Evaluate the line integral ∫⋅ for the vector field =sin() 2 cos() along the curve given by ()=3 2 2,1≤≤3.

The required answer is every 5 times we roll the dice and don't get a sum of 7, we can expect to get a sum of 7 once.

To find the probability of odds against a sum of 7 when rolling a pair of dice one time, we need to first determine the number of ways to get a sum of 7 versus the number of ways to get any other sum.
There are a total of 36 possible outcomes when rolling a pair of dice, as there are six possible outcomes for each die (1, 2, 3, 4, 5, or 6). To get a sum of 7, there are 6 possible combinations: 1+6, 2+5, 3+4, 4+3, 5+2, and 6+1. Therefore, the probability of rolling a sum of 7 is 6/36 or 1/6.

To find the odds against rolling a sum of 7, we can use the formula:
Odds against = (number of ways it won't happen) : (number of ways it will happen)
So the number of ways it won't happen (i.e. rolling any sum other than 7) is 36-6, or 30. Therefore, the odds against rolling a sum of 7 are:
Odds against = 30 : 6
Simplifying, we get:
Odds against = 5 : 1
This means that for every 5 times we roll the dice and don't get a sum of 7, we can expect to get a sum of 7 once.

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The amount of fertilizer required for a 200 square-foot garden is 5 pounds.

According to the given data, the directions say to use 4 pounds of fertilizer for 160 square feet of soil. Then, for 1 square foot of soil, Omar needs 4/160 = 0.025 pounds of fertilizer.So, to find the amount of fertilizer needed for 200 square feet of soil, we will multiply the amount of fertilizer for 1 square foot of soil with the area of Omar's garden.i.e., 0.025 × 200 = 5 pounds of fertilizer.
So, Omar needs 5 pounds of fertilizer for a 200 square-foot garden.

Therefore, the amount of fertilizer required for a 200 square-foot garden is 5 pounds.

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To raise 4 to the 13th power using regular exponentiation, a total of 12 multiplications are required.

To raise 4 to the 13th power using regular exponentiation, we can start by multiplying 4 by itself 13 times. However, this would require a total of 13 multiplications, which is not the most efficient way to calculate 4^13.

Instead, we can use a method called "exponentiation by squaring", which reduces the number of multiplications required. Here's how it works:

Start by writing the exponent (13) in binary form: 13 = 1101 (in binary).

Starting with the base (4), square it repeatedly, each time moving from right to left in the binary representation of the exponent.

Whenever we encounter a "1" in the binary representation of the exponent, we multiply the current result by the base.

Using this method, we can calculate 4^13 with the following steps:

So, using exponentiation by squaring, we only needed a total of 7 multiplications instead of 13, which is much more efficient.

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Soil salinity is the main factor that limits the seaward distribution of Iva in the marsh.

Iva is a plant that can tolerate a range of soil conditions, but high salinity levels make it difficult for the plant to grow and survive. As the marsh gets closer to the sea, the soil salinity increases, making it less favorable for Iva growth. Additionally, the presence of other herbivores can also limit the growth of Iva by reducing the availability of nutrients and resources. Soil oxygen levels and Juncus pressce can also affect Iva growth, but salinity has the most significant impact.

In conclusion, high soil salinity is the main factor that limits the seaward distribution of Iva in the marsh.

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The temperature of the compound increased by 3/4 of a degree in one hour. Conversion of 2 1/3 hours into a mixed number: 2 1/3 = 7/3 hours.

To find the rate of increase in temperature per hour, we will convert 1 hour into 3/7 hours as follows;

1 hour = 3/7 hours.

Thus, the temperature of the compound rose by 1 3/4 degrees every 2 1/3 hours or 7/3 hours:

= (1 3/4) / (7/3)

= (7/4) x (3/7)

= 21/28

= 3/4 of a degree per hour.

We are given that for a certain time, the temperature of a compound increased by 1 3/4 degrees every 2 1/3 hours. We are required to find how much the temperature of the compound rose in one hour. Let's begin by converting 2 1/3 hours into a mixed number.2 1/3 = 7/3 hours.

Now, to find the rate of increase in temperature per hour, we will convert 1 hour into 3/7 hours. Thus,

1 hour = 3/7 hours.

We can now find the temperature of the compound that rose per hour by dividing the temperature that rose in 7/3 hours by 7/3 hours and multiplying the result by 3/7. Let's substitute the temperature into the formula:

= (1 3/4) / (7/3)

= (7/4) x (3/7)

= 21/28

= 3/4 of a degree per hour.

Therefore, the temperature of the compound increased by 3/4 of a degree in one hour.

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The limit is less than 1, the series converges by the ratio test. The given series ∑(n=1 to infinity) 5n / [(2n^2

To determine the convergence or divergence of the series ∑(n=1 to infinity) 5n / [(2n^2 - 5)], we can use the limit comparison test or the ratio test.

Let's start with the limit comparison test. We choose a known convergent series with positive terms, say ∑(n=1 to infinity) 1/n^2.

First, let's calculate the limit of the ratio of the two series:

lim (n→∞) (5n / [(2n^2 - 5)]) / (1/n^2)

To simplify this expression, let's multiply the numerator and denominator by n^2:

lim (n→∞) [(5n * n^2) / (2n^2 - 5)] / 1

Simplifying further:

lim (n→∞) (5n^3) / (2n^2 - 5)

Since the degree of the numerator is greater than the degree of the denominator, we can divide both the numerator and denominator by n^2:

lim (n→∞) (5n^3 / n^2) / (2n^2 / n^2 - 5 / n^2)

= lim (n→∞) (5n) / (2 - 5/n^2)

As n approaches infinity, the term 5/n^2 approaches 0. Therefore:

lim (n→∞) (5n) / (2 - 5/n^2) = lim (n→∞) (5n) / 2

This limit is equal to infinity. Since the limit of the ratio of the two series is not finite (it diverges), we cannot use the limit comparison test to determine convergence.

Next, let's use the ratio test:

Using the ratio test, we calculate:

lim (n→∞) |(5(n+1) / [(2(n+1)^2 - 5)]) / (5n / [(2n^2 - 5)])|

Simplifying:

lim (n→∞) |(5(n+1) * [(2n^2 - 5)]) / (5n * [(2(n+1)^2 - 5)])|

Again, dividing the numerator and denominator by n^2:

lim (n→∞) |[(5(n+1) * (2n^2 - 5)) / (5n * (2(n+1)^2 - 5))] * (n^2 / n^2)

= lim (n→∞) |(5(n+1) * (2 - 5/n^2)) / (5 * (2(n+1)^2/n^2 - 5/n^2))|

As n approaches infinity, the term 5/n^2 approaches 0. Therefore:

lim (n→∞) |(5(n+1) * (2 - 5/n^2)) / (5 * (2(n+1)^2/n^2))|

= lim (n→∞) |(5(n+1) * 2) / (5 * 2(n+1)^2/n^2)|

= lim (n→∞) |(n+1) / (n+1)^2|

Taking the absolute value, we have:

lim (n→∞) |1 / (n+1)| = 0

Since the limit is less than 1, the series converges by the ratio test.

Therefore, the given series ∑(n=1 to infinity) 5n / [(2n^2

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Answer:

3.

Step-by-step explanation:

The mode is what number appears the most. Hope this helps!

The Curiosity Rover has traveled approximately distance covered 14.43 miles.

Given that the circumference of the Curiosity Rover's wheels is 157.1 cm and the wheels are rotated 14,756.8 times,

we need to find the distance covered by the Curiosity Rover.

Let us first convert the circumference from centimeters to miles:

1 mile = 160934.4 cm

Circumference in miles = 157.1/160934.4 miles

Circumference in miles = 0.000976615 miles

We know that distance covered is equal to the product of circumference and the number of revolutions. Thus,

Distance covered = Circumference * Number of revolutions

Distance covered = 0.000976615 miles * 14,756.8

Distance covered = 14.426192 miles

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After rotating the end table counterclockwise 180° about vertex J, the new coordinates of the vertices will be J(4,4), K(6,2), and L(7,2).

To rotate a point counterclockwise 180° about a fixed point, we can use the following transformation rules:

1. Translate the fixed point to the origin by subtracting its coordinates from all points.

2. Rotate the translated points counterclockwise 180° about the origin.

3. Translate the rotated points back to their original position by adding the coordinates of the fixed point.

In this case, the fixed point is J(4,4). Let's apply these transformation rules to find the new coordinates of the vertices:

1. Translate: Subtract 4 from the x-coordinates and 4 from the y-coordinates of all points:

  J(4-4, 4-4) = J(0,0)

  K(4-4, 6-4) = K(0,2)

  L(1-4, 6-4) = L(-3,2)

2. Rotate: Rotate the translated points counterclockwise 180° about the origin:

  J(0,0) remains unchanged

  K(0,2) rotates to (-0, -2) = (0,-2)

  L(-3,2) rotates to (3,-2)

3. Translate back: Add 4 to the x-coordinates and 4 to the y-coordinates of all points:

  J(0+4, 0+4) = J(4,4)

  K(0+4, -2+4) = K(4,2)

  L(3+4, -2+4) = L(7,2)

Therefore, after rotating the end table counterclockwise 180° about vertex J, the new coordinates of the vertices are J(4,4), K(4,2), and L(7,2).

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Main answer: The car's value after one year was $15,000.

Supporting explanation:

The cost of Mr. Deaver's new car was $20,000. After one year, the car's value decreased by 25%. Therefore, the car's value after one year can be found by subtracting the 25% decrease from the original cost of the car:

25% of $20,000 = 0.25 × $20,000 = $5,000

Subtracting $5,000 from $20,000 gives us the car's value after one year:

$20,000 - $5,000 = $15,000

Therefore, the car's value after one year was $15,000.

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The standard deviation of the resulting data would be 7. To understand why the standard deviation would be 7, let's consider the effect of multiplying each value in the data set by 1.75.

When we multiply each value by a constant, the mean of the data set is also multiplied by that constant. In this case, since multiplying by 1.75 increases the scale of the data, the mean is also multiplied by 1.75.

Now, the standard deviation measures the dispersion or spread of the data around the mean. When we multiply each value by 1.75, the spread of the data increases because the values are further away from the mean. Since the original standard deviation was 4 and each value is multiplied by 1.75, the resulting standard deviation is 4 * 1.75 = 7.

Therefore, the standard deviation of the resulting data is 7.

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Given:  Sylvan drove 128.6 km each day for 8 days. He drove 44.3 km each day for 12 days.To find:The total distance Sylvan drove.

Solution: Let's find the distance that Sylvan covered for the first 8 days.He covered 128.6 km each day, and as he covered this distance for 8 days, the total distance that he covered in 8 days would be:Distance covered = 128.6 km/day × 8 days= 1028.8 km Now,

let's find the distance that he covered in the next 12 days.He covered 44.3 km each day for 12 days, so the total distance covered would be:Distance covered = 44.3 km/day × 12 days= 531.6 km Now,

let's find the total distance that Sylvan drove:

Total distance = distance covered in the first 8 days + distance covered in the next 12 days= 1028.8 km + 531.6 km= 1560.4 km Hence, the total distance Sylvan drove is 1560.4 km.

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The value of the finite sum equation is,

⇒ S = 5 (3ⁿ - 1)

We have to given that;

Sequence is,

⇒ 10 + 30 + 90 + ..... = 2657200

Now, We get;

Common ratio = 30/10 = 3

Hence, Sequence is in geometric.

So, The sum of geometric sequence is,

⇒ S = a (rⁿ- 1)/ (r - 1)

Here, a = 10

r = 3

Hence, We get;

⇒ S = 10 (3ⁿ - 1) / (3 - 1)

⇒ S = 10 (3ⁿ - 1) / 2

⇒ S = 5 (3ⁿ - 1)

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the eigenvalues of A are λ = 2 and μ = -2/3, with algebraic multiplicities 1 and 2, respectively.

We know that the trace of a matrix is the sum of its eigenvalues and the determinant is the product of its eigenvalues. Let the two distinct eigenvalues of A be λ and μ. Then, we have:

tr(A) = λ + μ + λ or μ (since the eigenvalues are distinct)

-3 = 2λ + μ ...(1)

det(A) = λμ(λ + μ)

-28 = λμ(λ + μ) ...(2)

We can solve this system of equations to find λ and μ.

From equation (1), we can write μ = -3 - 2λ. Substituting this into equation (2), we get:

-28 = λ(-3 - 2λ)(λ - 3)

-28 = -λ(2λ^2 - 9λ + 9)

2λ^3 - 9λ^2 + 9λ - 28 = 0

We can use polynomial long division or synthetic division to find that λ = 2 and λ = -2/3 are roots of this polynomial. Therefore, the eigenvalues of A are 2 and -2/3, and their algebraic multiplicities can be found by considering the dimensions of the eigenspaces.

Let's find the algebraic multiplicity of λ = 2. Since tr(A) = -3, we know that the sum of the eigenvalues is -3, which means that the other eigenvalue must be -5. We can find the eigenvector corresponding to λ = 2 by solving the system of equations (A - 2I)x = 0, where I is the 3 x 3 identity matrix. This gives:

|1-2 2 1| |x1| |0|

|2 1-2 1| |x2| = |0|

|1 1 1-2| |x3| |0|

Solving this system, we get x1 = -x2 - x3, which means that the eigenspace corresponding to λ = 2 is one-dimensional. Therefore, the algebraic multiplicity of λ = 2 is 1.

Similarly, we can find the algebraic multiplicity of λ = -2/3 by considering the eigenvector corresponding to μ = -3 - 2λ = 4/3. This gives:

|-1/3 2 1| |x1| |0|

| 2 -5/3 1| |x2| = |0|

| 1 1 5/3| |x3| |0|

Solving this system, we get x1 = -7x2/6 - x3/6, which means that the eigenspace corresponding to λ = -2/3 is two-dimensional. Therefore, the algebraic multiplicity of λ = -2/3 is 2.

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You can use the following formula to calculate the surface area of the right rectangular prism:

[tex]\sf SA=2(wl+lh+hw)[/tex]

Where "w" is the width, "l" is the length, and "h" is the height.

Knowing that this right rectangular prism  has a length of 8 centimeters, a width of 3 centimeters and a height of 5 centimeters, you can substitute these values into the formula.

Then, the surface of the right rectangular prism is:

[tex]\sf SA=[(3 \ cm\times 8 \ cm)+( 8 \ cm\times 5 \ cm)+(5 \ cm\times3 \ cm)][/tex]

[tex]\Rightarrow\sf SA=158 \ cm^2[/tex]

If the coefficient of correlation is -0.4, then the slope of the regression line must be negative.(C)

The coefficient of correlation, denoted as 'r', measures the strength and direction of the linear relationship between two variables. In this case, r = -0.4, indicating a negative relationship.

The slope of the regression line, denoted as 'a', represents the change in the dependent variable for a unit change in the independent variable. Since the correlation coefficient is negative, the slope of the regression line must also be negative, as the variables move in opposite directions.

This means that as one variable increases, the other decreases. Thus, the correct answer is (c) the slope of the regression line must be negative.

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Assuming that there are 365 days in a year (ignoring leap years) and that all dates are equally likely, we can use the Pigeonhole Principle to determine the minimum number of teenagers needed to ensure that 4 of them were born on the same date.

There are 365 possible days in a year on which a person could be born. Therefore, if we select k teenagers, the total number of possible birthdates is 365k.

To guarantee that 4 of them were born on the exact same date, we need to find the smallest value of k for which 365k is greater than or equal to 4 times the number of possible birthdates. In other words:365k ≥ 4(365)

Simplifying this inequality, we get: k ≥ 4

Therefore, we need to select at least 4 + 1 = 5 teenagers to ensure that 4 of them were born on the exact same date.

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The inequality that correctly compares Hailey's account values is: $117.39 > -$121.06.

To correctly compare the account values, we can use the inequality symbol.

Since Hailey has $117.39 in her savings account and -$121.06 in her checking account, the correct inequality to compare the values is:

Savings account value > Checking account value

Therefore, the correct inequality is:

$117.39 > -$121.06

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The sales tax for the pair of jeans is $1.47.

We are given that;

Cost=$24.50

Percentage=6%

Now,

Step 1: Convert the sales tax rate to a decimal

6% = 6/100 = 0.06

Step 2: Multiply the cost of the jeans by the sales tax rate

24.50 x 0.06 = 1.47

Step 3: Round the sales tax amount to the nearest cent

1.47 is already rounded to the nearest cent

Therefore, by the percentage the answer will be $1.47.

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The amount of golden delicious apples than red delicious apples that Mr. Myers picked would be 14 1/8.

The extra amount of golden delicious apples that Mr. Myers picked in comparison to the red delicious apples that Mr. Myers picked would be gotten by subtracting the amount of golden delicious apples from red delicious apples as follows:

27 2/8 - 13 1/8

= 14 1/8

So, the amount with which the number of golden delicious apples that Mr. Myers got was greater than the red delicious apples is 14 1/8

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Complete Question:

Mr.Myers picked 13 1/8 bushels of red delicious apples and 27 2/8 bushels of golden delicious apples. How many bushels of golden delicious apples than of red delicious apples did he pick?

The solution for 8 cos 2 ( t ) − 2 sin ( t ) − 7 = 0 for all solutions 0 ≤ t < 2 π is

t ≈ 0.896 rad and t ≈ 5.387 rad.

We can use the trigonometric identity:

cos(2t) = 2cos²t - 1, to rewrite the equation as:

8(2cos²t - 1) - 2sint - 7 = 0

Simplifying and rearranging terms, we get:

16cos²t - 2sint - 15 = 0

Using the identity sin²(t) + cos²(t) = 1, we can substitute sin(t) = ±√(1 - cos²(t)) and get a quadratic equation in terms of cos(t):

16cos²(t) - 2(±√(1 - cos²(t))) - 15 = 0

Solving for cos(t), we get:

cos(t) = ±√(17)/4

Since 0 ≤ t < 2π, we can use the inverse cosine function to find the solutions in this interval:

t = cos⁻¹(√(17)/4) and t = 2π - cos⁻¹(√(17)/4)

Therefore, the solutions are:

t ≈ 0.896 rad and t ≈ 5.387 rad.

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To calculate the probability of choosing a c2 card from the first deck (Deck A) or a c6 card from the second deck (Deck B):

First, calculate the probability of choosing a c2 card from Deck A:

P(c2) = Number of c2 cards in Deck A / Total number of cards in Deck A

= 4/20

= 1/5

Next, calculate the probability of choosing a c6 card from Deck B:

P(c6) = Number of c6 cards in Deck B / Total number of cards in Deck B

= 2/10

= 1/5

Since the events of choosing a c2 card and a c6 card are mutually exclusive, the probability of both events occurring together (P(c2 and c6)) is zero.

Therefore, the probability of choosing a c2 card from Deck A or a c6 card from Deck B can be found by adding these probabilities:

P(c2 or c6) = P(c2) + P(c6) - P(c2 and c6)

= 1/5 + 1/5 - 0

= 2/5

So, the probability of choosing a c2 card from Deck A or a c6 card from Deck B is 2/5.

Now, let's calculate the probability of choosing a c3 card from Deck A or a c5 card from Deck B:

P(c3) = Number of c3 cards in Deck A / Total number of cards in Deck A

= 5/20

= 1/4

P(c5) = Number of c5 cards in Deck B / Total number of cards in Deck B

= 1/10

Therefore, the probability of choosing a c3 card from Deck A or a c5 card from Deck B is:

P(c3 or c5) = P(c3) + P(c5)

= 1/4 + 1/10

= 3/10

So, the probability of choosing a c3 card from Deck A or a c5 card from Deck B is 3/10.

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The probability that a randomly chosen value is greater than 3 is 0.8413.

(a) Let X be a random variable with a normal distribution with mean μ = 10 and standard deviation σ = 7. We want to find the proportion of the population that is less than 21, or P(X < 21).

Using the standard normal distribution, we can find the z-score corresponding to 21:

z = (21 - μ) / σ = (21 - 10) / 7 = 1.57

Looking up the corresponding probability in the standard normal distribution table, we find that P(Z < 1.57) = 0.9418.

Therefore, P(X < 21) = P(Z < 1.57) = 0.9418.

(b) We want to find the probability that a randomly chosen value is greater than 3, or P(X > 3).

Again, we can use the standard normal distribution and find the z-score corresponding to 3:

z = (3 - μ) / σ = (3 - 10) / 7 = -1

Using the standard normal distribution table, we find that P(Z > -1) = P(Z < 1) = 0.8413.

Therefore, P(X > 3) = 1 - P(X < 3) = 1 - P(Z < -1) = 1 - 0.1587 = 0.8413.

So the probability that a randomly chosen value is greater than 3 is 0.8413.

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Without access to Exercise 16.2, I'm unable to provide the regression equation.

However, I can provide a general framework for predicting sales using a regression equation with a given advertising budget and confidence interval. To predict sales with a 90% confidence interval, you would first need to input the advertising budget value of $90,000 into the regression equation. The resulting value would be your point estimate for the sales with that budget. Next, you would need to calculate the margin of error using the standard error of the estimate, which is a measure of the variability of the predicted sales around the regression line. The margin of error is equal to the critical value (which depends on the sample size and confidence level) times the standard error of the estimate. Finally, you would calculate the confidence interval by adding and subtracting the margin of error from the point estimate. The resulting interval would provide a range of values that you can be 90% confident includes the true sales value for the given advertising budget.

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Use the regression equation in Exercise 16.2 to predict with 90% confidence the sales when the advertising budget is $90,000.

The value of x in the function is 65 degrees

From the question, we have the following parameters that can be used in our computation:

sin 25° = cos x°

if the angles are in a right triangle, then we have tehe following theorem

if sin a° = cos b°, then a + b = 90

Using the above as a guide, we have the following:

25 + x = 90

When the like terms are evaluated, we have

x = 65

Hence, the value of x is 65 degrees

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The correct answer is (D) 0.0854. This means that if the significance level of the test is 0.05, we would fail to reject the null hypothesis, as the p-value (0.0854) is greater than the significance level (0.05).

We need to find the p-value associated with the given test statistic to determine the significance level of the claim. The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the observed one under the null hypothesis.

Assuming that the null hypothesis is that the mean systolic blood pressure of women aged 40-50 in the U.S. is equal to 126 mmHg, and the alternative hypothesis is that it is not equal to 126 mmHg, we can use a two-tailed test.

Looking up the z-score table or using a calculator, we find that the area to the right of z = 1.72 is 0.0427. Since this is a two-tailed test, the area in both tails is 0.0427 x 2 = 0.0854.

Therefore, the correct answer is (D) 0.0854. This means that if the significance level of the test is 0.05, we would fail to reject the null hypothesis, as the p-value (0.0854) is greater than the significance level (0.05).

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To make the best prediction of the number of students who wake up after 6:30 am, we can use the information provided by the survey.

Out of the 20 students surveyed:

3 students woke up before 6 am.

13 students woke up between 6 am and 6:30 am.

4 students woke up after 6:30 am.

Since the survey sample consists of 20 students, we can assume that the proportions observed in the sample are representative of the larger population of 500 students. To estimate the number of students who wake up after 6:30 am among the 500 students, we can use proportional reasoning.

We can calculate the proportion of students who woke up after 6:30 am in the sample and apply that proportion to the larger population.

The proportion of students who woke up after 6:30 am in the sample is 4/20 or 0.2.

To estimate the number of students who wake up after 6:30 am in the larger population of 500 students, we multiply the proportion by the total population size:

0.2 * 500 = 100

Based on this estimation, the best prediction would be that approximately 100 students wake up after 6:30 am among the 500 surveyed students.

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The average price per pound for all the coffee sold is $5.52 per pound, when 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound.

Suppose that 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound. We have to find the average price per pound for all the coffee sold.

Average price is equal to the total cost of coffee sold divided by the total number of pounds sold. We can use the following formula:

Average price per pound = (total revenue / total pounds sold)

In this case, the total revenue is the sum of the revenue from selling 650 pounds at $4 per pound and the revenue from selling 400 pounds at $8 per pound. That is:

total revenue = (650 lb * $4/lb) + (400 lb * $8/lb)

= $2600 + $3200

= $5800

The total pounds sold is simply the sum of 650 pounds and 400 pounds, which is 1050 pounds. That is:

total pounds sold = 650 lb + 400 lb

= 1050 lb

Using the formula above, we can calculate the average price per pound:

Average price per pound = total revenue / total pounds sold= $5800 / 1050

lb= $5.52 per pound

Therefore, the average price per pound for all the coffee sold is $5.52 per pound, when 650 lb of coffee are sold when the price is $4 per pound, and 400 lb are sold at $8 per pound.

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Let X be a geometric random variable with parameter p = and let Y be a Poisson random variable with parameter A 4. Assuming X and Y independent, then the expected value of the perimeter of the rectangle is 2( + 5), and the expected value of the area is 5.

For the expected values of the perimeter and area of the rectangle, we need to calculate the expected values of X and Y first, as well as their respective distributions.

We have,

X is a geometric random variable with parameter p =

Y is a Poisson random variable with parameter λ = 4

X and Y are independent

For a geometric random variable with parameter p, the expected value is given by E(X) = 1/p. In this case, E(X) = 1/p = 1/.

For a Poisson random variable with parameter λ, the expected value is equal to the parameter itself, so E(Y) = λ = 4.

Now, let's calculate the expected values of the perimeter and area of the rectangle using the given side lengths X and Y + 1.

Perimeter = 2(X + Y + 1)

Area = X(Y + 1)

To find the expected value of the perimeter, we substitute the expected values of X and Y into the equation:

E(Perimeter) = 2(E(X) + E(Y) + 1)

            = 2( + 4 + 1)

            = 2( + 5)

To find the expected value of the area, we substitute the expected values of X and Y into the equation:

E(Area) = E(X)(E(Y) + 1)

       = ( )(4 + 1)

       = 5

Therefore, the expected value of the perimeter of the rectangle is 2( + 5), and the expected value of the area is 5.

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The surface with the given vector equation is a paraboloid.

We are given the vector equation of a surface in terms of two parameters s and t:

r(s,t) = (ssin(2t), s^2, scos(2t))

To identify the surface, we need to eliminate the parameters s and t from this equation and obtain a simpler equation in terms of the Cartesian coordinates x, y, and z.

To eliminate t, we can take the ratio of the first and third components of r(s,t):

x/z = sin(2t)/cos(2t) = tan(2t)

Solving for t, we get:

t = 1/2 * atan(x/z)

Substituting this expression for t back into r(s,t), we get:

r(s,x,z) = (sx/sqrt(x^2 + z^2), s^2, sz/sqrt(x^2 + z^2))

To eliminate s, we can set s = sqrt(y) and obtain:

r(x,y,z) = (x/sqrt(1 + z^2/y), y, z/sqrt(1 + z^2/y))

This is the Cartesian equation of a paraboloid, which opens along the y-axis. Specifically, it is a circular paraboloid, since the x and z coordinates appear symmetrically.

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