3) and substitute them in the equation above.

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The factor or variable that is manipulated in an experiment is called the independent variable. Therefore, the correct answer is b. the independent variable.

The independent variable is a factor or variable that is manipulated or changed by the researcher in an experiment to observe its effect on the dependent variable. It is also called the predictor variable because it is used to predict changes in the dependent variable. The researcher controls the independent variable and can vary it as needed to test different hypotheses. In contrast, the dependent variable is the factor or variable that is being measured or observed in response to the changes made in the independent variable.

For example, in an experiment to test the effect of different doses of medication on blood pressure, the independent variable is the medication dosage, while the dependent variable is the blood pressure readings. The researcher can manipulate the dosage of the medication and measure the effect on the blood pressure to determine the optimal dosage for treating high blood pressure.

It is important to carefully choose and control the independent variable in an experiment to ensure accurate and reliable results. Any extraneous or confounding variables that could affect the dependent variable must be controlled or eliminated to isolate the effect of the independent variable.

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The simplified expression is just the set s. First, we can use the associative property of intersection to rearrange the parentheses: (r∩s) ∩(s∩(r∩s)) = (r∩s) ∩((r∩s)∩s)

We can use the commutative property of intersection to switch the order of r and s in the first set:

(r∩s) ∩((r∩s)∩s) = (s∩r) ∩((r∩s)∩s)

Now, we can use the distributive property of intersection over intersection to expand (r∩s)∩s:

(s∩r) ∩((r∩s)∩s) = (s∩r) ∩r∩s

Finally, we can use the associative and commutative properties of intersection to rearrange the sets again and simplify:

(s∩r) ∩r∩s = s∩r∩r∩s = s∩(r∩r)∩s = s∩s = s

Therefore, the simplified expression is just the set s.

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Answer:

67.5%

Step-by-step explanation:

To calculate the percentage decrease in the share price of Bitcoin, we can use the following formula:

Percentage decrease = ((original value - new value) / original value) x 100%

Here, the original value is the share price before the crash, which is $60,000, and the new value is the share price after the crash, which is $19,500.

Substituting these values into the formula, we get:

Percentage decrease = ((60,000 - 19,500) / 60,000) x 100%

= 40,500 / 60,000 x 100%

= 67.5%

Therefore, the percentage decrease in the share price of Bitcoin is 67.5%.

To calculate the time it would take to reach 50,000 pairs, we can use the average doubling time provided in the table. The shape of the curve is a J-shaped exponential curve.

which means that the number of pairs increases at an accelerating rate over time. Since the average doubling time is 11.05 years, we can determine the number of times we need to double the initial number of breeding pairs (791) to reach 50,000 pairs. By successively doubling the number of pairs and keeping track of the years passed, we reach 50,000 pairs in 235 years (as given in the table).

However, there are unrealistic assumptions made in predicting the time to reach 50,000 pairs. The main assumption is that the doubling time remains constant, while in reality, factors such as environmental limitations, availability of resources, and human intervention could cause the rate of growth to change over time.

Additionally, the model assumes that there are no significant negative events (such as disease outbreaks or natural disasters) that could negatively impact the bald eagle population during this period.

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The answer is non-conventional cash flow .

Cash flow refers to the movement of money in and out of a business or individual's financial accounts over a specific period of time. It represents the inflow and outflow of actual cash, as opposed to accounting profit or loss, which may include non-cash items such as depreciation or accruals .

When the algebraic signs on the net cash flows change more than once, the cash flow sequence is called non-conventional cash flow .

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The statement "defects are additive in a multi-step manufacturing process" is generally false.

It depends on the type of defects and the specific manufacturing process. In general, defects are not necessarily additive in a multi-step manufacturing process.

Some defects may be additive, meaning that they can accumulate or worsen as the manufacturing process progresses. For example, if a part is slightly out of tolerance in one step of the process, subsequent steps may exacerbate the deviation, leading to a larger defect in the final product.

On the other hand, some defects may be independent or even compensatory, meaning that they do not accumulate or cancel each other out as the process progresses. For example, if one step of the process introduces a defect in one dimension of a part, another step may correct the defect in another dimension, resulting in a final product that meets specifications.

Therefore, the statement "defects are additive in a multi-step manufacturing process" is generally false, as it depends on the specific types of defects and the manufacturing process being considered.

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The statement "defects are additive in a multi-step manufacturing process" is generally false.

It depends on the type of defects and the specific manufacturing process. In general, defects are not necessarily additive in a multi-step manufacturing process.

Some defects may be additive, meaning that they can accumulate or worsen as the manufacturing process progresses. For example, if a part is slightly out of tolerance in one step of the process, subsequent steps may exacerbate the deviation, leading to a larger defect in the final product.

On the other hand, some defects may be independent or even compensatory, meaning that they do not accumulate or cancel each other out as the process progresses. For example, if one step of the process introduces a defect in one dimension of a part, another step may correct the defect in another dimension, resulting in a final product that meets specifications.

Therefore, the statement "defects are additive in a multi-step manufacturing process" is generally false, as it depends on the specific types of defects and the manufacturing process being considered.

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Thus, the 4th coordinate of the rectangle with the given coordinates is found as : D(8, -5). Area of the rectangle ABCD = 21 sq. units.

The Pythagorean theorem serves as the foundation for the distance formula. A line connecting two sites of interest is the hypotenuse of a right triangle, and this particular line connects the two points of interest.

The neighbouring side is obtained by joining the x-coordinates of a two points in a horizontal line, whereas the opposing side is obtained by joining the y-coordinates.

d=√((x2 - x1)²+(y2 - y1)²)

Given data:

vertices of a rectangle ABCD -

A(5, 2), B(8, 2), C(5, -5)

Let the 4 the vertex be D(x,y).

Plot the coordinates on the graph.

Now, we know that - opposite sides of the rectangle are equal.

Thus,

AB = CD

From graph,D(x,y).

x - (5 + 3) = 8

y - (2 - 7) = -5

Thus, the 4th coordinate of the rectangle with the given coordinates is found as : D(8, -5).

Area of the rectangle ABCD = length x breadth

length = 2 + 5 = 7 units

width = 8 - 5 = 3 units

Area of the rectangle ABCD = 7 x 3

Area of the rectangle ABCD = 21 sq. units.

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To sketch the straight-line Bode plot of the gain only for the voltage transfer function T(S) = 20s/ S2 + 58s + 400, we first need to break it down into its constituent parts. The numerator is simply a constant gain of 20, while the denominator can be factored into two second-order terms:

T(S) = 20s/ (S+20)(S+20)

Using the standard Bode plot rules for second-order systems, we can plot each term separately and then combine them to get the overall plot. For each term, we need to find the resonant frequency, damping ratio, and gain at low and high frequencies.

For the first term (S+20), the resonant frequency is 20, the damping ratio is 1/2, and the low-frequency gain is 0 dB. At high frequencies, the gain rolls off at a rate of -20 dB/decade.

For the second term (S+20), the resonant frequency is also 20, the damping ratio is 1/2, and the low-frequency gain is 0 dB. However, at high frequencies, the gain rolls off at a rate of -40 dB/decade due to the double pole.

To combine these two plots, we simply add the gains at each frequency and use the steeper roll-off rate for the second term. The result is a straight-line Bode plot with a gain of 20 dB at low frequencies, a resonant peak at 20 rad/s, and a steep roll-off at high frequencies.

The plot will cross the 0 dB line at two points, one before and one after the resonant peak, due to the double pole in the transfer function.

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Step-by-step explanation:

26 is 355 plus the product of 265 and p

26 = 355 + 265p

The Extreme Value Theorem does not guarantee the existence of an absolute maximum and minimum for f(x) on this interval.

The extreme value theorem states that if a function is continuous over a closed interval, then it must have at least one absolute maximum and one absolute minimum on that interval. In the case of f(x) = cot(x) 10 over the interval [−π,π3], this function is continuous over the interval since it is the composition of two continuous functions (cot(x) and 10). Therefore, the extreme value theorem guarantees that there must be at least one absolute maximum and one absolute minimum for f(x) on this interval.
The Extreme Value Theorem states that if a function is continuous on a closed interval, then it has an absolute maximum and minimum on that interval. The function f(x) = cot(x) is not continuous over the interval [-π, π/3] due to the presence of vertical asymptotes, where the function is undefined. Therefore, the Extreme Value Theorem does not guarantee the existence of an absolute maximum and minimum for f(x) on this interval.

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The simplified form of the surd is 1 - 1/3√5

Here are the general steps to follow when rationalizing a surd:

Identify the surd in the denominator of the fraction.

Multiply the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate is obtained by changing the sign of the surd term in the denominator.

Simplify the resulting expression by expanding the brackets and collecting like terms.

If there is still a surd in the denominator, repeat the process until no surds remain in the denominator.

Given that;

√2 - √10/√2 + √10

Then;

√2 - √10/ √2 + √10  * √2 - √10/√2 - √10

2 -√20 - √20  + 10/2 -√20 + √20 + 10

2 - 2√20 + 10/2 + 10

12 - 2√20/12

1 - 1/3√5

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The exact length x of the diagonal of the rectangle is the exact length of the diagonal is 4sqrt(5).

We can use the Pythagorean theorem to solve for the diagonal:

The Pythagorean Theorem is a fundamental principle in mathematics that describes the relationship between the sides of a right triangle.

It states that the sum of the squares of the two shorter sides (the legs) of a right triangle is equal to the square of the length of the longest side (the hypotenuse).

x^2 = 4^2 + 8^2

x^2 = 16 + 64

x^2 = 80

x = sqrt(80)

x = 4sqrt(5)

Thus, the exact length of the diagonal is 4sqrt(5).

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A particular solution is y_p(t) = -1/2sin(t) + 1/2cos(t), and the general solution is: y(t) = y_h(t) + y_p(t) = c1e^(t/2)cos(√3t/2) + c2e^(t/2)sin(√3t/2) - 1/2sin(t) + 1/2cos(t).

a. For the equation y'' - y' + y = sin(t), the characteristic equation is r^2 - r + 1 = 0, which has complex roots r = (1 ± i√3)/2. Therefore, the homogeneous solution is y_h(t) = c1e^(t/2)cos(√3t/2) + c2e^(t/2)sin(√3t/2).

To find a particular solution, we assume it has the form y_p(t) = Asin(t) + Bcos(t), where A and B are unknown constants. Taking the derivatives, we get y_p'(t) = Acos(t) - Bsin(t) and y_p''(t) = -Asin(t) - Bcos(t). Substituting these into the original equation, we get:

(-Asin(t) - Bcos(t)) - (Acos(t) - Bsin(t)) + Asin(t) + Bcos(t) = sin(t)

Simplifying, we get:

2B = 1

Therefore, B = 1/2. Substituting this into the equation above, we get:

-Acos(t) + 1/2sin(t) + Acos(t) - 1/2sin(t) = sin(t)

Simplifying, we get:

A = -1/2

Therefore, a particular solution is y_p(t) = -1/2sin(t) + 1/2cos(t), and the general solution is:

y(t) = y_h(t) + y_p(t) = c1e^(t/2)cos(√3t/2) + c2e^(t/2)sin(√3t/2) - 1/2sin(t) + 1/2cos(t).

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The above is a logic puzzle. Logic puzzles challenge the mind and enhance critical thinking.

According to the clues given, Jane was seen checking out an action book after leaving either a Biology or History class. It was also determined that Jayson is enrolled in Biology and the student who checked out a fantasy book, Jose, has an English class immediately following Jenny's.

Furthermore, we were able to deduce that the individual who left a History class was the same person who checked out a mystery novel while the student studying French must have been present during 1st period.

Jaden, who is currently enrolled in Algebra class, is observed browsing through a Manga novel. It can be noted that while studying for academic subjects like Math, the temptation to deviate towards leisure reading material can often pose as a distraction.

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The numerical values of the coefficients A and B are:

A = -1/6
B = 1/6

To perform partial fraction decomposition, we need to break down the fraction into simpler terms.

The form of the partial fraction decomposition of the given function is:

5/(5x - 72) = A/(5x - 66) + B/(5x - 72)

Here, A and B are the coefficients we need to find. We can find them by cross-multiplying and equating the numerators of both sides of the equation:

5 = A(5x - 72) + B(5x - 66)

Now, we can substitute some values of x to get two equations in terms of A and B:

For x = 14:

5 = A(5(14) - 72) + B(5(14) - 66)

Simplifying and solving for A and B, we get:

A = 1/6
B = -1/6

For x = 12:

5 = A(5(12) - 72) + B(5(12) - 66)

Simplifying and solving for A and B again, we get:

A = -1/6
B = 1/6



Generally, for a function f(x) with a rational expression in the integral, we can use partial fraction decomposition to rewrite the expression as a sum of simpler fractions. This makes it easier to find the integral.

The coefficients A and B are constants in the simpler fractions, and their values can be determined by solving a system of linear equations.

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