Find the MacLaurin series for f(x) = arctan(x) and its radius of convergence/

The conditional PDF of X given Y = y is a geometric distribution with parameter 1-p.

Let X be a geometric random variable with parameter p, and let Y be a uniform random variable on the interval [0,1], which means the PDF of Y is fY(y) = 1 for 0 ≤ y ≤ 1 and 0 otherwise. We want to find the conditional PDF of X given Y = y.

By Bayes' theorem, the conditional PDF of X given Y = y is given by:

fX|Y(x|y) = fY|X(y|x) fX(x) / fY(y)

where fX(x) is the PDF of X, which is given by fX(x) = (1-p)^(x-1) p for x = 1, 2, 3, ..., and fY|X(y|x) is the PDF of Y given X = x, which is given by fY|X(y|x) = 1 for 0 ≤ y ≤ p and 0 otherwise.

To find fY(y), we use the law of total probability:

fY(y) = ∑ fX(x) fY|X(y|x) for all x

Plugging in the values of fX(x) and fY|X(y|x), we get:

fY(y) = ∑ (1-p)^(x-1) p for 0 ≤ y ≤ p and 0 otherwise.

Since Y is uniform on [0,1], we have fY(y) = 1 for 0 ≤ y ≤ 1 and 0 otherwise. Therefore, the above sum simplifies to:

∑ (1-p)^(x-1) p = p / (1 - (1-p)) = 1

Now we can plug in the values of fY(y) and fX(x|y) into the formula for the conditional PDF of X given Y = y:

fX|Y(x|y) = fY|X(y|x) fX(x) / fY(y)

fX|Y(x|y) = (1/p) (1-p)^(x-1) p / 1 = (1-p)^(x-1)

Thus, the conditional PDF of X given Y = y is a geometric distribution with parameter 1-p.

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Probability of getting a 7 when a card is picked randomly is 1/3.

Here, given that

A card is picked at random.

Possible outcomes = {5, 6, 7}

Number of possible outcomes = 3

Favorable outcomes = {7}

Number of favorable outcomes = 1

Probability = Number of favorable outcomes / Total outcomes

                  = 1/3

Hence the required probability is 1/3.

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The approximate probability that a randomly selected student spends at most 175 hours on the project is 0.7734, rounded to four decimal places.

To compute the approximate probability that a randomly selected student spends at most 175 hours on the project, we can use the normal approximation to the gamma distribution.

First, we need to find the mean and variance of the gamma distribution:

Mean = a×B = 40×4 = 160

Variance = a×B² = 40*4² = 640

Next, we can use the following formula to standardize the gamma distribution:

Z = (X - Mean) / √(Variance)

where X is the random variable following the gamma distribution.

For X <= 175 hours, we have:

Z = (175 - 160) / √(640) = 0.750

Using a standard normal distribution table or calculator, we can find the probability that Z is less than or equal to 0.750:

P(Z <= 0.750) = 0.7734

Therefore, the approximate probability that a randomly selected student spends at most 175 hours on the project is 0.7734, rounded to four decimal places.

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We have transformed the left side into the right side using trigonometric identities. We start with the left side of the equation:

(1 + cos 0) (1 – sin 0)

Expanding the product, we get:

1 - sin 0 + cos 0 - sin 0 cos 0

Using the identity sin² θ + cos² θ = 1, we can replace sin² θ with 1 - cos²θ:

1 - (1 - cos² θ) + cos θ - (1 - cos² θ) cos θ

Simplifying, we get:

2 cos² θ - cos θ - 1

Now we use the identity sin² θ + cos² θ = 1 again to replace cos² θ with 1 - sin²θ:

2(1 - sin² θ) - cos θ - 1

2 - 2 sin²θ - cos θ - 1

1 - 2 sin² θ - cos θ

Finally, using the identity sin 2θ = 2 sin θ cos θ, we can write:

1 - sin 2θ - cos θ

Which is the right side of the equation. Therefore, we have transformed the left side into the right side using trigonometric identities.

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Answer:

¡Hola! Con gusto te ayudaré a resolver este problema de matemáticas. Primero, tenemos que calcular las horas totales de trabajo de cada uno de ellos:

Pedro: 10 días x 8 horas/día = 80 horas

Luis: 14 días x 7 horas/día = 98 horas

Jose: 24 días x 9 horas/día = 216 horas

Luego, multiplicamos las horas de trabajo de cada uno por el precio de la hora de trabajo:

Pedro: 80 horas x S/ 5/hora = S/ 400

Luis: 98 horas x S/ 5/hora = S/ 490

Jose: 216 horas x S/ 5/hora = S/ 1080

Finalmente, para obtener el importe total del trabajo de los tres, sumamos los montos obtenidos para cada uno:

S/ 400 + S/ 490 + S/ 1080 = S/ (800/9) + S/ (980/9) + S/ (2160/9) = S/ (800/9 + 980/9 + 2160/9) = S/ (3940/9)

Por lo tanto, el importe total del trabajo de los tres es de S/ (3940/9) nuevos soles. Espero que esto te ayude. ¡No dudes en preguntar si tienes alguna otra duda!

Step-by-step explanation:

Hello! I'll be happy to help you solve this math problem. First, we need to calculate the total hours of work for each person:

Pedro: 10 days x 8 hours/day = 80 hours

Luis: 14 days x 7 hours/day = 98 hours

Jose: 24 days x 9 hours/day = 216 hours

Next, we multiply each person's hours of work by the hourly rate:

Pedro: 80 hours x S/ 5/hour = S/ 400

Luis: 98 hours x S/ 5/hour = S/ 490

Jose: 216 hours x S/ 5/hour = S/ 1080

Finally, to get the total cost of work for all three, we add up the amounts we calculated for each person:

S/ 400 + S/ 490 + S/ 1080 = S/ (800/9) + S/ (980/9) + S/ (2160/9) = S/ (800/9 + 980/9 + 2160/9) = S/ (3940/9)

Therefore, the total cost of work for all three is S/ (3940/9) nuevos soles. I hope this helps! Feel free to ask if you have any other questions.

The possible zeros of the polynomial are given as follows:

± 1/3, ± 2/3, ± 1, ±4/3, ± 2, ±8/3, ±3, ± 4, ± 6, ± 8, ± 12, ± 24.

To obtain the possible rational zeros of the function, we use the Rational Zero Theroem.

The rational zero theorem states that all the possible rational zeros of a function are given by plus/minus the factors of the constant by the factors of the leading coefficient.

The parameters for this function are given as follows:

The factors are given as follows:

Hence the possible zeros are given as follows:

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The quadratic polynomial is x² - 6x + 31/4

A quadratic polynomial is a polynomial in which the highest power of the unknown is 2.

To form a quadratic polynomial whose zeroes are 3 + √5/2 and 3 - √5/2, we proceed as follows.

Since the zeroes are

Then its factors are

So, to get the quadratic polynomial p(x), we multiply the factors together.

So, we have that

p(x) =  [(x - 3) - √5/2][(x - 3) + √5/2]

= (x - 3)² - (√5/2)²

= x² - 6x + 9 - 5/4

= x² - 6x + (36 - 5)/4

= x² - 6x + 31/4

So, the polynomial is x² - 6x + 31/4

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Since W satisfies all three conditions, it is a subspace of R4. And since we have shown that W = Span{u, v}, we can choose answer (C): "Since u and v are in R4 and W = Span{u, v}, W is a subspace of R4."

What is sub space?

In mathematics, a subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication as the original space.

To show that W is a subspace of R4, we need to show that it satisfies three conditions:

The zero vector is in W.

W is closed under vector addition.

W is closed under scalar multiplication.

First, let's find vectors u and v such that W = Span{u,v}. We are given that a vector B in W has the form:

B = (2s + 5s + 3t, 4s - 5t, 2t, 45-50)

We can rewrite this as:

B = (7s, 4s, 0, 45-50) + (3t, -5t, 2t, 0)

So, we can take u = (7, 4, 0, -5) and v = (3, -5, 2, 0) to span W.

Now, let's check the three conditions:

The zero vector is in W:

Setting s = t = 0 in the expression for B gives us the vector (0, 0, 0, -5). This vector is in W, so the zero vector is in W.

W is closed under vector addition:

Let B1 and B2 be two vectors in W. Then, we have:

B1 = su1 + tv1 = a1u + b1v

B2 = su2 + tv2 = a2u + b2v

where a1, b1, a2, b2 are scalars.

Then, B1 + B2 is given by:

B1 + B2 = su1 + tv1 + su2 + tv2

= (a1u + b1v) + (a2u + b2v)

= (a1 + a2)u + (b1 + b2)v

which is also in W, since it can be expressed as a linear combination of u and v.

W is closed under scalar multiplication:

Let B be a vector in W and let k be a scalar. Then, we have:

B = su + tv = au + bv

where a, b are scalars.

Then, kB is given by:

kB = k(su + tv)

= (ks)u + (kt)v

which is also in W, since it can be expressed as a linear combination of u and v.

Therefore, since W satisfies all three conditions, it is a subspace of R4. And since we have shown that W = Span{u, v}, we can choose answer (C): "Since u and v are in R4 and W = Span{u, v}, W is a subspace of R4."

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Answer: (-7,4) is 11 units away.

Step-by-step explanation:

First we can see that (-7,4) is farther away on the coordinate plane.

Next, if we count the number of units from (-7,4) to (4,4) we count 11 units

There fore (-7,4) is 11 units away

Answer:

To find the volume of the container, we need to use the formula for the volume of a cylinder, which is:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

Since the diameter of the ball is 9.60 inches, the radius is half of that, or 4.80 inches.

Since the height of the container is the same as the diameter of the ball, the height is also 9.60 inches.

Substituting the values into the formula, we get:

V = π(4.80)^2(9.60)

V ≈ 661.95 cubic inches

Therefore, the volume of the container is approximately 661.95 cubic inches.

The given statement "The complement of the false positive rate is the sensitivity of a test" is true because the false positive rate is the proportion of negative instances incorrectly classified as positive, while sensitivity is the proportion of positive instances correctly identified.

False positive rate (FPR) is the proportion of negative instances incorrectly classified as positive, while sensitivity (also known as true positive rate or recall) is the proportion of positive instances correctly identified.

The complement of FPR is 1 - FPR, which is also known as specificity.

Specificity measures the proportion of negative instances correctly identified.

However, the statement would be false if it claimed that the complement of FPR is specificity.

The correct statement would be: the complement of the false positive rate is the specificity of a test.

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The given statement "The complement of the false positive rate is the sensitivity of a test" is true because the false positive rate is the proportion of negative instances incorrectly classified as positive, while sensitivity is the proportion of positive instances correctly identified.

False positive rate (FPR) is the proportion of negative instances incorrectly classified as positive, while sensitivity (also known as true positive rate or recall) is the proportion of positive instances correctly identified.

The complement of FPR is 1 - FPR, which is also known as specificity.

Specificity measures the proportion of negative instances correctly identified.

However, the statement would be false if it claimed that the complement of FPR is specificity.

The correct statement would be: the complement of the false positive rate is the specificity of a test.

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Step-by-step explanation:

See image

a) We can say that the point P(6,5) lies on the hyperbola.

b) The quantity (F1P − F2P)2 is approximately 122.5.

(a) To verify that the point P(6,5) lies on the hyperbola, we need to substitute x=6 and y=5 into the equation of the hyperbola and see if the equation holds true.

So, substituting x=6 and y=5, we get:

5(6)^2 - 4(5)^2 = 80

180 - 100 = 80

80 = 80

Since the equation holds true, we can say that the point P(6,5) lies on the hyperbola.

(b) To compute (F1P − F2P)2, we need to first find the coordinates of the foci F1 and F2.

5x^2 - 4y^2 = 80 can be rewritten as (x^2)/(16) - (y^2)/(20) = 1, where a^2=16 and b^2=20.

The distance between the center (0,0) and the foci is c=√(a^2+b^2)=√(336)/2. So, the foci lie on the x-axis and have coordinates (±c,0).

Therefore, F1 has coordinates (√(336)/2,0) and F2 has coordinates (-√(336)/2,0).

Now, we can calculate the distance between P(6,5) and each focus using the distance formula.

F1P = √((6-√(336)/2)^2 + (5-0)^2) ≈ 3.26

F2P = √((6+√(336)/2)^2 + (5-0)^2) ≈ 13.92

So, (F1P − F2P)^2 = (3.26 - 13.92)^2 ≈ 122.5.

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The limit of the sequence is 0.

To determine the limit of the sequence, we can use the Limit Laws and theorems. We will start by simplifying the expression:

cn=ln((5n-7)/(12n+4))

cn=ln(5n-7)-ln(12n+4)

Now we can use the Limit Laws:

lim n→∞ ln(5n-7) = ∞ (since ln(x) → ∞ as x → ∞)

lim n→∞ ln(12n+4) = ∞ (since ln(x) → ∞ as x → ∞)

Therefore, we have:

lim n→∞ cn = lim n→∞ (ln(5n-7)-ln(12n+4))

= lim n→∞ ln(5n-7) - lim n→∞ ln(12n+4)

= ∞ - ∞ (which is an indeterminate form)

To evaluate this limit, we can use L'Hopital's Rule:

lim n→∞ ln(5n-7) - ln(12n+4) = lim n→∞ [ln((5n-7)/(12n+4))]

= lim n→∞ [(5/12)/(5/n - 3/4n²)]

Since the denominator goes to ∞ and the numerator is constant, we have:

lim n→∞ [(5/12)/(5/n - 3/4n²)] = 0

Therefore, we have:

lim n→∞ cn = 0

So the limit of the sequence is 0.

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X can be expressed as a linear combination of v1 and v2 with the coordinates (a', b') in the basis B.  The coordinate vector [x]B:

[x]B = (a', b')

To determine whether the vector x is in the span V of vectors v1 and v2, we need to check if there exist scalar coefficients a and b such that:

x = a × v1 + b × v2

Given that x = [23 29], v1 = [46 58], and v2 = [61 67], the equation can be written as:

[23 29] = a × [46 58] + b × [61 67]

This equation can be represented in the form of a matrix:

| 46 61 | | a | = | 23 |
| 58 67 | | b | = | 29 |

We can now find the reduced row-echelon form of the augmented matrix to solve for a and b:

| 46 61 23 |
| 58 67 29 |

After row-reducing the matrix, we get:

| 1 0 a' |
| 0 1 b' |

Since the system has a unique solution, x is in the span V of vectors v1 and v2. We can now find the coordinates of x with respect to the basis B=(v1, v2) and write the coordinate vector [x]B:

[x]B = (a', b')

Therefore, x can be expressed as a linear combination of v1 and v2 with the coordinates (a', b') in the basis B.

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The expressions which are equivalent to (k^1/8)^-1 as required by virtue of the laws of indices are; k^-⅛, 1 / k^⅛ and 1 / ⁸√k.

It follows from the task content that the expressions which are equivalent to the given expression are to be determined.

Given; (k^1/8)^-1

By the power of power law of indices; we have;

= k^-⅛

Also, by the negative exponent rule; we have;

= 1 / k^⅛.

Also, by the rational exponent law of indices; we have;

= 1 / ⁸√k.

Ultimately, the equivalent expressions are; k^-⅛, 1 / k^⅛ and 1 / ⁸√k.

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The probability of number of strings in two digits followed by a letter is 2600,

The probability of the mean contents of the 625 sample cans being less than 11.994 ounces can be calculated using the Z-score formula.

This formula takes into account the mean and standard deviation of the sample and the size of the sample.

The formula is Z = (x - μ) / (σ / √n),

Where,

x is the value we are looking for,

μ is the mean of the sample,

σ is the standard deviation of the sample and

n is the size of the sample.

In this case, x = 11.994, μ = 12, σ = 0.12, and n = 625.

The Z-score is then calculated to be -0.166, which corresponds to a probability of 0.106.

This means that there is a 0.106 probability that the mean contents of the 625 sample cans is less than 11.994 ounces.

The number of bit strings of length nine:

[tex]2^9[/tex] = 512 (Answer: 1)
The number of functions from a set with five elements to a set with four elements:

[tex]4^5[/tex] = 1024 (Answer: 2)
The number of one-to-one functions from a set with three elements to a set with eight elements:

8P3 = 8*7*6

= 336 (Answer: 3)
The number of strings of two digits followed by a letter:

10 X 10 X 26 = 2600 (Answer: 4)
So the correct matching is:
1 -> 1,
2 -> 2,
3 -> 3,
4 -> 4.

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The measures of each term are; WS=39, WY=78, WZ=58, ZR=29, ZT=16 and YZ=32.

WE are given that Z is the centroid of triangle. Since centroid is the centre point of the object. The point in which the three medians of the triangle intersect is the centroid of a triangle.

Given WR=87 SY=39 and YT=48

WS=39

As WS=WR

WY=WS+SY

WY=39+39=78

WZ=58

Now, ZR=WR-WZ

ZR=87-48=29

ZT=16

Similalry;

YZ=YT-ZT

=48-16=32

YZ=32

Hence, the measures are; WS=39, WY=78, WZ=58, ZR=29, ZT=16 and YZ=32

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Step-by-step explanation:

Could you give a little more clearer explanation? I would be glad to help!

The simplified expression is 3a² + 3b² + 14a - 14b - 6ab -5.

We have,

3(a - b)² + 14(a - b)-5​

Simplifying the Expression as

Using Algebraic Identity

(a-b)² = a² -2ab + b²

So, 3 (a² -2ab + b²) + 14 (a-b) -5

= 3a² -6ab + 3b² + 14a - 14b -5

= 3a² + 3b² + 14a - 14b - 6ab -5

Thus, the simplified expression is 3a² + 3b² + 14a - 14b - 6ab -5.

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The Romberg array is a table of values that is used to estimate the value of a definite integral. To compute the Romberg array, we use the Richardson extrapolation method, which is a process of successive approximation.

To compute the eight rows and columns of the Romberg array, we begin by splitting the integration interval into two equal-length subintervals. The trapezoidal method is then applied to each subinterval to produce two estimates of the integral. The Richardson extrapolation method is then used to get a better estimate of the integral based on these two estimations. This operation is continued, splitting the subintervals into smaller and smaller subintervals, until the Romberg array has the necessary number of rows and columns.

The Romberg array's general formula is as follows:

R(m,n) = (4^n R(m,n-1) - R(m-1,n-1)) / (4^n - 1)

where R(m,n) is the value of the integral estimate at row m and column n in the Romberg array.

The first column of the Romberg array contains the estimates obtained by the trapezoidal rule, while the subsequent columns are obtained by applying the Richardson extrapolation method using the values in the previous column.

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Answer:

S₁₀ = - 838860

Step-by-step explanation:

the first term a₁ = 4

r = [tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{-16}{4}[/tex] = - 4

substitute these values into [tex]S_{n}[/tex] , then

S₁₀ = [tex]\frac{4-4(-4)^{10} }{1-(-4)}[/tex]

     = [tex]\frac{4-4(1048576)}{1+4}[/tex]

     = [tex]\frac{4-4194304}{5}[/tex]

     = [tex]\frac{-4194300}{5}[/tex]

     = - 838860

To find the rate of change of y with respect to time t, we need to take the derivative of the function y = 500(1-0.2)^t with respect to t:

dy/dt = 500*(-0.2)*(1-0.2)^(t-1)

Simplifying this expression, we get:

dy/dt = -100(0.8)^t

Therefore, the rate of change of y with respect to t is given by -100(0.8)^t. This means that the rate of change of y decreases exponentially over time, and approaches zero as t becomes large.

The right answer is:

1. The amount of carbon dioxide in the atmosphere is increasing (or growing).

2. Political conflicts (disagreements) occur over control of these resources.

As the population continues to grow, the demand for energy will also increase, further exacerbating this problem.

The increase in human population has led to an increase in energy consumption, which is largely met by the burning of fossil fuels such as coal, oil, and gas.

As fossil fuels become increasingly scarce, there may be greater competition and conflicts over their control and distribution. This can lead to geopolitical tensions and instability in some regions.

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The area under the graph of f(x) = 1/x+1 over the interval [0,4] is approximately 0.9375.

In mathematics, "area" refers to the measure of the amount of space enclosed by a two-dimensional shape or region. It is a quantitative measure of the extent or size of a shape in terms of its length squared. Area is typically expressed in square units, such as square meters (m^2), square feet (ft^2), or square centimeters (cm^2), depending on the system of measurement used.

A rectangle is a quadrilateral with four right angles, where opposite sides are parallel and equal in length.

To estimate the area under the graph of the function f(x) = 1/(x+1) over the interval [0,4], we can use numerical integration methods such as the trapezoidal rule or Simpson's rule.

Let's use the trapezoidal rule, which approximates the area under a curve by dividing the interval into smaller trapezoids and summing their areas.

Divide the interval [0,4] into n equal subintervals.

Let's choose n = 4 for this example, which means we will have 4 subintervals of equal width. The width of each subinterval is given by Δx = (4-0)/4 = 1.

Compute the sum of the areas of the trapezoids.

The area of each trapezoid is given by the formula: (h/2) * (f(x_i) + f(x_{i+1})), where h is the width of the subinterval, f(x_i) is the value of the function at the lower endpoint, and f(x_{i+1}) is the value of the function at the upper endpoint.

Using the trapezoidal rule, we can estimate the area under the curve as follows:

Area ≈ (1/2) * (f(0) + f(1)) * 1 + (1/2) * (f(1) + f(2)) * 1 + (1/2) * (f(2) + f(3)) * 1 + (1/2) * (f(3) + f(4)) * 1

Plugging in the function f(x) = 1/(x+1) and evaluating at the endpoints, we get:

Area ≈ (1/2) * (1 + 1/2) * 1 + (1/2) * (1/2 + 1/3) * 1 + (1/2) * (1/3 + 1/4) * 1 + (1/2) * (1/4 + 1/5) * 1

Simplifying further, we get:

Area ≈ 0.9375

So, the estimated area under the graph of the function f(x) = 1/(x+1) over the interval [0,4] using the trapezoidal rule is approximately 0.9375 square units.

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Answer:

Step-by-step explanation:

[tex]\frac{2^{-3}}{3^{-3} }[/tex]

=(-8)/(-27)

= 8/27

Check the picture below.

so we have a cube with a pyramidical hole, so let's just get the volume of the whole cube and subtract the volume of the pyramid, what's leftover is the part we didn't subtract, the cube with the hole in it.

[tex]\textit{volume of a pyramid}\\\\ V=\cfrac{Bh}{3} ~~ \begin{cases} B=\stackrel{base's}{area}\\ h=height\\[-0.5em] \hrulefill\\ B=\stackrel{6\sqrt{2}\times 6\sqrt{2}}{72}\\ h=12 \end{cases}\implies V=\cfrac{72\cdot 12}{3}\implies 288 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{\LARGE volumes} }{\stackrel{ cube }{12^3}~~ - ~~\stackrel{ pyramid }{288}}\implies \text{\LARGE 1440}~in^3[/tex]

For the first equation, we start by assuming a solution of the form y(t) = t^r. Then, we can take the derivative of y(t) twice to get:

y'(t) = rt^(r-1)
y''(t) = r(r-1)t^(r-2)

Substituting these into the original equation, we get:

3t^2(r(r-1)t^(r-2)) - 15t(rt^(r-1)) + 27t^r = 0

Simplifying, we can divide through by t^r and factor out a common factor of 3r(r-1):

3r(r-1) - 15r + 27 = 0

This simplifies to:

r^2 - 5r + 9 = 0

Using the quadratic formula, we find that r = (5 +/- sqrt(7)i)/2. Since the equation is homogeneous, we know that the general solution must be a linear combination of the two independent solutions:

y(t) = c1*t^(5/2) + c2*t^(3/2)

However, since t < 0, we need to use the absolute value of t to get the general solution:

y(t) = c1*|t|^(5/2) + c2*|t|^(3/2)

Finally, we can simplify this to:

y(t) = -t^3 [c1 + c2 ln(-t)]

For the second equation, we can use the same method of assuming a solution of the form y(x) = x^r and taking derivatives to get:

y'(x) = rx^(r-1)
y''(x) = r(r-1)x^(r-2)

Substituting these into the original equation, we get:

x^2(r(r-1)x^(r-2)) - x(rx^(r-1)) + 5x^r = 0

Simplifying, we can divide through by x^r and factor out a common factor of r(r-1):

r(r-1) - r/x + 5 = 0

This simplifies to:

r^2 - r(1/x) + 5 = 0

Using the quadratic formula, we find that r = (1/x +/- sqrt(4-20x^2))/2. Since the equation is homogeneous, we know that the general solution must be a linear combination of the two independent solutions:

y(x) = c1*x^(1/2 + sqrt(4-20x^2)/2) + c2*x^(1/2 - sqrt(4-20x^2)/2)

We can simplify this to:

y(x) = x [c1 cos (2 ln x) + c2 sin (2 ln x)]

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The order of a in (z/(pq))× is exactly (p-1)(q-1), as desired.

To prove that (z/(pq))× has order (p − 1)(q − 1), we need to show that the least positive integer n such that (z/(pq))×n = 1 is (p − 1)(q − 1).

First, let's define (z/(pq))× as the set of all integers a such that gcd(a,pq) = 1 (i.e., a is relatively prime to pq) and a mod pq is also relatively prime to pq.

Now, we know that the order of an element a in a group is the smallest positive integer n such that a^n = 1. Therefore, we need to find the order of an arbitrary element a in (z/(pq))×.

Let's assume that a is an arbitrary element in (z/(pq))×. Since gcd(a,pq) = 1, we know that a has a multiplicative inverse modulo pq, denoted by a^-1. Therefore, we can write:

a * a^-1 ≡ 1 (mod pq)

Now, let's consider the order of a. Since gcd(a,pq) = 1, we know that a^(p-1) is congruent to 1 modulo p by Fermat's Little Theorem. Similarly, we can show that a^(q-1) is congruent to 1 modulo q. Therefore, we have:

a^(p-1) ≡ 1 (mod p)
a^(q-1) ≡ 1 (mod q)

Now, we can use the Chinese Remainder Theorem to combine these congruences and get:

a^(p-1)(q-1) ≡ 1 (mod pq)

Therefore, we know that the order of a must divide (p-1)(q-1).

To show that the order of a is exactly (p-1)(q-1), we need to show that a^k is not congruent to 1 modulo pq for any positive integer k such that 1 ≤ k < (p-1)(q-1).

Assume for contradiction that there exists such a k. Then, we have:

a^k ≡ 1 (mod pq)

This means that a^k is a multiple of pq, which implies that gcd(a^k, pq) ≥ pq. However, since gcd(a,pq) = 1, we know that gcd(a^k, pq) = gcd(a,pq)^k = 1. This is a contradiction, and therefore our assumption must be false.

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The order of a in (z/(pq))× is exactly (p-1)(q-1), as desired.

To prove that (z/(pq))× has order (p − 1)(q − 1), we need to show that the least positive integer n such that (z/(pq))×n = 1 is (p − 1)(q − 1).

First, let's define (z/(pq))× as the set of all integers a such that gcd(a,pq) = 1 (i.e., a is relatively prime to pq) and a mod pq is also relatively prime to pq.

Now, we know that the order of an element a in a group is the smallest positive integer n such that a^n = 1. Therefore, we need to find the order of an arbitrary element a in (z/(pq))×.

Let's assume that a is an arbitrary element in (z/(pq))×. Since gcd(a,pq) = 1, we know that a has a multiplicative inverse modulo pq, denoted by a^-1. Therefore, we can write:

a * a^-1 ≡ 1 (mod pq)

Now, let's consider the order of a. Since gcd(a,pq) = 1, we know that a^(p-1) is congruent to 1 modulo p by Fermat's Little Theorem. Similarly, we can show that a^(q-1) is congruent to 1 modulo q. Therefore, we have:

a^(p-1) ≡ 1 (mod p)
a^(q-1) ≡ 1 (mod q)

Now, we can use the Chinese Remainder Theorem to combine these congruences and get:

a^(p-1)(q-1) ≡ 1 (mod pq)

Therefore, we know that the order of a must divide (p-1)(q-1).

To show that the order of a is exactly (p-1)(q-1), we need to show that a^k is not congruent to 1 modulo pq for any positive integer k such that 1 ≤ k < (p-1)(q-1).

Assume for contradiction that there exists such a k. Then, we have:

a^k ≡ 1 (mod pq)

This means that a^k is a multiple of pq, which implies that gcd(a^k, pq) ≥ pq. However, since gcd(a,pq) = 1, we know that gcd(a^k, pq) = gcd(a,pq)^k = 1. This is a contradiction, and therefore our assumption must be false.

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