If there are seven ​multiple-choice questions on an​ exam, each having four possible​ answers, how many different sequences of answers are​ there?

Therefore, there are 210 different combinations of sequences that have 4 heads and 6 tails in 10 coin flips.

If heads come up four out of ten times that you flip a coin, this means that we have 4 heads and 6 tails in the sequence of 10 coin flips. The order of the heads and tails is important, so we are counting the number of possible sequences.

To calculate the number of possible sequences, we can use the formula for combinations:

C(n, r) = n! / (r! * (n - r)!)

here n is the total number of items (in this case, 10 coin flips), and r is the number of items we want to choose (in this case, the 4 heads).

So the number of different combinations of sequences with 4 heads and 6 tails is:

C(10, 4) = 10! / (4! * (10 - 4)!) = 210

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The solution to the proportional equation in this problem is given as follows:

x = 75/7.

The proportional equation in the context of this problem is defined as follows:

25/x = 7/3.

The equation is proportional, meaning that we can obtain the value of x applying cross multiplication as follows:

7x = 25 x 3

7x = 75

x = 75/7.

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a) The minimum number of wins he needs is 15. b) The standard deviation of X is σ = sqrt(Var(X)) ≈ 3.641. c) Standard normal table ≈ 0.411.

In part (a), we can use the formula for a binomial distribution to find the minimum number of wins Sam needs to break even. Let X be the number of wins in 520 spins, then X ~ Bin(520, 1/38). To break even, Sam needs to win at least $520, which means he needs at least m wins where 35m ≥ 520, or m ≥ 14.86. Since m must be an integer, the minimum number of wins he needs is 15.

In part (b), we can use the mean and variance of a binomial distribution to find the mean and standard deviation of X. The mean of X is E(X) = np = 520*(1/38) ≈ 13.684, and the variance of X is Var(X) = np(1-p) = 520*(1/38)*(37/38) ≈ 13.255. Therefore, the standard deviation of X is σ = sqrt(Var(X)) ≈ 3.641.

In part (c), we can use the Normal approximation to the binomial with the continuity correction to find P(X ≥ 15). Using the continuity correction, we can convert the discrete probability P(X ≥ 15) to a continuous probability P(X > 14.5). Standardizing X, we get Z = (14.5 - 13.684) / 3.641 ≈ 0.224. Using a standard normal table, we can find that P(Z > 0.224) ≈ 0.411. Therefore, P(X > 14.5) ≈ 0.411.

This answer is closer to the exact probability of 0.396 than the previous estimate of 0.10 too large, but it still overestimates the probability slightly. This could be due to the fact that the Normal approximation to the binomial assumes a continuous distribution, while the binomial distribution is discrete. Nonetheless, the Normal approximation with continuity correction is a useful tool for approximating probabilities in situations where the sample size is large.

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According to research studies, these types of rapes are not uncommon, and in fact, as many as 25% or more of all reported rapes may involve "multiple offenders."

Multiple offender rapes, also referred to as gang rapes, involve two or more perpetrators who sexually assault a victim.

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a) the mean score difference for the University of Southern California in the 2004-05 football season is 22.08. b) The median score difference is 31.

(a) To find the mean score difference, we add up all the differences and divide by the total number of games:

11 + 49 + 32 + 3 + 6 + 38 + 38 + 30 + 8 + 40 + 31 + 5 + 36 = 287

So, the mean score difference is:

287/13 = 22.08

Therefore, the mean score difference for the University of Southern California in the 2004-05 football season is 22.08.

(b) To find the median score difference, we need to arrange the differences in order from smallest to largest:

3, 5, 6, 8, 11, 30, 31, 32, 36, 38, 38, 40, 49

Since there are an odd number of games played (13), the median is the middle number. In this case, the median is:

Median = 30

Therefore, the median score difference for the University of Southern California in the 2004-05 football season is 30.

To find (a) the mean score difference and (b) the median score difference for the University of Southern California's 2004-05 football season, follow these steps:

1. Arrange the score differences in ascending order:
3, 5, 6, 8, 11, 30, 31, 32, 36, 38, 38, 40, 49

2. Calculate the mean score difference by adding all the score differences and dividing by the number of games (13):
(3+5+6+8+11+30+31+32+36+38+38+40+49) / 13 = 327 / 13 = 25.15

(a) The mean score difference is 25.15.

3. To find the median score difference, identify the middle value in the ordered list:
Since there are 13 games, the middle value is the 7th value in the ordered list: 31

(b) The median score difference is 31.

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The rate at which the height of the water in the tank increases is 40 millimeters per second.

To find the rate at which the height of the water in the water tank increases, we need to first calculate the volume of the tank.

Since the tank has the shape of a rectangular prism, its volume can be calculated by multiplying its base (50 cm2) with its height (h) and length (l).
Volume of the tank = base x height x length
V = 50 x h x l
We also know that the tank is being filled at the rate of 12 liters per minute.

Since 1 liter is equal to 1000 cubic centimeters (cc), the rate at which the volume of water in the tank increases can be calculated as follows:
Rate of increase of volume of water = 12 x 1000 cc/min
= 12000 cc/min
Now, to find the rate at which the height of the water in the tank increases, we need to differentiate the volume of the tank with respect to time (t).

This gives us the following formula:
dV/dt = 50 x dh/dt x l
Where dV/dt is the rate of increase of volume of water (12000 cc/min), dh/dt is the rate at which the height of the water in the tank increases (in mm/s), and l is the length of the tank (which we don't need to know).
Substituting the values we know, we get:
12000 = 50 x dh/dt x l
dh/dt = 12000 / (50 x l)
Since we want the answer in millimeters per second, we need to convert the base area from square centimeters to square millimeters. 1 square centimeter is equal to 100 square millimeters, so the base area of the tank is:
50 cm2 = 50 x 100 mm2 = 5000 mm2
Substituting this value for the base area, we get:
dh/dt = 12000 / (50 x 5000 x l)
dh/dt = 0.048 mm/s
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The volume of the rectangular pyramid is 5000 cubic meters. This is calculated using the formula V = (1/3) * base area * height, with a base area of 200 square meters and a height of 75 meters.

The formula for the volume of a rectangular pyramid is

V = (1/3) * base area * height

We are given that the base area is 200 square meters and the height is 75 meters. Substituting these values into the formula, we get

V = (1/3) * 200 * 75

V = 5000 cubic meters

Therefore, the volume of the rectangular pyramid is 5000 cubic meters.

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The rate at which the area of the rectangle is changing is -64 cm²/s.

To find the rate at which the area of the rectangle is changing, we'll need to use the given information and differentiate the area function with respect to time.

Step 1: Identify the given rates and measurements
- Length (L) is increasing at a rate of 15 cm/s (dL/dt = 15)
- Width (W) is decreasing at a rate of 8 cm/s (dW/dt = -8)
- At the specific moment we are interested in, L = 38 cm and W = 16 cm

Step 2: Write the equation for the area of the rectangle
- Area (A) = L * W

Step 3: Differentiate the area equation with respect to time (t)
- dA/dt = d(L * W)/dt = (dL/dt * W) + (L * dW/dt)

Step 4: Substitute the given information into the differentiated equation
- dA/dt = (15 * 16) + (38 * -8)

Step 5: Calculate the result
- dA/dt = 240 - 304 = -64 cm²/s

This negative value indicates that the area is decreasing. It's due to the fact that the width is decreasing faster than the length is increasing at the given moment.

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The point that is closest to (-8, -5) is (-8, 6), and its distance is 11 units.
Option A is the correct option.

We have,

The point that is closest to (-8, -5) is the one with the shortest distance.

To find the distance between two points, we can use the distance formula:

distance = √((x2 - x1)² + (y2 - y1)²)

Let's calculate the distance between (-8, -5) and each of the other points:

Distance between (-8, -5) and (-8, 6):

= √((-8 - (-8))² + (6 - (-5))²) = √(11²) = 11

Distance between (-8, -5) and (6, -5):

= √((6 - (-8))² + (-5 - (-5))²) = √(14²) = 14

Distance between (-8, -5) and (6, -5):

= √((6 - (-8))² + (-5 - (-5))²) = √(14²) = 14

Therefore,

The point that is closest to (-8, -5) is (-8, 6), and its distance is 11 units.

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The feasible solution space for an integer programming model is typically smaller than the feasible solution space for a linear programming version of the same model.

This is because integer programming models restrict the decision variables to integer values, while linear programming models allow for continuous values.

For example, consider a transportation problem where we want to determine the optimal way to transport goods from multiple factories to multiple warehouses. In a linear programming version of this problem, the decision variables representing the amount of goods transported between each factory and warehouse can take on any real value.

However, in an integer programming version of this problem, the decision variables must take on integer values representing the number of units of goods transported.

Since the integer programming model restricts the values of the decision variables, the feasible solution space is typically smaller than the feasible solution space for the linear programming version. This can make it more difficult to find an optimal solution for the integer programming model, as there may be fewer feasible solutions to choose from.

However, the integer programming model may be necessary in cases where decision variables must take on integer values, such as in inventory management or scheduling problems.

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The shape of a curve is determined by the relationship between the variables it represents. For example, a linear curve has a constant rate of change between the two variables, while an exponential curve shows an increasing rate of change.

In general, curves can have various shapes depending on the specific data set and the underlying relationship between the variables. The shape of the curve can provide insights into the nature of the relationship between the variables, such as whether it is linear or nonlinear, whether it has a positive or negative correlation, and whether it is symmetrical or skewed.

Therefore, to explain the shape of a particular curve, we need to look at the data set and the relationship between the variables it represents. We can then use qualitative terms to describe the curve, such as whether it is concave or convex, whether it has inflection points or asymptotes, and whether it has a plateau or a steep slope. By analyzing the shape of the curve, we can gain a better understanding of the relationship between the variables and make informed decisions based on the data.

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Approximately 46 adult tickets and 92 children tickets were sold during the recent showing at the movie theater.

To determine the number of adult and children tickets sold during a recent showing at the movie theater, where the ticket prices for adults and children are given, we can solve a system of equations based on the total number of tickets sold and the total revenue generated.

Let's assume the number of adult tickets sold is "a" and the number of children tickets sold is "c". Given that an adult ticket costs $7 and a children ticket costs $4.25, we can set up the following equations based on the total number of tickets sold and the total revenue generated:

Equation 1: a + c = 139 (equation representing the total number of tickets sold)

Equation 2: 7a + 4.25c = 720 (equation representing the total revenue generated)

To solve this system of equations, we can use substitution or elimination methods. Let's use the elimination method as an example:

Multiply Equation 1 by 4.25 to make the coefficients of "c" in both equations equal:

4.25a + 4.25c = 591.75

Subtract Equation 2 from the above equation:

(4.25a + 4.25c) - (7a + 4.25c) = 591.75 - 720

-2.75a = -128.25

Divide both sides of the equation by -2.75:

a = 46.8

Substitute the value of "a" back into Equation 1 to find "c":

46.8 + c = 139

c = 139 - 46.8

c = 92.2

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Answer:

The monthly mean temperature is calculated by adding together the daily means for each day of the month and dividing by the number of days in the month.

Step-by-step explanation:

The monthly mean temperature is calculated by adding together the daily means for each day of the month and dividing by the number of days in the month.

The monthly mean temperature is a measure of the average temperature of a month. It is calculated by adding together the daily mean temperatures for each day of the month and then dividing by the number of days in the month.

The daily mean temperature is the average temperature for a 24-hour period, typically measured at the midpoint of that period (usually at noon or midnight).

By calculating the monthly mean temperature, we can get a better sense of the overall temperature pattern of a particular month, which can be useful for monitoring climate changes, forecasting weather conditions, or analyzing weather data over time.

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The value of the probability P(Blue) will be; 0.50

We have the following parameters that can be used in our computation:

Coin = Two-sided coin

Colors = Orange and Blue

Using the above as a guide, we have:

P(Blue) = Number of blue/Number of sides

Substitute the known values in the above equation,

P(Blue) = 1/2

Evaluate;

P(Blue) = 0.5

Hence, the probability is 0.50

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The experimental probability of dealing a black cards on the basis of given dealt cards is 10/19.

The dealt cards are,

Spades: 3, 4, 6, J, Q, K that is six spade cards

Clubs: A, 2, 5, 7, J, K that is six clubs cards

Hearts: A, 2, 5 that is three hearts cards

Diamonds: A, 2, 3, 6, K that is four cards

So total number of dealt cards = 6 + 6 + 3 + 4 = 19 cards

Here number of black cards = Spade cards + Diamond Cards = 6 + 4 = 10 cards.

The probability of dealing a black card = Number of dealt black cards/ Total number of dealt cards = 10/19

Hence the experimental probability of dealing a black card is 10/19.

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The approximate probability that the total number of tickets given out during a 5-day week is between 195 and 275 is 0.8804 or 88.04%.

We need to know the mean and standard deviation of the distribution to calculate the approximate probability that the total number of tickets given out during a 5-day week is between 195 and 275.

Let's assume that the mean number of tickets given out per day is 50 and the standard deviation is 10 (these are just hypothetical values).

The total number of tickets given out during a 5-day week follows a normal distribution with mean 250 (= 5 days x 50 tickets per day) and standard deviation of the square root of 500 (= 5 days x 10²).

To find the probability that the total number of tickets given out during a 5-day week is between 195 and 275, we need to standardize the values using the z-score formula: z = (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation.

For x = 195: z = (195 - 250) / sqrt(500) = -2.46

For x = 275: z = (275 - 250) / sqrt(500) = 1.56

Using a calculator, the probability that z is between -2.46 and 1.56 is approximately 0.8804.

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The probability of the ball landing on a red slot on the next spin is still 18/38 or approximately 0.4737 or 47.37%.

The probability of the ball landing on a red slot on a single spin of a standard roulette wheel is 18/38 or approximately 0.4737 or 47.37%. This is because there are 18 red slots out of a total of 38 slots on the wheel.

The outcome of the previous 210 spins has no effect on the probability of the ball landing on a red slot on the next spin. Each spin is an independent event, and the probability of the ball landing on a red slot remains the same for each spin.

Therefore, even though the ball has landed on a red slot for the past 210 spins, the probability of it landing on a red slot on the next spin is still 18/38 or approximately 0.4737 or 47.37%.

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The image of the triangle after dilation centered at the origin with a scale factor of 5/3 is shown in following graph.

We know that the scale factor is nothing but the ratio of the size of the transformed image to the size of the original image.

From the attached figure the coordinates of the original triangle are:

(6, 9), (9, 9) and (9, 6)

And the scale factor is k = 3/5

Using above definition of scale factor, the coordiantes of the dilated triangle would be,

5/3 × (6, 9) = (10, 15)

5/3 × (9, 9) = (15, 15)

5/3 × (9, 6) = (15, 10)

Thus, the image of the triangle after dilation is shown in following graph.

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Find the complete question below.

There are 84,878 oranges in the stack.

We can solve it by using arithmetic series.

The first level of the stack has a rectangular base of 55 oranges by 88 oranges, which means there are 55 x 88 = 4840 oranges in the first level.

Each orange above the first level rests in a pocket formed by four oranges below, so the second level has 54 oranges by 87 oranges (one less on each side), which means there are 54 x 87 = 4698 oranges in the second level.

Similarly, the third level has 53 oranges by 86 oranges, which means there are 53 x 86 = 4558 oranges in the third level.

We can continue this pattern until we reach the top level, which has a single orange.

Therefore, the total number of oranges in the stack is:

4840 + 4698 + 4558 + ... + 1

This is an arithmetic series with a first term (a) of 4840, a common difference (d) of -142, and a number of terms (n) of 34 (since there are 34 levels in the stack).

Using the formula for the sum of an arithmetic series:

S = (n/2)(a + l)

where S is the sum, a is the first term, l is the last term, and n is the number of terms.

We can find the last term (l) using the formula for the nth term of an arithmetic series:

l = a + (n - 1)d

Substituting the values we have:

l = 4840 + (34 - 1)(-142) = 4840 - 4686 = 154

So the sum of the oranges in the stack is:

S = (34/2)(4840 + 154) = 17 x 4994 = 84878

Therefore, there are 84,878 oranges in the stack.

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There are 5 choices left), and 5 ways to choose the remaining element of Y. This gives us a total of 7 × 5 × 5 = 175 ordered pairs (X,Y).

Let's first consider the possible values of |X ∪ Y|.

If |X ∪ Y| = 1, it means that X and Y have no elements in common, and each set has only one element. There are 7 such sets: {1},{2},{3},{4},{5},{6},{7}.

If |X ∪ Y| = 2, it means that X and Y have one element in common. There are 7 ways to choose the common element, and 6 ways to choose the remaining element of X (it cannot be the same as the common element, so there are only 6 choices left), and 6 ways to choose the remaining element of Y (again, it cannot be the same as the common element or the element of X, so there are only 6 choices left). This gives us a total of 7 × 6 × 6 = 252 ordered pairs (X,Y).

If |X ∪ Y| = 3, it means that X and Y have two elements in common. There are 7 ways to choose the common elements, and 5 ways to choose the remaining element of X (it cannot be any of the common elements, so there are 5 choices left), and 5 ways to choose the remaining element of Y. This gives us a total of 7 × 5 × 5 = 175 ordered pairs (X,Y).

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The Kruskal-Wallis test is a nonparametric test that is used to determine whether three or more samples came from populations with the same distributions.

This test is particularly useful when the data does not meet the assumptions required for parametric tests, such as normality or equal variances.

To perform the Kruskal-Wallis test, follow these steps:

1. Combine all the data from the different samples into one dataset.
2. Rank the combined dataset in ascending order, assigning a rank of 1 to the lowest value, 2 to the next lowest, and so on.
3. Calculate the sum of ranks for each sample.
4. Calculate the test statistic, H, using the following formula:

  H = (12 / (N * (N + 1))) * Σ(Ri^2 / ni) - (3 * (N + 1))

  Where N is the total number of observations in all samples, Ri is the sum of ranks for each sample i, and ni is the number of observations in each sample i.

5. Determine the degrees of freedom, which is equal to the number of samples minus 1.
6. Compare the calculated H value with the critical value from the Chi-square distribution table at a chosen significance level (e.g., 0.05) and the calculated degrees of freedom.
7. If the calculated H value is greater than the critical value, reject the null hypothesis, which states that all the samples come from populations with the same distribution.

In summary, the Kruskal-Wallis test is a powerful nonparametric method for comparing three or more samples to determine if they come from populations with the same distribution. This test is particularly useful when parametric assumptions cannot be met, allowing for more robust and accurate statistical analysis.

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Based on the given information, we can perform a one-sample t-test to determine if there is sufficient evidence to support the claim that the chocolate chip bags are underfilled.

The null hypothesis (H0) states that the mean weight of the bags is equal to 416 grams, while the alternative hypothesis (H1) states that the mean weight is less than 416 grams.

Given the sample mean of 412 grams, standard deviation of 11 grams, and a sample size of 12 bags, we can calculate the t-statistic using the formula: t = (sample mean - population mean) / (standard deviation / √sample size).

The critical t-value for a one-tailed test at a 0.1 level of significance and 11 degrees of freedom (n-1) can be found in a t-distribution table. Comparing the calculated t-statistic to the critical t-value will help us determine whether to accept or reject the null hypothesis.

If the calculated t-statistic is less than the critical t-value, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the bags are underfilled at the 416-gram setting. If the t-statistic is greater than or equal to the critical t-value, we fail to reject the null hypothesis and cannot conclude that the bags are underfilled.

Remember to always consider the level of significance and the assumptions of the test (such as normality) when interpreting the results of a statistical hypothesis test.

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The area enclosed between f(x) = 0.2x^2 + 3 and g(x) = x from x = -2 to x = 4 is 28.536 square units.

To find the area enclosed between two curves, we need to find the definite integral of the difference between the two curves. In this case, we need to find:

∫[-2,4] (f(x) - g(x)) dx

Where f(x) = 0.2x^2 + 3 and g(x) = x. Substituting these into the integral, we get:

∫[-2,4] (0.2x^2 + 3 - x) dx

To solve this integral, we need to first distribute the negative sign:

∫[-2,4] (0.2x^2 - x + 3) dx

Then, we can integrate each term separately:

∫[-2,4] 0.2x^2 dx - ∫[-2,4] x dx + ∫[-2,4] 3 dx

Using the power rule of integration, we get:

[0.067x^3]_[-2,4] - [0.5x^2]_[-2,4] + [3x]_[-2,4]

Substituting the limits of integration, we get:

[0.067(4)^3 - 0.067(-2)^3] - [0.5(4)^2 - 0.5(-2)^2] + [3(4) - 3(-2)]

Simplifying, we get:

16.536 - 6 + 18

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Answer:

(x + 5)² + (y + 2)² = 120

Step-by-step explanation:

You need two pieces of information to write the equation of a circle, the center and the radius. This was given in the question so you can just use the following fill-in-the-blank formula to write the equation.

If the center is (h, k) and the radius is r, fill them in here:

(x - h)² + (y - k)² = r²

For your question the center is (-5, -2) and r is√120.

You do need to already know that "minus-a-negative" IS the same as "plus-a-positive" (that's why the final answer has + inside the parentheses) ALSO, you need to know that square and squareroot un-do each other. So if you square sqrt120, you just get "plain" 120. That is, (sqrt120)² is 120.

Fill in the center and radius:

(x - h)² + (y - k)² = r²

(x - -5)² + (y - -2)² = (√120)²

Simplify.

(x + 5)² + (y + 2)² = 120

Taaa-daaa! that's it! Don't you think more people would hate formulas less if they were sold as "fill-in-the-blank" and "shortcuts" !?!  I think so!

x_bar = (1/V)∫(∫xdA)*dy

y_bar = (1/V)∫(∫ydA)*dy

where V is the volume of the solid and dA is the differential area element.

To evaluate the integrals, we need to convert the equations of the curves into polar coordinates. From y^2 = x, we have x = y^2, and since y = 3 is a horizontal line, we can write y = 3cosθ. Thus, the region can be described by:

0 ≤ θ ≤ π/2

0 ≤ r ≤ 3cosθ

0 ≤ z ≤ r^2sinθ

The volume of the solid can be computed as follows:

V = ∫(∫(r^2sinθ)rdr)*dθ from 0 to π/2

= (1/3)*[r^4sinθ] from 0 to π/2

= (1/3)*[81 - 0] = 27

Now we can compute the x-coordinate of the centroid:

x_bar = (1/V)∫(∫xdA)*dy

= (1/27)∫(∫r^3cosθsinθrdr)dθdy

= (1/27)∫(∫r^4cosθsinθ)dθdy from 0 to 3cosθ

= (1/27)∫(∫r^4cosθsinθ)3cosθdθ from 0 to π/2

= (1/27)[∫(sinθcosθ)[81/5r^5] from 0 to 3cosθ] dθ from 0 to π/2

= (1/27)[27/5(4/5)] = 8/25

Therefore, the x-coordinate of the centroid is 8/25.

To find the y-coordinate of the centroid, we use the formula:

y_bar = (1/V)∫(∫ydA)*dy

= (1/27)∫(∫r^3cosθsinθrdr)dθdy

= (1/27)∫(∫r^4cosθsinθ)dθdy from 0 to 3cosθ

= (1/27)∫(∫r^4cosθsinθ)3cosθdθ from 0 to π/2

= (1/27)[∫(sinθcosθ)[81/5r^5] from 0 to 3cosθ] dθ from 0 to π/2

= (1/27)[27/5(27/8)] = 9/40

Therefore, the y-coordinate of the centroid is 9/40.

Hence, the centroid of the solid generated by revolving the region y^2 = x, y = 3, and x = 0 about the y-axis is (8/25, 9/40, 0).

2. To find the centroid of the solid generated by revolving the region bounded by the curves x^2 = y and x = 3 about the y-axis, we again need to use the formula:

x_bar = (1/V)∫(∫xdA

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The mean of the given frequency table is: 1.7

The steps to calculate the mean from a frequency table is as follows:

Step 1: Multiply the number values by the frequencies.

Step 2: Find the totals.

Step 3: Divide the total by n.

The formula for average mean here when given frequency of occurrence of each number is:

x' = Σfx/Σf

Thus:

x' = [(1 * 9) + (2 * 8) + (3 * 3)]/(9 + 8 + 3)

x' = (9 + 16 + 9)/20

x' = 34/20

x' = 1.7

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The $2005$th term of the sequence is $\boxed{145}$

We start by finding the second term in the sequence.

Since the first term is $2005$, the second term is equal to the sum of the cubes of the digits of $2005$,

which is

[tex]$2^3 + 0^3 + 0^3 + 5^3 = 133$[/tex]

To find the third term, we take the sum of the cubes of the digits of $133$, which is[tex]$1^3 + 3^3 + 3^3 = 55$.[/tex]

Continuing in this way, we can find the fourth term:[tex]$5^3 + 5^3 = 250$[/tex].

The fifth term is then[tex]$2^3 + 5^3 + 0^3 = 133$.[/tex]

Notice that the sequence now starts to repeat, since we have found $133$ as the sum of the cubes of the digits of both the second and fifth terms.

Thus, the sequence will continue to repeat every four terms.

Since the[tex]${2005}^{\text{th}}$[/tex] term is larger than $2005$, we can divide $2005$ by $4$ to find the remainder.

We get a remainder of $1$, which means that the[tex]${2005}^{\text{th}}$[/tex] term is the second term in the sequence, which is [tex]$\boxed{133}$[/tex]

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