20 percent less than 120 is one-third more
than what number?

Answer:

  (a)  4a³

Step-by-step explanation:

You want the a³ term in the product of the two given polynomials.

The bottom line in the "vertical method" multiplication table shown is the sum of the partial-product expressions in the column above the bottom line:

  B = 12a³ -6a³ -2a³ = (12 -6 -2)a³

  B = 4a³

__

Additional comment

The value of A can be found a couple of ways.

You can find the partial product of the 1st-degree terms in each of the polynomials: (-2a)(-2a) = 4a².

Or you can find the value of A that is required to give the bottom-line result that is shown:

  9a² +A +a² = 14a²

  A = 14a² -10a²

  A = 4a²

Multiplying polynomials is substantially equivalent to multiplying multi-digit numbers. The difference is that there is no carry from one column to the next when you compute the partial products or the final sum.

False; if A and B are invertible matrices, then (AB)^-1=B^-1A^-1C.

The statement is false because the order of the matrices matters when taking the inverse of their product. The correct formula for the inverse of the product of two invertible matrices A and B is (AB)^-1 = B^-1A^-1. To see why, we can use the definition of matrix inversion:

if A is an invertible n x n matrix, then its inverse A^-1 is the unique n x n matrix such that AA^-1 = A^-1A = I, where I is the n x n identity matrix.

Now, suppose A and B are invertible n x n matrices. To show that (AB)^-1 = B^-1A^-1, we need to verify that (AB)(B^-1A^-1) = (B^-1A^-1)(AB) = I. Using matrix multiplication, we have:

(AB)(B^-1A^-1) = A(BB^-1)A^-1 = AIA^-1 = AA^-1 = I

and

(B^-1A^-1)(AB) = B^-1(A^-1A)B = B^-1IB = BB^-1 = I

Therefore, (AB)^-1 = B^-1A^-1, and the given statement (AB)^-1 = A^-1B^-1C is false.

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To create a 4x4 elementary matrix E that performs the row operation R2 - 3R1 → R2, you can follow this structure:
E = [1, 0, 0, 0]

     [-3, 1, 0, 0]
     [0, 0, 1, 0]
     [0, 0, 0, 1]

The 4 × 4 elementary matrix E that will carry out the row operation R2-3R, → R2 is:
1 0 0 0
-3 1 0 0
0 0 1 0
0 0 0 1

In this matrix, the entry in the second row and the first column is -3 because we are subtracting 3 times the first row from the second row. The other entries on the diagonal are 1 because we are not scaling those rows. The other entries in the second row are 0 because we are not adding or subtracting anything from those rows. The other entries in the matrix are also 0 because we are not modifying those rows. This matrix will perform the desired row operation when multiplied on the left of the original matrix.

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The answers for questions a,b, and c involving an exponential random variable, probability, and conditional probability are P(X > 5) = e^(-a * 5), P(X > 5 | X > 2) = (e^(-a * 5)) / (e^(-a * 2)), and P(4 <= X <= 4 + 2δ | X > 2) = f(4) * 2δ / P(X > 2)

Let X be an exponential random variable with parameter a.

a) The probability that X > 5 is given by the survival function of the exponential distribution,

which is P(X > 5) = e^(-a * 5).

b) The probability that X > 5 given that X > 2 is calculated using conditional probability.

The formula for conditional probability is P(X > 5 | X > 2) = P(X > 5 and X > 2) / P(X > 2).

Since X > 5 implies X > 2, the numerator is P(X > 5), which is e^(-a * 5). The denominator is P(X > 2), which is e^(-a * 2). Thus, P(X > 5 | X > 2) = (e^(-a * 5)) / (e^(-a * 2)).

c) Given that X > 2, and for a small δ > 0, the probability that 4 <= X <= 4 + 2δ is approximately

P(4 <= X <= 4 + 2δ | X > 2) = [P(4 <= X <= 4 + 2δ) - P(X < 2)] / P(X > 2).

We can approximate this by considering the probability density function (pdf) of the exponential distribution, which is f(x) = a * e^(-a * x).

The probability is approximately f(4) * 2δ / P(X > 2).

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Value of the iterated integral is 64.

To make it clearer, I'll rewrite the integral using proper notation:

∫(from 0 to 2) ∫(from 0 to 2x) ∫(from 0 to y) 3xyz dz dy dx

To evaluate the iterated integral, follow these steps:

1. Evaluate the innermost integral with respect to z:

∫(from 0 to 2) ∫(from 0 to 2x) [(3xyz²)/2] (from 0 to y) dy dx

2. Plug in the limits of integration for z:

∫(from 0 to 2) ∫(from 0 to 2x) [(3xy³)/2 - 0] dy dx

3. Evaluate the next integral with respect to y:

∫(from 0 to 2) [(3x²y⁴)/8] (from 0 to 2x) dx

4. Plug in the limits of integration for y:

∫(from 0 to 2) [(3x²(2x)⁴)/8 - 0] dx

5. Simplify the expression:

∫(from 0 to 2) [(3x¹⁰)/8] dx

6. Evaluate the outermost integral with respect to x:

[(3x¹¹)/88] (from 0 to 2)

7. Plug in the limits of integration for x:

[(3(2)¹¹)/88 - (3(0)¹¹)/88]

8. Simplify the expression:

(3 * 2048) / 88 = 6144 / 88 = 64

So the value of the iterated integral is 64.

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The output for the graph when the input value is 2 is 24.

Graph is a data structure consisting of vertices (nodes) connected by edges (lines). Graphs are used to represent data in a wide variety of applications, including social networks, routing, scheduling, and data visualization. It can be used to model relationships between people, objects, and other entities. Graphs can also be used to represent abstract data such as the flow of control in a program or the flow of data in a computer network. Graphs can be directed or undirected, weighted or unweighted, and labeled or unlabeled. Graphs are an important tool in computer science, mathematics, and many other disciplines.

The output for the graph when the input value is 2 is y = 24. This can be calculated using the equation y = 12x - 8, where x is the input value.

To calculate the output, we will substitute the input value of 2 into the equation. This gives us the equation 12(2) - 8 = 24. Simplifying the equation gives us y = 24. Therefore, the output for the graph when the input value is 2 is 24.

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Answer:

D: x=3

Step-by-step explanation:

The volume of the prism is determined as  103.0 unit³.

The volume of the triangular prism is calculated by applying the following formula as shown below;

V = ¹/₂bhl

where;

The base of the prism is calculated as follows;

tan 25 = 4/b

b = 4/tan (25)

b = 8.58 units

The volume of the prism is calculated as follows;

V = ¹/₂ x 8.58 x 6 x 4

V = 103.0 unit³

,

Thus, the volume of the prism is a function of its base, height and length.

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So when x = 6 , y = 4.

Given

x=3 when y=8

To Find

The function that models the inverse variation.

Solution

if  x and y vary inversely, we can use the formula:

xy = k

where k is a constant. We can solve for k using the initial condition x = 3 when y = 8:

3(8) = k

k = 24

So the equation that models the inverse variation is:

xy = 24

We can use this equation to find the value of y for a given value of x, or the value of x for a given value of y. For example, if we want to find y when x = 6:

(6)y = 24

y = 4

So when x = 6, y = 4

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(a) The velocity of sphere A after the collision is 1.67 m/s to the right.

(b) The collision is inelastic.

(c) The velocity of sphere C after the collision is 1.13 m/s at 7.71° to the left of the initial direction of sphere B.

(d) The impulse imparted to sphere B by sphere C is 0.38 kg m/s at 172.3° to the left of the initial direction of sphere B.

(e) The second collision is inelastic.

(f) The velocity of the center of mass of the system of three spheres after the second collision is 1.54 m/s to the right. This can be calculated using the conservation of momentum and the fact that the center of mass of the system moves at a constant velocity if there are no external forces acting on it.

To determine if the collision is elastic or inelastic, we can check if kinetic energy is conserved. The initial kinetic energy of the system is (1/2)(0.6 kg)(4 m/s)² + (1/2)(1.8 kg)(2 m/s)² = 8.64 J. The final kinetic energy of the system is (1/2)(0.6 kg)(0.8 m/s)² + (1/2)(1.8 kg)(3 m/s)² = 19.44 J. Since the final kinetic energy is greater than the initial kinetic energy, we know that the collision is inelastic.

The impulse imparted to sphere B by sphere C is equal to the change in momentum of sphere B. This can be found using the final and initial momenta of sphere B: (1.8 kg)(3 m/s) - (1.8 kg)(cos(19°))(1.4 m/s) = 4.54 kg⋅m/s to the right.

Since kinetic energy is not conserved in the collision between sphere B and sphere C, we know that this collision is also inelastic.

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The complete question is:

Sphere A, of mass 0.600 kg. is initially moving to the right at 4.00 m/s. Sphere B of mass 1.80 kg, is initially to the right of sphere A and moving to the right at 2.00 m/s. After the two spheres collide, sphere B is moving at 3.00 m/s in the same direction as before. (a) What is the velocity (magnitude and direction) of sphere A after this collision? (b) Is this collision elastic or inelastic? (c) Sphere B then has an off-center collision with sphere C, which has mass 1.60 kg and is initially at rest. After this collision, sphere B is moving at 19.0° to its initial direction at 1.40 m/s. What is the velocity (magnitude and direction) of sphere C after this collision? (d) What is the impulse (magnitude and direction) imparted to sphere B by sphere C when they collide? (e) Is this second collision elastic or inelastic? (f)What is the velocity (magnitude and direction) of the center of mass of the system of three spheres (A, B, and C) after the second collision? No external forces act on any of the spheres in this problem.

The expression [tex](x^{2} - 16)(x +2)[/tex] which meets the equation [tex](x^{2} - 16)(x +2) = 0[/tex] for x = 4, x = -4, and x = -2.

In mathematics, an expression is a phrase that has at least two numbers or variables and at least one math operation. Addition, subtraction, multiplication, or division are all examples of math operations. An expression's structure is as follows: (Number/variable, Math Operator, Number/variable) is an expression.

When either the factor ([tex]x^{2}[/tex] - 16) or the factor (x + 2) is equal to zero, or both are equal to zero, the equation[tex](x^{2} - 16)(x + 2) = 0[/tex] is satisfied.

As a result, we must answer the following two equations:

[tex]x^{2}[/tex] - 16 = 0  and x + 2 = 0

To begin, we solve the equation x2 - 16 = 0:

[tex]x^{2}[/tex] - 16 = 0

(x - 4)(x + 4) = 0

x -4 = 0 or x + 4 = 0

x = 4 and x = -4

The equation x + 2 = 0 is then solved:

x + 2 = 0

x = -2

As a result, the x values that meet the equation ([tex]x^{2}[/tex] - 16)(x + 2) = 0 are:

x = 4, x = -4, and x = -2.

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The conclusion is that the distances bird travel is independent of their size. The Option A is correct.

The table shows the size of each bird in grams and the distance each bird traveled in kilometers. Based on the data, the conclusion that the student can make is that the distances bird travel is independent of their size.

The data shows that the smallest bird weighing only 3.0 grams traveled a much greater distance of 276 kilometers compared to the largest bird weighing 25.0 grams which only traveled a distance of 1 kilometer.  Therefore, it is concluded that the size of a bird does not necessarily determine how far it will travel during migration.

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Answer: Zahid obtained 170 marks.

Step-by-step explanation:

Let's start with the basic rules of the question.

We know that for each question answered correctly, 5 marks will be given. And for each incorrect answer, 3 marks will be deducted. Now the problem says that Zahid's marks have been deducted by 18. There are 3 marks deducted for each wrong answer so we'll divide 18 by 3, which gives us 6. Zahid got 6 questions wrong. However, there are 40 questions in the exam, so if we assume that the only ones he answered incorrectly are the 6 questions, then we should subtract 6 from 40. This leaves us with only the correct answers left which is 34. Now again, we know that for each correct answer 5 marks will be given. Assuming that Zahid answered the rest of the questions correctly, we should multiply 34 by 5, which gives us 170.

In numbers your workings might look like this:

18 ÷ 3 = 6

40 - 6 = 34

34 × 5 = 170

I hope this helped you answer your problem. Please let me know if you need any further explanation :)

(a) R is an equivalence relation, we need to prove that it satisfies the following three properties: reflexivity, symmetry, and transitivity.

(b) Reflexivity: For any a ∈ R+, we have aRa, since atb = Vab is equivalent to [tex]a^2 = a^2[/tex], which is true for any positive real number a.

a. Symmetry: For any a, b ∈ R+, if aRb, then bRa. This is because if atb = Vab, then bt a = Vab, which can be rearranged as atb = Vab, showing that bRa.

Transitivity: For any a, b, c ∈ R+, if aRb and bRc, then aRc. This is because if atb = Vab and btc = Vbc, then we can multiply these equations to get atb btc = Vab Vbc, which simplifies to atc = Vabbc. But by the commutativity of multiplication, Vabbc =  [tex]Vabc^2[/tex]. , so we have atc = [tex]Vabc^2[/tex]. Taking the square root of both sides gives atc = Vabc, which shows that aRc.

(b) The distinct equivalence classes resulting from R are the sets of positive real numbers whose arithmetic mean equals their geometric mean. Let us denote one such equivalence class as [a], where a is a positive real number that belongs to the class. Then, for any b ∈ [a], we have atb = Vab, which implies that b =  [tex]a^2/t[/tex]. Thus, every element of [a] is of the form [tex]a^2/t[/tex], where t is a positive real number.

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The area of the region of points satisfying the inequalities x ≤ 0, y ≤ 0, and y ≥ |x+4| - 5 is 4.5 square units.

if you graph the v shape on a graph, V , wiith vertex at (-4, -5) you can then make two triangles using the axis as a border.

The left triangle will have area 25/2

The right triangle witch will be smaller as it is below a rectangle will have area 8 and the rectangle will have area 4

Thus the total area is 49/2

To visualize the region of points satisfying the given inequalities, we can start by graphing the line y = |x+4| - 5.

That |x+4| is equal to x+4 when x is greater than or equal to -4, and -x-4 when x is less than -4.

Therefore, the equation of the line can be expressed as:

y = { x+9, for x ≤ -4 , -x-1, for x > -4

If you square both sides, then you get x+5 = 4[tex]x^2[/tex]

Which becomes polynomial 4[tex]x^2[/tex] -x -5

Factor to (4x-5)(x+1)

x = -1 and x = [tex]\frac{5}{4}[/tex]

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In order to calculate the surface area of a prism, it is necessary to sum up the areas of all its sides. One can obtain this number by using the ensuing formula:

Surface Area = 2B + Ph

The value B represents the area of the base of the prism, P refers to the perimeter, and h pertains to its height. To find the amount of space on the outside of a cylinder, one needs to add up the areas of its curved exterior, along with both circular tops.

The following method may be employed for such a computational process:

Surface Area = 2πr² + 2πrh

In this context, r indicates the radius of the circular foundation, whereas h denotes its altitude measurement.

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The Mean will decrease by 20% and the mode or median may or may not have any impact.

Each style of the candle will cost 20% less if the candle store is having a 20% off deal.

The mean, median, and mode cost per kind of candle will be impacted in the following ways assuming that each candle has a distinct price:

Mean: There will be a 20% decrease in the mean cost of each type of candle. This is so that a lower mean cost per kind of candle may be achieved. The mean is the sum of all prices divided by the total number of candles, thus if each price is decreased by 20%, the sum of prices will also be decreased by 20%.

Median: The sale may or may not have an impact on the median price for each type of candle. This is true because the median, which represents the middle value in a group of data, will not change if the order of the prices is not affected by the sale price.

The median, however, could change to a different number if the sale price results in a change in the ranking of the values.

Mode: The sale may or may not have an impact on the average price for each type of candle. This is true because the mode—the value that appears the most frequently in a set of data—remains same if the sale price does not alter the frequency of the prices.

The mode, however, can change to a different value if the selling price results in a change in the frequency of the prices.

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g'(4) = 18.75.

To find g'(4) given that f(4)=3, f'(4)=9, and g(x)=sqrt(xf(x)), follow these steps:

1. Write down the given information: f(4) = 3, f'(4) = 9, and g(x) = sqrt(xf(x)).
2. Differentiate g(x) using the product rule and chain rule: g'(x) = d(sqrt(xf(x)))/dx.
3. Apply the product rule: g'(x) = (d(sqrt(x))/dx) * (f(x)) + (sqrt(x)) * (df(x)/dx).
4. Differentiate sqrt(x) using the chain rule: d(sqrt(x))/dx = (1/2) * (x^(-1/2)).
5. Plug in the given values of f(4) and f'(4) into the equation: g'(4) = (1/2) * (4^(-1/2)) * (3) + (sqrt(4)) * (9).
6. Simplify the expression: g'(4) = (1/2) * (1/2) * (3) + (2) * (9).
7. Calculate the final result: g'(4) = (3/4) + 18.

So, g'(4) = 18.75.

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Answer: 6π cm

Step-by-step explanation:

The formula for the area of a circle is:

A = πr²

where A is the area and r is the radius.

Given that the area of the circle is 9π cm², we can solve for the radius as follows:

9π = πr²

Dividing both sides by π, we get:

r² = 9

Taking the square root of both sides, we get:

r = 3

Therefore, the radius of the circle is 3 cm.

The formula for the circumference of a circle is:

C = 2πr

Substituting the value of r, we get:

C = 2π(3) = 6π

Therefore, the circumference of the circle is 6π cm.

The following parts can be answered by the concept of Congruent.

For part (a), we need to check whether 37 = 3 (mod 7). Since 37 divided by 7 has remainder 2, the answer is NO.

For part (b), we already know that 66 = 3 (mod 7) because 66 divided by 7 has remainder 3.

For part (c), we need to check whether -17 = 3 (mod 7). To do this, we can add 7 to -17 until we get a positive number that is congruent to -17 modulo 7. We have -17 + 7 = -10, -10 + 7 = -3, and -3 + 7 = 4. Therefore, -17 is congruent to 4 (mod 7) and the answer is NO.

For part (d), we need to check whether -67 = 3 (mod 7). To do this, we can add 7 to -67 until we get a positive number that is congruent to -67 modulo 7. We have -67 + 7 = -60, -60 + 7 = -53, -53 + 7 = -46, -46 + 7 = -39, -39 + 7 = -32, -32 + 7 = -25, -25 + 7 = -18, -18 + 7 = -11, and -11 + 7 = -4.

Therefore, -67 is congruent to -4 (mod 7) and the answer is NO.

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Answer:

it can be written as 0.44y=40

44% can be written as 0.44

Step-by-step explanation:

to solve for y divide both sides by 0.44

to get y is equal to 100

The recursive formulas for each sequence are listed below:

Case 1: aₙ = 4 · aₙ₋₁ + 6

Case 2: aₙ = aₙ₋₁ · 2ⁿ

Case 3: aₙ = aₙ₋₁ + 99

Case 4: aₙ = aₙ₋₁ + n

Case 5: aₙ = aₙ₋₁ · (- 14)

Case 6: aₙ = aₙ₋₁ · n²

In this problem we find six sequences, whose recursive formulas must be determined. This can be done by a trial-and-error approach, this is, using the first element of the sequence and any of the six given sequences.

Case 1: 10, 46, 190, 766

aₙ = 4 · aₙ₋₁ + 6

a₁ = 10

a₂ = 4 · 10 + 6

a₂ = 46

a₃ = 4 · 46  + 6

a₃ = 184 + 6

a₃ = 190

a₄ = 4 · 190 + 6

a₄ = 766

Case 2: 4, 16, 128, 2048, 65536

aₙ = aₙ₋₁ · 2ⁿ

a₁ = 4

a₂ = 4 · 2²

a₂ = 16

a₃ = 16 · 2³

a₃ = 128

a₄ = 128 · 2⁴

a₄ = 2048

a₅ = 2048 · 2⁵

a₅ = 65536

Case 3: - 100, - 1, 98, 197, 296

aₙ = aₙ₋₁ + 99

a₁ = - 100

a₂ = - 100 + 99

a₂ = - 1

a₃ = - 1 + 99

a₃ = 98

a₄ = 98 + 99

a₄ = 197

a₅ = 197 + 99

a₅ = 296

Case 4: 17, 19, 22, 26, 31

aₙ = aₙ₋₁ + n

a₁ = 17

a₂ = 17 + 2

a₂ = 19

a₃ = 19 + 3

a₃ = 22

a₄ = 22 + 4

a₄ = 26

a₅ = 26 + 5

a₅ = 31

Case 5:

aₙ = aₙ₋₁ · (- 14)

a₁ = - 7

a₂ = (- 7) · (- 14)

a₂ = 98

a₃ = 98 · (- 14)

a₃ = - 1372

a₄ = (- 1372) · (- 14)

a₄ = 19208

Case 6: 7, 28, 252, 4032

aₙ = aₙ₋₁ · n²

a₁ = 7

a₂ = 7 · 2²

a₂ = 28

a₃ = 28 · 3²

a₃ = 252

a₄ = 252 · 4²

a₄ = 4032

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x1 = 2; x2 = 3; x3 = -1; x4 = -4

We can write the system of direct equations in stoked matrix form as

(0 3 0 1|-7)

(1 0 2 0|-4)

(3 0 1 2| 12)

(1 1 5 0| 26)

To break the system using row reductions, we perform a series of abecedarian row operations to transfigure the matrix into row stratum form and also into reduced row stratum form. We aim to gain a matrix of the form

(1 * * *| *)

(0 1 * *| *)

(0 0 1 *| *)

(0 0 0 0| 1)

where the non-zero entries in the last column indicate an inconsistency.

Performing the row operations, we get

( 1 0 0 0| 2)

(0 1 0 0| 3)

(0 0 1 0|-1)

(0 0 0 1|-4)

thus, the result of the system of direct equations is

x1 = 2

x2 = 3

x3 = -1

x4 = -4

Since we've attained a unique result, the system of direct equations is harmonious.

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Answer:

id go 48 The circumference is 16π cm, about 50.27 cm.

Step-by-step explanation:

diameter: 16 cm

circumference: 16π cm ≈ 50.27 cm

Step-by-step explanation:

The diameter is twice the radius:

 d = 2r = 2(8 cm)

 d = 16 cm

The diameter is 16 cm.

__

The circumference is pi times the diameter.

 C = πd

 C = π(16 cm)

 C = 16π cm ≈ 50.27 cm

The vertices of the resulting image, Figure C’D’E’F’ are; C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

WE are given that coordinates of the vertices of quadrilateral CDEF are C(6, 6), D(6, 8), E(8, 10), and F(10, 8). The figure is rotated 90° about the origin.

WE can take that point in any of the two surrounding quadrants. Example, if the point is on positive x axis, then it can taken as of first quadrant or fourth quadrant.

On origin, No effect as we assumed rotation is being with respect to origin.

If the figure is rotated clockwise as

C'(6, -6); D'(8, -6); E'(10,-8); F'(8, -10)

If the figure is rotated counterclockwise as

C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

Since clockwise rotation 90 degrees about the origin transforms a point (x, y) to (y, -x).

Also, counterclockwise rotation 90 degrees about the origin transforms a point (x, y) to (-y, x).

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The vertices of the resulting image, Figure C’D’E’F’ are; C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

WE are given that coordinates of the vertices of quadrilateral CDEF are C(6, 6), D(6, 8), E(8, 10), and F(10, 8). The figure is rotated 90° about the origin.

WE can take that point in any of the two surrounding quadrants. Example, if the point is on positive x axis, then it can taken as of first quadrant or fourth quadrant.

On origin, No effect as we assumed rotation is being with respect to origin.

If the figure is rotated clockwise as

C'(6, -6); D'(8, -6); E'(10,-8); F'(8, -10)

If the figure is rotated counterclockwise as

C'(-6, 6); D'(-8, 6); E'(-10, 8); F'(-8, 10)

Since clockwise rotation 90 degrees about the origin transforms a point (x, y) to (y, -x).

Also, counterclockwise rotation 90 degrees about the origin transforms a point (x, y) to (-y, x).

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In 5 possible schedules, John can play basketball on Wednesday or Friday or both.

There are two possible scenarios:

1) John plays basketball on Wednesday only or Friday only, but not both.
- If John plays basketball on Wednesday, he has 2 options left to play on Friday (either play or not play), so there are 2 possibilities.
- If John plays basketball on Friday, he has 2 options left to play on Wednesday, so there are 2 possibilities.
Therefore, there are a total of 2+2=4 possibilities for playing basketball on either Wednesday or Friday, but not both.

2) John plays basketball on both Wednesday and Friday.

In this case, there are only 1 possibility.
So, the total number of possible schedules to play basketball on Wednesday or Friday or both is 4+1=5.

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In 5 possible schedules, John can play basketball on Wednesday or Friday or both.

There are two possible scenarios:

1) John plays basketball on Wednesday only or Friday only, but not both.
- If John plays basketball on Wednesday, he has 2 options left to play on Friday (either play or not play), so there are 2 possibilities.
- If John plays basketball on Friday, he has 2 options left to play on Wednesday, so there are 2 possibilities.
Therefore, there are a total of 2+2=4 possibilities for playing basketball on either Wednesday or Friday, but not both.

2) John plays basketball on both Wednesday and Friday.

In this case, there are only 1 possibility.
So, the total number of possible schedules to play basketball on Wednesday or Friday or both is 4+1=5.

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The particular solution is: ln|y| = (x^2)/6 + ln|4|, Or, alternatively: y = 4*exp((x^2)/6)

To answer your question, let's first discuss the key terms involved:

1. Differential equation: dy/dx = xy/3
2. Table of values
3. Slope field
4. Particular solution with initial condition f(0) = 4

Now let's address your question step by step:

1. We are given the first-order differential equation dy/dx = xy/3.

2. To complete the table of values, you will need to select a set of points (x,y) and calculate the corresponding slopes using the given equation. For example, if you choose the point (1,1), the slope at that point will be dy/dx = (1*1)/3 = 1/3.

3. A slope field is a graphical representation of the slopes at various points on the coordinate plane. To sketch a slope field, draw short line segments at each point in the table with the corresponding slope calculated in step 2.

4. To find the particular solution with the initial condition f(0) = 4, we need to solve the given differential equation. Separate the variables by dividing both sides by y and multiplying both sides by dx:

(dy/y) = (x/3)dx

Now, integrate both sides with respect to their respective variables:

∫(1/y)dy = ∫(x/3)dx + C

ln|y| = (x^2)/6 + C

To find the constant C, use the initial condition f(0) = 4:

ln|4| = (0^2)/6 + C => C = ln|4|

Thus, the particular solution is:

ln|y| = (x^2)/6 + ln|4|

Or, alternatively:

y = 4*exp((x^2)/6)

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a) The sum ∑_(j∈s) j is equal to 1+3+5+7, which equals 16.
b) The sum ∑_(j∈s) j^2 is equal to 1^2+3^2+5^2+7^2, which equals 84.
c) The sum ∑_(j∈s) (1/j) is equal to 1/1+1/3+1/5+1/7, which cannot be simplified further.
d) The sum ∑_(j∈s) 1 is simply the number of elements in s, which is 4.

Given the set s = {1, 3, 5, 7}, here are the values for each sum:
a) ∑_(j∈s) j: This is the sum of all elements in the set. 1 + 3 + 5 + 7 = 16.
b) ∑_(j∈s) j^2: This is the sum of the squares of all elements in the set. 1^2 + 3^2 + 5^2 + 7^2 = 1 + 9 + 25 + 49 = 84.
c) ∑_(j∈s) (1/j): This is the sum of the reciprocals of all elements in the set. 1/1 + 1/3 + 1/5 + 1/7 ≈ 0.271 (rounded to three decimal places).
d) ∑_(j∈s) 1: This sum is asking for the sum of the number 1 repeated the same number of times as there are elements in the set. Since there are 4 elements in s, the sum is 1 + 1 + 1 + 1 = 4.

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The given series [tex]\sigma_n=1^\infty 5/n^3[/tex] is convergent.

The series given is:

Σₙ= [tex]1^\infty[/tex] 5/n³

The nth partial sum of this series is given by:

[tex]S_n = \sigma_k=1^n 5/k^3[/tex]

To calculate the first eight terms of the sequence of partial sums, we substitute n = 1, 2, 3, ..., 8 in the expression for Sₙ:

Rounding each partial sum to four decimal places, we get:

Based on these partial sums, it appears that the series is convergent. As we compute more and more terms of the sequence of partial sums, we observe that the sums increase, but at a decreasing rate, which suggests convergence.

To show that the series is convergent, we need to show that the sequence of partial sums approaches a finite limit as n approaches infinity.

We can use the Integral Test to show that the series converges. According to the Integral Test, if the series [tex]\sigma_n=1^\infty a_n[/tex]  is a series of non-negative terms and the integral from 1 to infinity of a continuous, positive, decreasing function f(x) is finite, then the series converges.

In this case, we can use f(x) = 5/x³ as the function and integrate from 1 to infinity:

Integral from 1 to infinity of 5/x³ dx = [5/(-2x²)] from 1 to infinity

= [tex]-5/2 \lim (x- > \infty)[1/x^2 - 1/1][/tex]

= 5/2

Since the integral of f(x) is finite, the series [tex]\sigma_n=1^\infty 5/n^3[/tex] converges by the Integral Test.

Therefore, the given series is convergent, as observed from the partial sums, and the sum of the series can be found by taking the limit of the sequence of partial sums as n approaches infinity.

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