HELP ME PLS EXPLAIN I DONT KNOW HOW TO DO THIS

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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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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To estimate the displacement of a moving object when given its velocity function, you can use the Definite Integral and Riemann Sum approaches.

Steps:

1. You're given the object's velocity function, which represents the rate of change of its position with respect to time.
2. To find the displacement, you need to calculate the total change in position over a given time interval. This can be done by finding the area under the velocity function curve within that time interval.
3. The Definite Integral approach allows you to find the exact area under the curve by integrating the velocity function over the specified time interval.
4. The Riemann Sum approach provides an approximation of the area under the curve by dividing the interval into smaller subintervals and summing up the areas of rectangles formed using the velocity function values at certain points within these subintervals.

Both of these approaches can help estimate the object's displacement, but the Definite Integral will give you a more accurate result, while the Riemann Sum provides an approximation that gets better as the number of subintervals increases. U-substitution is a method used for finding integrals, but it's not a direct approach to estimate displacement; it could be a part of the process if the velocity function requires it for integration.

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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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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.

Step-by-step explanation:

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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 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 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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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.

Joan should invest  $4720 annually in her sinking fund to have $250,000 when she retires

We can use the future value formula for an annuity to solve this problem:

FV = PMT * [(1 + r)^n - 1] / r

Where:

We want to find PMT, so we can rearrange the formula:

PMT = FV * r / [(1 + r)^n - 1]

Plugging in the values we know:

FV = $250,000

r = 0.04

n = 29

PMT = $250,000 * 0.04 / [(1 + 0.04)^29 - 1]

PMT = $250,000 * 0.04 / 22.718

PMT = $4720

So Joan should invest approximately $4720 annually in her sinking fund to have $250,000 when she retires in 29 years, assuming an interest rate of 4% compounded annually.

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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

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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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 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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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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To find the radius of convergence, we can use the ratio test.

Let's apply the ratio test to the series:

sigma^[infinity]_n=1 4(−1)^n nx^n

The ratio test tells us that the series converges if the limit of the absolute value of the ratio of the (n+1)th term to the nth term is less than 1:

lim n -> infinity |a_n+1 / a_n| < 1

where a_n = 4*(-1)^n * n * x^n.

Let's compute the ratio of the (n+1)th term to the nth term:

|a_n+1 / a_n| = |4*(-1)^(n+1) * (n+1) * x^(n+1) / (4*(-1)^n * n * x^n)|

             = |(n+1) / n| * |x|

             = (n+1) / n * |x|

We want to find the values of x for which the above limit is less than 1. So we need to solve the inequality:

lim n -> infinity (n+1) / n * |x| < 1

Taking the limit as n approaches infinity, we get

lim n -> infinity (n+1) / n * |x| = |x|

lim n -> infinity (n+1) / n * |x| = |x|

So the inequality reduces to:

|x| < 1

Therefore, the radius of convergence R is 1.

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Step-by-step explanation:

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

The total number of arrangements of the letters in the word "REPETITION" where the first occurrence of the letter E is before the first occurrence of the letter T is 120,960.. This can be answered by the concept of Combination.

To solve this problem, we can use the concept of permutations. The word "REPETITION" has a total of 10 letters.

Step 1: Calculate the total number of arrangements of all the letters without any restrictions.

The total number of arrangements of 10 letters without any restrictions can be calculated using the formula for permutations, which is n! (n factorial), where n is the number of items to be arranged. In this case, the total number of arrangements without any restrictions is 10! (10 factorial), which is equal to 3,628,800.

Step 2: Consider the restriction where the first occurrence of the letter E is before the first occurrence of the letter T.

In order to satisfy this restriction, we need to consider the positions of E and T in the arrangements. There are three possible cases:

Case 1: E is in the first position and T is in the second position.

In this case, we have fixed the positions of E and T, and the remaining 8 letters can be arranged in 8! (8 factorial) ways.

Case 2: E is in the first position and T is in a position other than the second.

In this case, we have fixed the position of E, and the position of T can be any of the remaining 8 positions. The remaining 8 letters can be arranged in 8! ways.

Case 3: E is not in the first position and T is in a position after E.

In this case, the position of T is fixed, and the position of E can be any of the positions before T. The remaining 8 letters can be arranged in 8! ways.

Step 3: Calculate the total number of arrangements that satisfy the restriction.

We add the total number of arrangements from each case calculated in Step 2:

8! + 8! + 8!

Step 4: Simplify the expression.

We can factor out 8! from the sum:

8! + 8! + 8! = 3 x 8!

Step 5: Calculate the final answer.

Substitute the value of 8! (8 factorial) into the expression:

3 x (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) = 3 x 40,320 = 120,960

Therefore, the total number of arrangements of the letters in the word "REPETITION" where the first occurrence of the letter E is before the first occurrence of the letter T is 120,960.

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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 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 given series is a geometric series with the formula: ∑ (1 + 6)^n / 7^n (from n=1 to infinity) In a geometric series,

The convergence or divergence is determined by the common ratio (r). In this case, the common ratio r is (1 + 6) / 7, which simplifies to 1.

Since the absolute value of the common ratio |r| is equal to 1, the series is inconclusive regarding convergence or divergence. Therefore, we need to use another test.

Notice that (1+6)/7 = 7/6 > 1. This means that the terms of the series do not approach zero as n approaches infinity. Therefore, the series sigma^[infinity]_n=1 (1+6)^n/7^n diverges by the divergence test. Therefore, the series does not have a sum.

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Step-by-step explanation:

they are similar, that means for our case here that they're is one central scaling factor for all sides between the 2 triangles.

by looking at the forms of both triangles, we see that

ET corresponds to TR.

FT corresponds to TS.

FE corresponds to RS.

for TE and TR we have the length information :

25 and 40

so, the scaling factor between these 2 corresponding sides can then be used for the other pairs of corresponding sides.

the scaling factor to go from the large to the small triangle is

25/40 = 5/8

therefore,

FT = TS × 5/8 = 22 × 5/8 = 11×5/4 = 55/4 = 13.75

FE = RS × 5/8 = 54 × 5/8 = 27×5/4 = 33.75

the perimeter of TFE is therefore

25 + 13.75 + 33.75 = 72.5

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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Answer:

Step-by-step explanation:

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

=(-8)/(-27)

= 8/27

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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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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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]