The pressure distribution on the 1-m-diameter circular disk in the figure below is given in the table. Determine the drag on the disk Note. Apply the right endpoint approximation

The length of the pathway that runs along the diagonal of the play area is approximately 36 meters.

Given: Length of the rectangular play area = 30 meters Width of the rectangular play area = 20 meters To find: The length of a pathway that runs along the diagonal of the play area.

Formula to find diagonal of rectangle is as follows:d = √(l² + w²)Where,d = diagonal of the rectangular play areal = length of the rectangular play areaw = width of the rectangular play area.

Substituting the given values in the above formula,d = √(30² + 20²)d = √(900 + 400)d = √1300d = 36.0555 m (approx)

Therefore, the length of the pathway that runs along the diagonal of the play area is approximately 36 meters (rounded to the nearest meter).

Note: Here, we use the square root of 1300 in a calculator to find the exact value of the diagonal and rounded it off to the nearest meter.

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The length of the pathway along the diagonal of the play area is approximately 36 meters.

The length of the pathway that runs along the diagonal of the play area can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the length is the hypotenuse, while the 30-meter side and the 20-meter side are the other two sides.

Applying the Pythagorean theorem, we have:

a2 + b2 = c2

where a = 30 meters and b = 20 meters. Solving for c, the length of the pathway:

c2 = a2 + b2

c2 = 302 + 202

c2 = 900 + 400

c2 = 1300

Next, we take the square root of both sides to find the length of the pathway:

c = √1300

c ≈ √1296

c ≈ 36 meters

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The answer is `70/1` or simply `70`.

Given that the line plot records the weight of each box, it can be observed that the weight of the boxes ranges from 40 to 110. Let us find the weight of one of the heaviest boxes and one of the lightest boxes.Heaviest box: 110Lightest box: 40The difference between the weight of the heaviest box and the lightest box = 110 - 40= 70Therefore, one of the heaviest boxes weighs 70 more than one of the lightest boxes. So, the required fraction is `70/1`.Hence, the answer is `70/1` or simply `70`.

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The average speed of Mike for the entire race is 4.54 m/s.

To find out the average speed of Mike during the entire race, we need to have the total distance and the total time taken. Now, the distance covered by Mike is given in two parts, the first 25 meters and the last 25 meters.

So, the total distance covered by Mike is 25+25 = 50 meters.

The time taken by Mike to cover the first 25 meters is 4.2 seconds.

And, the time taken by Mike to cover the last 25 meters is 6.8 seconds.

Therefore, the total time taken by Mike is 4.2+6.8 = 11 seconds.

To find out the average speed of Mike, we use the formula:

Speed = Distance / Time

Average speed = Total distance covered / Total time taken

Therefore, the average speed of Mike for the entire race is given as:

Average speed = Total distance covered / Total time taken

= 50 meters / 11 seconds

= 4.54 m/s

Therefore, the average speed of Mike for the entire race is 4.54 m/s.

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The expression is rationalize to give C  [tex]= \frac{\sqrt{Em} }{m}[/tex]

From the information given, we have that the surd form is expressed as;

C=[tex](\frac{E}m} )^(^1^/^2^)[/tex]

This is represented as;

C =[tex]= \frac{\sqrt{E} }{\sqrt{m} }[/tex]

We need to know that the process of simplifying a fraction by removing surds (such as square roots or cube roots) from its denominator is known as rationalization of surds. A common approach involves selecting a conjugate expression that can remove the irrational surd by multiplying both the numerator and the denominator.

Then, we have;

C = [tex]= \frac{\sqrt{E} * \sqrt{m} }{\sqrt{m} * \sqrt{m} }[/tex]

multiply the values, we have;

C = [tex]\frac{\sqrt{Em} }{m}[/tex]

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When working with polar coordinates, it's important to remember that r represents the distance from the origin to the point in question. Since we're looking for a positive value of r, we'll choose the positive square root when solving for r. This ensures that we're measuring the distance in a positive direction, away from the origin.

For example, let's say we have a point in Cartesian coordinates (3, -4). To find the polar coordinates with r > 0, we first need to find the value of r. We can use the Pythagorean theorem to do this:

r^2 = x^2 + y^2
r^2 = 3^2 + (-4)^2
r^2 = 9 + 16
r^2 = 25

Now we can take the positive square root to solve for r:

r = sqrt(25)
r = 5

So the distance from the origin to the point (3, -4) is 5. To find the angle (theta) in polar coordinates, we can use the inverse tangent function:

theta = arctan(y/x)
theta = arctan(-4/3)

Note that we use the negative value for y because the point is in the third quadrant, where y values are negative.

So the polar coordinates for the point (3, -4) with r > 0 are (5, arctan(-4/3)).

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The side of the pentagonal koi pond with the deck around it is (3x/2) feet where x is the length of each interior side.

Let the side of the pentagon be x feet.

Since there are five sides, the sum of all the interior angles is (5 – 2) × 180 = 540°.

Each angle of the pentagon is given by 540°/5 = 108°.

The deck of equal width is provided around the pond, so let the width be w feet.

Therefore, the side of the pentagon with the deck around it has length (x + 2w) feet.

The length of the exterior side of the pentagon is equal to the length of the corresponding interior side plus the width of the deck.

Therefore, the length of the exterior side of the pentagon is (x + 3w) feet.

We know that the lengths of the exterior sides of the pentagon are equal.

Therefore, the length of each exterior side is (x + 3w) feet.

So,

(x + 3w) × 5 = 5x.

Solving this equation gives 2w = x/2.

So, the side of the pentagon with the deck around it is (x + x/2) feet or (3x/2) feet.

Therefore, the side of the pentagonal koi pond with the deck around it is (3x/2) feet where x is the length of each interior side.

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The given statement, "A 99.8% confidence interval for the population mean is 54.4", is false. The correct interval is (56.05, 60.95).

Part 2 of 2:

We can use the following formula to find the confidence interval for the population mean:

CI = x ± z*(s/√n)

where x is the sample mean, s is the sample standard deviation, n is the sample size, z is the z-score corresponding to the desired level of confidence, and CI is the confidence interval.

For a 99.8% confidence interval, we need to find the z-score that corresponds to an area of 0.001 on each tail of the standard normal distribution. Using a standard normal distribution table or a calculator, we find that the z-score is approximately 3.090.

Substituting the given values into the formula, we have:

CI = 58.5 ± 3.090*(9.5/√57)

Simplifying this expression, we get:

CI = 58.5 ± 2.45

Therefore, the 99.8% confidence interval for the population mean is (58.5 - 2.45, 58.5 + 2.45), or (56.05, 60.95), rounded to one decimal place.

So the given statement, "A 99.8% confidence interval for the population mean is 54.4", is false. The correct interval is (56.05, 60.95).

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The value of the line integral of a vector field F along the path C is (10, 24). No, the line integral of F along C does not depend on the joining (0,0) to (1,6).

To evaluate the line integral of F along the path C, we need to parameterize the path. Since the path is given by y=6x^2 and it goes from (0,0) to (1,6), we can parameterize it as follows:

r(t) = (t, 6t^2), 0 ≤ t ≤ 1

The differential of r(t) is dr/dt = (1, 12t), so we can write:

F(r(t)).dr = (5t(6t^2), 8(6t^2))(1, 12t)dt

= (30t^2, 96t^3)dt

Now we can integrate this expression over the range of t from 0 to 1:

∫[0,1] (30t^2, 96t^3)dt = (10, 24)

Therefore, the value of the line integral of F along C is (10, 24).

The answer to whether the integral depends on the joining (0,0) to (1,6) is no. This is because the line integral only depends on the values of the vector field F and the path C, and not on the specific points used to parameterize the path.

As long as the path C is the same, the line integral will have the same value regardless of the choice of points used to define the path.

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1. An advantage of offering more choices for something is that it gives customers a greater range of options to choose from, which can increase customer satisfaction and loyalty. Offering 50 flavors of ice cream instead of four can attract a wider range of customers with different preferences, leading to increased sales and revenue. Additionally, having more options can help differentiate the store from competitors, as customers may be more likely to choose a store that offers more variety.

2. An advantage of offering less for something is that it can simplify the decision-making process for customers. This can be particularly helpful for customers who are indecisive or overwhelmed by too many options. Offering only three flavors such as vanilla, chocolate, and swirl can make the decision-making process easier for customers, leading to a faster transaction and potentially increased customer satisfaction. Additionally, offering less can help the store to streamline its operations by reducing the number of ingredients and supplies needed, which can lead to cost savings.

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The value of the expression decreases because there is less of `n` in the expression.

When the value of n decreases in the expression `n+15n+15`, the value of the entire expression also decreases.

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

The expression `n+15n+15` can be simplified as follows:Combine like terms, which are the two terms that contain `n`. `n` and `15n` add up to `16n`.

Thus, the expression can be rewritten as `16n + 15`.When `n` decreases, the value of the expression decreases because there is less of `n` in the expression.

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If "A" denotes the event that student takes statistics and B denotes event that the student is senior, the probability of P(A' or B) is (c) 0.82.

To find P(A' or B), we want to find the probability that a student is not a senior or take statistics (or both).

We know that the total number of students surveyed is 100, and out of those students : 15 seniors take statistics; 35 seniors take calculus

18 juniors take statistics,  32 juniors take calculus.

The probability P(A' or B) is written as P(A') + P(B) - P(A' and B);

To find the probability of a student not taking statistics, we add the number of students who take calculus (seniors and juniors) and divide by the total number of students:

⇒ P(A') = (35 + 32) / 100 = 0.67;

The probability of student being a senior,

⇒ P(B) = (15 + 35)/100 = 0.50,

Next, to find probability of student who is not take statistics and is a senior, which are 35 students,

So, P(A' and B) = 35/100 = 0.35;

Substituting the values,

We get,

P(A' or B) = 0.67 + 0.50 - 0.35 = 0.82;

Therefore, the correct option is (c).

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The given question is incomplete, the complete question is

A student surveyed 100 students and determined the number of students who take statistics or calculus among seniors and juniors. Here are the results.

              Statistics   Calculus

Senior           15              35

Junior           18               32

Let A be the event that the student takes statistics and B be the event that the student is a senior.

What is P(A' or B)?

(a) 0.18

(b) 0.68

(c) 0.82

(d) 0.97

The f-intercept of the function f(t) = (t-5)(t^7)(t-6) is 0, and the t-intercepts are t=5, t=0 (with multiplicity 7), and t=6.

To find the f-intercept of the function f(t) = (t-5)(t^7)(t-6), we need to find the value of f(t) when t=0. To do this, we substitute 0 for t in the function and simplify:

f(0) = (0-5)(0^7)(0-6) = 0

Therefore, the f-intercept of the function is 0.

To find the t-intercepts of the function, we need to set f(t) equal to 0 and solve for t. We can do this by using the zero product property, which states that if ab=0, then either a=0, b=0, or both.

So, setting f(t) = (t-5)(t^7)(t-6) = 0, we have three factors that could be equal to 0:

t-5=0, which gives us t=5
t^7=0, which gives us t=0 (this is a repeated root)
t-6=0, which gives us t=6

Therefore, the t-intercepts of the function are t=5, t=0 (with multiplicity 7), and t=6.

In summary, the f-intercept of the function f(t) = (t-5)(t^7)(t-6) is 0, and the t-intercepts are t=5, t=0 (with multiplicity 7), and t=6.

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Remember to convert degrees to radians if required. Rounded to three decimal places, we have:

1st set: (5.831, 3.678 radians)
2nd set: (5.831, 9.960 radians)

It appears that there is a small typo in the coordinates you provided. Assuming the correct coordinates are (-5, -3), I can help you find the polar coordinates.

First, let's calculate the radial distance (r) and the angle (θ) for the point (-5, -3).

To find r, use the formula: r = √(x² + y²)
r = √((-5)² + (-3)²) = √(25 + 9) = √34

Now, we can find the angle (θ) using the arctangent formula: θ = arctan(y/x)
θ = arctan(-3/-5) = arctan(0.6)

Now, convert θ from radians to degrees: θ ≈ 30.964°

Since the point is in the third quadrant, add 180° (or π radians) to the angle:
θ = 30.964° + 180° ≈ 210.964°

Now, we have our first set of polar coordinates: (r, θ) ≈ (5.831, 210.964°)

To find the second set of polar coordinates, simply add 360° (or 2π radians) to the angle:
θ₂ = 210.964° + 360° ≈ 570.964°

The second set of polar coordinates is: (r, θ) ≈ (5.831, 570.964°)

Remember to convert degrees to radians if required. Rounded to three decimal places, we have:

1st set: (5.831, 3.678 radians)
2nd set: (5.831, 9.960 radians)

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The line integral of F=xyi+yzj+xzk from (0,0,0) to (1,1,1) over the path C given by r=ti+t^2j+t^4k for 0≤t≤1 is 1/5.

To solve for the line integral, we first need to parameterize the curve. From the given equation, we have r(t) = ti + t^2j + t^4k.

Next, we need to find the differential of r(t) with respect to t: dr/dt = i + 2tj + 4t^3k.

Now we can substitute r(t) and dr/dt into the line integral formula:

∫[0,1] F(r(t)) · (dr/dt) dt = ∫[0,1] (t^3)(t^2)i + (t^5)(t)j + (t^2)(t^4)k · (i + 2tj + 4t^3k) dt

Simplifying this expression, we get:

∫[0,1] (t^5 + 2t^6 + 4t^9) dt

Integrating from 0 to 1, we get:

[1/6 t^6 + 2/7 t^7 + 4/10 t^10]_0^1 = 1/6 + 2/7 + 2/5 = 107/210

Therefore, the line integral is 107/210.

However, we need to evaluate the line integral from (0,0,0) to (1,1,1), not just from t=0 to t=1.

To do this, we can substitute r(t) into F=xyi+yzj+xzk, giving us F(r(t)) = t^3 i + t^3 j + t^5 k.

Then, we can substitute t=0 and t=1 into the integral expression we just found, and subtract the results to get the line integral over the given path:

∫[0,1] F(r(t)) · (dr/dt) dt = (107/210)t |_0^1 = 107/210

Therefore, the line integral of F over the path C is 1/5.

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The linear and nonlinear Green-Lagrange strains can be determined by calculating the derivatives of the displacement field.

To determine the linear and nonlinear Green-Lagrange strains, we need to calculate the derivatives of the displacement field with respect to the spatial coordinates. The Green-Lagrange strain tensor represents the infinitesimal deformation experienced by a material point in a body.

The linear Green-Lagrange strain tensor is obtained by taking the symmetric part of the displacement gradient tensor, while the nonlinear Green-Lagrange strain tensor involves additional terms resulting from the nonlinearity of the displacement field.

By differentiating the given displacement field expression with respect to the spatial coordinates, we can obtain the necessary derivatives and calculate both the linear and nonlinear Green-Lagrange strains. The linear and nonlinear Green-Lagrange strains can be found by calculating the derivatives of the displacement field with respect to the spatial coordinates.

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The series converges for n = 1 (−1)n − 1 n4 7n

To test the series for convergence or divergence, we can use the alternating series test.

First, we need to check that the terms of the series are decreasing in absolute value. Taking the absolute value of the general term, we get:

|(-1)ⁿ-1/n4⁴ * 7n| = 7/n³

Since 7/n³ is a decreasing function for n >= 1, the terms of the series are decreasing in absolute value.

Next, we need to check that the limit of the absolute value of the general term as n approaches infinity is zero:

lim(n->∞) |(-1)ⁿ-1/n⁴ * 7n| = lim(n->∞) 7/n³ = 0

Since the limit is zero, the alternating series test tells us that the series converges.

Therefore, the series converges.

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To find the area between two z-scores on a calculator, we use the command "normalcdf" on most scientific calculators.

This command calculates the area under the normal distribution curve between two specified z-scores. We need to input the two z-scores and the mean and standard deviation of the normal distribution, which can be obtained from the problem statement or by calculating them from the given data.

Another command that is used in conjunction with "normalcdf" is "invNorm". This command can be used to find the z-score corresponding to a given area under the normal distribution curve. It is used when we are given the area and we need to find the corresponding z-score.

Together, these two commands are useful for solving problems that involve normal distributions, such as finding probabilities, finding critical values, or constructing confidence intervals. It is important to understand how to use these commands properly in order to perform accurate and efficient calculations.

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Okay, let's break this down step-by-step:

* The curve is y = sqrt(x) (1)

* The limits of integration are: x = 1 to x = 4 (2)

* We need to integrate y with respect to x over these limits (3)

* Substitute the curve equation (1) into the integral:

∫4 sqrt(x) dx (4)

* Integrate: (√4)3/2 - (√1)3/2 (5) = 43/2 - 13/2 (6) = 15 (7)

* The volume of a solid generated by revolving a region about an axis is:

Volume = 2*π*15 (8) = 30*π (9)

Therefore, the volume of the solid generated when the region is revolved about the y-axis is 30*π.

Let me know if you have any other questions!

The volume of the solid generated is approximately 77.74 cubic units.

To find the volume of the solid generated when the region enclosed by y=sqrt(x), x=1, x=4, and the x-axis is revolved about the y-axis, follow these steps:

Step 1: Identify the given functions and limits.

y = sqrt(x) is the function we will use, with limits x=1 and x=4.

Step 2: Set up the integral using the shell method.
Since we are revolving around the y-axis, we will use the shell method formula for volume:
V = 2 * pi * ∫[x * f(x)]dx from a to b, where f(x) is the function and [a, b] are the limits.

Step 3: Plug the function and limits into the integral.
V = 2 * pi * ∫[x * sqrt(x)]dx from 1 to 4

Step 4: Evaluate the integral.
First, rewrite the integral as:
V = 2 * pi * ∫[x^(3/2)]dx from 1 to 4

Now, find the antiderivative of x^(3/2):
Antiderivative = (2/5)x^(5/2)

Step 5: Apply the Fundamental Theorem of Calculus.
Evaluate the antiderivative at the limits 4 and 1:
(2/5)(4^(5/2)) - (2/5)(1^(5/2))

Step 6: Simplify and calculate the volume.
V = 2 * pi * [(2/5)(32 - 1)]
V = (4 * pi * 31) / 5
V ≈ 77.74 cubic units

So, The volume of the solid generated is approximately 77.74 cubic units.

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The lengths of the sides of triangle PQR are as follows:

Side PQ: 3 units

Side QR: approximately 6.71 units

Side RP: 6 units

To find the lengths of the sides of triangle PQR, we can utilize the distance formula, which states that the distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space is given by:

d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Now, let's proceed to find the lengths of the sides of triangle PQR.

Side PQ:

The coordinates of points P and Q are P(3, 6, 5) and Q(5, 4, 4) respectively. Applying the distance formula, we have:

PQ = √((5 - 3)² + (4 - 6)² + (4 - 5)²)

= √(2² + (-2)² + (-1)²)

= √(4 + 4 + 1)

= √9

= 3

Therefore, the length of side PQ is 3 units.

Side QR:

The coordinates of points Q and R are Q(5, 4, 4) and R(5, 10, 1) respectively. Using the distance formula, we can calculate the length of side QR:

QR = √((5 - 5)² + (10 - 4)² + (1 - 4)²)

= √(0² + 6² + (-3)²)

= √(0 + 36 + 9)

= √45

≈ 6.71

Hence, the length of side QR is approximately 6.71 units.

Side RP:

To find the length of side RP, we need to calculate the distance between points R(5, 10, 1) and P(3, 6, 5). By applying the distance formula, we get:

RP = √((3 - 5)² + (6 - 10)² + (5 - 1)²)

= √((-2)² + (-4)² + 4²)

= √(4 + 16 + 16)

= √36

= 6

Therefore, the length of side RP is 6 units.

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The numbers in least to greatest order are: 0.10, 0.111, 0.125, 0.13.

To solve 1/8, 13%, 0.10 and 1/9 in least to greatest step-by-step, we first need to convert them into the same form of numbers. Here's how:1. Convert 1/8 into a decimal number:1/8 = 0.1252. Convert 13% into a decimal number:13% = 0.13 (by dividing 13 by 100)3. Convert 1/9 into a decimal number:1/9 ≈ 0.111 (rounded to the nearest thousandth)So, the given numbers in decimal form are:0.125, 0.13, 0.10, 0.111Now, we can put them in order from least to greatest:0.10, 0.111, 0.125, 0.13Therefore, the numbers in least to greatest order are: 0.10, 0.111, 0.125, 0.13.

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There are 1,680 different ways to select the officers for your club.

To determine the number of ways you can select officers for your club, you'll need to use the concept of permutations.

In this case, there are 8 members and you need to choose 4 positions (president, vice president, secretary, and treasurer).

The number of ways to arrange 8 items into 4 positions is given by the formula:

P(n, r) = n! / (n-r)!

where P(n, r) represents the number of permutations, n is the total number of items, r is the number of positions, and ! denotes a factorial.

For your situation:

P(8, 4) = 8! / (8-4)! = 8! / 4! = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (4 × 3 × 2 × 1) = (8 × 7 × 6 × 5) = 1,680

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Answer:

a

Step-by-step explanation:

Answer:

the answer would be the first one

Step-by-step explanation:

The total number of different types of jeans available is 30. The correct answer is e. 30.

Since each design can be made with either short or long length, and there are 3 designs in total, there are 2 options for length for each design.

Additionally, there are 5 color patterns available for each design and length combination.

Therefore, the total number of different types of jeans available can be calculated as follows:

2 (options for length) x 3 (designs) x 5 (color patterns) = 30.

Therefore, there are 30 different types of jeans offered in all.

Hence, the correct answer is an option (e).

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(a) Let X be the length of an eel in the lake. Then X ~ N(84, 18^2). The probability that an eel is a giant (i.e., X > 120) is:

P(X > 120) = P(Z > (120-84)/18) = P(Z > 2) = 0.0228 (using standard normal distribution table)

Therefore, the probability that an eel is a giant is 0.0228 or about 2.28%.

(b) Let Y be the number of giants in a sample of 100 eels. Then Y follows a binomial distribution with parameters n = 100 and p = P(X > 120) = 0.0228. The expected number of giants in a sample of 100 eels is:

E(Y) = np = 100(0.0228) = 2.28

Therefore, we expect about 2.28 giants in a sample of 100 eels.

(c) To find the proportion of eels between 75 cm and 90 cm, we need to standardize these values using the mean and standard deviation of the population:

P(75 < X < 90) = P[(75-84)/18 < (X-84)/18 < (90-84)/18]

= P(-0.5 < Z < 0.33)

= 0.3736 - 0.3085

= 0.0651

Therefore, about 6.51% of eels are between 75 cm and 90 cm.

(d) The distribution of sample means follows a normal distribution with mean μ = 84 and standard error σ/sqrt(n) = 18/sqrt(100) = 1.8 (by Central Limit Theorem). Therefore, the distribution of sample means is N(84, 1.8^2).

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The Jensen's Alpha for your portfolio is -1.88%.

To calculate Jensen's Alpha, follow these steps:

1. Determine the actual return of your portfolio, which is 4.39%.
2. Determine the expected return based on the CAPM formula, which is 6.27%.
3. Subtract the expected return from the actual return: 4.39% - 6.27% = -1.88%.

Jensen's Alpha measures the portfolio's excess return compared to the expected return based on its risk level (beta) and the market return.

In this case, your portfolio underperformed by 1.88% compared to the expected return. It is important to note that the portfolio's standard deviation and beta do not affect the calculation of Jensen's Alpha directly, but they do play a role in the CAPM formula for determining the expected return.

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We can use the binomial distribution to solve this problem.

Let X be the number of people out of 15 who used an online travel website to book their hotel. Then, X follows a binomial distribution with n = 15 and p = 0.7.

The expected value of X is given by:

E(X) = n × p

Substituting the values given in the problem, we get:

E(X) = 15 × 0.7 = 10.5

Therefore, we would expect 10 people (rounding down 10.5 to the nearest whole person) out of 15 to use an online travel website to book their hotel.

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The general solution of the system of differential equations is given by the two equations:

y = ±e^(4x+C1)

x = ±e^(-y/2+C2)

where the ± signs indicate the two possible solutions depending on the initial conditions.

To solve the system of differential equation, we first use the given equations to find the general solution for each variable separately.

This is done by isolating the variables on one side of the equation and integrating both sides with respect to the other variable.

Once we have the general solutions for each variable, we can combine them to form the general solution for the system of differential equations.

This is done by substituting the general solution for one variable into the other equation and solving for the other variable.

The resulting general solution contains two possible solutions, each with its own constant of integration. The choice of which solution to use depends on the initial conditions of the problem.

To solve the system of differential equations:

dy/dx = 4y

dx/dy = -x/2

The first equation can be written as:

dy/y = 4dx

Integrating both sides:

ln|y| = 4x + C1

where C1 is the constant of integration.

Taking the exponential of both sides:

|y| = e^(4x+C1)

Simplifying by removing the absolute value:

y = ±e^(4x+C1)

where ± represents the two possible solutions depending on the initial conditions.

The second equation can be written as:

dx/x = -dy/2

Integrating both sides:

ln|x| = -y/2 + C2

where C2 is the constant of integration.

Taking the exponential of both sides:

|x| = e^(-y/2+C2)

Simplifying by removing the absolute value:

x = ±e^(-y/2+C2)

where ± represents the two possible solutions depending on the initial conditions.

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The complete equation of (5x + ....)² = ....*x² + 70xy +  ....  is 25² + 70xy + 49y²

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

(5x + ....)² = ....*x² + 70xy +  ....

Rewrite the expression as

(5x + ay)² = ....*x² + 70xy +  ....

When expanded, we have

(5x + ay)² = 25x² + 2 * 5x * ay + (ay)²

Evaluate the products

So, we have

(5x + ay)² = 25x² + 10axy + (ay)²

This means that

10axy = 70xy

So, we have

a = 7

The equation becomes

(5x + ay)² = 25x² + 10 * 7xy + (7y)²

Evaluate

(5x + ay)² = 25x² + 70xy + 49y²

Hence, the complete equation is 25² + 70xy + 49y²

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Answer: 485 dolls approximately,

Tom should be able to assemble 485 dolls after 90 days if he continues to work at the same rate as before, according to the given information.  This means that y = 5(90) + 35, and solving it gives y = 485.The scatter plot showed that as the days went by, Tom assembled more dolls. He collected data on how many dolls he assembled per day and placed the data on a scatter plot. He labeled the r-axis "days" and the y-axis "dolls assembled." He found a line of best fit for the data, which has the equation y = 5x +35. This equation allows us to estimate the number of dolls that Tom could assemble after any number of days. We were asked to find the number of dolls that Tom should be able to assemble after 90 days, and the answer is 485 dolls.

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The answer is D. 49,062.50 cubic millimeters vanilla ice cream in one chocolate dip cone holds when filled to be level with the top of the cone.

To calculate the amount of vanilla ice cream that one chocolate dip cone can hold when filled to the top, we need to find the volume of the cone-shaped space inside the cone. The formula for the volume of a cone is V = (1/3)πr^2h, where V is the volume, π is approximately 3.14, r is the radius of the cone's top, and h is the height of the cone.

Given that the radius of the inside of the cone top is 25 millimeters and the height of the inside of the cone is 102 millimeters, we can substitute these values into the volume formula.

V = (1/3) × 3.14 × 25^2 × 102

 = (1/3) × 3.14 × 625 × 102

 = 0.3333 × 3.14 × 625 × 102

 ≈ 49,062.50 cubic millimeters

Therefore, one chocolate dip cone will hold approximately 49,062.50 cubic millimeters of vanilla ice cream when filled to be level with the top of the cone.

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