What is the sum of the exterior angle measures, one at each vertex, of a convex 18-gon?
°
help literally

Answer:

We can start by using the formula:

distance = speed x time

We know that Oliver traveled 66 miles in 4 hours, so we can use this information to find his speed:

speed = distance / time

speed = 66 miles / 4 hours

speed = 16.5 miles per hour

Now that we know Oliver's speed, we can use the same formula to find how many miles he could travel in 14 hours:

distance = speed x time

distance = 16.5 miles per hour x 14 hours

distance = 231 miles

Therefore, Oliver could travel 231 miles in 14 hours at this speed.

√111: Irrational number

0: Whole number, Rational number

Π (Pi): Irrational number

71: Natural number, Whole number, Rational number

√-81: Non-real number

√111: Irrational number - The square root of 111 is an irrational number because it cannot be expressed as a fraction or a terminating or repeating decimal.

0: Whole number, Rational number - Zero is a whole number because it is a non-negative integer. It is also a rational number because it can be expressed as the ratio 0/1.

Π (Pi): Irrational number - Pi is an irrational number because it is a non-repeating, non-terminating decimal. It cannot be expressed as a fraction.

71: Natural number, Whole number, Rational number - 71 is a natural number because it is a positive integer. It is also a whole number and a rational number because it can be expressed as the ratio 71/1.

√-81: Non-real number - The square root of -81 is a non-real number because it involves the square root of a negative number. It cannot be expressed as a real number.

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a. Covariance = 10.11
b. Correlation coefficient = 0.376

Considering the following sample data:

x 12 18 20 22 25

y 15 20 25 22 27

a. Calculation of covariance

Covariance can be calculated by the formula:

Cov (X, Y) = [Σ(X - μx) (Y - μy)] / n

where, Σ denotes the sum of, X and Y are the variables, μx and μy are the means of X and Y respectively, and n is the sample size.

x  y  x-μx  y-μy  (x-μx)(y-μy) (-)^2  (-)^2

12  15  -6.6  -5.6  37.12  43.56  31.36

18  20  -0.6  -0.6  0.36  0.36  0.36

20  25  1.4  4.4  6.16           1.96  19.36

22  22  3.4  -2.6  -8.84  11.56  6.76

25  27  6.4  2.4  15.36  41.16  5.76

Σx = 97, Σy = 109, Σ(x-μx) = 5.00, Σ(y-μy) = -2.00, Σ(x-μx)(y-μy) = 50.56

Covariance is: Cov (X, Y) = [Σ(X - μx) (Y - μy)] / n= 50.56/5= 10.11

Thus, the covariance between the variables is 10.11.

b-1. Calculation of correlation coefficient.

Correlation coefficient is a statistical measure that measures the degree to which two random variables are associated. It can be calculated by the formula:

= Cov (X, Y) /  where, Cov (X, Y) is the covariance between X and Y, σX and σY are the standard deviations of X and Y respectively.

σx2 = [Σ(x-μx)2] / (n-1)σy2 = [Σ(y-μy)2] / (n-1)σx = √[Σ(x-μx)2] / (n-1)σy = √[Σ(y-μy)2] / (n-1)

x  y  (x-μx)  (y-μy)  (x-μx)2  (y-μy)2  (-)(-)

12  15  -6.6          -5.6         43.56  31.36               1  

18  -0.6  -0.6          0.36         0.36  0.32                5

20  25    1.4          4.4          1.96          19.36               22  

22  3.4  -2.6          11.56  6.76          -8.84                25  

27  6.4  2.4          41.16  5.76          15.36

Σx = 97, Σy = 109, Σ(x-μx) = 5.00, Σ(y-μy) = -2.00, Σ(x-μx)(y-μy) = 50.56

σx2 = 30.70

σy2 = 25.70

σx = √30.70 = 5.54

σy = √25.70 = 5.07

Correlation coefficient is:

= Cov (X, Y) /  = 10.11 / (5.54*5.07)= 0.376

Thus, the correlation coefficient is 0.376.

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Answer:

Step-by-step explanation:

[tex]y=x^2+10x+8=(x+5)^{2}-17[/tex]

Answer:

y = (x + 5)^2 - 17

Step-by-step explanation:

To write the quadratic function y = x^2 + 10x + 8 in vertex form, we need to complete the square. We start by adding and subtracting the square of half of the coefficient of x, which is (10/2)^2 = 25:

y = x^2 + 10x + 8

= (x^2 + 10x + 25) - 25 + 8

= (x + 5)^2 - 17

Therefore, the quadratic function in vertex form is:

y = (x + 5)^2 - 17

The vertex of this parabola is at the point (-5, -17), and the axis of symmetry is the vertical line x = -5. The term (-17) represents the minimum value of the function.

On solving the inequalities 2(5-x) > 6 and 22 > 2(9+x) the correct comparison is "The inequalities have the same solutions."

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

To solve the inequality 2(5-x) > 6, we can simplify as follows -

2(5-x) > 6

10 - 2x > 6

-2x > 6 - 10

-2x > -4

x < 2

So the solution to this inequality is x < 2.

To solve the inequality 22 > 2(9+x), we can simplify as follows -

22 > 2(9+x)

22 > 18 + 2x

4 > 2x

2 > x

So the solution to this inequality is x < 2.

Both inequalities have the same solution, x < 2.

Therefore, the correct comparison of solutions is option D.

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The probabilities regarding a person spending are given as follows:

a) More than 28: 0.2033 = 20.33%.

b) Between 9 and 21: 0.3608 = 36.08%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 23, \sigma = 6[/tex]

The probability of a person spending more than 28 is one subtracted by the p-value of Z when X = 28, hence:

Z = (28 - 23)/6

Z = 0.83

Z = 0.83 has a p-value of 0.7967.

1 - 0.7967 = 0.2033 = 20.33%.

The probability of spending between 9 and 21 is the p-value of Z when X = 21 subtracted by the p-value of Z when X = 9, hence:

Z = (21 - 23)/6

Z = -0.33

Z = -0.33 has a p-value of 0.3707.

Z = (9 - 23)/6

Z = -2.33

Z = -2.33 has a p-value of 0.0099.

Hence:

0.3707 - 0.0099 = 0.3608 = 36.08%.

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The most likely measurement of the distance from the earth to the moon would be 1.3 x 10^8 ft.

Therefore the answer is A)1. 3 x 10⁸ ft.

This is because the distance from the earth to the moon is approximately 238,900 miles, which is equivalent to approximately 1.3 x 10^8 feet. Options B, C, and D are all significantly larger or smaller than this value and do not reflect the actual distance from the earth to the moon.

In general, measurements of distance can be expressed in a variety of units, such as feet, meters, or miles. It's important to use the correct unit when making calculations or comparisons to ensure that the results are accurate and meaningful. When dealing with very large or very small distances, scientific notation can be a useful way to express the measurement in a compact and standardized form.

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--The question is incomplete, answering to the question below--

"which value would be the most likely measurement of the distance from the earth to the moon?

A)1. 3 x 10⁸ ft.

B)1. 3 x 10⁻⁹ ft.

C)1. 3 x 10¹⁰⁰ ft.

D)1. 3 x 10² ft."

Answer:

3.8

Step-by-step explanation:

using sinθ = opp/hypo

sin(25) = x/9

0.4226 = x/9

x = 9(0.4226) = 3.8

The constraints of inequalities are 3x + 4y ≤ 640 and 75x + 60y ≤ 12900

Represent the types of cellphones with x and y

Using the problem statements, we have the following table of values

                         x             y       Available

Labor (hours)   3            4         640

Materials ($)     75         60       12900

From the above, we have the following constraints of inequalities:

3x + 4y ≤ 640              

75x + 60y ≤ 12900

The graph of the inequalities and the shaded region are added as an attachment

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Answer:

x  = -26/7

Step-by-step explanation:

Cancel terms that are in both the numerator and denominator

Multiply the numbers

Distribute

Subtract the numbers

Rearrange terms

Subtract the numbers

Multiply the numbers

Answer:

To solve this equation, we need to use the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, we need to simplify the expression inside the parentheses:

6 - 3/3(7x + 2) = 6(8 - 3)

6 - 1(7x + 2) = 6(5)

6 - 7x - 2 = 30

4 - 7x = 30

Next, we need to isolate the variable (x) on one side of the equation. We can do this by subtracting 4 from both sides:

4 - 7x - 4 = 30 - 4

-7x = 26

Finally, we can solve for x by dividing both sides by -7:

x = -26/7

Therefore, the solution to the equation is x = -26/7.

The factored form of 196x²- y² is (14x + y)(14x - y).

A factored form is a parenthesized algebraic expression. In effect a factored form is a product of sums of products, or a sum of products of sums. Any logic function can be represented by a factored form, and any factored form is a representation of some logic function.

The slope intercept form of a straight line is one of the most common forms used to represent the equation of a line. The slope intercept formula can be used to find the equation of a line when given the slope of the straight line and the y-intercept. The formula is y=mx+b.

The expression 196x² - y² can be factored using the difference of squares formula, which states that:

a²- b² = (a + b)(a - b)

In this case, we have a = 14x and b = y, so we can write:

196x² - y² = (14x + y)(14x - y)

Therefore, the factored form of 196x²- y² is (14x + y)(14x - y).

The expression (14x + y)(14x - y) is the factored form of a quadratic expression and does not represent a linear equation that can be written in slope-intercept form.

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The distance from first base to dead center field in a certain baseball stadium is 338 feet.

Explanation:

The distance from first base to dead center field in a certain baseball stadium is 338 feet. Given,The distance from home plate to dead center field in a certain baseball stadium is 407 feet.A baseball diamond is a square with a distance from home plate to first base of 90 feet.

To find,How far is it from first base to dead center field?

Solution:Given that the distance from home plate to dead center field is 407 feet.The baseball diamond is a square with a distance from home plate to first base of 90 feet.Now we have to find the distance from first base to dead center field.We can find the distance by using the Pythagorean theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.Let us consider a right triangle ABC where AB represents the distance from home plate to first base, AC represents the distance from home plate to dead center field, and BC represents the distance from first base to dead center field.

As per the Pythagorean theorem, we have

AC² = AB² + BC²

Putting the values, we have

AC² = (90)² + BC²AC² = 8100 + BC²AC² - BC² = 8100

Taking the square root on both sides, we getAC = √(8100 + BC²)

Now we have AC = 407 ft,AB = 90 ftAC² = AB² + BC²407² = 90² + BC²BC² = 407² - 90²BC² = 165649BC = √165649BC = 407 ft - 90 ft

BC = 338 ft

Therefore, the distance from first base to dead center field in a certain baseball stadium is 338 feet.

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The values of the variables can be proved by solving the the equations to get;

8. y = 3

9. k = -2

10. w = 14

11. x = -9

An equation is a statement that indicates that two expressions are equivalent by joining them with the '=' sign.

The method used to prove the value of the variable is by solving the equations as follows;

8. (5·y - 1)/2 = 7

Therefore;

2 × 7 = 5·y - 1

14 = 5·y - 1

5·y = 14 + 1 = 15

y = 15/5 = 3

y = 3

9. 10·k - 4 = 2·k - 20

Therefore;

10·k - 2·k = 8·k = 4 - 20 = -16

8·k = -16

k = -16/8 = -2

Therefore;

k = -2

10. -8·(w + 1) = -5·(w + 10)

-8·w - 8 = -5·w - 50

Therefore;

-5·w + 8·w = 50 - 8 = 42

3·w = 42

w = 42/3 = 14

Therefore;

w = 14

11. 14 - 2·(x + 8) = 5·x - (3·x - 34)

Therefore;

14 - 2·x - 16 = 5·x - 3·x + 34

14 - 2·x - 16 = -2·x - 2

5·x - 3·x + 34 = 2·x + 34

Therefore;

-2·x - 2 = 2·x + 34

2·x + 2·x = -2 - 34 = -36

4·x = -36

x = -36/4 = -9

x = -9

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The basketball will reach its maximum height 1.735 seconds after it has been thrown vertically into the air.

A function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. A function can be represented as an equation, a graph, a table of values, or a verbal description. For example, the function f(x) = 2x + 1 represents a relationship between the input x and the output 2x + 1.

To find the maximum height of the basketball, we need to find the vertex of the parabola represented by the function f(x). The vertex of x-coordinate is:

x = -b/2a

The coefficients of the quadratic equation a[tex]x^2[/tex] + b[tex]x[/tex] + c are a, b, and c. In this case, a = -4.9 and b = 17, so:

x = -17/(2*(-4.9)) = 1.735 (rounded to 3 decimal places)

Therefore, the basketball will reach its maximum height 1.735 seconds after it has been thrown vertically into the air.

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Answer: 20%

Step-by-step explanation:

150,000 - 125,000 = 25,000

20 percent of 125,000 = 25k

Answer:

D

Step-by-step explanation:

every term of one expression gets multiplied with every term of the other expression.

(6x² - 3x - 6)(4x² + 5x + 4) =

= 6×4x²×x² + 6×5x²×x + 6×4x² - 3×4x×x² - 3×5x×x -

3×4x - 6×4x² - 6×5x - 6×4

3 terms × 3 terms = 9 terms.

now we combine similar factors for the 9 terms

24x⁴ + 30x³ + 24x² - 12x³ - 15x² - 12x - 24x² - 30x - 24

and now we combine similar terms

24x⁴ + 18x³ - 15x² - 42x - 24

Answer: -6(p - 6) can be simplified using the distributive property of multiplication:

-6(p - 6) = -6 * p - (-6 * 6)

= -6p - (-36)

= -6p + 36

Therefore, -6(p - 6) is equivalent to -6p + 36.

Step-by-step explanation:

Answer:

Option A.

Step-by-step explanation:

A 30° - 60° - 90° triangle is a Right Triangle that has special side measures.

Let's summarize the sides in a ratio.

[tex]Short \ Leg: Long \ Leg: Hypotenuse\\x:x \sqrt{3} : 2x[/tex]

The short leg is just x.

The long leg is multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is double of the short leg.

For example, if the short leg is 2;

[tex]Short \ Leg = 2\\Long \ Leg = 2\sqrt{3} \\Hypotenuse = 4[/tex]

Let's look at the 4 options provided. We should check if the values of the sides match with a 30° - 60° - 90° triangle.

Option A has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option A is correct!

Option B has a short leg with the value of 5.

The long leg is correct, it's multiplied by [tex]\sqrt{3}[/tex].

The hypotenuse is incorrect, it's triple of the short leg.

Option B is incorrect.

Option C has a short leg with the value of 5.

The long leg is incorrect, it's multiplied by [tex]2\sqrt{3}[/tex].

The hypotenuse is correct, it's double of the short leg.

Option C is incorrect.

Option D has a short leg with the value of 10.

The long leg is incorrect, it's multiplied by [tex]\frac{1}{2} \sqrt{3}[/tex].

The hypotenuse is incorrect, it's [tex]1 \frac{1}{2}[/tex] of the short leg.

Option D is incorrect.

Our only 30° - 60° - 90° triangle is Option A.

Answer:

250 cubic cm

Step-by-step explanation:

Side length = x

[tex]2x^{2} + 4x(10) = 120[/tex]

[tex]x^{2} +2x - 30 =0[/tex]
After factorization, we will get (x+6) ( x-5) = 0

side length should be positive, so we take x to be 5.
Dimensions will be 5 x 5 x 10 = 250 cubic centimeters.

The coordinates of point Y are (10 - x1, 4y1 - 10), where (x1, y1) are the coordinates of point T.

If W is the midpoint of segment TY, then the coordinates of W are the average of the coordinates of T and Y. Using the midpoint formula, we can find the coordinates of Y:

Let the coordinates of T be (x1, y1) and the coordinates of Y be (x2, y2).

x-coordinate of W = (x-coordinate of T + x-coordinate of Y) / 2

y-coordinate of W = (y-coordinate of T + y-coordinate of Y) / 2

Putting in the coordinates of W and T, we get:  5 = (x1 + x2) / 2

y-coordinate of Y = 2y1 - y-coordinate of W

y-coordinate of Y = 2y1 - (y1 + y2) / 2

Simplifying these equations, we get:

x1 + x2 = 10

y2 = 4y1 - 10

From the first equation, we can solve for x2:  x2 = 10 - x1

Putting this into the second equation, we get:  y2 = 4y1 - 10

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Answer:

Farrah borrowed $155 from her brother and has paid back $15 so far. She plans to pay back $35 each month until the debt is paid off.

To determine the number of months it will take to pay off the debt, we need to solve the equation:

x * 35 + 15 = 155

where x is the number of months it will take to pay off the debt.

Simplifying the equation, we get:

x * 35 = 155 - 15

x * 35 = 140

x = 4

Therefore, it will take 4 months to pay off the debt.

Options that describe the number of months it will take to pay off the debt are:

Step-by-step explanation:

Answer:

vro I guess this is locating root 5

Step-by-step explanation:

mark me BRAINLIST

The mass of 5600 paper clips is 5.6 kilograms.

Here we use the unitary method to solve the problem. The unitary method is a mathematical technique used to solve problems.

It involves finding the value of one unit of a given quantity and then using that value to determine the value of other units of the same or different quantities.

Here we have

One paperclip has a mass of 1 gram.  

Mass of 1000 paperclips = 1 kilogram  

The mass of 1 paperclip in kilogram = 1/1000 = 0.001 kg

Similarly

Mass of 5600 paperclips = 0.001 kg × 5600 = 5.6 kg

Therefore,

The mass of 5600 paper clips is 5.6 kilograms.

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the number that is not part of the solution set is A) 0.

How to find and what does variable mean?

To solve the inequality x + 2 ≥ 3, we need to isolate the variable x.

x + 2 ≥ 3

Subtract 2 from both sides:

x ≥ 1

This means that any value of x that is greater than or equal to 1 is part of the solution set.

To check which of the given numbers is not part of the solution set, we need to substitute each of them in the inequality and see if it is true or false.

A) 0 + 2 ≥ 3 --> 2 ≥ 3 (False)

B) 2 + 2 ≥ 3 --> 4 ≥ 3 (True)

C) 3 + 2 ≥ 3 --> 5 ≥ 3 (True)

D) 6 + 2 ≥ 3 --> 8 ≥ 3 (True)

Therefore, the number that is not part of the solution set is A) 0.

In mathematics, a variable is a symbol or letter that represents a value or a quantity that can vary or change. It is often used to represent unknown or undefined values or quantities, and is commonly denoted by letters such as x, y, z, a, b, and c.

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The correct option is C. Green image of triangle ABC.

How to find the rotated shape or coordinates of image about origin?

As we know that triangle ABC, is rotated 270° counterclockwise about the origin. So, the algebraic rule for a figure is,

90° Clockwise or 270° counter clockwise, (x,y) = (y,-x)

As, C(-3,3) which is equal to (3,3) from the algebraic rule.

And (3,3) belongs to 1st quadrant i.e. green image is the correct answer.

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Triangle below represents is  option  C. Green image of triangle ABC.

A triangle is a geometric shape with three sides and three angles. It is one of the most fundamental shapes in geometry and is used extensively in mathematics, physics, engineering, and many other fields. Triangles are often classified based on their angles and sides.

As we know that triangle ABC, is rotated 270° counterclockwise about the origin. So, the algebraic rule for a figure is,

90° Clockwise or 270° counter clockwise, (x,y) = (y,-x)

As, C(-3,3) which is equal to (3,3) from the algebraic rule.

And (3,3) belongs to 1st quadrant i.e. green image is the correct answer.

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6.2y - 3.7 = 6.2y + n n = -3.7 is the value we use to put this equal to and then solve for n. Hence, -3.7 is the value of p that equalises the two equations.

We need to simplify both equations and set them equal to one another in order to get the value of p that makes the expressions comparable.

Putting the left half of the equation first: Group like words to get (-y + 5.3) + (7.2y - 9) as -y + 7.2y - 9 + 5.3.

We will now put this equal to 6.2y + n and get n: 6.2y - 3.7 = 6.2y + n \sn = -3.7

Hence, -3.7 is the value of p that renders the equations equal.

A statement proving the equivalence of two mathematical expressions, sometimes incorporating one or more unknown variables, is known as an equation. Usually, an equal sign is used to denote it.

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The results of the composition of two functions are listed below:

Case 1:  

a) f[g(2)] = 3

b) f[g(x)] = 18 · x² - 60 · x + 51

c) g[f(x)] = 6 · x² - 2

d) g[g(x)] = 9 · x - 20

e) f[f(- 2)] = 163

Case 13: g[f(x)] = (√x + 2)² + 3, Dom {g[f(x)]} = [0, + ∞)

Case 15: g[f(x)] = (3√x + 1) / [27 · (√x)³], Dom {g[f(x)]} = (0, + ∞)

Case 17: g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2, Dom {g[f(x)]} = All real numbers.

Case 21: g[f(x)] = - 3 / (√2 - 4 · x), Dom {g[f(x)]} = All real numbers except x = √2 / 4.

In this problem we must determine, analyze and evaluate the composition of two functions, whose definition is shown below:

f ° g (x) = f [g (x)]

Now we proceed to determine the composition of functions:

Case 1: f(x) = 2 · x² + 1, g(x) = 3 · x - 5

a) f[g(2)] = 2 · (3 · 2 - 5)² + 1 = 2 · 1² + 1 = 2 + 1 = 3

b) f[g(x)] = 2 · (3 · x - 5)² + 1 = 2 · (9 · x² - 30 · x + 25) + 1 = 18 · x² - 60 · x + 51

c) g[f(x)] = 3 · (2 · x² + 1) - 5 = 6 · x² - 2

d) g[g(x)] = 3 · (3 · x - 5) - 5 = 9 · x - 20

e) f[f(- 2)] = 2 · [2 · (- 2)² + 1]² + 1 = 2 · (2 · 4 + 1)² + 1 = 2 · 9² + 1 = 162 + 1 = 163

Case 13: f(x) = √x + 2, g(x) = x² + 3

g[f(x)] = (√x + 2)² + 3

Dom {g[f(x)]} = [0, + ∞)

Case 15: f(x) = 3√x, g(x) = (x + 1) / x³

g[f(x)] = (3√x + 1) / (3√x)³ = (3√x + 1) / [27 · (√x)³]

Dom {g[f(x)]} = (0, + ∞)

Case 17: f(x) = 1 / (x - 4), g(x) = (2 / x) + 4

g[f(x)] = 1 / [[(2 / x) + 4] - 4] = x / 2

Dom {g[f(x)]} = All real numbers.

Case 21: f(x) = √2 - 4 · x, g(x) = - 3 / x

g[f(x)] = - 3 / (√2 - 4 · x)

Dom {g[f(x)]} = All real numbers except x = √2 / 4.

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The value of y for x ranging from -4 to 4 are 34, 19, 6, -3, -2, 1, 10, 23 and 38, respectively.

To find the values of y for the given range of x, we can substitute each value of x into the equation y = 2x² - x - 2 and simplify. The completed table of values is:

(The table is attached below)

The given relation is y = 2x² - x - 2. To create a table of values for this relation, we can substitute values of x in the given range of -4 to 4 and calculate the corresponding values of y using the equation. In the first table, we substitute values of x ranging from -4 to 0 and calculate the corresponding values of y. In the second table, we substitute values of x ranging from 0 to 4 and calculate the corresponding values of y. These tables show how the values of y change as x changes within the given range, and provide a way to graph the relation as well.

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The x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

What is derivative?

In calculus, the derivative of a function is a measure of how much the function changes as its input variable changes. More specifically, the derivative of a function f(x) at a particular value of x, denoted by f'(x), is defined as the limit of the ratio of the change in the function value to the change in the input variable as the change in the input variable approaches zero.

To find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, pi], we need to find the derivative of f(x) and set it equal to 0.

f(x) = 2x + sin(4x)

f'(x) = 2 + 4cos(4x)

Setting f'(x) equal to 0, we get:

2 + 4cos(4x) = 0

cos(4x) = -1/2

We know that cos(4x) = -1/2 has solutions at 4x = 2π/3 and 4x = 4π/3 (plus any multiple of 2π), because these are the solutions to cosθ = -1/2 in the interval [0,2π). So, we can write:

4x = 2π/3 or 4x = 4π/3

Solving for x in each equation, we get:

x = π/6 or x = π/3

However, we need to check that these solutions are in the interval [0, pi].

π/6 is in the interval [0, pi], but π/3 is not.

Therefore, the only x-coordinate where f'(x) = 0 and x is in the interval [0, pi] is:

x = π/6

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The length of XZ is 5.5 m.

The length of side XZ is calculated by applying the following cosine and sine rule.

If the length of WY is 7 m, then ∠WYZ is calculated as follows;

cos Y = (z² + w² - y² ) / (2zw)

where;

cos Y = ( 7² + 5.1² - 3² ) / ( 2 x 7 x 5.1 )

cos Y = 0.9245

Y = cos⁻¹ (0.9245)

Y = 22.4⁰

The value of ∠WYX is calculated as follows;

cos Y = (x² + w² - y² ) / (2xw)

cos Y = ( 7² + 5² - 4.8² ) / ( 2 x 7 x 5)

cos Y = 0.728

Y = cos⁻¹ (0.728)

Y = 43.28⁰

The value of ∠ZYX = 43.28⁰ + 22.4⁰ = 65.68⁰

The length of XZ is calculated by using the following cosine rule.

|XZ|² = |XY|² + |ZY|² - (2 x |XY| x |XY|) cos Y

|XZ|² = 5² + 5.1²  -  (2 x 5 x 5.1 ) x cos (65.68)

|XZ|² = 30

|XZ| = √30

|XZ| = 5.5 m

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