What is 2/18 in simplest form

The fraction of the total of paper clips Cecily put in the jar and drawer is [tex]\frac{97}{100}[/tex] (see the attachment below).

Paper clips in the jar: 37/100, which means Cecily put 37 clips in the jar.

Paper clips in the drawer: 6/10, now let's find out the number of clips this fraction is equivalent to:

100 (total clips) / 10 x 6 = 60

Total clips: 60 +37= 97. This number of clips can be expressed as [tex]\frac{97}{100}[/tex] .

To represent this in a grid color a total of 97 squares in a grid with 100 squares as it is shown below.

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Answer: We can calculate the size of the lot in acres and then compare it to the minimum size of 3/4 of an acre that the family is looking for.

1 acre = 43,560 square feet

So, a 30,000 sq-ft lot is equivalent to:

30,000 sq-ft / 43,560 sq-ft/acre = 0.6864 acres

Since 0.6864 acres is less than 3/4 of an acre, the family's agent will be wasting her time by showing them the house with a 30,000 sq-ft lot. The lot is not big enough for the family's needs.

Step-by-step explanation:

A sequence of transformations is applied to a polygon. [ The following statements represent a sequence of transformations where the resulting polygon is similar to the original polygon but have a smaller area than the original polygon is that: a dilation about the origin by a scale factor of 2 3 followed by a rotation of 90° counterclockwise about the origin and a translation 5 units left.

The engineer's balance sheet would show a monthly income of $9,002.60 ($4,501.30 x 2), total expenses of $4,001.36, and a balance of $499.94.

Balance sheet is a financial statement that contains details of a company's assets or liabilities at a specific point in time.

First we can use the given information to create the following balance sheet for the civil engineer:

INCOME:

Bi-weekly net pay: $2,250.65 x 2 = $4,501.30 (this assumes two pay periods per month)

EXPENSES:

     TOTAL EXPENSES: $4,001.36

BALANCE: $4,501.30 - $4,001.36 = $499.94

Therefore, the engineer's balance sheet would show a monthly income of $9,002.60 ($4,501.30 x 2), total expenses of $4,001.36, and a balance of $499.94.

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The value of C is 108°

Given [tex]a=33,\ \ b=26, \ \ \angle B=31^0 .[/tex]

We have to find [tex]\angle C, \ \angle A \ and\ c[/tex]

We can use the cosine formula to find these:

[tex]cosB=\frac{a^2+c^2-b^2}{2ac}\\cos31^o=\frac{33^2+c^2-26^2}{2(33)(c)}\\0.8572=\frac{413+c^2}{66c}\\56.5752c=413+c^2\\i.e. c^2-56.5752c+413=0\\\Rightarrow c=47.9647, \ 8.6104\\So, \mathbf{c=48 \ or \ c=9}\\Now if c=48:\\cosA=\frac{b^2+c^2-a^2}{2bc}\\=\frac{26^2+48^2-33^2}{2(26)(48)}\\[/tex]

[tex]=\frac{1891}{2496}\\=0.75761\\\therefore A=cos^{-1}(0.75761)=40.746^0\approx 41^0\\i.e. \mathbf{\angle A= 41^0}\\cosC=\frac{a^2+b^2-c^2}{2ab}\\=\frac{33^2+26^2-48^2}{2(33)(26)}\\=\frac{-539}{1716}\\=-0.3141\\\therefore C=cos^{-1}(-0.3141)=108.306^0\approx 108^0[/tex]

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The solution is [tex]x_1[/tex] = 3 and [tex]x_2[/tex]= 0.

Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem.

Given:

We transfer the row with the resolving element from the previous table into the current table, elementwise dividing its values ​​into the resolving element:

[tex]P_2[/tex] = [tex]P_2[/tex] / [tex]x_{2,1[/tex] = 12 / 4 = 3;

[tex]x_{2,1[/tex] = [tex]x_{2,1[/tex] / [tex]x_{2,1[/tex] = 4 / 4 = 1;

[tex]x_{2,2[/tex] = [tex]x_{2,2[/tex] / [tex]x_{2,1[/tex] = 3 / 4 = 0.75;

[tex]x_{2,3[/tex] = [tex]x_{2,3[/tex] / [tex]x_{2,1[/tex] = 0 / 4 = 0;

[tex]x_{2,4[/tex] = [tex]x_{2,4[/tex] / [tex]x_{2,1[/tex] = 1 / 4 = 0.25;

The remaining empty cells, except for the row of estimates and the column Q, are calculated using the rectangle method, relative to the resolving element:

[tex]P_1[/tex] = ([tex]P_1[/tex] . [tex]x_{2,1[/tex]) - ([tex]x_{1,1[/tex] . [tex]P_2[/tex]) / [tex]x_{2,1[/tex]= ((6 x 4) - (1 x 12)) / 4 = 3

[tex]x_{1,1[/tex]= (([tex]x_{1,1[/tex]. [tex]x_{2,1[/tex]) - ([tex]x_{1,1[/tex] . [tex]x_{2,1[/tex])) / [tex]x_{2,1[/tex]= ((1 x 4) - (1 x 4)) / 4 = 0

[tex]x_{1,3[/tex]= (([tex]x_{1, 3[/tex]. [tex]x_{2,1[/tex]) - ([tex]x_{1,1[/tex] . [tex]x_{2,3[/tex])) / [tex]x_{2,1[/tex]= ((1 x 4) - (1 x 0)) / 4 = 1

[tex]x_{1,4[/tex] = (([tex]x_{1, 4[/tex]. [tex]x_{2,1[/tex]) - ([tex]x_{1,1[/tex]. [tex]x_{2,4[/tex])) / [tex]x_{2,1[/tex]= ((0 x 4) - (1 x 1)) / 4 = -0.25

Objective function value

We calculate the value of the objective function by elementwise multiplying the column [tex]C_b[/tex] by the column P, adding the results of the products.

Max P = (C[tex]b_1[/tex] x [tex]P_{01[/tex]) + (C[tex]b_{11[/tex] x [tex]P_2[/tex])= (0 x 3) + (7 x 3) = 21;

Evaluated Control Variables

We calculate the estimates for each controlled variable, by element-wise multiplying the value from the variable column, by the value from the Cb column, summing up the results of the products, and subtracting the coefficient of the objective function from their sum, with this variable.

Max [tex]x_1[/tex]= ((C[tex]b_1[/tex] . [tex]x_{1,1[/tex]) + (C[tex]b_2[/tex] . [tex]x_{2,1[/tex]) ) - k[tex]x_1[/tex] = ((0 x 0) + (7 x 1) ) - 7 = 0

Max [tex]x_2[/tex]= ((C[tex]b_1[/tex] . [tex]x_{1,2[/tex]) + (C[tex]b_2[/tex] . [tex]x_{2,2[/tex]) ) - k[tex]x_2[/tex] = ((0 x 1.25) + (7 x 0.75) ) - 5 = 0.25

Max [tex]x_3[/tex]= ((C[tex]b_1[/tex] . [tex]x_{1,3[/tex]) + (C[tex]b_2[/tex] . [tex]x_{2,3[/tex]) ) - k[tex]x_3[/tex] = ((0 x 1) + (7 x 0) ) - 0 = 0

Max [tex]x_4[/tex]= ((C[tex]b_1[/tex] . [tex]x_{1,4[/tex]) + (C[tex]b_2[/tex] . [tex]x_{2,4[/tex]) ) - k[tex]x_4[/tex] = ((0 x -0.25) + (7 x 0.25) ) - 0 = 1.75

Since there are no negative values ​​among the estimates of the controlled variables, the current table has an optimal solution.

The value of the objective function:

F = 21

The variables that are present in the basis are equal to the corresponding cells of the column P, all other variables are equal to zero:

[tex]x_1[/tex] = 3;

[tex]x_2[/tex]= 0;

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Answer:

= 4y + x - 16 = 0.

Step-by-step explanation:

Point are (-8,6)

where

x1 = -8 and y1 = 6

the equation o the line is given by the formula

[tex] = \frac{y - y1}{x - x1} = m[/tex]

where (m) is the gradient

the gradient (m) = -1/4

therefore the equation of the line

[tex] \frac{y - y1}{x - x1} = m \\ = \frac{y - 6}{x - ( - 8)} = \frac{ - 1}{4} \\ = \frac{y - 6}{x + 8} = \frac{ - 1}{4} \\ = 4(y - 6) = - 1(x + 8) \\ = 4y - 24 = - x - 8 \\ = 4y = - x - 8 + 24 \\ = 4y = - x + 16 \\ = 4y + x - 16 = 0.[/tex]

therefore the equation of the line is = 4y + x - 16 = 0.

note// y and x where not given any value.

Hi! The answer would be 445.272727273

Glad I can help :)

The solution of the inequality using the coordinate pair is 5.

Linear inequalities are those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠ and the value of the exponent of the variable is 1.

Given linear inequality is,

y ≤ -2x + 3

We have a coordinate pair (-1, 5).

So substitute x = -1 and y = 5 in the given inequality.

5 ≤ (-2 × -1) + 3

5 ≤ 2 + 3

5 ≤ 5

Hence the missing numbers are found.

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The conditions on a, b, and c for which the Linear equations has solutions are det(A) ≠ 0,  Rank[A|B] = Rank(A).

(a) 3x + 2y - z = a

x + y + 2z = b

5x + 4y + 32 = c

For this system to have solutions, the augmented matrix [A|B]  must have a unique solution. This is equivalent to determinant of the coefficient matrix A is non-zero, and the system is consistent (the row rank of the augmented matrix is equal to the row rank of the coefficient matrix). Therefore, the conditions on a, b, and c for this system to have solutions are:

det(A) ≠ 0

Rank[A|B] = Rank(A)

(b) -3x + 2y + 4z = a

-x - 2y + 3z = b

-x - 6y + 232 = 0

For this system to have solutions, the conditions are the same as in (a):

det(A) ≠ 0

Rank[A|B] = Rank(A)

(c) 4x - 2y + 3z = a

2x - 3y – 2z = b

4x - 2y + 32 = c

For this system to have solutions, the conditions are the same as in (a) and (b):

det(A) ≠ 0

Rank[A|B] = Rank(A)

For each system of linear equations to have solutions, the number of equations must be equal to the number of variables, and the determinant of the coefficient matrix must be non-zero.

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The final answers are:

a) P(A) = 7/12

b) P(B) = 11/36

To compute the probabilities of Events A and B, we can use a table to list all possible outcomes and their corresponding sums:

Die 1 Die 2 Sum

1         1         2

1         2         3

1         3         4

1         4        5

1         5        6

1         6        7

2         1        3

2         2        4

2         3        5

2         4        6

2         5        7

2         6        8

3         1        4

3         2        5

3         3        6

3         4        7

3         5        8

3         6        9

4         1        5

4         2        6

4         3        7

4         4        8

4         5        9

4         6        10

5         1         6

5         2         7

5         3         8

5         4         9

5         5         10

5         6         11

6         1          7

6         2          8

6         3               9

6           4          10

6         5          11

6         6          12

a) Event A: The sum is greater than 6. From the table, we can see that there are 21 outcomes where the sum is greater than 6 (highlighted in bold). Thus, the probability of Event A is:

P(A) = 21/36 = 7/12

b) Event B: The sum is divisible by 3 or 5 (or both). From the table, we can see that the sums that are divisible by 3 or 5 (or both) are:

3, 5, 6, 9, 10, 12

There are 11 outcomes with these sums (highlighted in bold). Thus, the probability of Event B is:

P(B) = 11/36

Therefore, the final answers are:

a) P(A) = 7/12

b) P(B) = 11/36

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To measure the magnitude of the orbital angular momentum L and the projection of the angular momentum along a particular axis (say, the z-axis), we need to use the mathematical formalism of quantum mechanics.

In quantum mechanics, the orbital angular momentum L of a particle is an operator that acts on the wave function describing the particle's motion. Similarly, the z-component of the angular momentum Lz is also an operator.

The magnitude of the orbital angular momentum L can be calculated from the components of the angular momentum operator using the expression:

L^2 = L₁^2 + L₂^2 + L₃^2

where L₁, L₂, and L₃ are the x, y, and z components of the angular momentum operator, respectively. To measure the magnitude of the orbital angular momentum, we would need to measure the values of L₁ L₂, and L₃ and use them to calculate L^2.

The projection of the angular momentum along a particular axis (say, the z-axis) is given by the operator L₃. To measure the z-component of the angular momentum, we would need to measure the value of L₃ for a particular state of the system.

In practice, these measurements are often carried out using experiments involving the interaction of particles with magnetic fields. The behavior of the particles in the magnetic field allows us to infer information about the angular momentum of the particles.

The measurement of the magnitude and projection of angular momentum is an important part of many areas of physics, including quantum mechanics and solid-state physics.

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The distance traveled in the time interval 0 ≤ t ≤ 6, given the velocity function v(t) = t^2 - 6t - 16, can be determined by calculating the definite integral of the absolute value of the velocity function over the given time interval. The result is 128 units of distance.

To find the distance traveled in the given time interval, we need to integrate the absolute value of the velocity function v(t) = t^2 - 6t - 16 over the interval 0 ≤ t ≤ 6. The reason for taking the absolute value is that distance is a scalar quantity and does not depend on the direction of motion.

Taking the integral of the absolute value of the velocity function, we have:

∫|v(t)| dt = ∫|t^2 - 6t - 16| dt

To evaluate this integral, we need to split it into intervals where the velocity function is positive and negative. The absolute value function essentially removes the negative sign from the expression inside the absolute value brackets.

Next, we find the points where the velocity function changes sign by setting v(t) = 0:

t^2 - 6t - 16 = 0

Solving this quadratic equation, we find t = -2 and t = 8 as the points where the velocity changes sign.

Now, we evaluate the integral over the intervals [0, 2] and [2, 6] separately, considering the absolute value of the velocity function within each interval.

∫|v(t)| dt = ∫(t^2 - 6t - 16) dt over [0, 2] + ∫(-(t^2 - 6t - 16)) dt over [2, 6]

Evaluating the definite integrals, we obtain:

(128/3) + (128/3) = 256/3

Therefore, the distance traveled in the time interval 0 ≤ t ≤ 6 is 256/3 units of distance, which is approximately 85.33 units of distance.

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A)  We can draw a line perpendicular to side ED from the center of the hexagon, which will bisect side ED and create two additional triangles.

B) Area of each triangle = 31.16 square inches (rounded to two decimal places)

C) Total surface area = 375.87 square inches (rounded to two decimal places)

Any geometric shape with three dimensions can have its surface area determined. The area or region that an object's surface occupies is known as its surface area.

Now in the given question,

Part A: To decompose this shape into triangles, we can draw lines connecting the center of the hexagon to each of its vertices, creating six triangles. We can also draw a line perpendicular to side ED from the center of the hexagon, which will bisect side ED and create two additional triangles.

Part B: Each of the six triangles created by connecting the center of the hexagon to its vertices is an equilateral triangle, because all sides of the hexagon are equal in length. The area of an equilateral triangle can be calculated using the formula:

Area = (√3 / 4) x side²

where side is the length of one side of the equilateral triangle. In this case, the length of each side of the equilateral triangle is also 12 inches, so we have:

Area = (√3 / 4) x 12²

Area = (√3 / 4) x 144

Area = 36√3 square inches

Each of the two triangles created by drawing a perpendicular from the center of the hexagon to side ED is a right triangle, because one of its angles is 90 degrees. We can use the Pythagorean theorem to calculate the length of the other two sides of the right triangle. We know that one side has length 10.4 inches and the hypotenuse (which is also a side of the hexagon) has length 12 inches. Let x be the length of the other side of the right triangle. Then we have:

x² + 10.4² = 12²

x²+ 108.16 = 144

x² = 35.84

x = √35.84

x = 5.99 inches (rounded to two decimal places)

The area of each right triangle can be calculated using the formula:

Area = (1/2) x base x height

where the base is 5.99 inches (the length of the side opposite the 90 degree angle) and the height is 10.4 inches (the length of the side adjacent to the 90 degree angle). We have:

Area = (1/2) x 5.99 x 10.4

Area = 31.16 square inches (rounded to two decimal places)

Part C: To determine the area of the table's surface, we need to add up the areas of all eight triangles. There are six equilateral triangles, each with an area of 36sqrt(3) square inches, and two right triangles, each with an area of 31.16 square inches. Therefore, the total area of the table's surface is:

Total area = 6 x 36√3 + 2 x 31.16

Total area = 216√3 + 62.32

Total area = 375.87 square inches (rounded to two decimal places)

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Answer: The ratio $42/81$ can be simplified by dividing both the numerator and denominator by the greatest common factor of 42 and 81, which is 9.

$42/81 = (42 \div 9) / (81 \div 9) = 4/9$

So, the ratio $42/81$ can be expressed as the simplified fraction $\frac{4}{9}$.

Step-by-step explanation:

The value of covariance cov(y1, y2) is -0.222

                      y1      

y2         0         1         2          Total

0          1/9      2/9     1/9          4/9

1          2/9      2/9      0            4/9

2          1/9        0       0            1/9

Total   4/9       4/9     1/9          1      

  marginal distribution of y2:

y2      P(y2)      y2P(y2)      y2^2P(y2)

0         4/9        0.0000        0.0000

1          4/9         0.4444        0.4444

2          1/9         0.2222        0.4444

total      1          0.66667       0.88889

E(y2) = 0.6667

E(y2^2) = 0.8889

Var(y2) = E(y2^2)-(E(y2))^2=0.4444

marginal distribution of y1  

y1       P(y1)       y1P(y1)      y1^2P(y1)

0         4/9       0.0000      0.0000

1          4/9        0.4444       0.4444

2          1/9         0.2222      0.4444

total  1.00          0.6667      0.8889

E(y1) = 0.6667

E(y1^2) = 0.8889

Var(y1)= σy = E(y1^2)-(E(y1))^2 = 0.4444

E(XY) =∑xyP(x,y)=0.2222

Cov(y1,y2) = -0.222

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The question is geometric distribution. We are concerned with just the first success. That means there is no fixed number of trials

Before a situation can model binomial distribution.

For question a.

The probability of success is not the same for each trial. Since we are not putting any back. So it cannot be modeled as binomial distribution

For question b.

This has many outcomes not just success and failure.

Lan needs to record Pluto bar, jolly farm, husky kisses, finish fish, and hair heads.

For question c.

The question is geometric distribution. We are concerned with just the first success. That means there is no fixed number of trials

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Answer:

35 + 83) − (72 − 63) = 109

Step-by-step explanation:

To subtract the polynomials (35 + 83) − (72 − 63), we'll simplify each term in the polynomials first and then subtract the corresponding terms:

(35 + 83) = 118

(72 - 63) = 9

Now we can subtract the corresponding terms:

118 - 9 = 109

So the result of (35 + 83) − (72 − 63) is 109.

Aɳʂɯҽɾҽԃ Ⴆყ ɠσԃKEY ꦿ

Answer:

Okay first do what is in the parenthesis:

35 + 83 = 118

72 - 63 = 9

Now subtract:

118 - 9 = 109

The answer is: 109

Step-by-step explanation:

Hope it helps!

For answering 5 questions wrong Hunter's score is 75 and for for answering x question wrong his score will be  f(x)=100-5*x.

A function is defined as the relationship between input and output, where each input has exactly one output. The inputs are the elements in the domain and the outputs are elements in the co-domain.

Hunter is taking a multiple choice test with a total of 100 points available.

Each question is worth exactly 5 points,

From that the number of questions in the test is ,[tex]\frac{100}{5}=20[/tex]questions

Hunter's test score (out of 100) if he got 5 questions wrong,

20-5=15 questions correct.

Each question carries 5 marks, then 15*5= 75 marks.

Hunter score will be 75 marks.

And again each question carries 5 marks, If hunter answers x questions wrong his score will be f(x)=100-5*x.

It is a linear equation.

Hence, for answering 5 questions wrong Hunter's score is 75 and for for answering x question wrong his score will be f(x)=100-5*x.

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(a) The diver hit the water after 2.5 seconds

(b) The diver hit the water with a velocity of 80 ft/s

Since the cliff diver plunges from a height of 100 ft above the water surface and the distance the diver falls in seconds is given by the function d(t) = 16t². We can say d(t) = 100ft. So we have:

d(t) = 16t² = 100

t² = 100/16

t² = √(100/16)

t = 10/4

t = 2.5 seconds

The derivative of the distance will give us the velocity. That is:

v(t) = d'(t) = 32t

Substitute t =2.5  into v(t):

v(2.5) = 32(2.5)

v(2.5) = 80 ft/s

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The value of x and y is 31° and 101°.  Value is the measure of what something is worth, either financially or in terms of importance.

[tex]$\angle \mathrm{PRT}+\angle \mathrm{RTP}+\angle \mathrm{TPR}=180^{\circ} \text { (angle sum property of triangle) } \\[/tex]

[tex]& \Rightarrow \mathrm{x}+\left(180^{\circ}-\angle \mathrm{RTQ}\right)+60^{\circ}=180^{\circ} \text { (linear pair) } \\[/tex]

[tex]& \Rightarrow \mathrm{x}+\left(180^{\circ}-97^{\circ}\right)+60^{\circ}=180^{\circ} \\[/tex]

We get,

[tex]& \Rightarrow \mathrm{x}=31^{\circ} \\[/tex]

[tex]& \text { Now } \angle \mathrm{PRT}+\angle \mathrm{TRQ}+\angle \mathrm{QRS}=180^{\circ} \text { (angle of straight line) } \\[/tex]

[tex]& \Rightarrow \mathrm{x}+48^{\circ}+\mathrm{y}=180^{\circ} \\[/tex]

Simplifying,

[tex]& \Rightarrow 31^{\circ}+48^{\circ}+\mathrm{y}=180^{\circ} \\[/tex]

Then we get,

[tex]& \Rightarrow \mathrm{y}=101^{\circ}[/tex]

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The vector operations defined for a position vector a and a velocity vector b are vector addition and scalar multiplication.

Simplified Vector expression is (3u - 29) x u + (3u - 29) x v.

False, as the process that follows, solving for θ by dividing by the magnitude of v and taking the inverse sine, is only valid if v x a = 0.

1.28: The vector operations defined for a position vector a and a velocity vector b are vector addition and scalar multiplication.

1.29: Using the properties of the cross-product, the simplified expression is:

3u x u + 3u x v - 29 x u - 29 x v

Using the fact that the cross-product is distributive over vector addition, we can write this as:

3u x u + 3u x v - 29 x u - 29 x v = (3u - 29) x u + (3u - 29) x v

So the simplified vector expression is (3u - 29) x u + (3u - 29) x v.

1.30: False.

The given equation,

v x (a x R) = 0,

can be simplified to a scalar triple product,

(v x a) · R = 0.

Since the scalar triple product is equal to the determinant of a matrix formed by the three vectors,

this equation implies that either v x a = 0 or R = 0.

Therefore, the process that follows, solving for θ by dividing by the magnitude of v and taking the inverse sine, is only valid if v x a = 0.

The vector operations defined for a position vector a and a velocity vector b are vector addition and scalar multiplication.

Simplified Vector expression is (3u - 29) x u + (3u - 29) x v.

False as the process that follows, solving for θ by dividing by the magnitude of v and taking the inverse sine, is only valid if v x a = 0.

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The value of x is given as follows:

x = 34.

The angle measures are given as follows:

In a parallelogram, consecutive angles are supplementary, meaning that the sum of their measures is of 180º.

Angles A and B are consecutive, hence the value of x is obtained as follows:

2x + 10 + 3x = 180

5x = 170

x = 34.

Then the angle measures are given as follows:

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a) the population variance and sd use n in the denominator instead of n-1.  - false

b) the sample variance and sd inherit the strengths and weaknesses of the sample mean. - True

c) if we did not square the distances in the numerator, we would end up with 0. -false

d) the sample variance and sd are robust to outliers and/or extreme points.-false

a) False. The population variance and standard deviation use N in the denominator, where N is the size of the population. The sample variance and standard deviation use n-1 in the denominator, where n is the size of the sample.

b) True. The sample variance and standard deviation inherit the strengths and weaknesses of the sample mean, since they are based on the deviations of the sample values from the sample mean.

c) False. If we did not square the distances in the numerator, we would end up with the mean deviation, which is a valid measure of dispersion but not the variance.

d) False. The sample variance and standard deviation are not robust to outliers and/or extreme points, as they are sensitive to the values of the data points and can be significantly influenced by the presence of outliers.

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The average rate of change over the interval (-2,0) is -2.

The average rate of change describes how quickly one quantity changes in comparison to another. It indicates how much the function changed per unit during the specified interval.

Given that the interval is (-2,0). The coordinate of the point from the interval -2,0 is ( -2 , 4 ).

The average rate of change will be calculated as:-

Rate of change = ΔY / ΔX

Rate of change = ( 4 - 0 ) ( -2 - 0 )

Rate of change = -2

Therefore, the average rate of change will be -2.

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Since there are 20 professors and each professor is on exactly 3 committees, there must be a total of 60 committee positions.

Let's assume that it is not possible to select a distinct representative from each committee. This means that there must be some committee that does not have a distinct representative.

Let's consider the professors on this committee. Since each professor is on exactly 3 committees, each of the professors on this committee must also be on 2 other committees.

Let's consider the 2 other committees for each of the professors on this committee. Since there are 6 professors on the committee, there are a total of 12 other committees to consider.

Each of these 12 committees must have a representative chosen from the remaining 14 professors, since we have assumed that it is not possible to select a distinct representative from the original 10 committees.

However, since there are only 14 professors remaining, and each professor can only be on 3 committees, it is not possible to choose a representative from all 12 of these committees without choosing at least one professor to be on 4 committees. This is a contradiction, since we have assumed that each professor is on exactly 3 committees.

Therefore, our assumption that it is not possible to select a distinct representative from each committee must be false, and it is indeed possible to do so.

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