questions 5 and 6 please!
will give brainliest to whoever answers
90 points

After considering the given interval we reach the conclusion that the coefficient of t² is 1/2(3w² - 1), under the condition that the given expression is [tex](1-2 t w+t^{2})^{-1/2}[/tex] with range of (t<<1)

To evaluate the value for the given expression we have to apply the principles of binomial theorem

Then

[tex](1-2 t w+t^{2})^{-1/2} = (1 + (-2 t w + t^{2})/2 + (-2 t w + t^{2})^{2/8} + (-2 t w + t^{2})^{3/16} + ...)[/tex]

= 1 - t w + 3/8 * t² * w² - 5/16 * t³ * w³ +

The coefficient of t is the coefficient of the first term with a power of t.

Therefore, the coefficient of t is 1/2(3w² - 1).

The binomial theorem refers to the statement regarding any positive integer n, the nth power of the sum of two numbers a and b could be expressed as the sum of n + 1 terms of the form. The binomial theorem is applied in algebra and probability theory.

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Given statement solution is :- Regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

Let's analyze the two given statements:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)

To prove this statement, we can use logical equivalences and deductions:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Given)

¬{[¬Q ∨ (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{[(¬Q ∨ ¬S) ∧ (¬Q ∨ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{(¬Q ∨ ¬S ∨ ¬P) ∧ (¬Q ∨ R ∨ ¬P)} → (¬P ↔ P) (De Morgan's Law)

(Q ∧ S ∧ P) ∨ (¬Q ∧ P) → (¬P ↔ P) (De Morgan's Law)

At this point, the formula ¬P ↔ P is a contradiction. The left side (¬P) states that P is false, while the right side (P) states that P is true. Therefore, the whole statement is always true, regardless of the values of Q, S, and P.

Since the statement is always true, we can conclude ¬(Q ∧ P) ∨ (T ∧ R) is true as well, making the second part of the question valid.

[(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]

In this statement, we have two identical sub-statements connected by a logical OR operator.

If we assume that (T ∨ Q) → (T ∨ S) is true, then the whole statement is true, regardless of the values of T, Q, and S.

If we assume that (T ∨ Q) → (T ∨ S) is false, then we need to check the other part of the statement.

Assuming (T ∨ Q) → (T ∨ S) is false means that (T ∨ Q) is true, but (T ∨ S) is false. In this case, the second part of the statement [(T ∨ Q) → (T ∨ S)] is true.

Therefore, regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

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Answer:

Distance between points is 25 Units.

Step-by-step explanation:

To find the distance between two points, we can use the distance formula. The formula is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Given the two points: (12, 5) and (-12, -2), we can assign the values as follows:

(x1, y1) = (12, 5)

(x2, y2) = (-12, -2)

Plugging the values into the distance formula, we get:

d = sqrt((-12 - 12)^2 + (-2 - 5)^2)

= sqrt((-24)^2 + (-7)^2)

= sqrt(576 + 49)

= sqrt(625)

= 25

Therefore, the distance between the points (12, 5) and (-12, -2) is 25 units.

The average rate of change of the function over the interval is 12

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

f(x) = x² + 2x + 1

The interval is given as

From x = 4 to x = 6

The function is a quadratic function

This means that it does not have a constant average rate of change

So, we have

f(4) = 4² + 2(4) + 1 = 25

f(6) = 6² + 2(6) + 1 = 49

Next, we have

Rate = (49 - 25)/(6 - 4)

Evaluate

Rate = 12

Hence, the rate is 12

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Answer:

Step-by-step explanation:

148 people

The probability of picking a blue marble and then a green marble without replacing the blue one is 2/15.

From the figure we can write

Number of Blue marbles = 2

Number of Green marbles = 6

Number of Purple marbles = 2

Total number of marbles = 2 + 6 + 2 = 10

Now, Probability of picking a blue marble

= Number of Blue marbles / Total number of marbles

= 2 / 10

= 1/5

After removing the blue marble, the total number of marbles becomes 9

Probability of picking a green marble

= 6 / 9

= 2/3

So, the probability of picking a blue marble and then a green marble

Probability = (1/5) x (2/3) = 2/15.

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The values of the composite functions are

(1) (f + g)(x) = 4x² + 2x - 2

(2) (fg)(x) = 8x³ - 4x² - 2x + 1

(3) (fⁿ)(x) = (4x² - 1)ⁿ

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

f(x) = 4x² - 1

g(x) = 2x - 1

Using the above as a guide, we have the following:

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 4x² - 1 + 2x - 1

Evaluate the like terms

(f + g)(x) = 4x² + 2x - 2

(fg)(x) = f(x) * g(x)

(fg)(x) = (4x² - 1) * (2x - 1)

Evaluate the products

(fg)(x) = 8x³ - 4x² - 2x + 1

(fⁿ)(x) = (f(x))ⁿ

So, we have

(fⁿ)(x) = (4x² - 1)ⁿ

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g is a function because each student is uniquely mapped to each course the student takes in a school year.

In Mathematics and Geometry, a function refers to any mathematical equation which is typically used to define and represent a relationship that exists between two or more variables such as an ordered pair, points on a graph or table.

Based on the description of g, we can reasonably infer and logically deduce that the relation g represent a function because the input values are uniquely mapped to the output values.

In this context, we can conclude that the description of g represents a function because a student represent the input values that is being to each course (output value) the student takes in a school year.

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The translation used to create the image A'B'C'D', from the pre image ABCD is T(7, 0), which corresponds with the option;

7 units to the right

The coordinates of the vertices of the polygon ABCD are;

A(-2, -1), B(0, -4), C(4, -4), D(2, -1)

The coordinates of the vertices of the polygon A'B'C'D' are;

A'(5, -1), B'(7, -4), C'(11, -4), D'(9, -1)

The coordinates of the vertices of the image indicates;

The difference between the coordinates are;

A'(5, -1) - A(-2, -1) = (5 - (-2), -1 - (-1)) = (7, 0)

B'(7, -4) - B(0, -4) = (7 - 0, -4 - (-4)) = (7, 0)

C'(11, -4) - C(4, -4) = (11 - 4, -4 - (-4)) = (7, 0)

D'(9, -1) - D(2, -1) = (9 - 2, -1 - (-1)) = (7, 0)

The translation that is used to create the image is therefore;

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Answer:

  See attached

Step-by-step explanation:

You want the given numbers classified as to Natural, Whole, Integer, Rational, Irrational, and/or Real.

All the given numbers are real.

The number √2 is irrational. All the others are rational.

The values √16 = 4, 0, and -5 are integer values. Whole numbers are non-negative integers, so exclude -5. Natural numbers are positive integers, so exclude 0 and -5.

The Xs indicate the categories each number belongs to.

The probability of the digestive tract disease, given a positive test result, is 45%.

A. Two-Way Frequency Table:

Let's create a two-way frequency table to represent the situation:

                                      Test Positive             Test Negative           Total

Disease Present                   A                                 B              5,000

Disease Not Present           C                                 D              95,000

Total                                   5,000                     95,000     100,000

In this table:

B. Probability of Disease Given Positive Test Result:

P(Disease | Positive)

= (P(Positive | Disease) x P(Disease)) / P(Positive)

= 0.94  x 0.05 / P(Positive)

To calculate P(Positive), we can use the law of total probability:

P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease)

So, P(Positive | No Disease) = 1 - P(Test Negative | No Disease)

= 1 - 0.94 = 0.06

P(No Disease) = 1 - P(Disease) = 1 - 0.05 = 0.95

Now, P(Positive) = (0.94 x 0.05) + (0.06 x 0.95)

= 0.047 + 0.057

= 0.104

Finally, we can calculate P(Disease | Positive):

P(Disease | Positive) = (0.94 x 0.05) / 0.104

= 0.047 / 0.104

≈ 0.452

Therefore, the probability of the digestive tract disease, given a positive test result, is 45%.

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The difference in height is 400 metres.

Height refers to vertical distance from the top to the object's base. Occasionally, it is also labeled as an altitude which measures from down to top of a surface.

To get difference in height, we will subtract the height after the eruption from the height before the eruption:

Given:

Height before eruption = 2.95 kilometers

Height after eruption = 2.55 kilometers

Difference in height = Height before eruption - Height after eruption

Difference in height = 2.95 km - 2.55 km

Difference in height = 0.4 kilometers

0.4 kilometers to metres will be:

= 0.4 * 1000 metres

= 400 metres.

Full question:

Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Find the difference in height of Mount St. Helens before and after the eruption.

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The missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.

Local government sector's revenue in 2017/18 (L): Missing value (to be calculated)

Provincial sector's revenue in 2017/18 (P): 410.6 (billion)

According to the information provided, L is 20.12% of P. Mathematically, we can write this as:

L = 20.12/100 * P

Substituting the known value of P, we get:

L = 20.12/100 * 410.6

L ≈ 82.61 (rounded to two decimal places)

Therefore, the missing value E, the total revenue for the period 2017/18, is approximately 82.61 billion.

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Answer: To determine which statement is true about lines r and s, we need to compare their slopes.

The slope of a line can be calculated using the formula:

slope = (change in y-coordinates) / (change in x-coordinates)

For line r, using the points (-5, 2) and (-3, 8):

slope_r = (8 - 2) / (-3 - (-5))

= 6 / 2

= 3

For line s, using the points (-6, 4) and (2, 12):

slope_s = (12 - 4) / (2 - (-6))

= 8 / 8

= 1

Now, let's analyze the statements:

Line r is steeper than line s. (False)

Since the slope of line r (3) is greater than the slope of line s (1), this statement is true.

Line r is steeper than line (Incomplete statement)

This statement is incomplete.

Line s has a negative slope. (False)

The slope of line s (1) is positive, not negative. So, this statement is false.

Line s is steeper than line r. (False)

Since the slope of line s (1) is less than the slope of line r (3), this statement is false.

Line r has a negative slope. (False)

The slope of line r (3) is positive, not negative. So, this statement is false.

Based on the analysis, the true statement about lines r and s is:

Line r is steeper than line s.

With the coordinate provided, the lengths of the triangle are Side AB =  5 units, Side BC = 10 units and Side CA = 12.4 units.

To find the side lengths of the triangle using the coordinates provided, we use the distance formul.

The distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).

Side AB: Distance between A(-7, 3) and B(-3, 6)

= √((-3 - -7)² + (6 - 3)²)

= √((4)² + (3)²)

= √(16 + 9)

= √25

= 5 units

Side BC: Distance between B(-3, 6) and C(5, 0)

= √((5 - -3)² + (0 - 6)²)

= √((8)² + (-6)²)

= √(64 + 36)

= √100

= 10 units

Side CA: Distance between C(5, 0) and A(-7, 3)

= √((-7 - 5)² + (3 - 0)²)

= √((-12)² + 3²)

= √(144 + 9)

= √153

= 12.4 units

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Answer:

Step-by-step explanation:

area of a rectangular piece of cloth 3 1/4m by 25 cm

the area of ​​the rectangle is found by making the base by the height, we convert the meters into centimeters

3 1/4 m = 3.25m = 325 cm

we find the area

325 × 25 = 8125 cm² (or 0.8125 m²)

The probability that either event will occur is 0.62

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

Event A = 6 + 6 = 12

Event B = 20 + 6 = 26

Both A and B = 6

Other Events = 20

Using the above as a guide, we have the following:

Total = A + B + C + Others - Both

So, we have

Total = 12 + 26 - 6 + 20

Evaluate

Total = 52

So, we have

P(A) = 12/52

P(B) = 26/52

Both A and B = 6/52

For either events, we have

P(A or B) = (12 + 26 - 6)/52

Evaluate

P(A or B) = 0.62

Hence, the probability that either event will occur is 0.62

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Between 40 and 60 minutes the target heart rate strictly increasing then strictly decreasing. Therefore, option B is the correct answer.

From the given graph, in the first curve between the interval 0-10 speed is increasing,  between the interval 10-30 speed is constant and  between the interval 30-40 speed is decreasing.

In the second curve between the interval 40-50 speed is increasing,  between the interval 50-60 speed is decreasing, between the interval 60-65 speed is increasing and between the interval 65-73 speed is constant.

In the third curve between the interval 73-85 speed is increasing,  between the interval 85-92 speed is constant and  between the interval 92-100 speed is decreasing.

Therefore, option B is the correct answer.

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There are 75 triangles that can be formed using these 9 vertices.

Here we have,

We have 9 vertices arranged in a pattern where 5 vertices are above and 4 vertices are below in a parallel direction.

And we want to know how many triangles can be formed using these vertices.

In order to this,

Count the number of ways to choose 3 vertices from the 9 vertices.

This can be done using the combination formula, which is:

[tex]^{9}C_{3}[/tex] = 9! / (3! * (9-3)!)

      = 84.

Since,

A degenerate triangle is one where all three vertices lie on the same line. To count the number of degenerate triangles,

We need to count how many ways we can choose 3 vertices that lie on the same line.

There are 5 lines with 3 points each (the 5 lines with the upper vertices). So the number of degenerate triangles is:

5 [tex]^{3}C_{3}[/tex] + 4  [tex]^{3}C_{3}[/tex] = 9.

Subtract the number of degenerate triangles from the total number of triangles to get the number of non-degenerate triangles.

Therefore, the number of non-degenerate triangles is:

84 - 9 = 75.

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his pay per workday will be $127.45.

Thus, the formula for calculating simple interest is J = C i t, where J is interest, C is principal, i is interest rate, and t is time. That is, J represents the amount added to the initial value, the capital is the calculation basis, the interest rate is the percentage applied on the capital and time is the capitalization period.

Knowing that:

The total weekends in a year:

52 * 2 = 104 weekend days.

Total paid vacation days, including holidays that fall on weekends:

10 - 4 = 6 paid vacation days.

Total non-working days:

104 + 6 = 110 non-working days.

Total working days in a year:

365 - 110 = 255 working days.

Pay per workday = Annual salary / Total working days

= $32,500 / 255= $127.45

Therefore, the librarian's pay per workday is approximately $127.45.

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Answer:

48in.²

Step-by-step explanation:

The formula for the area of a triangle is 1/2 x b x h

Since the b = 8 and the h = 12, substitute 8 and 12 for b and h.

1/2 x 8 x 12

4 x 12

48in.²

Answer:

d

Step-by-step explanation:

area= ½ × b × h

= ½ × 8 × 12

= ½ × 96

= ⁹⁶/²

= 48in²

Answer:3rd one

Step-by-step explanation:

Answer:

The other sides = 5 centimeters.

Step-by-step explanation:

Let's assume the length of the other two sides of the rectangle is "x" centimeters.

A rectangle has two pairs of equal sides. Therefore, we can set up the equation for the perimeter of the rectangle:

Perimeter = 2(length + width)

Given:

Length = 6 centimeters

Perimeter = 22 centimeters

Using the equation, we can substitute the given values:

22 = 2(6 + x)

Simplifying further:

22 = 12 + 2x

Subtracting 12 from both sides:

10 = 2x

Dividing both sides by 2:

5 = x

Therefore, the length of the other two sides of the rectangle is 5 centimeters.

To find the area of a tent, including the floor, we need to measure or determine the dimensions of both the tent's floor and any additional areas such as vestibules or extensions.

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The solution to the inequality expression is -8/9 ≤ x ≥ 4/9

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

|3x + 6/9| ≥ 2

Expand the inequality expression

So, we have

-2 ≤ 3x + 6/9 ≥ 2

Subtract 6/9 from both sides

So, we have

-24/9 ≤ 3x ≥ 12/9

Divide through the equation by 3

-8/9 ≤ x ≥ 4/9

Hence, the solution to the inequality expression is -8/9 ≤ x ≥ 4/9

The graph is attached

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Answer:

The correct answer is (d): (x - 5) is a factor of f(x).

Step-by-step explanation:

The range of the function g(x) = f(x+2)-5 is:

R = [0, ∞)

For a function y = f(x), we define the range as the set of possible outputs of the function.

Here we want to find the range of:

g(x) = f(x+2)-5

Where f(x) = x²

Replacing f(x) in the rule for g, we will get the composition:

g(x) = (x + 2)² - 5

The first part can only be a positive number or 0, then the minimum of the range is -5

And then the function only grows, then the range of g(x) is:

R = [0, ∞)

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The graph of the function is attached and the amplitude and the period are 5 and 2π

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

f(x) = 5cos(θ)

A sinusoidal function is represented as

f(x) = Acos(B(x + C)) + D

Where

So, we have

A = 5

Period = 2π/1

Evaluate

A = 5

Period = 2π

Hence, the amplitude is 5 and the period is 2π

The graph of the function is attached

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Answer: 59,048

Step-by-step explanation:

To find the sum of the given finite series:

Σ2· (3)n-1

n=1

where n ranges from 1 to 10, we can use the formula for the sum of a geometric series.

The formula for the sum of a geometric series is:

S = a * (r^n - 1) / (r - 1)

In this case, a = 2 (the first term of the series) and r = 3 (the common ratio).

Using the formula, we can calculate the sum:

S = 2 * (3^10 - 1) / (3 - 1)

= 2 * (59049 - 1) / 2

= (118098 - 2) / 2

= 118096 / 2

= 59048

Therefore, the sum of the given finite series is 59,048.

Answer: i think c

Step-by-step explanation: