Is 20/√40 the same as 1/√2

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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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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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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1.  we can use the first equation to find y = 3x + 1 = 3(9) + 1 = 28 So, the larger integer is 28, (2).  The smaller integer is 9. The larger integer is 10, which is the next consecutive integer.

An integer is a whole number ,neither a fraction or a decimal, in mathematics. Integers can be zero, positive, or negative.

Integer examples include -3, -2, -1, 0, 1, 2, 3, and so forth.

The letter "Z" stands for the set of integers.

In mathematics and allied disciplines such as computer science, physics, and finance, integers are utilized extensively.

A whole number is a non-negative integer in mathematics, which indicates that it is either a positive integer or zero.

Negative and fractional numbers are not included in whole numbers. The full integers 0, 1, 2, 3, 4, and so on are examples.

In many different branches of mathematics as well as in other disciplines, such as arithmetic, algebra, and geometry, whole numbers are used. They are also employed in daily tasks like quantity calculations and object counts.

According to question :-

Let's call the smaller integer x and the larger integer y. We know that:

y = 3x + 1 (the largest of the two integers is one more than three times the smaller)

x + y = 37 (the sum of the two integers is 37)

We can use substitution to solve for y in terms of x:

x + (3x + 1) = 37

4x + 1 = 37

4x = 36

x = 9

Now we can use the first equation to find y:

y = 3x + 1 = 3(9) + 1 = 28

So the larger integer is 28.

Let's call the smaller integer x and the larger integer y. We know that:

y = x + 1 (the integers are consecutive)

x + 2y = 29 (the sum of the smaller and twice the larger is 29)

We can substitute y = x + 1 in the second equation:

x + 2(x + 1) = 29

Simplifying and solving for x:

3x + 2 = 29

3x = 27

x = 9

So the smaller integer is 9. The larger integer is 10, which is the next consecutive integer.

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When solving an oblique triangle given three sides, use the inverse cosine form of the Law of cosines to solve for an angle.

When solving an "oblique-triangle" given three sides, we would use the inverse cosine, form of the Law of Cosines to solve for an angle.

The "Inverse-Cosine" function allows us to find the measure of an angle when given the ratio of the lengths of the triangle's sides.

The "Law-of-Cosines" relates the lengths of the sides of a triangle to the cosine of one of its angles.

The "Law-of-Cosines" states that for any triangle with sides of lengths a, b, and c, and opposite angles A, B, and C, respectively:

⇒ c² = a² + b² - 2ab × cos(C),

To solve for an angle, we would rearrange the equation to find the cosine of the angle,

⇒ Cos(C) = (a² + b² - c²)/(2ab),

Then, we will take the inverse cosine of both sides of the equation to find the value of the angle,

⇒ C = cos⁻¹((a² + b² - c²)/(2ab)),

This helps us to find the measure of angle C, depending on the units used in the original triangle sides.

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The given question is incomplete, the complete question is

When solving an oblique triangle given three sides, use the _____ form of the Law of cosines to solve for an angle.

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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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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The slope of the given line is 1/3 for the line passes through (0, 2) and (3, 3).

The slope of a line represents the rate at which it rises or falls as it moves horizontally from left to right.

It is calculated by taking the difference in the y-coordinates (vertical change) and dividing it by the difference in the x-coordinates (horizontal change) of two points on the line.

The slope can also be thought of as the tangent of the angle formed by the line and the positive x-axis.

In this problem, we are given that the slope of the line is 1/3.

We can use this information to find the slope between any two points on the line.

Let's consider the points (0, 2) and (3, 3) that lie on the line.

The change in y-coordinates is 3 - 2 = 1, and the change in x-coordinates is 3 - 0 = 3.

Therefore, the slope between these two points is 1/3.

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The question is -

What is the slope of the line?

Derive Linear Equations-Quiz-Level H

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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The probability of examining the five people till first patient has athlete's foot is given by (0. 84)^4(0. 16)^1.

Percent of people have athlete foot = 16%

Let us consider the probability of success represented by p = 0.16.

Using geometric distribution,

Probability of finding first patient with athlete's foot on kth examination is given by geometric probability distribution,

P(X = k) = (1 - p)^(k - 1) × p

where X is random variable representing number of trials needed until  first success.

Probability of finding the first patient with athlete's foot on the fifth examination k = 5,

⇒ P(X = 5) = (1 - p)^(5 - 1) × p

⇒ P(X = 5) = (0.84)^4 × 0.16

Simplify it we get,

⇒ P(X = 5) = 0.0473

Therefore, the probability of a podiatrist examining five people until he finds the first patient who has athlete's foot is (0. 84)^4(0. 16)^1.

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The resultant value of the given expression x² + 10x + 24 when x = 3 is (C) 63.

A finite collection of symbols that are properly created in line with context-dependent criteria is referred to as an expression, sometimes known as a mathematical expression.

Expressions in writing are made using mathematical operators such as addition, subtraction, multiplication, and division.

For instance, "4 added to 2" will have the mathematical formula 2+4.

So, we have the expression:

= x² + 10x + 24

Now, solve when x = 3 as follows:

= x² + 10x + 24

= 3² + 10(3) + 24

= 9 + 30 + 24

= 63

Therefore, the resultant value of the given expression x² + 10x + 24 when x = 3 is (C) 63.

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Correct question:

Evaluate the expression when x = 3.

x² + 10x + 24

a. 81

b. 86

c. 63

d. 60

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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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]\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]

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 communication is a one-way process in which a message is conveyed from a sender to a receiver is one of the assumptions made by the linear model of communication.

The straightforward linear model of communication presents communication as a process with distinct beginnings and endings.

But this model ignores the fact that communication is frequently a more complex and involved process that needs context, feedback, and a variety of channels.

The linear model presumes that the message is clearly understood by the recipient and that it is received by the recipient free from distortion or interference.

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An assumption of the linear model of communication is the belief that communication is a one-way process where a sender transmits a message to a receiver through a channel. This model assumes that communication is a linear process where the sender encodes a message, which is then sent through a channel to the receiver who decodes the message.

In the linear model, the sender encodes a message, sends it through a channel, and the receiver decodes the message. This model does not account for feedback or interaction between the sender and receiver, assuming that communication is a straightforward and direct transmission of information without any complexities or interruptions.

This model assumes that there is little to no feedback or interaction between the sender and receiver, and that the message is transmitted and received accurately without any distortion. However, this assumption is often criticized for oversimplifying the complex nature of communication and ignoring the role of context and feedback in the communication process.

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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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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]

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:

Data collected from several entities over a period of time (minutes, hours, days, etc.) are called time series data.

Time series data is a type of data that is collected over a period of time, with measurements taken at regular intervals. The data is often used to track changes in a particular variable over time and to analyze patterns or trends in the data.

Examples of time series data include stock prices, temperature readings, and daily sales figures. Time series data can be used to make forecasts or predictions about future trends based on past data, and it is often analyzed using statistical methods such as trend analysis, seasonality analysis, and autocorrelation analysis.

Time series data can be represented graphically using a line chart or a scatterplot, with time on the x-axis and the variable of interest on the y-axis. Time series data can be univariate, meaning that there is only one variable being measured, or multivariate, meaning that there are multiple variables being measured at the same time.

Statistical methods are a set of techniques and procedures used to analyze and interpret data in order to draw meaningful conclusions from it. These methods involve the collection, analysis, and interpretation of data in order to identify patterns, trends, and relationships between variables.

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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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Joshua started his own lawn care business. He donates 10% of his income to a local charity, which is recognized by the IRS as a qualified charitable organization. Joshua can claim expenses from his business and charitable donations as tax reductions, and they will reduce the amount of taxable income.

The first blank should be filled with "tax reductions" and the second blank should be filled with "amount of taxable income."

Hence, the correct statement would be: "Joshua can claim expenses from his business and charitable donations as tax reductions, and they will reduce the amount of taxable income."

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The number "Six hundred six and four thousand, three hundred fifteen ten-thousandths" is equal to 606.4315 in numerical form.

Given that

Six hundred six and four thousand, three hundred fifteen ten-thousandths

To write this number in numerical form, we need to add the values of each of its digits:

6 × 100 + 0 × 10 + 6 × 1 + 4 × 0.1 + 3 × 0.01 + 1 × 0.005 = 606.4315

Therefore, the number "Six hundred six and four thousand, three hundred fifteen ten-thousandths" is equal to 606.4315 in numerical form.

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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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Answer:  Elaine is 135cm tall

Step-by-step explanation:

Elaine's height is 75% of her father's height.

We can calculate Elaine's height by multiplying her father's height by 75% or 0.75.

Elaine's height = 0.75 * 180 cm

                        = 135 cm

Therefore, Elaine is 135 cm tall.

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Since the vertex is (0, -3), the quadratic function can be written in vertex form as:

f(x) = a(x - 0)^2 - 3

Where 'a' is a constant that determines the shape of the parabola. Since the end behavior of the function is y --> - Infinite as x --> - infinite and y --> - Infinite as x --> + infinite, the leading coefficient 'a' must be negative.

So, f(x) = -a(x^2 - 0x) - 3

Now, using the given point (1, -7) on the parabola, we can substitute the coordinates into the function and solve for 'a'.

-7 = -a(1^2 - 0(1)) - 3

-7 = -a - 3

a = 10

Therefore, the quadratic function that satisfies the given characteristics is:

f(x) = -10x^2 - 3

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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 MAD of the hourly wages given would be $ 0.48. The range would be $ 2.00. Q1 would be $8.25. Q3 would then be $9.25. The IQR would be $1.00

Calculate the MAD:

First, find the mean of the data set:

mean = (sum of all values) / (number of values)

mean = (8.25 + 8.50 + 9.25 + 8.00 + 10.00 + 8.75 + 8.25 + 9.50 + 8.50 + 9.00) / 10

mean = 88.00 / 10 = 8.80

Then, find the mean of these absolute deviations:

MAD = (sum of absolute deviations) / (number of values)

MAD = (0.55 + 0.30 + 0.45 + 0.80 + 1.20 + 0.05 + 0.55 + 0.70 + 0.30 + 0.20) / 10

MAD = 4.10 / 10 = 0.41

Calculate the range:

range = maximum value - minimum value

range = 10.00 - 8.00 = 2.00

Find Q1 and Q3:

{8.00, 8.25, 8.25, 8.50, 8.50, 8.75, 9.00, 9.25, 9.50, 10.00}

Q1 is the median of the lower half, and Q3 is the median of the upper half.

Lower half: {8.00, 8.25, 8.25, 8.50, 8.50}

Upper half: {8.75, 9.00, 9.25, 9.50, 10.00}

Q1 = median of lower half = 8.25

Q3 = median of upper half = 9.25

Calculate the IQR:

IQR = Q3 - Q1

IQR = 9.25 - 8.25 = 1.00

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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]

Learn more about coordinate here:

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