: 1. A committee consists of seven computer science (CS) and five computer engineering (CE). A subcommittee of three CS and two CE students is to be formed. In how many ways can his be done if (a) any CS and any CE students can be included? (b) one particular CS student must be on the committee? (c) two particular CE students cannot be on the committee? 2. Draw Venn diagrams and shade the areas corresponding to the following sets: (a) (AUBUC) n(AnBnC) (b) (b) (AUB) n(AUC) (c) (c) [(AUB) nC]U (ANC)

answer is option C

AC =~= BD

Answer: AC = BD this is because these two are of the same and exact equal lengths

A model is a representation of reality or a real-life situation. The correct answer is E.

The correct answer is "model."A model is a representation of reality or a real-life situation. It can be used to simulate or describe the behavior of a system, process, or phenomenon.

Models are often used in various fields such as science, engineering, economics, and computer science to understand and analyze complex systems or make predictions about real-world scenarios.

A model is a simplified or abstract representation of a real-life situation or system. It captures the essential features or characteristics of the system while ignoring or simplifying irrelevant details. Models can be physical, conceptual, or mathematical in nature.

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The marginal distribution of students that are in the 10th Grade is 28%

In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset.

The P(10th grade) is determined by

∈P(A)= P( A and B₁) + P(A and B₂) + .....+ P(A and Bₓ) whereas B₁, B₂ and Bₓ are mutually exclusive and collective exhaustive events.

⇒100/870 + 32/870 + 108/870

= 240/870

reducing to lowest terms we have

8/29

≈28%

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a. The values of a and b are;

b.ff(x) = x implies x² - x - 2 = 0.

To find the value of a and b, we use the given information to form two equations in a and b:

f(b) = b gives b/(b-a) = b,

a = b/2

f(2a) = 2a gives (b/(2a-a)) = 2a

b = 4a²

To show that ff(x) = x implies x² - x - 2 = 0, we substitute ff(x) into the equation:

ff(x) = x

f(f(x)) = x

f(b/(x-a)) = x

b/(b/(x-a)-a) = x

b(x-a)/(b-(x-a)a) = x

bx - ba = bx - x² + a²x - a²a

x² - x - 2 = 0

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The value of u* in problem min x2 – (x1 – 2)3 + 3 subject to X2 > 1  is 2.

To find the value of u* in the given problem, we can utilize the Karush-Kuhn-Tucker (KKT) conditions.

First, let's set up the Lagrangian function:

L(x, u) = x2 - (x1 - 2)3 + 3 - u(x2 - 1)

The KKT conditions are as follows:

Stationarity condition: ∇f(x) - u∇g(x) = 0

∂L/∂x1 = -3(x1 - 2)² = -u∂g/∂x1

∂L/∂x2 = 2x2 - u = -u∂g/∂x2

Primal feasibility condition: g(x) ≤ 0

x2 - 1 > 0

Dual feasibility condition: u ≥ 0

Complementary slackness condition: u * (x2 - 1) = 0

From the stationarity condition, we can deduce that 2x2 - u = 0. Combining this with the complementary slackness condition, we have two possible cases:

Case 1: u = 0

From 2x2 - u = 0, we get x2 = 0. However, this contradicts the constraint x2 > 1, so this case is not feasible.

Case 2: x2 - 1 = 0

In this case, we have x2 = 1, and from 2x2 - u = 0, we find u = 2.

Therefore, the value of u* in this problem is 2.

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The linear function in the context of this problem is defined as follows:

y = 18 - 2x.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

The coefficients m and b represent the slope and the intercept, respectively, and are explained as follows:

When the input is of zero, the output is of 18, hence the intercept b is given as follows:

b = 18.

When the input increases by one, the output decreases by two, hence the slope m is given as follows:

m = -2.

Then the function is given as follows:

y = 18 - 2x.

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Answer:

H shot R

explain:

Because the R is next to the hoop

The values of the missing parts are;

WX = 2.2

<X = 24. 6 degrees

<W = 65.4 degrees

Using the Pythagorean theorem, we have that;

WX² = 2² + 1²

Find the value

WX² = 4 + 1

Add the values

WX = √5

WX = 2.2

Using the sine identity, we get;

sin θ = opposite/hypotenuse

substitute the values

sin W = 2/2.2

Divide the values

sin W = 0. 9090

Find the inverse

W = 65. 4 degrees

Then, we get;

X = 180 - 90 - 65.4

Subtract the values

X = 24. 6 degrees

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The local extrema points of the function f(x,y) = 2x³ + xy² + 5x² + y² are (0, 0) and (-5/3, 0). Saddle points are (-1, 2) and (-1, -2).

The given function is :

f(x,y) = 2x³ + xy² + 5x² + y²

[tex]f_x(x,y)[/tex] = 6x² + y² + 10x

[tex]f_y(x,y)[/tex] = 2xy +2y

Let both the partial derivatives equal 0.

6x² + y² + 10x = 0

and

2xy +2y = 0

⇒ y(x + 1) = 0

⇒ y = 0 and x = -1

Substitute x = -1 into the equation of 6x² + y² + 10x = 0.

6(-1)² + y² + 10(-1) = 0

-4 + y² = 0

y = +2 or -2

Substitute y = 0 into the equation of 6x² + y² + 10x = 0.

equation of 6x² + 10x = 0.

6x = -10

x = -5/3 or x = 0

So the critical points are :

(-1, 2), (-1, -2), (0, 0) and (-5/3, 0).

[tex]f_{xx}(x,y)[/tex] = 12x + 10

[tex]f_{yy}(x,y)[/tex] = 2x + 2

[tex]f_{xy}(x,y)[/tex] = 2y

Now,

D = [tex]f_{xx}(x,y)f_{yy}(x,y)-[f_{xy}(x,y)]^2[/tex]

So at (0, 0) :

D > 0 and [tex]f_{xx}[/tex] > 0, so it is a local minimum point.

At (-5/3, 0) :

D > 0 and [tex]f_{xx}[/tex] < 0, so it is a local maximum point.

At (-1, 2) :

D < 0 and it is saddle point.

At (-1, -2) :

D < 0 and it is saddle point.

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Answer: The mid-point of the given line segment is (4,2,10).

Step-by-step explanation: We can find the mid-point of the line segment given to us by using the formula:

x=[tex]\frac{x_{1}+x_{2} }2}[/tex]

y=[tex]\frac{y_{1}+y_{2}}{2}[/tex]

z=[tex]\frac{z_{1}+z_{2}}{2}[/tex]

where x,y, and z are the coordinates of the mid-point of the line segment.

Now, [tex]{x_{1}[/tex]=2,[tex]{x_{2}[/tex]=6,[tex]{y_{1}[/tex]=0,[tex]{y_{2}[/tex]=4,[tex]{z_{1}[/tex]=-6,[tex]{z_{2}[/tex]=26

By substituting the values in the above formula we get,

x=(2+6)/2=4

y=(0+4)/2=2

z=(-6+26)/2=10

Thus, the mid-point of the given line segment is: (4,2,10)

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The value of d is 8 units.

Given is a setup of a circular wheel with radius AC of 5 units and a tangent AB of 12 units, we need to find the value of d,

We know that the tangents are perpendicular to the circle,

So, ΔCAB is a right triangle, using the Pythagoras theorem,

BC² = AB² + AC²

BC² = 5² + 12²

BC² = 169

BC = 13

d = 13-5

d = 8

Hence the value of d is 8 units.

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Answer: Andiswa is 14 and Jonas is 11

Answer:

Heya Let's Calculate.

Step-by-step explanation:

Let's form equation.

Let the age of Jonas be x.

Then, the age of Andiswa is x+3

Equation=

x+x+3=25

⇒2x+3=25

⇒2x=25-3

⇒x=25/5

x=5

The expected number of students in the selected student's class, E(X), is approximately 21.91.

To find the expected value (E) of the random variable X, which represents the number of students in the selected student's class, we need to calculate the weighted average of X over all possible values of X.

Let's denote the number of students in each class as X1, X2, and X3, where X1 = 20, X2 = 22, and X3 = 25.

The probability of selecting a student from class 1 is P(X = X1) = 20/67, as there are 20 students in class 1 out of a total of 67 students.

Similarly, the probabilities for class 2 and class 3 are P(X = X2) = 22/67 and P(X = X3) = 25/67, respectively.

Now, we can calculate the expected value E(X):

E(X) = X1 * P(X = X1) + X2 * P(X = X2) + X3 * P(X = X3)

= 20 * (20/67) + 22 * (22/67) + 25 * (25/67)

≈ 21.91

Therefore, the expected number of students in the selected student's class, E(X), is approximately 21.91.

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The equation of the plane passing through the point (1,5,4) is 23y - z = 111

Given data ,

To find the equation of the plane passing through the point (1, 5, 4) and perpendicular to the line x = 1 + 7t, y = t, z = 23t, we can use the following approach:

To find the direction vector of the line, which is the coefficients of t in each coordinate. In this case, the direction vector is (7, 1, 23).

Since the plane is perpendicular to the line, the normal vector of the plane will be orthogonal to the direction vector. We can take the direction vector and find two other vectors that are orthogonal to it to determine the normal vector.

The two orthogonal vectors to (7, 1, 23) is to take the cross product of (7, 1, 23) with two arbitrary vectors that are not parallel to each other. Let's choose the vectors (1, 0, 0) and (0, 1, 0).

Cross product 1: (7, 1, 23) x (1, 0, 0)

= (0, 23, -1)

Cross product 2: (7, 1, 23) x (0, 1, 0)

= (-23, 0, -7)

So, the two vectors that are orthogonal to the direction vector (7, 1, 23).

Now, the equation of the plane using the normal vector and the given point (1, 5, 4).

The equation of the plane is given by the dot product of the normal vector and the vector connecting the given point to any point (x, y, z) lying on the plane:

(0, 23, -1) · (x - 1, y - 5, z - 4) = 0

Expanding the dot product, we have:

0(x - 1) + 23(y - 5) + (-1)(z - 4) = 0

23(y - 5) - (z - 4) = 0

Hence , the equation of the plane passing through the point (1, 5, 4) and perpendicular to the line x = 1 + 7t, y = t, z = 23t is 23y - z = 111.

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The complete question is attached below:

Find the equation of the plane which passes through the point (1,5,4) and is perpendicular to the line x=1+7t, y=t, z=23t

The predicted y value for each of the following x scores are:

To find the predicted y value (y) for each of the given x score using the regression equation y = 2x - 7, we substitute the x values into the equation and calculate the corresponding y values.

For x = 0:

y = 2(0) - 7

= -7

The predicted y value for x = 0 is -7.

For x = 1:

y = 2(1) - 7

= -5

The predicted y value for x = 1 is -5.

For x = 3:

y = 2(3) - 7

= -1

The predicted y value for x = 3 is -1.

For x = -2:

y = 2(-2) - 7

= -11

The predicted y value for x = -2 is -11.

So, the predicted y values for the given x scores are:

For x = 0, y = -7

For x = 1, y = -5

For x = 3, y = -1

For x = -2, y = -11

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To find an orthonormal basis of the nullspace of A, we first need to find the nullspace of A. The nullspace of a matrix A consists of all vectors x such that Ax = 0. We can find the nullspace of A by solving the system of linear equations Ax = 0.

Using row operations, we can reduce A to row echelon form:

[ 1 0 1 1 0 -1 6 4 ] -> [ 1 0 1 1 0 -1 0 1 ]

The pivot variables are x1, x3, and x4. We can express the non-pivot variables in terms of the pivot variables:

x2 = -x1

x5 = -x1 + x7

x6 = x1 - 6x7

Thus, the general solution to Ax = 0 is:

[ x1, -x1, x3, x4, -x1 + x7, x1 - 6x7, x7 ]

To find an orthonormal basis, we need to orthogonalize this set of vectors and then normalize them. One way to do this is to use the Gram-Schmidt process.

Using the Gram-Schmidt process, we can orthogonalize the set of vectors by subtracting their projections onto previously orthogonalized vectors. We can start by normalizing the first vector:

v1 = [ 1, 0, 1, 1, 0, -1, 0, 0 ] / sqrt(3)

Next, we can subtract the projection of the second vector onto v1:

v2 = [ 0, -1, 0, 0, 1, -6, 1, 0 ] - proj[0,-1,0,0,1,-6,1,0]v1

v2 = [ 0, -1, 0, 0, 1, -6, 1, 0 ] + [ 0, 1/sqrt(3), 0, 0, -1/sqrt(3), 2/sqrt(3), -1/sqrt(3), 0 ]

v2 = [ 0, -1/sqrt(3), 0, 0, 1/sqrt(3), -4/sqrt(3), 2/sqrt(3), 0 ]

Finally, we can normalize v2:

v2 = [ 0.41, -0.41, 0, 0, 0.41, -0.82, 0.41, 0 ]

Thus, the columns of the matrix [ 0.41, 0, -0.41, 0.82, -1, 0, 0, 0 ] form an orthonormal basis of the nullspace of A.

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If deviations are positive, it means that the observations are above the mean. This is option 1.

Positive deviations indicate that the individual observations are greater than the mean of the dataset. Deviation is a measure of how far each observation deviates from the mean. When the deviations are positive, it implies that the observations are higher than the average or central value represented by the mean.

Linear regression is a statistical technique used to model the relationship between variables. It is not directly related to the sign of deviations. Therefore, option 2 is incorrect.

Option 3, which states that positive deviations indicate that observations are below the mean, is incorrect. Positive deviations indicate observations above the mean, not below.

Option 4, stating that SSE (Sum of Squared Errors) will be negative, is not necessarily true. SSE is a measure of the discrepancy between the observed values and the values predicted by a regression model. It is calculated by summing the squared differences between the observed and predicted values. The SSE can be positive or zero, but it cannot be negative.

In summary, when deviations are positive, it indicates that the observations are above the mean, not below. The statement that SSE will be negative is incorrect.

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The volume would be:

Volume = (1/3)(1 x 1)(2)

Volume ≈ 0.67 cubic feet

To determine a good ratio for comparing the actual pyramid to a piñata-sized pyramid, we need more information about the dimensions of the actual pyramid and the desired size of the piñata. Once we have that information, we can calculate the ratio by comparing the height, base, and volume of the two pyramids.

Assuming we have the necessary information, let's say the actual pyramid has a height of 10 feet and a base of 8 feet by 8 feet, and we want to create a piñata-sized pyramid with a height of 2 feet and a base of 1 foot by 1 foot. In this case, the ratio would be:

1: (2/10) or 1:5

To calculate the surface area of the piñata pyramid, we can use the formula:

Surface Area = (base x base) + 2(base x slant height)

Using the dimensions given, the surface area would be:

Surface Area = (1 x 1) + 2(1 x sqrt(0.5^2 + 2^2))

Surface Area ≈ 6.83 square feet

To calculate the volume of the piñata pyramid, we can use the formula:

Volume = (1/3)(base x base)(height)

Using the dimensions given, the volume would be:

Volume = (1/3)(1 x 1)(2)

Volume ≈ 0.67 cubic feet

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The factored form of the quadratic equation 4x²+ 100x + 255 = 0 is (2x + 17)(2x + 15) = 0.

We have the equation:

4x² + 100x + 255 = 0

Now, factorizing

4x² + 34x + 30x + 255 = 0

Now, we group the terms and factor by grouping:

(4x² + 34x) + (30x + 255) = 0

2x(2x + 17) + 15(2x + 17) = 0

(2x + 17)(2x + 15) = 0

Now, we set each factor equal to zero and solve for x:

2x + 17 = 0 --> 2x = -17 --> x = -17/2

2x + 15 = 0 --> 2x = -15 --> x = -15/2

The factored form of the quadratic equation 4x²+ 100x + 255 = 0 is (2x + 17)(2x + 15) = 0 and the solutions for x are x = -17/2 and x = -15/2.

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The probability that the sum of the throw is divisible by 4 but greater than 7 is 17 / 36.

There are 36 possible outcomes when two dice are tossed.

The sum of the dice can be any number from 2 to 12. The sum is divisible by 4 when the sum is 4, 8, or 12. The sum is greater than 7 when the sum is 8 , 9 , 10 , 11, or 12.

There are 3 ways to get a sum of 4, 3 ways to get a sum of 8, 2 ways to get a sum of 12, 4 ways to get a sum of 9, 4 ways to get a sum of 10, and 1 way to get a sum of 11.

The probability of getting a sum that is divisible by 4 or greater than 7 is:

= ( 3 + 3 + 2 + 4 + 4 + 1 ) / 36

= 17 / 36

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(a) To find the area of the region R bounded by the curves y = -x^2 - 3x + 10 and y = 5 + x, we need to find the points of intersection of the two curves and integrate the difference in y-values.

First, let's find the points of intersection by setting the two equations equal to each other: -x^2 - 3x + 10 = 5 + x.  Rearranging and simplifying: x^2 + 4x + 5 = 0. The quadratic equation has no real solutions, which means the two curves do not intersect. Since there are no points of intersection, the region R does not exist, and the area of R is equal to 0. (b) To find the exact volume of the solid of revolution obtained when R is rotated 2π radians about the x-axis using the disk/washer method, we need to integrate the cross-sectional areas of the disks or washers formed.The region R is bounded by the curves y = -x^2 - 3x + 10 and y = 5 + x. To determine the limits of integration, we need to find the x-values where the curves intersect.

Setting the two equations equal to each other: -x^2 - 3x + 10 = 5 + x. Rearranging and simplifying: x^2 + 4x + 5 = 0. Solving this quadratic equation, we find the solutions: x = -2 ± √(4 - 4(1)(5)) / 2. x = -2 ± √(-16) / 2. Since the discriminant is negative, the quadratic equation has no real solutions. Therefore, the curves y = -x^2 - 3x + 10 and y = 5 + x do not intersect. As there are no points of intersection, the volume of the solid of revolution is 0.(c) To find the exact volume of the solid of revolution obtained when R is rotated 2π radians about the line x = 3 using the method of cylindrical shells, we need to integrate the product of the circumference of the shells and their heights. The region R is bounded by the curves y = -x^2 - 3x + 10 and y = 5 + x. The line x = 3 is the axis of rotation. To determine the limits of integration, we need to find the y-values where the curves intersect.

Setting the two equations equal to each other: -x^2 - 3x + 10 = 5 + x. Rearranging and simplifying: x^2 + 4x + 5 = 0. Solving this quadratic equation, we find the solutions: x = -2 ± √(4 - 4(1)(5)) / 2. x = -2 ± √(-16) / 2. Since the discriminant is negative, the quadratic equation has no real solutions. Therefore, the curves y = -x^2 - 3x + 10 and y = 5 + x do not intersect. As there are no points of intersection, the volume of the solid of revolution is 0.

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The given sequent "((P-Q) & (RS)) ⊢ =(P&R) Qvs" is proved.

To prove the sequent ((P-Q) & (RS)) ⊢ =(P&R) Qvs, we will use natural deduction rules.

1. Assume ((P-Q) & (RS)) as the premise.

2. From (1), apply the ∧-elimination rule to get (P-Q).

3. From (1), apply the ∧-elimination rule to get (RS).

4. Assume P as a sub-derivation.

5. From (4), apply the →-intro rule to get (P→R).

6. From (5) and (2), apply the →-Elim rule to get R.

7. From (6), apply the ∧-intro rule to get (P&R).

8. Assume Q as a sub-derivation.

9. From (9) and (3), apply the →-intro rule to get (Q→S).

10. From (10) and (2), apply the →-Elim rule to get S.

11. From (11), apply the ∨-intro rule to get (Qvs).

12. From (7) and (12), apply the =-intro rule to get =(P&R) Qvs.

13. Discharge the assumptions made in steps 4 and 8.

14. Finally, discharge the assumption made in step 1.

Therefore, we have proven the sequent ((P-Q) & (RS)) ⊢ =(P&R) Qvs.

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The expected number of times one needs to flip a fair coin to get two consecutive heads is 6.

To find the expected number of times one needs to flip a fair coin to get two consecutive heads, we can use the concept of conditional probability.

Let E be the expected number of flips needed to get two consecutive heads.

On the first flip, there is a probability of 1/4 of getting two consecutive heads. If this does not happen, we have to start over and flip the coin again.

On the second flip, there is a probability of 1/4 of getting two consecutive heads, but we have already used one flip, so the expected number of additional flips needed is E.

Therefore, we can write the following equation:

E = (1/4)2 + (1/2)(E+1) + (1/4)(E+2)

E = (1/2)( 1 + E + 1 + E/2 + 1)

E = (1/2)(3 + 3E/2)

E = 3/2 + 3E/4

E/4 = 3/2

E = 6

Therefore, the expected number of times one needs to flip a fair coin to get two consecutive heads is 6.

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To calculate the limits below, we can use basic limit rules and arithmetic operations.

lim_(x->2) [f(x) + g(x)]

= lim_(x->2) f(x) + lim_(x->2) g(x) (by limit laws)

= 1 + (-4)

= -3

We can use the limit laws to add the limits of f(x) and g(x) since they are both approaching the same point, 2. Then, we can simply add the values of the limits to get the limit of their sum.

lim_(x->2) [g(x) - f(x)h(x)]

= lim_(x->2) g(x) - lim_(x->2) [f(x)h(x)] (by limit laws)

= -4 - [lim_(x->2) f(x)] [lim_(x->2) h(x)] (by limit laws and arithmetic operations)

= -4 - 1(0)

= -4



Again, we can use the limit laws to subtract the limit of f(x) multiplied by h(x) from the limit of g(x). To find the limit of f(x) multiplied by h(x), we can use the product rule for limits and multiply the limits of f(x) and h(x) together.

lim_(x->2) [g(x)/f(x)]

= [lim_(x->2) g(x)] / [lim_(x->2) f(x)] (by limit laws)

= -4 / 1

= -4



We can use the limit laws to divide the limit of g(x) by the limit of f(x) since they are both approaching the same point, 2.



The limits of [f(x) + g(x)], [g(x) - f(x)h(x)], and [g(x)/f(x)] as x approaches 2 are -3, -4, and -4, respectively.

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Given that x has a normal distribution with mean µ = 100 and standard deviation σ = 18, the probability P(x ≥ 120) is to be determined.

The standardized value of x can be calculated as follows: z = (x - µ) / σHere, x = 120, µ = 100, and σ = 18.∴ z = (120 - 100) / 18 = 1.11From the standard normal distribution table, the probability P(Z ≥ 1.11) = 0.1335 (approx.)Thus, the main answer is option A. 0.1335 Probability P(x ≥ 120) can be calculated by standardizing x as follows: z = (x - µ) / σwhere µ is the mean and σ is the standard deviation.

Here,

we have: µ = 100,

σ = 18, and

x = 120∴

z = (120 - 100) / 18

= 1.11

Now, we can calculate the probability P(x ≥ 120) by using the standard normal distribution table as follows

:P(x ≥ 120)

= P(Z ≥ 1.11)

From the standard normal distribution table, we get:

P(Z ≥ 1.11)

= 0.1335 (approx.)

Therefore, the probability P(x ≥ 120) is 0.1335 (approx.)Thus, the main answer is option A. 0.1335.

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hello

the answer is:

Sin A = BC/AB ----> Sin 32° = 14/AB ----> AB = 26.42

AB² = BC² + AC² ----> (26.42)² = (14)² + AC² ---->

AC² = (26.42)² - (14)² ----> AC² = 502.0164 ---->

AC = 22.40

OR

Cos A = AC/AB ----> Cos 32° = AC/(26.42) ---->

AC = 22.40

The series   [tex]$\sum_{n=1}^{\infty} \frac{n^2 + 1}{n^{10}}$[/tex] converges. The given series converges.

To test this series for convergence  [tex]$\sum_{n=1}^{\infty} \frac{n^2 + 1}{n^{10}}$[/tex] You could use the Limit Comparison Test, comparing it to the series

[tex]$\sum_{n=1}^{\infty} \frac{1}{n}$[/tex]  

where [tex]p=1 > 0$.[/tex]

Now, we will use the Limit Comparison Test to determine if the given series converges or diverges.According to the Limit Comparison Test,

if [tex]$\lim_{n\to\infty} \frac{a_n}{b_n}[/tex]  =[tex]c$[/tex] where [tex]$c > 0$[/tex],

then both [tex]$\sum_{n=1}^{\infty} a_n$[/tex]

and [tex]$\sum_{n=1}^{\infty} b_n$[/tex] converge or both diverge.

That is  , [tex]$\bullet$[/tex] If  [tex]$\sum_{n=1}^{\infty} b_n$[/tex] converges,

then  [tex]$\sum_{n=1}^{\infty} a_n$[/tex]  converges[tex].$\bullet$[/tex]

If [tex]$\sum_{n=1}^{\infty} b_n$[/tex] diverges,

then [tex]$\sum_{n=1}^{\infty} a_n$[/tex]  diverges.

Let [tex]$a_n = \frac{n^2 + 1}{n^{10}}$[/tex] and

[tex]$b_n = \frac{1}{n}$[/tex]

Then, [tex]$\lim_{n\to\infty} \frac{a_n}{b_n}[/tex] =  [tex]\lim_{n\to\infty} \frac{n^2 + 1}{n^{10}} \cdot \frac{n}{1}[/tex]

= [tex]\lim_{n\to\infty} \frac{n^3 + n}{n^{10}}[/tex]

=[tex]\lim_{n\to\infty} \frac{1}{n^6}+ \lim_{n\to\infty} \frac{1}{n^9}[/tex]

=[tex]0$.[/tex]

Since [tex]\lim_{n\to\infty} \frac{a_n}{b_n} = 0$,[/tex]

which is a finite positive number.

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The limit of a(n) / b(n) is infinity, and b(n) is a known convergent series, we can conclude that the original series [tex]\sum (1/n^2 + 1/n)[/tex] also converges. The statement "Converges" is the correct answer.

To test the series [tex]\sum(1/n^2 + 1/n)[/tex] for convergence, we can use the Limit Comparison Test.

We will compare it to the series Σ(1/n),

which is a known series that converges.

Let's denote the original series as [tex]a(n) = 1/n^2 + 1/n[/tex],

and the comparison series as b(n) = 1/n.

We need to calculate the limit of the ratio of the terms of the two series as n approaches infinity:

[tex]\lim_{n \to \infty} a(n)/b(n)\\ \\ \lim_{n \to \infty} [(1/n^2 + 1/n) / (1/n)]\\\\ \lim_{n \to \infty} (n+1)[/tex]

As n approaches infinity, the limit of (n + 1) is infinity.

Since the limit of a(n) / b(n) is infinity, and b(n) is a known convergent series, we can conclude that the original series [tex]\sum(1/n^2 + 1/n)[/tex] also converges.

Therefore, the statement "Converges" is the correct answer.

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The area of given regular hexagon is  509.22   square units.

For the given polygon,

Number of sides = 6

Since we know that,

A regular hexagon is a polygon with six equal sides and six equal angles. All of the sides and angles of a regular polygon are equal. A regular pentagon, for example, has 5 equal sides, whereas a regular octagon has 8 equal sides. When such prerequisites are not satisfied, polygons can take on the appearance of a variety of irregular forms. When six equilateral triangles are placed side by side, a regular hexagon is formed. The area of the regular hexagon is thus six times the size of the identical triangle.

Therefore,

It is called regular hexagon.

Since we know that,

Area of regular hexagon = (3√3/2)a²

Here we have a = 14

Therefore,

Area of given hexagon = (3√3/2)14²

                                       = 509.22   square units.

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The direction of the rescue boat, obtained by representing the location of the sailboat using vectors is N 8.7° E

A vector is a quantity that posses the properties magnitude and direction.

The vector form of the motion of the sailboat can be presented as follows;

d = 8 × sin(12)·i + 8 × cos(18)·j

The next direction of the sailboat = 3 miles south = -3 j

Therefore, the distance and direction of the sailboat is; 8 × sin(12)·i + 8 × cos(18)·j + -3 j

The direction the rescue boat has to leave from the dock in order to intercept the sailboat is; arctan(3 + 8 × cos(18))/( 8 × sin(12)) ≈ 81.26°

The direction relative to the North = 90° - 81.26° ≈ 8.7°

The direction of the rescue both is N 8.7° E

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Rounding to three decimal places, the correlation coefficient between cost and miles is approximately -0.398.

To calculate the correlation coefficient between cost and miles, we need to first calculate the mean of the cost and miles, as well as the standard deviations of each variable. Then we can use the formula for the correlation coefficient, which is:

correlation coefficient (r) = Σ((x_i - x_mean) * (y_i - y_mean)) / (n * x_std * y_std)

Let's calculate it step by step:

Calculate the mean of cost (x) and miles (y):

x_mean = (171 + 397 + 270 + 88 + 438) / 5 = 272.8

y_mean = (941 + 3093 + 2003 + 433 + 3019) / 5 = 1581.8

Calculate the standard deviation of cost (x) and miles (y):

x_std = sqrt(((171 - 272.8)^2 + (397 - 272.8)^2 + (270 - 272.8)^2 + (88 - 272.8)^2 + (438 - 272.8)^2) / 5) ≈ 131.150

y_std = sqrt(((941 - 1581.8)^2 + (3093 - 1581.8)^2 + (2003 - 1581.8)^2 + (433 - 1581.8)^2 + (3019 - 1581.8)^2) / 5) ≈ 968.294

Calculate the correlation coefficient (r):

r = ((171 - 272.8) * (941 - 1581.8) + (397 - 272.8) * (3093 - 1581.8) + (270 - 272.8) * (2003 - 1581.8) + (88 - 272.8) * (433 - 1581.8) + (438 - 272.8) * (3019 - 1581.8)) / (5 * 131.150 * 968.294)

r ≈ -0.398

Rounding to three decimal places, the correlation coefficient between cost and miles is approximately -0.398.

After performing the calculations, we find that the correlation coefficient between cost and miles is approximately -0.398. This means that there is a weak negative correlation between the cost of flights and the distance in miles. In other words, as the distance increases, the cost tends to slightly decrease.

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