QUESTION 3 Read the extract below and answer the questions that follow. Cape Town- a thirteen-year-old-boy was arrested and released on warning after he was caught on video assaulting an assistant teacher on the body and on head using his closed fists. Published: Dec 7, 2021 By: Siphokazi Vuso The grade 12 pupil from Harmony Primary in Steenberg can be seen on the video assaulting an assistant teacher after she had confiscated a ball from another pupil. Spokesperson Captain Frederick Van Wyk said witnesses tried to keep the learner away from the teacher when he broke loose and assaulted her. He said the boy was arrested on December 3 and was released in the care of his parents on a warning to appear in the Wynberg Magistrate's Court on Wednesday. [Adapted from: https://www.iol.co.za Accessed on 02 January 2023] /15/ 3.1 Define the term 'conflict resolution' and explain how the above situation is an example of bad conflict resolution. (1+2)(3) 3.2 Suggest TWO strategies that learners and teachers could put in place to decrease the culture of violent within their schools.​

We may calculate the distance from Carlos and the tree as [tex]D = 781.23[/tex] ft using trigonometric relationships.

Many functions in trigonometry offer excellent illustrations of how calculus may be used in practical situations. By the way, basic trigonometry primarily deals with the algebraic features of circular functions, therefore basic calculus is not always simpler than basic trigonometry. On the other hand, calculus is fundamentally non-algebraic in nature.

The mnemonic SOHCAHTOA makes it simple for us to recall the three primary trigonometric ratios.

Then one of the cathetus of the triangle is equal to:

[tex]275ft - 6ft = 269ft[/tex]

That cathetus is the opposite cathetus to the angle of 19° (the elevation angle). And the adjacent cathetus is the distance between Carlos and the tree.

Then we can use the relation:

Where:

[tex]a=19^{0}[/tex]

opposite cathetus [tex]=269ft[/tex]

adjacent cathetus [tex]=D[/tex]

Replacing that, we get

[tex]tan(19^{0} ) = 269ft/D[/tex]

Solving for D, we get:

[tex]D = 269ft/tan(19^{0} ) = 781.2ft[/tex]

So we conclude that Carlos is at [tex]781.23[/tex] ft of the tree.

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The algebraic expressions for the given questions are found as:

1) g/4 + 15.

2) 6 - y + 3

3) d * 8

What are algebraic expressions?

A mathematical expression made up of variables, constant algebraic numbers, and algebraic operations are known as an algebraic expression. Everything with a variable value is anything whose value changes over time. Variables in expressions are often represented by letters of the alphabet. Finding the simplified term of the given expression is the goal of simplifying the algebraic expression. We must first understand how to join like terms, factor a number, and the order of operations before we can factorise or simplify the expression. When like terms are combined, the constant terms are divided for the purpose of simplicity and the variables with the same degree are grouped together.

1) The quotient of g and 4 is given by g/4.

Now, this is added to 15.

So the algebraic expression is g/4 + 15.

2) The difference between y and 3 is y - 3.

Now from 6, this difference is subtracted.

So the expression becomes,

6 - (y-3) = 6 - y + 3

3) The product of two numbers is found by multiplying the numbers.

  So the expression is :

      d * 8

Therefore the algebraic expressions for the given questions are found according to the given statements.

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Answer: To determine the total dollar amount of Finished Goods Inventory at the end of March, we need to calculate the total cost of the jobs that have been completed during March and have been transferred to the Finished Goods Inventory.

From the information given, we know that the company has no beginning Work in Process or Finished Goods inventories, so all the costs incurred during March are related to the jobs started during the month.

Let's start by calculating the total cost of Job 2, which is finished but not sold by the end of March. We will then use this cost to calculate the total cost of the jobs that have been completed during March and transferred to the Finished Goods Inventory.

Job 2:

Direct materials: Rs. 10,000

Direct labor: Rs. 8,000

Overhead applied: 60% x Rs. 10,000 = Rs. 6,000

Total cost of Job 2: Rs. 24,000

Now, let's calculate the total cost of the jobs that have been completed during March:

Job 1:

Direct materials: Rs. 6,000

Direct labor: Rs. 4,000

Overhead applied: 60% x Rs. 6,000 = Rs. 3,600

Total cost of Job 1: Rs. 13,600

Job 3:

Direct materials: Rs. 8,000

Direct labor: Rs. 5,000

Overhead applied: 60% x Rs. 8,000 = Rs. 4,800

Total cost of Job 3: Rs. 17,800

Total cost of jobs completed during March: Rs. 13,600 + Rs. 17,800 = Rs. 31,400

Since the company has no beginning Finished Goods Inventory, the total cost of the jobs completed during March and transferred to the Finished Goods Inventory is equal to the total cost of the Finished Goods Inventory at the end of March.

Therefore, the total dollar amount of Finished Goods Inventory at the end of March is Rs. 31,400.

Step-by-step explanation:

With the infοrmatiοn prοvided, the sοlutiοn is 2x + y = 9.

A mathematical equatiοn, including such 6 x 4 = 12 x 2, states that twο numbers οr values are equivalent. a meaningful nοun. An equatiοn is used when twο οr even mοre factοrs must be cοnsidered jοintly in οrder tο understand οr explain the whοle situatiοn.

Given:

Slοpe οr gradient οf line = -2

The lοcatiοn where the line passes = (1,7)

Equatiοn οf a line thrοugh (h,k) with a slοpe οf m:

(y - k) = (x -h) m

Equatiοn οf a line thrοugh (1,7) with a slοpe οf (-2): -

⇒ (y - 7) = (x - 1)(-2)

⇒ y - 7 = - 2x + 2

⇒ y = -2x + 2 + 7

⇒ y = -2x + 9

οr

⇒2x + y = 9

Verificatiοn οf respοnse:

We may check οur respοnse by substituting (1,7) in the fοrmula 2x+y=9.

⟹ LHS = (2×1)+7

⟹ LHS = 2+7

⟹ LHS = 9

Therefοre , LHS = RHS.

Alsο, yοu can use the equatiοn tο determine the gradient οr slοpe and cοnfirm yοur sοlutiοn.

⟹ Slοpe is equal tο -(cοefficient οf x)/ (cοefficient οf y)

⟹ Slοpe = - 2 /1

⟹ slοpe = -2

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Answer:

b,c,e

Step-by-step explanation: a: 5+0=5

b:15x0 =0

c: 1.4+2.7=4.1

d:2/3 x 5/9

e:4& 2/3 =5-1/3

Answer:

6.25 seconds

Step-by-step explanation:

Using the the equation

h = 100t − 16t^2

The ball hits the ground, when h = 0, solving below equation:

100t − 16t^2 = 0

4t(25 - 4t) = 0

t = 0 seconds, this is the starting point when ball was hit

25 - 4t = 0 ⇒ 4t = 25 ⇒ t= 25/4 = 6.25 seconds later the ball hits the ground

Answer:

Tahlia was correct in remembering that you can find the slope of a line using the slope formula.

She was also correct that the slope is 0.5.

However, if you use the slope-intercept formula y = mx + b, you still have to find b (the y-intercept) by plugging in one of the two points and the slope.

So, if we plug in the first point and the slope into the line, we can find b and thus the full equation of the line:

[tex]-7.5=0.5(-3)+b\\-7.5=-1.5+b\\b=-6\\\\y=0.5x-6[/tex]

The function g(x) = 3/(x-2) + 5 is a transformation of the parent function f(x) = 1/x with a horizontal shift of 2 units to the right, a vertical stretch by a factor of 3, and a vertical shift of 5 units up.

The function g(x) = 3/(x-2) + 5 is a transformation of the parent function f(x) = 1/x.

Here are the transformations that were applied to f(x) to obtain g(x):

1. Horizontal shift: The denominator of f(x) is x, which means that the vertical asymptote is at x = 0. By subtracting 2 from x in the denominator of g(x), we are shifting the vertical asymptote to x = 2. This is a horizontal shift of 2 units to the right.

2. Vertical stretch: The coefficient 3 in front of the fraction means that the function is stretched vertically by a factor of 3 compared to f(x). This means that the distance between the horizontal asymptotes (which are at y = 0 for f(x) and y = 5 for g(x)) is three times larger for g(x) than for f(x).

3. Vertical shift: The constant 5 added to the fraction means that the entire function is shifted vertically by 5 units compared to f(x). This means that the horizontal asymptote of g(x) is at y = 5 instead of y = 0.

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The expression for the given Venn Diagrams are:

A Venn diagram employs circles or other shapes that overlap to show the logical connections between two or more groups of objects.

They frequently serve to visually group objects while highlighting how they differ and how they are similar.

The union of two sets A and B is a new set that contains all the elements that are in A or in B or in both. It is denoted by A ∪ B.

Example: Let A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

The intersection of two sets A and B is a new set that contains all the elements that are in both A and B. It is denoted by A ∩ B.

Example: Let A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.

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The greatest common factor of 4x³y² and 12x²y is: 4x²y.

The GCF is a term used to refer to the largest common factor between any two or more numbers (Greatest Common Factor). The GCF of two natural integers, x and y, is the largest possible number that equally and completely divides both x and y. GCF is typically calculated using the following three techniques: division, multiplication, and prime factorization.

Now in the given question,

4x³y² can be written as = 2×2×x×x×x×y×y

12x²y can be written as = 2×2×3×x×x×y

The common factors among them are:

2×2×x×x×y = 4x²y.

Therefore, the greatest common factor of 4x³y² and 12x²y is: 4x²y.

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The reference angle for A = -300° is given by R = 60°

Every angle is measured from the positive part of the x-axis to its terminal line (the line that determines the end of the angle) traveling counterclockwise. For angles in the first quadrant, i.e., angles less than or equal to 90 degrees, the reference angle is equal to the original angle.

The reference angle is always positive

0° to 90°: reference angle = angle

90° to 180°: reference angle = 180° - angle

180° to 270°: reference angle = angle - 180°

270° to 360°: reference angle = 360° - angle

Given data ,

Let the reference angle be represented as R

Now , let the given angle be A

A = -300°

And , for 270° to 360°: reference angle = 360° - angle

when the angle is negative and less than -270°

The reference angle is given by adding 360° to it and the angle lies in the first quadrant

So , the reference angle R = A + 360°

R = -300 + 360

R = 60°

Hence , the reference angle is 60°

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Therefore, the likelihood of a thunderstorm is roughly 0.2174, or a 21.74% chance that one will occur.

The study of random events or phenomena falls under the category of probability, which is a branch of mathematics. It is a way to quantify the possibility or likelihood of an event happening. A number between 0 and 1, where 0 denotes an improbable occurrence and 1 denotes a certain event, is used to express probability. By dividing the number of favorable outcomes by the total number of possible outcomes, one can determine the probability of an occurrence. In many disciplines, including statistics, economics, engineering, and science, probability is used to make predictions and well-informed choices in uncertain situations.

given

We must divide the amount of beneficial outcomes (chances of a thunderstorm) by the total number of possible outcomes in order to express the suggested degree of likelihood as a probability value between 0 and 1 inclusive.

When the likelihood of a thunderstorm is stated as "5 in 23," it indicates that out of a possible 23, 5 outcomes are likely to be favorable.

Therefore, the likelihood of a rainstorm is:

P (thunderstorm) is the ratio of the number of successful outcomes to all other possible outcomes.

Thunderstorm P(thunder) = 5 / 23

As a result, the probability value for the indicated degree of likelihood can be expressed as:

P(thunderstorm)= 5/230.2174

Therefore, the likelihood of a thunderstorm is roughly 0.2174, or a 21.74% chance that one will occur.

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Answer:

21 miles per gallon

Step-by-step explanation:

find the gradient of the line:

change in y = 105 - 21 = 84

change in x = 5 - 1 = 4

gradient is change in y/ change in x

= 84/4 = 21

The equation can be converted as: 2 (S - 1/2 u t) / t.

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

Let's start by simplifying the equation using the distributive property of multiplication:

S = 1/2 u t + 1/2 v t

Next, we can isolate the variable v by subtracting 1/2 u t from both sides of the equation:

S - 1/2 u t = 1/2 v t

Now we can multiply both sides of the equation by 2 to eliminate the fraction:

2 (S - 1/2 u t) = v t

Finally, we can solve for v by dividing both sides of the equation by t:

v = 2 (S - 1/2 u t) / t

Therefore, the solution for v is v = 2 (S - 1/2 u t) / t.

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Looking at the table again, we can see that f(4) is -6. Thus, f (g(2)) is -6.

Value is the worth of something, based on its importance, usefulness, or desirability. It is something that has a desired quality, such as excellence, utility, or desirability. Value can be physical, such as a commodity, or intangible, such as respect, loyalty, or knowledge. Value is determined by both the individual and the society in which they live. Value can be assigned to both tangible and intangible things, such as a house, an education, or a friendship. Value is a subjective measure and can vary greatly from person to person.

The answer is -6. To evaluate f (g(2)), we need to determine what g(2) is first. Looking at the table, we can see that g(2) is 4. Therefore, we need to find the value of f(4). Looking at the table again, we can see that f(4) is -6. Thus, f (g(2)) is -6.

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Answer:

(f-g)(x) = -5x^3 + 8x - 8.

(f-g)(-2) = 16.

Step-by-step explanation:

To find (f-g)(x), we need to subtract g(x) from f(x) and simplify:

(f-g)(x) = f(x) - g(x)

= (-5x^3 - 4x^2 + 8x) - (-4x^2 + 8)

= -5x^3 - 4x^2 + 8x + 4x^2 - 8

= -5x^3 + 8x - 8

Therefore, (f-g)(x) = -5x^3 + 8x - 8.

To find (f-g)(-2), we substitute x = -2 into the expression we just found:

(f-g)(-2) = -5(-2)^3 + 8(-2) - 8

= -5(-8) - 16 - 8

= 40 - 24

= 16

Therefore, (f-g)(-2) = 16.

By solving the equation we know that the correct value of (D) x is 4.

A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.

The four mathematical operations of addition, subtraction, multiplication, and division are also used in simple equations, either singly or in combination.

A formula would be 3x - 5 = 16, for instance.

When this equation is solved, we discover that the value of the variable x is 7.

So, solve for the given equation:

|-4+5x|=16

5x - 4 = 16

Now, by looking at the equation we can tell that the value of x needs to be 4 as follows:

5x - 4 = 16

5(4) - 4 = 16

20 - 4 = 16

16 = 16

Therefore, by solving the equation we know that the correct value of (D) x is 4.

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Complete question:

Solve : select all answers that applies

|-4+5x|=16

A. x= -11

B. x = -12/5

C. x= 4/5

D. x= 4

E. x = 3/4

If Shaun eats popping candy while drinking a can of soda, soda streams out of Shaun's nose and he gets embarrassed. It isn't the case that soda streams out of Shaun's nose and he gets embarrassed. So, it isn't the case that Shaun eats popping candy while drinking a can of soda. The logical form of this argument is denying the antecedent. The Option B is correct.

If we represent the argument in logical symbols, it would look like this:

The argument can be summarized as follows:

If P, then Q.

Not P.

Therefore, not Q.

This is the form of denying the antecedent, where the argument asserts that if P is true, then Q must also be true. However, the argument then proceeds to deny the antecedent by claiming that P is not true, and therefore, Q must not be true either.

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Answer:

Have a good day and time and I6ykjjuu7ttggfszd to the 5tjyyv

1) The volume of the first rectangular prism is 175 ft³

2) The volume of the second rectangular prism is 60ft³

3) The volume of the third rectangular prism 140ft³

4) The volume of the Fourth rectangular prism 216ft³

Note that in order to find the volume of area of a complex figure, you must deconstruct it into manageable proportions.

Shape 1 above, we deconstructed the rectangular prism into two 3 dimensional shapes with the following metrics:

a)

Length = 7ft

Height = 2ft

widht = 2ft

Volume = Volume = 7ft x 2ft x 2ft
Volume = 28  ft³

b)

Length = 7ft

Height = 3ft

width = 7ft
Volume = 147 ft³

Total Volume = 28  ft³ + 147 ft³

Volume for Shape 1 = 175 ft³

Shape 2 above, we deconstructed the rectangular prism into two 3-dimensional shapes with the following metrics:

a) Length = 3ft

Height = 2ft

width = 1ft

Volume = 6 ft³

b) Length = 6ft

Height = 3ft

width = 3ft

Volume = 54 ft³

Thus, total volume for shape 2 is:

Volume = 6 + 54

Volume = 40 ft³

Shape 3, above, we deconstructed the rectangular prism into two 3-dimensional shapes with the following metrics:

a) Length = 7ft

Height = 1ft

width = 2ft
Volume = 14ft³

b)
Length = 7ft

Height = 3ft

width = 6ft
Volume = 126ft³

Thus, the total volume fior shape 3, =

Volume = 14ft³ + 126ft³

Volume = 140ft³


Shape 4, above, we deconstructed the rectangular prism into two 3-dimensional shapes with the following metrics:

a) Length = 4ft

Height = 3ft

width = 2ft
Volume = 24 ft³


b) Length = 4ft

Height = 12ft

width = 4ft
Volume = 192 ft³

Thus, the total volume for Shape 4:

Volume =  24 ft³+ 192 ft³
Volume = 216ft³

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Answer: From part b, we know that the number of sides in my envelope is t + q = 10 + 6 = 16, and the number of sides in my partner's envelope is 2t - q = 20 - 6 = 14.

To determine the number of sides in both envelopes combined, we add these values: 16 + 14 = 30.

Therefore, the combined number of sides in both envelopes is 30.

Step-by-step explanation:

The inequality which represents the least number of days that Kelli needs to run to reach her goal is

2 1/4 + 1/3d ≥ 4.

The least number of days after Monday that Kelli needs to run to reach her goal is 5.25 days

Monday= 2 1/4 miles

Other days of the week = 1/3 miles

Her goal =™4 miles

Number of days = d

2 1/4 + 1/3d ≥ 4

subtract 2 1/4 from both sides

1/3d = 4 - 2 1/4

1/3d = 4 - 9/4

1/3d = 16-9 /4

1/3d = 7/4

divide both sides by 1/3

d = 7/4 ÷ 1/3

d = 7/4 × 3/1

= 21/4

d = 5.25 days

Hence, Kelli will reach her goal in at least 5.25 days.

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For the new height requirements, this branch of the military requires women's heights to be at least 2.52 in and at most 6.96in.

If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.

If we have

[tex]X \sim N(\mu, \sigma)[/tex]

(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex]  )

then it can be converted to standard normal distribution as

[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]

(Know the fact that in continuous distribution, probability of a single point is 0, so we can write

[tex]P(Z \leq z) = P(Z < z) )[/tex]

Also, know that if we look for Z = z in z tables, the p value we get is

[tex]P(Z \leq z) = \rm p \: value[/tex]

We are given that;

Mean= 63.8 in

Standard deviation= 2.3 in

Requirement of women height= 58in - 80in.

Now,

To find the minimum and maximum height requirements for this branch of the military, we need to use the standard normal distribution formula, which converts any normal distribution to a standard normal distribution with a mean of 0 and a standard deviation of 1.

We can find the z-scores for the minimum and maximum heights as follows:

For the minimum height requirement:

z = (58 - 63.8) / 2.3 = -2.52

For the maximum height requirement:

z = (80 - 63.8) / 2.3 = 6.96

Using a standard normal distribution table or calculator, we can find the corresponding probabilities for these z-scores:

For a z-score of -2.52, the probability is 0.0059 or 0.59%

For a z-score of 6.96, the probability is very close to 1, or 100%

Therefore, the new height requirements for this branch of the military are:

At least 58 inches, which corresponds to a z-score of -2.52

At most 80 inches, which corresponds to a z-score of 6.96

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Check the picture below.

1. A point less than 1 in Katrina's number line is [tex]\frac{3}{8}[/tex]

2. A point greater than 1 in Eric's number line is 1[tex]\frac{1}{4}[/tex]          

A number line is a straight line with numbers placed at equal distances from each other.

To identify a line less than 1 in a number line divided into 8, we need to know the value of what each point would be

Each values in Katrina's number line should be 0, [tex]\frac{1}{8}[/tex], [tex]\frac{2}{8}[/tex] , [tex]\frac{3}{8}[/tex],  [tex]\frac{4}{8}[/tex],  [tex]\frac{5}{8}[/tex],  [tex]\frac{6}{8}[/tex],  [tex]\frac{7}{8}[/tex]  1

To identify the line greater than 1 in a number line divided into 4, we need to know the values of what each points would be

Each values in Eric's number line should be 1, 1[tex]\frac{1}{4}[/tex], 1[tex]\frac{2}{4}[/tex], 1[tex]\frac{3}{4}[/tex], 2

Check the images below to get a proper view of how your number line should look like.

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Answer: y = 4/3x+1

Step-by-step explanation:

Answer:

y=4/3 + 1

Step-by-step explanation:

There are different ways to do this for finding slope, you can use the formula or you can do rise over run. How to find rise over run you count how many places up so our current place is (-3,-3) and we are trying to find (3,5) so if you were to start with rise, you rise 8 places then run you go either left or right in our case right, you move 6 places.  So our fraction would be 8/6 or simplified 4/3. But using the formula to make sure we got the right answer, our formula is y2-y1/x2-x1 so that would look like 5- -3/3- -3. To solve on our top side two negatives give us a positive so the top is 8 and then bottom is 6 so we get 8/6 or 4/3. But our slope intercept would be y= 4/3x +1.

To solve problems with integers, first, you must identify which operation you need to  first then, you need to think about which mathematical operations you need to use.

What is integers ?

Integers are a set of whole numbers that include both positive and negative numbers, as well as zero. Integers can be represented on a number line that extends infinitely in both directions.

To solve problems with integers, you need to understand the basic operations such as addition, subtraction, multiplication, and division. You also need to know the rules and properties of integers, such as the commutative, associative, and distributive properties, as well as the rules for adding, subtracting, multiplying, and dividing positive and negative integers.

When solving a problem with integers, you should first identify which operation you need to perform first based on the order of operations. This order is typically remembered with the acronym PEMDAS (parentheses, exponents, multiplication and division, and addition and subtraction). Once you have identified the operation, you can use the appropriate mathematical technique to solve the problem.

Therefore, to solve problems with integers, first, you must identify which operation you need to  first then, you need to think about which mathematical operations you need to use.

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The answers to all limits are:

(a) lim x → a [ f(x) + 2g(x)] =  -6,

(b) lim x → a [h(x) − 3g(x) + 1] = 13

(c) lim x → a [ f(x)g(x)] = -8

(d) lim x → a [g (x)]² = 16

(e) lim x → a 3√6 + f(x) / 2g(x)

The central limit theorem states that if you take sufficiently large samples from a population, the samples' means will be normally distributed, even if the population isn't normally distributed.

(a) lim x → a [ f(x) + 2g(x)]

= lim x → a f(x) + 2 lim x → a g(x) (using algebraic limit theorem)

= 2 + 2(-4)

= -6

(b) lim x → a [h(x) − 3g(x) + 1]

= lim x → a h(x) - 3 lim x → a g(x) + lim x → a 1 (using algebraic limit theorem)

= 0 - 3(-4) + 1

= 13

(c) lim x → a [ f(x)g(x)]

= lim x → a f(x) * lim x → a g(x) (using algebraic limit theorem)

= 2 * (-4)

= -8

(d) lim x → a [g(x)]²

= [lim x → a g(x)]^2 (using algebraic limit theorem)

= (-4)^2

= 16

(e) lim x → a 3√6 + f(x) / 2g(x)

= [3√6 + lim x → a f(x)] / [2 lim x → a g(x)] (using algebraic limit theorem)

= [3√6 + 2] / [2(-4)]

= (-3√6 - 2) / 8

Therefore, the answers to all limits are:

(a) lim x → a [ f(x) + 2g(x)] =  -6,

(b) lim x → a [h(x) − 3g(x) + 1] = 13

(c) lim x → a [ f(x)g(x)] = -8

(d) lim x → a [g (x)]² = 16

(e) lim x → a 3√6 + f(x) / 2g(x)

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As a result, x = pi/6 is the only answer in the range [0,2pi).

When quantifying the angles of trigonometry or periodic functions, radians are frequently taken into account. Radians are always expressed in units of pi, with pi equivalent to either 3.14 or 22/7.

We can start by simplifying the left-hand side of the equation using the identity:

csc x = 1/sin x

4 sec x csc x = 4(1/cos x)(1/sin x) = 4/(cos x sin x)

Substituting this back into the original equation, we get:

4/(cos x sin x) = 8 csc x

Multiplying both sides by cos x sin x, we get:

4 = 8 sin x

Dividing both sides by 8, we get:

sin x = 1/2

This means that x is either pi/6 or 5pi/6, since these are the two angles in the interval [0,2pi) where sin x = 1/2.

However, we need to check if these solutions satisfy the original equation.

For x = pi/6:

4 sec x csc x = 4 sec(pi/6) csc(pi/6) = 4(2) (2/√3) = 16/√3

8 csc x = 8 csc(pi/6) = 8(2/√3) = 16/√3

So, x = pi/6 is a solution.

For x = 5pi/6:

4 sec x csc x = 4 sec(5pi/6) csc(5pi/6) = 4(-2) (-2/√3) = 16/√3

8 csc x = 8 csc(5pi/6) = 8(-2/√3) = -16/√3

So, x = 5pi/6 is not a solution.

Therefore, the only solution in the interval [0,2pi) is x = pi/6.

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