determine the stability of the system, whose characteristic equation is D(s) = s^6+ 3s^5 + 2s^4 + 9s^3 + 5s^2 + 12s+ 20.

the perimeter of the rectangle is 56 feet.

To find the perimeter of a rectangle, we add up the lengths of all four sides.

In this case, the rectangle measures 10 feet by 18 feet. Therefore, the length of the top and bottom sides is 10 feet, and the length of the left and right sides is 18 feet.

So, the perimeter of the rectangle is:

2(10 feet) + 2(18 feet) = 20 feet + 36 feet = 56 feet

Therefore, the perimeter of the rectangle is 56 feet.
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According to the given function, the value of ∂Q / ∂x is 1, and the value of ∂P / ∂y is -1

In the given equation, Q = x + y and P = x − y, we can think of Q and P as functions of x and y. That is, for every combination of x and y, we get a corresponding value of Q and P.

Now, the partial derivative of Q with respect to x (denoted as ∂Q/∂x) tells us how Q changes when we vary x while keeping y constant. Similarly, the partial derivative of P with respect to y (denoted as ∂P/∂y) tells us how P changes when we vary y while keeping x constant.

In this case, ∂Q/∂x = 1, which means that if we increase x by a small amount, Q will also increase by the same amount. The value of y does not affect this relationship. Similarly, ∂P/∂y = -1, which means that if we increase y by a small amount, P will decrease by the same amount. The value of x does not affect this relationship.

In summary, functions are rules that assign outputs to inputs, and partial derivatives can help us understand how these outputs change as we vary the inputs.

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The boat traveled approximately 59.4 kilometers from its original position in a straight line through calm seas.

To determine the distance the boat traveled from its original position, we need to consider both the westward and southward distances and use the Pythagorus theorem. The question states: A boat traveled in a straight line through calm seas until it was 43 kilometers west and 41 kilometers south of its original position.
1: Identify the two legs of the right triangle. In this case, the westward distance is 43 km (one leg), and the southward distance is 41 km (the other leg).
2: Use the Pythagorean theorem to find the distance traveled (hypotenuse). The formula is a² + b² = c², where a and b are the legs, and c is the hypotenuse.
3: Substitute the given values into the formula:
(43 km)² + (41 km)² = c²
4: Calculate the squares of the values:
1849 km² + 1681 km² = c²
5: Add the squared values together:
3530 km² = c²
6: Take the square root of both sides of the equation to find the value of c:
√3530 km² = c
c ≈ 59.4 km

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When we talk about  hermite interpolation the points under its creation justify its cause, therefore the points are

When we talk about ordinary interpolation the points under its creation justify its cause, therefore the points are

When we talk about cubic spline inter- polant the points under its creation justify its cause, therefore the points are

When we talk about hermite cubic interpolant the points under its creation justify its cause, therefore the points are

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If the distribution of the scores of one variable changes across the categories of another variable then the variables are associated to some extent. The correct answer is 1.

When the distribution of scores for one variable changes across the categories of another variable, it implies that there is some relationship between the two variables.

This relationship doesn't necessarily mean that they are perfectly associated, have a cause-and-effect relationship, or are indicators of the same concept.

However, it does show that the variables are related in some way, hence they are associated to some extent. Option 1 is correct.

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So the area of the triangle is 3 square units.

To solve this problem, we first need to identify the base and height of the triangle.

The base of the triangle is the line segment connecting the points (5,4) and (8,-1). We can find the length of this base by using the distance formula:

Distance between two points (x1, y1) and (x2, y2) is given by:

[tex]d = \sqrt((x2 - x1)^2 + (y2 - y1)^2)[/tex]

Base length:

[tex]d = \sqrt((8 - 5)^2 + (-1 - 4)^2)[/tex]

[tex]= \sqrt(3^2 + (-5)^2)[/tex]

[tex]= \sqrt(9 + 25)[/tex]

[tex]= \sqrt(34)[/tex]

So the base length of the triangle is sqrt(34).

To find the height of the triangle, we need to find the perpendicular distance from the vertex (2,-1) to the base. We can use the formula for the distance from a point to a line to find the height:

Distance between a point (x1, y1) and a line [tex]Ax + By + C = 0[/tex] is given by:

[tex]h = |Ax1 + By1 + C| / \sqrt(A^2 + B^2)[/tex]

We can find the equation of the line containing the base by using the slope-intercept form:

Slope of the line containing the base:

[tex]m = (y2 - y1) / (x2 - x1)[/tex]

[tex]= (-1 - 4) / (8 - 5)[/tex]

[tex]= -5/3[/tex]

Intercept of the line containing the base:

[tex]y - y1 = m(x - x1)\\\\= y - 4 = (-5/3)(x - 5)\\\\= y = (-5/3)x + (35/3)[/tex]

So the equation of the line containing the base is:

5x + 3y - 35 = 0

Now we can substitute the coordinates of the vertex (2,-1) into the formula for the distance from a point to a line to find the height:

Height:

[tex]h = |5(2) + 3(-1) - 35| / \sqrt(5^2 + 3^2)[/tex]

[tex]= 6 / \sqrt(34)[/tex]

So the height of the triangle is 6/sqrt(34).

Finally, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Area:

Area = (1/2) * sqrt(34) * (6/sqrt(34))

= 3

So the area of the triangle is 3 square units.

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The directional derivative of f at point (2,1,1) in the direction of vector [tex]\hat{i} -2 \hat{j}+3\hat{k}[/tex] is [tex]-4/\sqrt{14}[/tex].

To find the directional derivative of f(x,y,z)=xy³+yz³ at point (2,1,1) in the direction of the vector [tex]\hat{i} -2 \hat{j}+3\hat{k}[/tex], we first need to find the unit vector in the direction of this vector.

The magnitude of the vector is [tex]\sqrt{(1^2+(-2)^2+3^2)} = \sqrt{14}[/tex], so the unit vector in the direction of [tex]\hat{i} -2 \hat{j}+3\hat{k}[/tex] is:
[tex](1/\sqrt{14})\hat{i} - (2/\sqrt{14})\hat{j} + (3/\sqrt{14})\hat{k}[/tex]

Next, we need to find the gradient of f at point (2,1,1):
grad(f) = (∂f/∂x, ∂f/∂y, ∂f/∂z) = (y³, 3xy² + z³, 3yz²)

Evaluated at (2,1,1), this becomes:

grad(f)(2,1,1) = (1, 7, 3)

Finally, we can find the directional derivative by taking the dot product of the unit vector in the direction of [tex]\hat{i} -2 \hat{j}+3\hat{k}[/tex] with the gradient of f:

D_∆u f(2,1,1) = grad(f)(2,1,1) · u = (1, 7, 3) · [tex](1/\sqrt{14})\hat{i} - (2/\sqrt{14})\hat{j} + (3/\sqrt{14})\hat{k}[/tex]

Simplifying, we get:

D_∆u f(2,1,1) = [tex](1/\sqrt{14}) - (14/\sqrt{14}) + (9/\sqrt{14}) = -4/\sqrt{14}[/tex]

Therefore, the directional derivative of f at point (2,1,1) in the direction of vector [tex]\hat{i} -2 \hat{j}+3\hat{k}[/tex] is [tex]-4/\sqrt{14}[/tex].

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If X has a mean of 6 and a Poisson distribution. We may determine that k = 4 results in a probability of 0.4098, a value less than 0.45, using a Poisson probability calculator.To guarantee that students have plenty to read during the flight, the professor must bring at least 5 papers.

The professor reads papers for an average of thirty minutes per paper, according to a Poisson process. This indicates that there is an exponential distribution with a mean reading interval of 30 minutes.

The professor will have a period of three hours (180 minutes) to study the papers throughout the journey. The number of papers the professor can read in this amount of time can be predicted using a distribution based on with a mean of 180/30 = 6.

In order to keep the probability that we'll run out of papers to read below 0.45, we need to determine how few pages the professor should bring. Accordingly, we must choose the smallest number k such that: P(X <= k) < 0.45

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The final answer based on the given information, the correct answer is D.

Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate when the response variable has a normal distribution.

The relationship between the data points on the normal probability plot is approximately linear, which suggests that the variable under consideration is not normally distributed. A normal probability plot of a normally distributed variable would show a straight line, but the plot shown in this question appears to have a slight curve. Therefore, the variable is probably not normally distributed.

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To find the average value of a function f(x) over an interval [a, b], we use the formula:

avg(f) = (1 / (b - a)) * ∫[a, b] f(x) dx

where ∫[a, b] f(x) dx represents the definite integral of f(x) over the interval [a, b].

a) For the function f(t) = 5 sin(3t), the interval is [0, 2π/3] because one period of sin(3t) is 2π/3.

avg(f) = (1 / (2π/3 - 0)) * ∫[0, 2π/3] 5 sin(3t) dt

Using integration by substitution, we get:

avg(f) = (1 / (2π/3)) * [-5/3 cos(3t)] |[0, 2π/3]

avg(f) = (1 / (2π/3)) * [-5/3 cos(2π) + 5/3 cos(0)]

avg(f) = (1 / (2π/3)) * (5/3 - (-5/3))

avg(f) = 5/2π

Therefore, the average value of f(t) = 5 sin(3t) over the interval [0, 2π/3] is 5/2π.

b) For the function g(t) = 4 cos(8t), the interval is [0, π/4] because one period of cos(8t) is π/4.

avg(g) = (1 / (π/4 - 0)) * ∫[0, π/4] 4 cos(8t) dt

Using integration by substitution, we get:

avg(g) = (1 / (π/4)) * [1/2 sin(8t)] |[0, π/4]

avg(g) = (1 / (π/4)) * [1/2 sin(2π) - 1/2 sin(0)]

avg(g) = (1 / (π/4)) * (0 - 0)

avg(g) = 0

Therefore, the average value of g(t) = 4 cos(8t) over the interval [0, π/4] is 0.

c) For the function h(t) = cos^2(2t), the interval is [0, π/4] because one period of cos^2(2t) is π/4.

avg(h) = (1 / (π/4 - 0)) * ∫[0, π/4] cos^2(2t) dt

Using the identity cos^2(x) = (1/2) + (1/2)cos(2x), we can write:

cos^2(2t) = (1/2) + (1/2)cos(4t)

Substituting this into the integral, we get:

avg(h) = (1 / (π/4)) * ∫[0, π/4] [(1/2) + (1/2)cos(4t)] dt

avg(h) = (1 / (π/4)) * [(1/2)t + (1/8) sin(4t)] |[0, π/4]

avg(h) = (1 / (π/4)) * [(1/2)(π/4) + (1/8) sin(π)]

avg(h) = (1 / (π/4)) * [(π/8) + 0]

avg(h) = 2/π

Therefore, the average value of h(t) = cos^2(2t) over the interval [0, π/4] is 2/π.

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a. 18

b. 6

c. -1

d. -3

A number system is described as as a system of writing to express numbers.

A number system is  the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner and also provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures.

a. (-2)(-9) = 18

b. 3(2) = 6

c. = (7 - 8) ÷ (-1) = -1

d. (-1)(3) = -3

In conclusion, the code to enter is: 18, 6, -1, -3

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The function F(s) can be rewritten as F(s) = (2e^(-2s)(s - 1)) / ((s - 1)^2 + 1). Now we can identify it as the Laplace transform of a function multiplied by an exponential, which corresponds to a shift in the time domain.

Let G(s) = 2(s - 1) / ((s - 1)^2 + 1), then F(s) = e^(-2s)G(s). To find the inverse Laplace transform of G(s), we can use the formula for the inverse Laplace transform of functions in the form of K(s - a) / ((s - a)^2 + b^2), which corresponds to K * e^(at) * sin(bt).

The inverse Laplace transform of G(s) = 2 * e^t * sin(t). Therefore, the inverse Laplace transform of F(s) is found by shifting the time domain of G(s) by 2 units, due to the e^(-2s) term.

To find the inverse Laplace transform of F(s) = [2(s − 1)e^−2s] / ( s^2 − 2s + 2 ), we first need to factor the denominator using the quadratic formula:

s = [2 ± sqrt(2^2 - 4(1)(2))] / 2
s = 1 ± j

So the denominator can be written as:

s^2 - 2s + 2 = (s - (1 + j))(s - (1 - j))

Now we can use partial fractions to simplify the expression:

[2(s − 1)e^−2s] / ( s^2 − 2s + 2 ) = A / (s - (1 + j)) + B / (s - (1 - j))

Multiplying both sides by (s - (1 + j))(s - (1 - j)), we get:

2(s - 1)e^(-2s) = A(s - (1 - j)) + B(s - (1 + j))

Setting s = 1 + j, we get:

2(1 + j - 1)e^(-2(1 + j)) = A(1 + j - (1 - j))

Simplifying and solving for A, we get:

A = -j e^(2 - 2j)

Setting s = 1 - j, we get:

2(1 - j - 1)e^(-2(1 - j)) = B(1 - j - (1 + j))

Simplifying and solving for B, we get:

B = j e^(2 + 2j)

Now we can rewrite F(s) as:

F(s) = -j e^(2 - 2j) / (s - (1 + j)) + j e^(2 + 2j) / (s - (1 - j))

Taking the inverse Laplace transform of each term, we get:

f(t) = -j e^(2 - 2j) e^(t - (1 + j)t) + j e^(2 + 2j) e^(t - (1 - j)t)

Simplifying, we get:

f(t) = e^(t - j*t) - e^(t + j*t)

Therefore, the inverse Laplace transform of F(s) is f(t) = e^(t - j*t) - e^(t + j*t).

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The number of students that are in each of the 6 bues are 46

To solve the problem, we need to divide the total number of students by the number of buses.

First, we need to subtract the 5 students who traveled by car from the total number of students:

281 - 5 = 276

Next, we divide 276 by the number of buses (6):

276 ÷ 6 = 46

Therefore, there were 46 students in each bus.

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To find h'(2), we need to use the chain rule. Let's consider each option:

for option (a), h(x) = 5f(x) − 2g(x)
Using the product rule, we have:
                  h'(x) = 5f'(x) - 2g'(x)

Therefore, at x=2:
h'(2) = 5f'(2) - 2g'(2) = 5(-1) - 2(5) = -15

for option (b), h(x) = f(x)g(x)
Using the product rule again, we have:
                h'(x) = f'(x)g(x) + f(x)g'(x)

Therefore, at x=2:
h'(2) = f'(2)g(2) + f(2)g'(2) = (-1)(3) + (-4)(5) = -23

for option (c), h(x) = f(x)/g(x)
Using the quotient rule, we have:
h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2

Therefore, at x=2:
h'(2) = [f'(2)g(2) - f(2)g'(2)] / [g(2)]^2 = [(-1)(3) - (-4)(5)] / 3^2 = 23/9

for option (d), h(x) = g(x)/ (1+f(x))
Using the quotient rule again, we have:
h'(x) = [(1+f(x))g'(x) - g(x)f'(x)] / [(1+f(x))^2]

Therefore, at x=2:
h'(2) = [(1+f(2))g'(2) - g(2)f'(2)] / [(1+f(2))^2] = [(1-4)(5) - 3(-1)] / (1-4)^2 = -8/9

Therefore, the answer is (d) h(x) = g(x)/ (1+f(x)) and h'(2) = -8/9.

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The answer is True.

Explanation: If all the diagonal entries of a square matrix are zero, then we can use the Laplace expansion along the first column to calculate the determinant. The Laplace expansion states that the determinant of a matrix A can be calculated by multiplying the entries of any row or column of A by their corresponding cofactors and then adding up these products.

Since all the diagonal entries of our matrix are zero, we can focus on the first column. The first entry of the first column is zero, so we don't need to consider it. For the second entry, we need to multiply it by its corresponding cofactor, which is the determinant of the (n-1)x(n-1) matrix obtained by deleting the second row and second column of the original matrix. But since all the diagonal entries of this smaller matrix are also zero, we can repeat the process and use the Laplace expansion along the second column.

Continuing in this way, we eventually reduce the problem to calculating the determinant of a 1x1 matrix, which is just its single entry. But since all the entries of our original matrix are zero, the determinant is also zero. Therefore, if all the diagonal entries of a square matrix are zero, then the determinant of that matrix must also be zero.

The answer is True

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The height of the flagpole when marci who is 5 feet tall is standing in lines of sight to the top and bottom of the pole is 20 feet.

To estimate the height of the flagpole, we can use trigonometry.

Let's call the height of the flagpole "h" and the distance from Marci to the base of the pole "d".

We can set up a right triangle with the flagpole as the hypotenuse, Marci's height as one leg, and the distance from Marci to the base of the pole as the other leg.

Using the tangent function, we can say:

tan(θ) = perpendicular/base

In this case, the opposite is Marci's height (5 feet) and the adjacent is the distance from Marci to the base of the pole (d).

We don't know the angle yet, but we can use the fact that the angle formed by Marci's lines of sight to the top and bottom of the pole is the same as the angle formed by the top of the pole, Marci's eye, and the base of the pole (since these are alternate interior angles).

Let's call this angle "x".
So, we have:
tan(x) = 5/d
To find the height of the pole, we need to use another trig function. Since we know the adjacent and the hypotenuse, we can use the cosine function:
cos(x) = base/hypotenuse

In this case, the adjacent is still d and the hypotenuse is h + 5 (since Marci's height is included in the overall height of the flagpole).

So we have:

cos(x) = d/(h + 5)

We can rearrange this equation to solve for h:

h = (d/cos(x)) - 5

Now we just need to find the value of x.

We know that the tangent of x is 5/d, so we can use the inverse tangent function (tan^-1) to find x:

x = tan^-1(5/d)

Plugging this into the equation for h, we get:

h = (d/cos(tan^-1(5/d))) - 5

We can simplify this a bit by using the identity:

cos(tan^-1(x)) = 1/√(1 + x^2)

So we have:

h = (d * √(1 + (5/d)²)) - 5

Now we just need to plug in the values given in the problem. Let's say that Marci stands 20 feet away from the base of the pole. Then we have:

h = (20 * √(1 + (5/20)²)) - 5

h = 19 feet (rounded to the nearest foot)

So the answer is (b) 20 feet.

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The general solution to the differential equation xy' + 3y = 0 is given by:

y = c/x^3

where c is a constant.

To verify that this is a solution, we can substitute it into the differential equation:

xy' + 3y = 0

[tex]x(d/dx)(c/x^3) + 3(c/x^3)[/tex]= 0

-c/x^3 + 3(c/x^3) = 0

The last step follows from the fact that the derivative of [tex]x^(-n)[/tex]is [tex]-nx^(-n-[/tex]1).

This simplifies to:

[tex]2c/x^3[/tex] = 0

which is true if and only if c = 0 or x = infinity. Since c can be any constant, this means that y = [tex]c/x^3[/tex] is a family of solutions to the differential equation.

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Diego and Darnell are roommates. They spend most of their time studying (of course), but they leave some time for their favorite activities: making pizza and brewing root beer. I recommend plotting the opportunity costs of both roommates on a graph to visualize the price range for trading pizza.

a. The opportunity cost of making a pizza for Diego is 2 hours of brewing root beer, while for Darnell it is 4 hours of brewing root beer. Therefore, Diego has the absolute advantage in making pizza as he takes less time to make it. However, Darnell has the comparative advantage in making pizza as his opportunity cost of making pizza is higher, meaning he gives up less root beer for every pizza made.
b. Diego will trade away pizza in exchange for root beer because his opportunity cost of making root beer is lower than Darnell's.
c. The highest price at which pizza can be traded that would make both roommates better off is when the price of pizza is equal to the opportunity cost of making it for Darnell, which is 6/4 = 1.5 gallons of root beer per pizza. At this price, Darnell would be willing to trade away a pizza for 1.5 gallons of root beer, while Diego would be willing to trade away 2 gallons of root beer for a pizza. The lowest price at which pizza can be traded that would make both roommates better off is when the price of pizza is equal to the opportunity cost of making it for Diego, which is 4/2 = 2 gallons of root beer per pizza. At this price, Diego would be willing to trade away a pizza for 2 gallons of root beer, while Darnell would be willing to trade away 4/1.5 = 2.67 gallons of root beer for a pizza.

Here is a graph of the trade-off between making pizza and brewing root beer for Diego and Darnell:
a) To calculate each roommate's opportunity cost of making a pizza, we need to find the ratio of the time it takes to make a pizza to the time it takes to brew a gallon of root beer for each roommate.
For Diego:
Opportunity cost of making a pizza = Time to make a pizza / Time to brew a gallon of root beer = 2 hours / 4 hours = 0.5 gallons of root beer
For Darnell:
Opportunity cost of making a pizza = Time to make a pizza / Time to brew a gallon of root beer = 4 hours / 6 hours = 0.67 gallons of root beer
Diego has the absolute advantage in making pizza, as he can make a pizza in less time than Darnell. Diego also has the comparative advantage in making pizza, as his opportunity cost (0.5 gallons of root beer) is lower than Darnell's (0.67 gallons of root beer).
b) Since Diego has the comparative advantage in making pizza, he should trade away pizza in exchange for root beer.
c) To find the highest and lowest prices at which pizza can be traded to make both roommates better off, we need to look at their opportunity costs. The price of pizza must fall between Diego's and Darnell's opportunity costs.

The highest price at which pizza can be traded is just below Darnell's opportunity cost, which is 0.67 gallons of root beer. The lowest price at which pizza can be traded is just above Diego's opportunity cost, which is 0.5 gallons of root beer. This price range ensures that both roommates benefit from the trade.

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If the two numbers are negative it gives only negative number.

It is always not true.

If we subtract a smaller negative integer from the larger one, the answer will be a positive integer.

But when we subtract larger negative integer with smaller one, the answer will be a negative integer.

Let us take two examples:

Let -1 and -2 be two integers, then the difference between them will be

-1 - (-2)

= -1 + 2

= 1[Positive integer]

But when we subtract -2 from -1, then

-2 - (-1)

= -2 + 1

= -1 [negative integer]

Thus, the difference of two negative numbers is not always negative.

false, (-1/4) + (-1/4) > -1 is a counterexample.

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Check the picture below.

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{10}\\ a=\stackrel{adjacent}{3x}\\ o=\stackrel{opposite}{4x} \end{cases} \\\\\\ (10)^2= (3x)^2 + (4x)^2\implies 100=9x^2+16x^2\implies 100=25x^2 \\\\\\ \cfrac{100}{25}=x^2\implies 4=x^2\implies \sqrt{4}=x\implies 2=x \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \stackrel{ 3(2) }{\text{\LARGE 6}}\hspace{5em}\stackrel{ 4(2) }{\text{\LARGE 8}}~\hfill[/tex]

The z-score for x = 90 is z = 1.50, which means that a value of 90 is 1.50 standard deviations above the mean.

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The sample standard deviation describes the (a) variability of the (b) scores. In this case, the standard distance between (c) scores and (d) the sample mean is 15 points.

To compute the estimated standard error for the sample mean, we can use the formula:
Standard Error = s / sqrt(n)
where s is the sample standard deviation and n is the sample size. Plugging in the given values, we get:
Standard Error = 15 / sqrt(25) = 3
The standard error provides a measure of (e) the standard distance between a (f) sample mean and the (g) population mean. It indicates how much the sample mean is likely to differ from the true population mean due to chance variation in the sample. A smaller standard error suggests that the sample mean is a more precise estimate of the population mean.

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The measure of the central angle indicated is 270 degrees

The term "diameter" refers to a straight line segment that passes through the center of a circle or a sphere, connecting two points on the circumference or surface of the circle or sphere. In other words, it is the longest distance that can be measured between two points on the edge of the circle or sphere, passing through its center.

To find the measure of the central angle indicated, we need to first identify the endpoints of the diameter that contains points W, T, and X. Let's assume that this diameter is WX. Then, we can find the measure of the central angle WTX by finding the measure of the arc WT and dividing it by 2.

We know that the diameter WX passes through the midpoint of segment VT, which we can find by averaging the coordinates of V and T. Using the coordinates given in the diagram, we have:

V: (9x-2, 15x+10)

T: (15x+10, 9x-2)

Midpoint of VT: ((9x-2 + 15x+10)/2, (15x+10 + 9x-2)/2)

= (12x + 4, 12x + 4)

Since this midpoint lies on the diameter WX, we can find the coordinates of point X by reflecting the midpoint across the y-axis:

X: (-12x - 4, 12x + 4)

Now we can find the measure of the arc WT by finding the difference between the angles formed by radii WT and WX. Let's call the center of the circle O:

m∠WOT = 90 degrees (since WT is a diameter)

m∠WOX = 180 degrees (since WX is a diameter)

m∠TOX = m∠WOT - m∠WOX = -90 degrees

To convert this angle to a positive measure, we can add 360 degrees:

m(arc WT) = m∠WOT - m∠TOX + 360 degrees = 90 degrees - (-90 degrees) + 360 degrees = 540 degrees

Finally, we can find the measure of the central angle WTX by dividing the measure of arc WT by 2:

m∠WTX = m(arc WT)/2 = 540 degrees/2 = 270 degrees

Therefore, the measure of the central angle indicated is 270 degrees

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To estimate the flux of the vector field F out of the small sphere of radius 0.01 centered at (4, 5, 6), we can use the divergence theorem:

Flux = ∫∫S F · dS = ∫∫∫V div F dV

where S is the surface of the sphere and V is the volume inside the sphere.

Since div F = 9, we have:

Flux = ∫∫∫V div F dV = 9 ∫∫∫V dV = 9 (4/3 π (0.01)^3) ≈ 0.000038

So the estimated flux of F out of the small sphere is 0.000038 (rounded to six decimal places).

To estimate div F at the point (4, 9, 11), we can use the same idea in reverse. We know that the flux of F out of the small cube of side 0.01 centered at (4, 9, 11) is 0.003, so by the divergence theorem:

Flux = ∫∫S F · dS = ∫∫∫V div F dV

where S is the surface of the cube and V is the volume inside the cube.

Since the flux is given as 0.003, we have:

0.003 = ∫∫∫V div F dV

And since the volume of the cube is (0.01)^3, we have:

div F ≈ 0.003 / (0.01)^3 = 3

So the estimated value of div F at the point (4, 9, 11) is 3.

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Answer:

Step-by-step explanation:

c+5=792

Miguel was asked to find the scale factor for the dilation of triangle ABC to triangle A'B'C'. Miguel's answer was 5/9 which was not accurate.

For illustration, assume we know that AB = 4, BC = 6, AC = 5, A'B' = 10, B'C' = 15, and A'C' = 12. At that point the scale calculate for the expansion of triangle ABC to A'B'C' can be calculated as takes after:

The proportion of the length of AB to A'B' is 4/10 = 2/5.

The proportion of the length of BC to B'C' is 6/15 = 2/5.

The proportion of the length of AC to A'C' is 5/12.

Since the proportions of comparing sides are equal, ready to take the normal of the proportions to induce the scale calculation:

(2/5 + 2/5 + 5/12) / 3 = (24/60 + 24/60 + 25/60) / 3 = 73/180

The scale figure is 73/180, which isn't break even with Miguel's reply of 5/9. Subsequently, Miguel's reply is inaccurate in this case.

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Using Gaussian elimination followed by back substitution, we get the solution of the system of equations x = -3z - 11, y = 2z + 2, z is free variable The row operations used were: R2 = R2 - 2R1, R3 = R3 + 2R1, R3 = R3 + 7R2. The corresponding elementary matrices were M1 = [1 0 0; -2 1 0; 0 0 1], M2 = [1 0 0; 0 1 2; 0 0 1], M3 = [1 0 0; 0 1 0; 0 7 1]. The product M3M2M1 was found to be  [1 -6 -8; 0 1 2; 0 0 1]. The matrix MA appears as the row-reduced echelon form of A. The matrix U has a triangular form and the matrix L has 1's along the diagonal and values below the diagonal that were used to eliminate the entries in U.

We perform the following row operations

R2 → R2 - 2R1

R3 → R3 + 2R1

R3 → R3 + 7R2

This gives us the following augmented matrix

[1  3  4 |  1]

[0 -1 -2 | -4]

[0  0  0 |  0]

Now, using back substitution, we get

z = any real number

-y - 2z = -4

x + 3y + 4z = 1

So, our solution is

x = -3z - 11

y = 2z + 2

z is free variable

The elementary matrices corresponding to the row operations used in part (a) are

M1 = [1 0 0; -2 1 0; 0 0 1]

M2 = [1 0 0; 0 1 0; 2 0 1]

M3 = [1 0 0; 0 1 0; 0 7 1]

We compute the product of the elementary matrices as

Mk...M2M1 = [1 -6 -8; 0 1 2; 0 0 1]

We have MA = [1 -6 -8; 0 1 2; 0 0 0], which appears during the row reduction process in part (a) when we have arrived at the row echelon form of the augmented matrix.

U is the upper triangular matrix obtained after performing Gaussian elimination on A, while L is the lower triangular matrix obtained from the product of the elementary matrices found in part (c). They have the property that LU = A, and the determinant of U is the product of the pivots of A (in this case, 1*(-1)*0 = 0).

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The probability that the proportion of tickets sold in a sample of 865 tickets would differ from the population proportion by greater than 3% would be 0.0456, rounded to four decimal places.

By using the following formula, we can find the standard error in the population:

SE = √(p(1-p)/n),

where p is population proportion and n is sample size.

Since p = 0.2 and n = 865 in this instance, Therefore,

SE = √(0.2 × 0.8 / 865) = 0.015.

The z-score is then determined by dividing (0.03 - 0) by 0.015, which is 2.

We can determine that the likelihood of receiving a z-score larger than 2 or less than -2 is around 0.0456 using a conventional normal distribution table.

So, the likelihood that the proportion of tickets sold in a sample of 865 tickets would deviate from the general proportion by more than 3% is roughly 0.0456, rounded to four decimal places.

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[tex]\longrightarrow\text{5x}^10-80\text{x}^2[/tex]

[tex]\longrightarrow\text{5(x}^10-16\text{x}^2)[/tex]

[tex]\longrightarrow\text{5(x}^2(\text{x}^8-16))[/tex]

[tex]\longrightarrow\text{5x}^2(\text{x}^8-16)[/tex]

Break down

[tex]\longrightarrow\text{5x}^2(\text{x}^4+4)(\text{x}^2+2)(\text{x}^2+2)[/tex]

x + 3, x > -3 is the piece which will be included in the given function          f(x) = [ x + 3 ]. This can be obtained by removing modulus and finding the value for x.

Each element of a non-empty set A has a function connecting it to at least one element of a second non-empty set B. A function f between two sets, A and B, is said to have a "domain" and a "co-domain" in mathematics. For any value of an or b, F = (a,b)| holds true.

Which piece will the function include?

Given that,  

f(x) = [ x + 3] which is an absolute value function.

Therefore,

| x+3 | > 0

By removing the modulus,

x+3 > 0

x > - 3

Hence x + 3, x > -3 is the piece which will be included in the given function:

f(x) = [ x + 3].

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