There are 56 female performers in a dance recital. The ratio of men to women is 3:8. How many men are in the dance recital?

The intersection over the indexed family {A_i} is nonempty, as there are no elements that belong to all three sets:
⋂(A_i) = A_1 ∩ A_2 ∩ A_3 = ∅ (empty set)

An indexed family of sets that is pairwise disjoint but has a nonempty intersection can be represented as follows:

Consider an indexed family of sets {A_i} where i belongs to the index set I, with I = {1, 2, 3}. Define the sets A_i as:

A_1 = {1, 2}
A_2 = {2, 3}
A_3 = {1, 3}

The sets are pairwise disjoint since no two sets share any common elements:
A_1 ∩ A_2 = {2} (not empty)
A_1 ∩ A_3 = {1} (not empty)
A_2 ∩ A_3 = {3} (not empty)

However, the intersection over the indexed family {A_i} is nonempty, as there are no elements that belong to all three sets:

⋂(A_i) = A_1 ∩ A_2 ∩ A_3 = ∅ (empty set)

In this example, the indexed family of sets is pairwise disjoint, but the intersection over the family is nonempty.

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The Fisher information for a Pareto(α) distribution is I(α) = nα² / (α - 1)².

To determine the Fisher information for a sample from a Pareto(α) distribution, follow these steps:

1. Recall the Pareto(α) probability density function (PDF): f(x) = αxᵃ⁺¹), where x ≥ 1 and α > 0.
2. Compute the log-likelihood function, L(α) = ln(f(x1,...,xn)) = ∑ ln(α) - (α+1)ln(xi) for i = 1 to n.
3. Differentiate L(α) with respect to α: dL/dα = ∑ (1/α) - ln(xi).
4. Differentiate dL/dα again: d²L/dα² = -∑ (1/α²).
5. The Fisher information is the negative expectation of the second derivative: I(α) = -E(d²L/dα²).
6. Apply the Pareto(α) distribution's expectation: I(α) = nα² / (α - 1)².

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The inverse of 'a' modulo 'm' is not possible for the given pairs (your day and month of birth: a=34, m=91) because 'a' and 'm' are not relatively prime.

To find the inverse of 'a' modulo 'm', we need to determine a number 'x' such that (a * x) % m = 1. This means that 'x' is the multiplicative inverse of 'a' modulo 'm'. However, for an inverse to exist, 'a' and 'm' must be relatively prime, meaning they do not have any common factors other than 1. In the given pair (a=34, m=91), 'a' and 'm' share a common factor of 13. Therefore, an inverse does not exist.

When 'a' and 'm' are not relatively prime, there is no integer 'x' that satisfies the equation (a * x) % m = 1. In this case, we cannot find the inverse of 'a' modulo 'm'. It is important to note that for an inverse to exist, 'm' must be a positive integer greater than 1, and 'a' must be a positive integer less than 'm'. In the given pair (34, 91), both conditions are met, but the lack of relative primality between 'a' and 'm' prevents the existence of an inverse.

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Therefore, the actual SAT score for the 29th percentile, use the SAT score percentiles chart, locate the 29th percentile, and identify the corresponding SAT score.

The percentile score indicates the percentage of students who scored lower than the student in question. To find the actual SAT score, we need to use a conversion table that correlates percentile scores with actual SAT scores. For example, if the conversion table shows that a percentile score of 29 corresponds to an actual SAT score of 1150, then the student's actual SAT score is 1150.

To find the actual SAT score of a student who receives a percentile score of 29, we need to use a conversion table that correlates percentile scores with actual SAT scores. The percentile score indicates the percentage of students who scored lower than the student in question. For example, if the conversion table shows that a percentile score of 29 corresponds to an actual SAT score of 1150, then the student's actual SAT score is 1150.
To find the actual SAT score corresponding to the 29th percentile, we'll use the SAT score percentiles chart. The chart maps percentiles to specific SAT scores. Percentiles represent the percentage of test-takers who scored at or below a particular score.

Step 1: Locate an official SAT score percentiles chart. You can find this on the College Board website or other reputable sources.
Step 2: Find the 29th percentile on the chart. Look for the row with "29" in the percentile column.
Step 3: Identify the corresponding SAT score in the same row. This score represents the 29th percentile.

Therefore, the actual SAT score for the 29th percentile, use the SAT score percentiles chart, locate the 29th percentile, and identify the corresponding SAT score.

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Answer:

Step-by-step explanation:

I think it would be 500

Answer:

The value of the car after 5 years is $12271.05

The present value of the car, PV = $16500

The rate of depreciation, r = 5.75%

r = 5.75/100

r = 0.0575

Step-by-step explanation:

The probability of any one of the eight arrangements is 0.125, or 12.5% when rounded to three decimal places.

There are eight possible arrangements of boys and girls when a couple plans to have three children. They are:

BBB (all boys)

BBG (two boys, one girl)

BGB (one boy, two girls)

BGG (one boy, two girls)

GBB (two boys, one girl)

GBG (one boy, two girls)

GGB (two boys, one girl)

GGG (all girls)

The probability of any one of these arrangements can be calculated using the formula for probability:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, since all eight arrangements are equally likely, the probability of any one of them is:

Probability = 1 / 8

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Answer:y=-4/2x

Step-by-step explanation:

B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).

To find the power series representation for g(x), we need to rewrite g(x) in terms of the given geometric series.

Notice that g(x) can be written as:
g(x) = x^3/1 - x = x^3 * (1/1-x)

We can now substitute the formula for the geometric series to get:

g(x) = x^3 * sigma^infinity_k = 0 x^k
= sigma^infinity_k = 0 (x^3 * x^k)
= sigma^infinity_k = 0 x^(k+3)

Therefore, the power series representation for g(x) is:
sigma^infinity_k = 0 x^(k+3)

The interval of convergence of this series is the same as that of the geometric series, which is |x| < 1.

In interval notation, this can be written as (-1, 1).

Therefore, the correct answer is B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).

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Kelly's rectangle has four square corners.

A rectangle is a quadrilateral with four sides and four angles. In a rectangle, opposite sides are equal in length, and all angles are right angles (90 degrees). A square is a special type of rectangle where all sides are equal in length

. Since a square is a type of rectangle, it also has four right angles, making all its corners square corners. Therefore, Kelly's rectangle, which is not specified as a square, may have different side lengths, but it will still have four right angles, resulting in four square corners.

These corners are formed by the intersection of the sides at right angles, creating a shape with sharp, 90-degree angles. So, regardless of the specific dimensions of Kelly's rectangle, it will always have four square corners.

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In the topical method, we focus on discussing different topics or subjects by considering various aspects and details related to them. This approach allows us to think about and communicate more effectively by addressing both general and specific points that might be relevant to the conversation.

The method of communication that involves reviewing possible things (both general and specific) that might be said. The correct answer is: e. Topical.

The method described in the question is a form of "topical" communication. This approach involves considering different topics or subjects that may need to be discussed and organizing thoughts and information around them. By reviewing possible things that may be said on each topic, one can prepare for a more effective and focused communication.

                             This method can be especially helpful in situations where there are multiple topics to cover or when discussing complex information that requires careful organization and planning.

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Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.

(a) To find the number of cars entering the parking garage over the interval 0 ≤ t ≤ 10, we need to integrate the rate of cars entering the garage with respect to time. ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars.
(b) To find ′′(5), we need to differentiate the rate of cars leaving the garage with respect to time twice. ′′(t) = -65cos(0.281) and ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour. This value represents the rate of change of the rate of cars leaving the garage at t = 5.
(c) To find the number of cars in the parking garage at time t = 10, we need to subtract the total number of cars leaving the garage from the total number of cars entering the garage from t = 0 to t = 10. This gives approximately 559 cars in the garage at t = 10.


Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.

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this function assumes that the baseball is thrown from ground level, and it does not take into account any external factors that may affect the trajectory of the ball (such as air resistance, wind, or spin).

Assuming that air resistance can be ignored, the height (in feet) of a baseball thrown upward with an initial velocity of 30 ft/s at time t (in seconds) can be modeled by the function:

h(t) = 30t - 16t^2

This function represents the position of the baseball above the ground, and it is a quadratic equation with a downward-facing parabolic shape. The initial velocity of 30 ft/s corresponds to the coefficient of the linear term, and the coefficient of the quadratic term (-16) is half the acceleration due to gravity (32 ft/s^2).

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The nth term test, also known as the Test for Divergence, is a useful tool for determining the divergence of a given series. All of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.

In order to use this test, you should analyze the limit of the sequence's terms as n approaches infinity. If the limit is not zero, then the series diverges.
For each of the series provided, let's apply the nth term test:
A) arctan(n), n=1 to infinity:
The limit as n approaches infinity of arctan(n) is π/2, which is not zero. Therefore, the series diverges.
B) 61:
Since the series consists of a constant term, the limit as n approaches infinity is 61, which is not zero. Therefore, the series diverges.
C) n(n+3)/(n+4), n=1 to infinity:
As n approaches infinity, the limit of n(n+3)/(n+4) is 1, which is not zero. Therefore, the series diverges.
D) ln(n), n=1 to infinity:
The limit as n approaches infinity of ln(n) is infinity, which is not zero. Therefore, the series diverges.
In conclusion, all of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.

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Answer : the solution to the initial-value problem is y(t) = 2e^(-3t) + 2e^(-27t), where t >= 0.

Initial-value problem using Laplace transforms, we'll follow these steps:

1. Take the Laplace transform of the differential equation.

2. Apply the initial conditions to obtain the transformed equation.

3. Solve the transformed equation for the Laplace transform of the unknown function.

4. Take the inverse Laplace transform to find the solution in the time domain.

Step 1: Taking the Laplace transform of the differential equation:

We have the differential equation: 2y'' + 36y' + 163y = 0

Taking the Laplace transform of each term using the properties of the Laplace transform, we get:

2[s^2Y(s) - sy(0) - y'(0)] + 36[sY(s) - y(0)] + 163Y(s) = 0

Step 2: Applying the initial conditions:

We are given y(0) = 2 and y'(0) = 0. Substituting these values into the transformed equation, we get:

2[s^2Y(s) - 2s] + 36[sY(s) - 2] + 163Y(s) = 0

Step 3: Solving the transformed equation for Y(s):

Rearranging the equation, we have:

(2s^2 + 36s + 163)Y(s) = 4s + 72

Dividing both sides by (2s^2 + 36s + 163), we obtain:

Y(s) = (4s + 72) / (2s^2 + 36s + 163)

Step 4: Taking the inverse Laplace transform:

To find the solution y(t) in the time domain, we need to compute the inverse Laplace transform of Y(s). However, the denominator of Y(s) is a quadratic expression, so we need to perform partial fraction decomposition.

The quadratic expression 2s^2 + 36s + 163 can be factored as (s + 3)(s + 27). Therefore, we can rewrite Y(s) as follows:

Y(s) = (4s + 72) / [(s + 3)(s + 27)]

Using partial fraction decomposition, we express Y(s) as:

Y(s) = A / (s + 3) + B / (s + 27)

To find A and B, we multiply both sides by the denominator and equate the numerators:

(4s + 72) = A(s + 27) + B(s + 3)

Expanding and collecting like terms:

4s + 72 = (A + B)s + 27A + 3B

By comparing coefficients, we get the following system of equations:

A + B = 4    ---(1)

27A + 3B = 72 ---(2)

Solving the system of equations, we find A = 2 and B = 2.

Now we can rewrite Y(s) as:

Y(s) = 2 / (s + 3) + 2 / (s + 27)

Taking the inverse Laplace transform of Y(s) using the table of Laplace transforms, we obtain the solution in the time domain:

y(t) = 2e^(-3t) + 2e^(-27t)

Therefore, the solution to the initial-value problem is y(t) = 2e^(-3t) + 2e^(-27t), where t >= 0.

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The weighted moving average formula with weights of 0.3 and 0.7 can be calculated using the AVERAGE and SUMPRODUCT functions in Excel. This formula can be used to calculate forecasted values for a range of years.

To use a 2-year weighted moving average to calculate forecasts for the years 1992-2002 with the weight of 0.7 assigned to the most recent year data, we can use the SUMPRODUCT function.
First, we need to create a table that includes the years 1990-2002 and their corresponding data points. Then, we can use the following formula to calculate the weighted moving average:
=(0.3*AVERAGE(B2:B3))+(0.7*B3)
This formula calculates the weighted moving average for each year by taking 30% of the average of the data for the previous two years (B2:B3) and 70% of the data for the most recent year (B3). We can then drag the formula down to calculate the forecasted values for the remaining years.
The SUMPRODUCT function can be used to simplify this calculation. The formula for the weighted moving average using SUMPRODUCT would be:
=SUMPRODUCT(B3:B4,{0.3,0.7})
This formula multiplies the data for the previous two years (B3:B4) by their respective weights (0.3 and 0.7) and then sums the products to calculate the weighted moving average for the most recent year. We can then drag the formula down to calculate the forecasted values for the remaining years.
In summary, the weighted moving average formula with weights of 0.3 and 0.7 can be calculated using the AVERAGE and SUMPRODUCT functions in Excel. This formula can be used to calculate forecasted values for a range of years.

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The series converges based on the limit comparison test.

To determine whether the given series converges or diverges, we can apply the limit comparison test. The limit comparison test states that if the limit of the ratio between the given series and a known convergent series is a finite positive value, then the given series converges. If the limit is zero or infinite, the given series diverges.

Let's consider the series ∑(9n^(3/2) - 10n - 34n) from n = 1 to infinity.

To apply the limit comparison test, we need to find a known convergent series to compare it with. A good choice is the p-series ∑(1/n^p), where p > 0.

Now, let's find the limit of the ratio of the two series:

lim(n→∞) [(9n^(3/2) - 10n - 34n) / (1/n^(3/2))]

= lim(n→∞) [(9n^(3/2) - 10n - 34n) * (n^(3/2))]

= lim(n→∞) [9n^3 - 10n^(5/2) - 34n^(5/2)]

To simplify the expression, divide all terms by n^(5/2):

= lim(n→∞) [(9n^3 / n^(5/2)) - (10n^(5/2) / n^(5/2)) - (34n^(5/2) / n^(5/2))]

= lim(n→∞) [9n^(3 - 5/2) - 10 - 34]

= lim(n→∞) [9n^(1/2) - 10 - 34]

= lim(n→∞) [9n^(1/2) - 44]

Since the limit is a finite value (-44), the ratio converges. Therefore, by the limit comparison test, the given series ∑(9n^(3/2) - 10n - 34n) converges.

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Panchito should mix 30 ml of cranberry juice with 120 ml of orange juice to make 150 ml of a 22% cranberry-orange juice blend.

To determine the amounts of cranberry juice and orange juice that Panchito should mix, we can set up a system of equations based on the given information. Let's assume Panchito mixes x ml of cranberry juice and y ml of orange juice.

The total volume of the mixture is 150 ml: x + y = 150.

The percentage of cranberry juice in the mixture is 22%: (0.05x) / 150 = 0.22.

Simplifying the second equation, we get:

0.05x = 0.22 * 150

0.05x = 33

x = 33 / 0.05

x = 660 ml

Substituting this value back into the first equation, we can solve for y:

660 + y = 150

y = 150 - 660

y = -510 ml

Since the solution for y is negative, it is not feasible. This indicates that there is no way to create a 22% cranberry-orange juice blend using a 5% cranberry juice and a 90% orange juice.

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The simplified expression √39(√6 + 7) is equal to √(39 * 6) + 7√(39 .

To simplify the expression √39(√6 + 7), we can follow these steps:

Step 1: Distribute the square-root (√) to both terms inside the parentheses:

  √39 * √6 + √39 * 7

Step 2: Simplify the square roots separately:

  √(39 * 6) + √(39 * 7)

Step 3: Calculate the products under the square roots:

  √234 + √273

Step 4: Simplify the square roots further:

  √(9 * 26) + √(9 * 3 * 3 * 3)

Step 5: Split the square root of a product into the product of the square roots:

  √9 * √26 + √9 * √(3 * 3 * 3)

Step 6: Simplify the square roots of perfect squares:

  3√26 + 3√(3 * 3 * 3)

Step 7: Multiply the numbers outside the square roots:

  3√26 + 9√3

Note that the simplified form is obtained by simplifying the square roots as much as possible and combining like terms.

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A) is correct answer. The inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.

The inequality that represents the range of possible pounds purchased is as follows:

0.41 < (3.40/x) < 0.50.

Let's discuss the given problem step-by-step.

Sarah pays $3.40 for x pounds of bananas.

The cost of one pound of bananas is greater than $0.41 and less than $0.50.

Therefore, the cost of x pounds of bananas can be written as:

3.40 < x(0.50) and 3.40 > x(0.41)

⇒ 0.41x < 3.40 < 0.50x

⇒ 0.41 < (3.40/x) < 0.50

Hence, the inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.

The answer is option A.

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The proportion of variation explained by the model is called the coefficient of determination, also denoted as R-squared.

It is a statistical measure that represents the percentage of the variance in the dependent variable that is explained by the independent variable(s) in the regression model. In other words, it measures the goodness of fit of the regression line to the observed data points. The coefficient of determination ranges from 0 to 1, where 0 indicates that the model does not explain any of the variance in the dependent variable, and 1 indicates that the model explains all of the variance in the dependent variable. The coefficient of determination is often used in regression analysis to evaluate the predictive power of the model and to compare the fit of different models.

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The number and the kind of roots of the equation, f(x) = x³ - x² - x + 1, is: D. 3 complex roots.

To determine the number and kind of roots of the equation f(x) = x³ - x² - x + 1, we can analyze the discriminant of the equation.

The discriminant, denoted as Δ, is given by:

Δ = b² - 4ac

In this case, the equation is in the form ax³ + bx² + cx + d = 0, where a = 1, b = -1, c = -1, and d = 1.

Calculating the discriminant:

Δ = (-1)² - 4(1)(-1)(-1) = 1 - 4(1)(1) = 1 - 4 = -3

The discriminant is negative (Δ < 0). This means that there are no real roots for the equation f(x) = x³ - x² - x + 1.

Therefore, the answer is:

D. 3 complex roots

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Answer:

21.5 cm

Step-by-step explanation:

a² + b² = c²

20² + 8² = c²

400 + 64 = c²

464 = c²

c = √464

c = 21.5

Answer: 21.5 cm

The appropriate symbols to use for the answer are: µ - z * (s / √n) < δ < µ + z * (s / √n)

To construct a confidence interval estimate for the mean body temperature of all healthy humans, we can use the symbol "µ" to represent the population mean.

A 99% confidence interval estimate for the mean body temperature can be represented as:

µ - z * (s / √n) < µ < µ + z * (s / √n)

In this expression:

"z" represents the critical value from the standard normal distribution corresponding to the desired confidence level (in this case, 99%).

"s" represents the sample standard deviation.

"n" represents the sample size.

Therefore, the appropriate symbols to use for the answer are:

µ - z * (s / √n) < δ < µ + z * (s / √n)

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The answer is option 13. 4 square centimeters.

Let's first find the length of the sides of each triangle. Since the perimeter of each triangle is 10 centimeters, and each triangle has 3 sides of equal length, the length of each side of the triangles is given by;

Side length = Perimeter ÷ Number of sides

= 10 ÷ 3= 3.33 (rounded to 2 decimal places)

The base of each triangle measures 2 centimeters, and the length of the side is 3.33 centimeters.

We can use the Pythagorean theorem to find the height of the triangles. Using Pythagorean theorem,

a² + b² = c²where a = 1, b = h and c = 3.33

From the formula above, we can find that:

h² = c² - a²

= 3.33² - 1²

≈ 10.77h

≈ √10.77

≈ 3.28

The area of each triangle is given by the formula;

Area = 1/2 x base x height

= 1/2 x 2 x 3.28

= 3.28 square centimeters (rounded to 2 decimal places)

Since there are 4 triangles, the total area of the plastic triangles on the spinner is approximately:

Total area = 4 x 3.28

= 13.12 square centimeters (rounded to 2 decimal places)

Therefore, the answer is option 13. 4 square centimeters.

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Let us assume that m is not prime. This means that there exists a prime factor p of m such that p ≤ √m. Since a is relatively prime to m, it must also be relatively prime to p.

Now, let's consider the order of a modulo p. We know that ordpa divides p-1, since p is prime. However, since a and p are relatively prime, we also know that ordpa cannot be equal to p-1, since this would imply that a is a primitive root modulo p, which is impossible since p is a prime factor of m and therefore does not have any primitive roots modulo p.
So, ordpa must divide p-1, but it cannot be equal to p-1. Therefore, ordpa must be strictly less than m-1 (since m has p as a factor, which means that m-1 has p-1 as a factor). However, we know that ordma = m-1. This means that ordpa cannot be equal to ordma.
This is a contradiction, since we assumed that ordma = m-1 and that ordpa divides m-1. Therefore, our initial assumption that m is not prime must be false. Therefore, m must be prime.
In conclusion, if m is a positive integer and a is an integer relatively prime to m such that ordma = m-1, then m must be prime.

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Answer:

$108.80

Step-by-step explanation:

89.25x0.06 = $5.36 tax

89.25 + 5.36= 94.61

94.61 x 0.15 = 14.91 tip

94.61 + 14.91 = 108.80 total

A faster way: 89.25*1.06*1.15=108.80

Answer :   ∫∫[Y > 3/X] (1/3) * (y^2/9) dx/dy.

(a) To calculate P(X/Y > 1), we need to find the probability that the ratio of X to Y is greater than 1.

The joint probability density function of X and Y, since they are independent, is given by f(X,Y) = fX(x) * fY(y).

Given that X has a Uniform distribution on (0, 3), the probability density function of X, fX(x), is:

fX(x) = 1/(3-0) = 1/3 for 0 < x < 3, and 0 otherwise.

The probability density function of Y, fY(y), is given as:

fY(y) = y^2/9 for 0 < y < 3, and 0 otherwise.

Now, we can calculate P(X/Y > 1) as follows:

P(X/Y > 1) = ∫∫[X/Y > 1] f(X,Y) dxdy

          = ∫∫[X > Y] fX(x) * fY(y) dxdy

          = ∫∫[X > Y] (1/3) * (y^2/9) dxdy

          = ∫[0,3] ∫[0,x] (1/3) * (y^2/9) dydx

          = (1/3) ∫[0,3] [(1/9) * (y^3/3)] evaluated from 0 to x dx

          = (1/3) ∫[0,3] (x^3/27) dx

          = (1/3) * [(1/108) * (x^4)] evaluated from 0 to 3

          = (1/3) * [(1/108) * (3^4 - 0^4)]

          = (1/3) * [(1/108) * 81]

          = 1/4.

Therefore, P(X/Y > 1) = 1/4.

(b) To calculate P(X + Y > 2), we need to find the probability that the sum of X and Y is greater than 2.

We can calculate this as follows:

P(X + Y > 2) = ∫∫[X + Y > 2] f(X,Y) dxdy

            = ∫∫[X > 2 - Y] fX(x) * fY(y) dxdy

            = ∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy.

To solve this integral, we can break it into two parts based on the range of Y:

For 0 < y < 2:

∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[0,2] ∫[2-y,3] (1/3) * (y^2/9) dxdy.

For 2 < y < 3:

∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[2,3] ∫[0,3] (1/3) * (y^2/9) dxdy.

Calculating these integrals will give us the desired probability.

(c) To calculate P(X * Y > 3), we need to find the probability that the product of X and Y is greater than 3.

Similarly, we can set up the

integral:

P(X * Y > 3) = ∫∫[X * Y > 3] f(X,Y) dxdy

            = ∫∫[Y > 3/X] fX(x) * fY(y) dxdy

            = ∫∫[Y > 3/X] (1/3) * (y^2/9) dxdy.

We can then evaluate this integral over the appropriate ranges to find the desired probability.

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The first digit must be 1. The remaining seven ones must be either 0 or 1.

Therefore, there can be formed [tex]2^7=128[/tex] different numbers.

To find the area enclosed by the polar curve r = 12sinθ, we can use the formula for the area of a polar curve: A = 1/2 * ∫(r^2)dθ. For r = 12sinθ, the integral limits are from 0 to π because the curve covers a full period of the sine function.

Let's evaluate the integral using angle identity:

A = 1/2 * ∫(r^2)dθ
A = 1/2 * ∫((12sinθ)^2)dθ, with θ from 0 to π

A = 1/2 * ∫(144sin^2θ)dθ

Now, we can use the double angle identity sin^2θ = (1 - cos(2θ))/2:

A = 1/2 * ∫(144(1 - cos(2θ))/2)dθ

A = 72 * ∫(1 - cos(2θ))dθ, with θ from 0 to π

Now, we can integrate:

A = 72 * [θ - 1/2 * sin(2θ)] from 0 to π

A = 72 * [π - 0 - (1/2 * sin(2π) - 1/2 * sin(0))]

A = 72 * π

The exact area enclosed by the polar curve r = 12sinθ is 72π square units.

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The subtended arc for the second circle is 31.2 cm long.

Given a circle with a radius of 6 cm and a subtended arc of 15.6 cm, we can find the angle's measure in radians using the formula: angle (in radians) = arc length/radius. Plugging in the values, we get angle = 15.6 cm / 6 cm = 2.6 radians.
For the second circle with a radius of 12 cm, we can find the subtended arc length by rearranging the formula: arc length = angle (in radians) * radius. Using the angle of 2.6 radians, we get arc length = 2.6 radians * 12 cm = 31.2 cm. Therefore, the subtended arc for the second circle is 31.2 cm long.

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