What is the width of the pool?

The function f is onto.

To determine if the function f is one-to-one, we need to check if each element in the domain maps to a unique element in the range. Looking at the function, we can see that f(a) = 1, f(b) = 2, f(c) = 3, and f(d) = 3. Since two elements in the domain (c and d) map to the same element in the range (3), the function f is not one-to-one.

To determine if the function f is onto, we need to check if every element in the range is mapped to by at least one element in the domain. Looking at the function, we can see that all four elements in the range (1, 2, 3, and 4) are mapped to by at least one element in the domain. Therefore, the function f is onto.

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Answer: This situation involves choosing a group of 5 players out of a total of 12 players, where the order in which the players are chosen does not matter. Therefore, this is an example of a combination problem.

The number of ways to choose a group of 5 players out of 12 is given by the formula for combinations:

n C r = n! / (r! * (n-r)!)

where n is the total number of players, r is the number of players being chosen, and "!" represents the factorial operation.

In this case, we have n = 12 and r = 5, so the number of different choices of starters is:

12 C 5 = 12! / (5! * (12-5)!)

= 792

Therefore, there are 792 different choices of starters that the coach can make from the team of 12 players.

Step-by-step explanation:

The company should charge the person $240.

The expected value of a discrete distribution is calculated as the sum of each outcome multiplied by it's respective probability.

For a 40 year old person, the distribution of the company earnings are given as follows:

For an expected value of 70, the value of x is obtained as follows:

-200000(0.00085) + 0.99915x = 70

x = (70 + 200000(0.00085))/0.99915

x = $240.

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Answer:

[tex]\large\boxed{\tt x \approx 95.6}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to solve for x by using \underline{Trigonometric Identities}.}[/tex]

[tex]\large\underline{\textsf{What are Trigonometric Identities?}}[/tex]

[tex]\boxed{\begin{minipage}{20 em} \\ \underline{\textsf{\large Trigonometric Identities;}} \\ \\ \textsf{Trigonometric Identities are trigonometric ratios determined with what's given in order to find a missing value. For a Right Triangle, the Trigonometric Identities are Sine, Cosine, and Tangent. These are used to find missing sides.} \\ \\ \tt Sine = \tt $ \tt \frac{Opposite}{Hypotenuse} \\ \\ Cosine = \frac{Adjacent}{Hypotenuse} \\ \\ Tangent = \frac{Opposite}{Adjacent} \end{minipage}}[/tex]

[tex]\textsf{We should determine whether Sine, Cosine, or Tangent will actually help us}[/tex]

[tex]\textsf{determine x. We are given a Right Triangle that has 1 15}^{\circ} \ \textsf{angle, and a side with}[/tex]

[tex]\textsf{a length of 99. Because this side is opposite of the right angle, this side is called}[/tex]

[tex]\textsf{the \underline{Hypotenuse}.}[/tex]

[tex]\textsf{The side labeled x is \underline{Adjacent}, which means that it's touching the given angle.}[/tex]

[tex]\textsf{Using what was given to us, we should use Cosine since we are asked for the}[/tex]

[tex]\textsf{Adjacent Angle when given the Hypotenuse.}[/tex]

[tex]\large\underline{\textsf{Solving;}}[/tex]

[tex]\textsf{Remember that;}[/tex]

[tex]\tt \cos(15^{\circ}) =\frac{Adjacent}{Hypotenuse}[/tex]

[tex]\textsf{We're given;}[/tex]

[tex]\tt \cos(15^{\circ}) =\frac{x}{99}[/tex]

[tex]\textsf{To find the value of x, we first should remove the fraction using cancellation.}[/tex]

[tex]\textsf{We are able to use the \underline{Multiplication Property of Equality} to prove that the}[/tex]

[tex]\textsf{equation remains equal.}[/tex]

[tex]\underline{\textsf{Multiply both expressions by 99;}}[/tex]

[tex]\tt 99 \cos(15^{\circ}) =\not{99} \frac{x}{\not{99}}[/tex]

[tex]\tt 99 \cos(15^{\circ}) =x[/tex]

[tex]\underline{\textsf{Evaluate;}}[/tex]

[tex]\tt 99 \cos(15^{\circ}) \approx \boxed{\tt 95.6}[/tex]

[tex]\large\boxed{\tt x \approx 95.6}[/tex]

Fred should use a critical value of 2.145 and a standard error of the mean of 2.84 to construct a 95% confidence interval for the average length of movies he watches

To construct a confidence interval for the population mean length of movies watched by Fred, we can use the following formula

Confidence interval = sample mean +/- (critical value) x (standard error)

where the standard error of the mean (SE) is calculated as:

SE = sample standard deviation / sqrt(sample size)

Since Fred's sample size is 15 and he wants a 95% confidence interval, we need to find the critical value for a t-distribution with 14 degrees of freedom (n-1), which can be obtained from a t-distribution table or calculator.

Using a t-distribution calculator with 14 degrees of freedom and a 95% confidence level, we find the critical value to be 2.145.

Next, we can calculate the standard error of the mean as

SE = 11 / sqrt(15) = 2.84

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The conditions will the linear equations have no solutions, one solution, infinitely many solutions are specified by variables and their coefficient matrix and states if the system is consistent or inconsistent.

Let us assume simple equations:

ax + by = c

ex + fy = g

Here a, b, c, e, f, and g are constants.

The different conditions are determined as:

1. No solutions: It is represented by D, If the determinant of the coefficient matrix is zero and the procedure is inconsistent, then we can assume that the system has no solutions.

2. One solution: If the coefficient matrix of any system is non-zero, then the system has one solution.

3. Infinitely many solutions: If the system is Consistent and the determinant of the coefficient matrix is zero, then the system has  Infinitely many solutions.

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The statement "Some twins are sisters. All twins are siblings. Therefore, some siblings are sisters" is true.

Usually an illustration of a substantial deductive contention in which the conclusion takes coherently from the premises.

The primary introduction states that a few twins are sisters, which suggests that they are female twins. The moment preface states that all twins are kin, which implies that they are related by blood.

In this manner, on the off chance that a few twins are sisters and all twins are kin, it consistently takes after that a few kin are sisters.

It is imperative to note that the conclusion isn't essentially genuine for all kin, as a few kin may be brothers or mixed-gender twins. Be that as it may, the contention is still consistently substantial since the conclusion takes after coherently from the premises 

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the guy wires are 850 feet long. We can use the Pythagorean theorem to solve this problem.

what is Pythagorean theorem  ?

The Pythagorean theorem is a mathematical concept that describes the relationship between the sides of a right triangle. It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the given question,

We can use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the tower, the guy wires, and the ground form a right triangle. Let's call the length of the guy wires "x". Then we can set up the following equation:

x² = 840² + (130)²

Simplifying and solving for x, we get:

x² = 705600 + 16900

x² = 722500

x = √722500

x = 850

Therefore, the guy wires are 850 feet long.

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In the given triangle the 3 trigonometric ratios are:

(A) Sinθ = 3/5, (B) Cosθ = 4/5, and (Tanθ = 3/4)

The trigonometric functions in mathematics are real functions that connect the right-angled triangle's angle to the ratios of its two side lengths.

They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.

In general, arcsine, arccosine, tangent, cotangent, secant, and cosecant functions are used to express the inverses of sine, cosine, tangent, cotangent, secant, and cosecant functions.

So, according to t the given triangle, the 3 trigonometric ratios would be:

Sinθ = B/H

Sinθ = 27/45

Sinθ = 3/5

Cosθ = P/H

Cosθ = 36/45

Cosθ = 4/5

Tanθ = B/P

Tanθ = 27/36

Tanθ = 3/4

Therefore, in the given triangle the 3 trigonometric ratios are:

(A) Sinθ = 3/5, (B) Cosθ = 4/5, and (Tanθ = 3/4)

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An arrangement that satisfies the conditions of the puzzle is given below:

 4  9  2

 3  5  7

 8  1  6

Here is one possible arrangement of the digits 1, 2, 3, 4, and 5 in the square, with each row, column, and diagonal summing up to 15:

   4  9  2

   3  5  7

   8  1  6

We can check that this arrangement works:

Rows: 4+9+2=15, 3+5+7=15, 8+1+6=15

Columns: 4+3+8=15, 9+5+1=15, 2+7+6=15

Diagonals: 4+5+6=15, 2+5+8=15

Therefore, this arrangement satisfies the conditions of the b.

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The length of the segment [tex]\overline{ST}[/tex], obtained using Thales Theorem is 18 2/3

Thales Theorem, also known as the triangle proportionality theorem states that if a segment is drawn such that it is parallel to a side of a triangle, and it also intersects the other two sides of the triangle at distinct points, than the other two sides are divided by the segment in the same ratio

Thales Theorem, also known as the triangle proportionality theorem indicates;

12/14 = 16/[tex]\overline{ST}[/tex]

Therefore;

[tex]\overline{ST}[/tex]/16 = 14/12

[tex]\overline{ST}[/tex] = 16 × 14/12 = 56/3 = 18 2/3

Segment [tex]\overline{ST}[/tex] is 18 2/3 units long

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The total number of holes Craigs need to have a score of 3 today is equal to 37.

On every every frisbee golf hole having a score 3 = one bonus.

Total scored while playing last week = 3 on 5 holes

Total extra bonus scored by Craig = 5

Getting extra bonus on total = 42 from score of 3

let us consider Craigs need 'x' holes on a score of 3 today

Required equation is,

x + 5 = 42

Subtract 5 from both the side of the equation we get,

⇒ x = 42 - 5

⇒ x = 37 holes

Therefore, Craigs need to have 37 holes to score a 3 today.

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To compute the covariance matrix of the variables in the data, the "matrix-operation" which should be used is ([tex]X^{t}[/tex] × X)/n.

The "Covariance" matrix is defined as a symmetric and positive semi-definite, with the entries representing the covariance between pairs of variables in the data.

The "diagonal-entries" represent the variances of individual variables, and the off-diagonal entries represent the covariances between pairs of variables.

Step(1) : Compute the transpose of the centered data matrix X, denoted as [tex]X^{t}[/tex]. The "transpose" of a matrix is found by inter-changing its rows and columns.

Step(2) : Compute the "dot-product" of [tex]X^{t}[/tex] with itself, denoted as [tex]X^{t}[/tex] × X.

The dot product of two matrices is computed by multiplying corresponding entries of the matrices and summing them up.

Step(3) : Divide the result obtained in step(2) by the number of samples in the data, denoted as "n", to get the covariance matrix.

This step scales the sum of the products by 1/n, which is equivalent to taking the average.

So, the covariance matrix "C" of variables in "centered-data" matrix X can be expressed as: C = ([tex]X^{t}[/tex] × X)/n.

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The given question is incomplete, the complete question is

Let X be a matrix of centered data with a column for each field in the data and a row for each sample. Then, not including a scalar multiple, how can we use matrix operations to compute the covariance matrix of the variables in the data?

Answer:

No, they are not equivalent expressions.

Step-by-step explanation:

2(x + 6) + x = 2x + 12 + x = 3x + 12.

3x + 6 = 3x + 6.

The two expressions are not equal because they have different coefficients of x and different constant terms.

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If the area of a kite is 35cm square, the area of the new kite with all diagonals multiplied by 2 is 70 cm².

If we multiply all the diagonals of a kite by 2, then the area of the new kite will be 4 times the area of the original kite.

The area of a kite is given by the formula:

Area = (diagonal 1 x diagonal 2)/2

Let the diagonals of the original kite be d1 and d2. Then, the area of the original kite can be expressed as:

Area = (d1 x d2)/2 = 35 cm²

If we multiply all the diagonals of the original kite by 2, then the new diagonals will be 2d1 and 2d2. The area of the new kite can be expressed as:

New area = (2d1 x 2d2)/2 = 2d1d2

Substituting the value of d1d2 from the original equation, we get:

New area = 2d1d2 = 2 x (d1 x d2) = 2 x Area = 2 x 35 cm² = 70 cm²

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By performing these statistical computations, you can analyze and interpret the central tendency of your dataset, which helps in understanding the overall pattern and distribution of the data.

It looks like you're seeking information on further statistical computation related to mean, mode, and median.

To calculate the mean, mode, and median of a dataset, follow these steps:

1. Mean: The mean is the average of all data points in a dataset.
  - Step 1: Add up all the data points.
  - Step 2: Divide the sum by the total number of data points.

2. Mode: The mode is the data point that occurs most frequently in a dataset.
  - Step 1: Count the frequency of each data point.
  - Step 2: Identify the data point(s) with the highest frequency.

3. Median: The median is the middle value in a dataset when the data points are arranged in ascending order.
  - Step 1: Arrange the data points in ascending order.
  - Step 2: If there is an odd number of data points, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values.

By performing these statistical computations, you can analyze and interpret the central tendency of your dataset, which helps in understanding the overall pattern and distribution of the data.

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a. The most typical case is desired: Mode. The mode is a useful measure of central tendency when the most typical or common value is of relevance since it denotes the value or category that occurs most frequently in a data collection.

b. The distribution is open-ended: Median. The median is the middle value in a data set when arranged in ascending or descending order. It is a suitable measure of central tendency when the distribution is open-ended or skewed, as it is less affected by extreme values compared to the mean.

c. The data collection has an extreme value: the median. The median is less sensitive to extreme values compared to the mean, making it a better measure of central tendency in data sets with extreme values or outliers.

d. The data are categorical: Mode. The mode is appropriate for categorical data, as it represents the most frequently occurring category or value in the data set.

e. Further statistical computations will be needed: This statement does not indicate a specific measure of central tendency. Further statistical computations may be needed to determine the appropriate measure of central tendency depending on the characteristics of the data and the specific objectives of the analysis.

f. The numbers should be split into two roughly equal groups, one of which should contain the higher values and the other should contain the smaller values: Median.  The median is the value that separates a data set into two equal halves, making it suitable for dividing data into two approximately equal groups based on their values.

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COMPLETE QUESTION-

For these situations, state which measure of central tendency - mean, median, or mode-should be used.

a. The most typical case is desired.

b. The distribution is open-ended.

c. There is an extreme value in the data set.

d. The data are categorical.

e. Further statistical computations will be needed.

f. The values are to be divided into two approximately equal groups, one group containing the larger values and one containing the smaller values.

We need approximately 12 servings of Designer Whey and 2 servings of Muscle Milk to obtain a 7-day supply that provides exactly 262 grams of protein and 54 grams of carbohydrates.

Let x be the number of servings of Designer Whey and y be the number of servings of Muscle Milk needed to obtain a 7-day supply that provides exactly 262 grams of protein and 54 grams of carbohydrates.

From the information given, we know that each serving of Designer Whey provides 20 grams of protein and 3 grams of carbohydrates, and each serving of Muscle Milk provides 16 grams of protein and 9 grams of carbohydrates.

Therefore, we can set up the following system of equations:

20x + 16y = 262

3x + 9y = 54

To solve for x and y, we can use any method of solving a system of equations. For example, we can use substitution:

From the second equation, we can solve for x in terms of y:

x = (54 - 9y)/3 = 18 - 3y

Substituting this into the first equation, we get:

20(18 - 3y) + 16y = 262

Simplifying, we get:

80y = 182

Solving for y, we get:

y = 2.275

Substituting this into the equation x = 18 - 3y, we get:

x = 12.175

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There are 4,320 ways to select six of the letters of the word algorithm and write them in a row if the first letter must be "a".

There are 7 letters in the word "algorithm", and we need to select 6 of them and arrange them in a row such that the first letter is "a". We can first choose the remaining 5 letters from the remaining 6 letters (excluding "a") in 6 choose 5 ways

⁶C₅ = 6!/5! = 6

Once we have chosen the 5 letters, we can arrange the 6 selected letters (including "a") in a row in 6! ways. Therefore, the total number of ways to select 6 letters and arrange them in a row with the first letter being "a" is

⁶C₅ × 6! = 6 × 720 = 4,320

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The measure of the unknown side from the given triangle is 14.48.

The given triangle is a right triangle with the following sides;

Hypotenuse = 15

Adjacent = x

Acute angle = 52 degrees

We are to determine the measure of the unknown side using trigonometry identity

Cos 15 = Adjacent/Hypotenuse

Cos 15 = x/15

x = 15cos15

x = 15(0.9659)

x = 14.48

Hence the measure of the unknown side is 14.48

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The side length x of the triangle to the nearest tenth is 170.3

The figures in the image are right-triangle.

From the diagram:

Angle θ = 20°

Opposite to angle θ = 62

Adjacent to angle θ = x

To find the value of x, we use the trigonometric ratio.

Note that: tangent = opposite / adjacent

Plug in the values

tan( 20 ) = 62 / x

Solve for x

x = 62 / tan( 20 )

x = 170.3

Therefore, the measure of side length labelled x is 170.3 units.

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The directionality of the alternative hypothesis and the support offered by the sample data determine whether the null hypothesis is likewise rejected at the 5% level of significance for a related one-tailed test.

When the two-tailed test rejects the null hypothesis, it means that the sample data, regardless of how we look at it, support the null hypothesis.. A one-tailed test, however, simply considers the evidence in one way. As a result, the null hypothesis should be used if the sample data only show evidence that the alternative hypothesis is true in one direction (for example, greater than).

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This is a contradiction, which means that there is no solution for the given equation. Therefore, the correct answer is option C, "No solution".

There is only one solution for the given equation.

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

Simplifying the equation, we get:

5 - 3x = 6 - 3x

Subtracting 6 from both sides, we get:

-1 - 3x = -3x

Adding 3x to both sides, we get:

-1 = 0

A linear equation is an equation that can be written in the form y = mx + b, where y and x are variables, m is the slope, and b is the y-intercept. It represents a straight line on a graph. Linear equations can be used to model a variety of real-world situations, such as the relationship between temperature and time, or the cost of producing a certain quantity of goods.

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We can see that both sides are equal, which means that the equation has infinitely many solutions.

Therefore, the answer is: D. Infinitely many solutions is correct.

To solve for the number of solutions of 5 - 3x = 4 + x + 2 -4x,

we first simplify the equation by combining like terms:

Combine like terms on both sides of the equation:

5 - 3x = 6 - 3x

Compare the coefficients of the x terms:

-3x = -3x

Since both sides of the equation have the same coefficients for the x terms, there are infinitely many solutions.

Therefore, the answer is: D. Infinitely many solutions is correct.

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The slope of the rope is 0.25 with the respective situation as the rope is attached to 6-foot high wall and the horizontal distance is 24 feet.

We need to determine the ratio of the vertical change to the horizontal change to establish the slope of the rope's line. To begin, we may use the Pythagorean theorem to calculate the length of the rope:

a² + b² = c²

where an is the height of the wall, b is the horizontal distance from the wall to the point where the rope is tied to the ground, and c is the length of the rope.

When we solve for c, we get:

c = √(6² + 24²)

c = √(36 + 576)

c = √612

c = around 24.73 feet

Consider a right triangle created by the wall, the point where the rope is fastened to the ground, and a point on the rope directly above the wall's top.

The vertical change is the wall's height, which is 6 feet.

The horizontal change is the distance of 24 feet between the place where the rope is tied to the ground and the wall.

As a result, the slope of the rope-representing line is:

vertical change / horizontal change = slope

slope = 6 / 24

slope = 0.25

As a result, the slope of the rope's line is 0.25.

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The size of the square to cut is 5/3 inches and the maximum volume of the box is 266.67 cubic inches.

To find the size of the square to cut and the maximum volume, we can follow these steps:

Let's call the length of each side of the square to be cut x inches. So the dimensions of the base of the box would be (16-2x) inches by (10-2x) inches.

The height of the box would be x inches since we are folding up the sides.

The volume of the box can be found by multiplying the length, width, and height: V = (16-2x)(10-2x)x.

To find the maximum volume, we can take the derivative of V with respect to x and set it equal to zero, since the maximum volume occurs at a critical point.

After taking the derivative and simplifying it, we get the equation 24x^2 - 520x + 1600 = 0.

Solving this quadratic equation, we get x = 5/3 or x = 20/3. Since x must be less than 5 (the length of the shorter side), the only feasible solution is x = 5/3 inches.

Plugging this value of x back into the equation for the volume, we get V = (16-2(5/3))(10-2(5/3))(5/3) = 266.67 cubic inches.

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Answer:

[tex]y = 2x - 1[/tex]

Step-by-step explanation:

We can represent this line in an equation in slope-intercept form:

[tex]y = mx + b[/tex],

where [tex]m[/tex] is the line's slope (rise over run), and [tex]b[/tex] is the y-coordinate of its y-intercept.

First, we can solve for the line's slope:

slope = rise / run = 2/1 = 2

Next, we can identify the y-coordinate of the y-intercept by looking at the vertical axis' value when the red line crosses it:

-1

Finally, we can put these two pieces of information together to form an equation in slope-intercept form:

[tex]\boxed{y = 2x - 1}[/tex]

To make 2 batches of cement, Jude will need 6 quarts of water and 8 packages of pre-mixed mortar.

To make 5 batches of cement, Jude will need 15 quarts of water and 20 packages of pre-mixed mortar.

Ratio and fractions are the main bases on which proportion is explained. Two ratios are equal when they are expressed as a fraction in the form of a/b, ratio a:b, and then a proportion. In this case, a and b can be any two integers. The ratio and proportion are important building blocks for understanding the numerous ideas in science and mathematics.

To make 2 batches of cement, Jude will need 6 quarts of water and 8 packages of pre-mixed mortar.

To make 5 batches of cement, Jude will need 15 quarts of water and 20 packages of pre-mixed mortar.

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Slot machines are a real-world example of a variable ratio schedule.

A variable-ratio schedule in operant conditioning is a partial reinforcement schedule where a response is reinforced after an arbitrary number of responses. 1 A consistent, high rate of response is produced by this schedule. A reward based on a variable-ratio schedule is one that can be found in gambling and lottery games.

The individual will continue to engage in the target behavior in variable ratio schedules because he is unsure of how many responses he must give before receiving reinforcement. This leads to highly stable rates and increases the behavior's resistance to extinction.

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Rotate triangle △C'D'E' counterclockwise by approximately -0.785 radians about the origin:

[tex]x1' = 1 \times cos(-0.785) - (-4) \times sin(-0.785) \approx 0.436[/tex]

[tex]y1' = 1 \times sin(-0.785) + (-4) \times cos(-0.785) \approx -3.678[/tex]

[tex]x2' = -7 \times cos(-0.785) - 8[/tex]

The given point [tex]s C(2, -8), D(-6, 4),[/tex] and [tex]E(0, 4)[/tex] form the triangle △CDE, and the points U(1, -4), V(-3, 2), and W(0, 2) form the triangle △UVW, with △CDE being the preimage of △UVW.

To represent the transformation algebraically, we can use a combination of translations and rotations.

Translation:

To translate a point (x, y) by a vector (h, k), we add h to the x-coordinate and k to the y-coordinate of the point.

To transform triangle △CDE to triangle △UVW, we can first translate triangle △CDE by a vector (h, k) to obtain triangle △C'D'E', where C' = C + (h, k), D' = D + (h, k), and E' = E + (h, k).

Since the coordinates of C are (2, -8) and the coordinates of U are (1, -4), we can calculate the translation vector (h, k) as follows:

[tex]h = 1 - 2 = -1[/tex]

[tex]k = -4 - (-8) = 4[/tex]

So the translation vector is [tex](-1, 4).[/tex]

Rotation:

To rotate a point (x, y) by an angle θ counterclockwise about the origin, we use the following formulas:

[tex]x' = x \times \cos(\theta) - y times \sin(\theta)[/tex]

[tex]y' = x \times \sin(\theta) + y \times \cos(\theta)[/tex]

To transform triangle △C'D'E' to triangle △UVW, we can apply a rotation of angle θ counterclockwise about the origin to triangle △C'D'E', where C' = (x1', y1'), D' = (x2', y2'), and E' = (x3', y3'). Since the coordinates of C' are (2, -8) after translation, and the coordinates of U are (1, -4), we can calculate the rotation angle θ as follows:

[tex]\theta = atan2(y1' - y2', x1' - x2') - atan2(y1 - y2, x1 - x2)= atan2((-8 + 4) - (-4), (2 + 1) - (-6 + 3)) - atan2((-8) - (-4), 2 - (-6))[/tex]

Using a calculator, we can find θ to be approximately -0.785 radians.

So, the algebraic representation of the transformation that maps triangle [tex]\triangle CDE[/tex] to triangle [tex]\triangle UVW[/tex]  is:

Translate triangle △CDE by the vector (-1, 4) to obtain triangle △C'D'E':

[tex]C' = (2, -8) + (-1, 4) = (1, -4)[/tex]

[tex]D' = (-6, 4) + (-1, 4) = (-7, 8)[/tex]

[tex]E' = (0, 4) + (-1, 4) = (-1, 8)[/tex]

Therefore, Rotate triangle △C'D'E' counterclockwise by approximately -0.785 radians about the origin:

[tex]x1' = 1 \times cos(-0.785) - (-4) \times sin(-0.785) \approx 0.436[/tex]

[tex]y1' = 1 \times sin(-0.785) + (-4) \times cos(-0.785) \approx -3.678[/tex]

[tex]x2' = -7 \times cos(-0.785) - 8[/tex]

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The missing angles of the diagram are:

∠1 = 118°

∠2 = 62°

∠3 = 118°

∠4 = 30°

∠5 = 32°

∠6 = 118°

∠7 = 30°

∠8 = 118°

Supplementary angles are defined as two angles that sum up to 180 degrees. Thus:

∠1 + 62° = 180°

∠1 = 180 - 62

∠1 = 118°

Now, opposite angles are congruent and ∠2 is an opposite angle to 62°. Thus: ∠2 = 62°.

Similarly: ∠3 = 118° because it is congruent to ∠1

Alternate angles are congruent and ∠5 is an alternate angle to 32°. Thus:

∠5 = 32°

Sum of angle 4 and 5 is a corresponding angle to ∠2 . Thus:

∠4 + ∠5 = 62

∠4 + 32 = 62

∠4 = 30°

This is an alternate angle to ∠7 and as such ∠7 = 30°

Sum of angles on a straight line is 180 degrees and as such:

∠8 = 180 - (30 + 32)

∠8 = 118° = ∠6 because they are alternate angles

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In the 8 Axioms, 4 and 6 axioms fails to holds and S is not a vector space. Rest of the axioms try to hold the vector space.

To demonstrate that S is not a vector space, we must demonstrate that at least one of the eight vector space axioms fails to hold. Let us examine each axiom in turn:

Therefore, axioms 4 and 6 fail to hold, and S is not a vector space.

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The correct question:

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by α(x₁, x₂) = (αx₁, αx₂); (x₁, x₂) ⊕ (y₁, y₂) = (x₁ + y₁, 0). We use the symbol ⊕ to denote the addition operation for this system in order to avoid confusion with the usual addition x + y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation ⊕, is not a vector space. Which of the eight axioms fail to hold?