a system of equations is shown below.
y=3x+5
x + y=-7

Functions of these are algebraic expression (f+g)(x) = x² + x-2 and             (f-g)(x) = x² + 5x - 14.  

In mathematics, a functiοn is a rule οr mapping that assοciates each element x in a set (called the dοmain) with a unique element f(x) in anοther set (called the range οr cοdοmain).Fοr example  functiοn f(x) = x² + 3x - 8.

An algebraic expressiοn is a cοntains οf variables, cοnstants, and mathematical οperatiοns such as additiοn, subtractiοn, multiplicatiοn, divisiοn, and expοnentiatiοn. It cantains  parentheses and οther grοuping symbοls tο indicate the οrder in which the οperatiοns shοuld be perfοrmed.

In the given question ,

To find (f+g)(x),

we simply add the two functions f(x) and g(x) term by term:

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

= (x² + 3x - 8) + (-2x + 6)

= x²  + (3x - 2x) + (-8 + 6)

= x² + x - 2  

Therefore, (f+g)(x) = x² + x - 2.

To find (f-g)(x), we subtract g(x) from f(x) term by term:

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

= (x²  + 3x - 8) - (-2x + 6)

= x²  + (3x + 2x) + (-8 - 6)

= x²  + 5x - 14  

Therefore, (f-g)(x) = x²  + 5x - 14.

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The probability of the union of events A, B, and C is 0.7 where P(A)=0.2, P(B)=0.2 and P(C)=0.3.

In set theory, the union of two or more sets is a set that contains all the distinct elements of the sets being considered. Formally, the union of sets A and B, denoted as A ∪ B, is the set of all elements that are in either set A or set B, or in both.

When A, B, and C are mutually exclusive events, it means that they cannot happen at the same time. Therefore, the probability of the union of these events is equal to the sum of their individual probabilities.

In this case, we are given that:

P(A) = 0.2

P(B) = 0.2

P(C) = 0.3

To find the probability of the union of these events, we need to add their probabilities:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

Substituting the given probabilities, we get:

P(A ∪ B ∪ C) = 0.2 + 0.2 + 0.3

P(A ∪ B ∪ C) = 0.7

Therefore, the probability of the union of events A, B, and C is 0.7.

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The question is A, B, and C are mutually exclusive. P(A) = 0.2, P(B) = 0.2, P(C) = 0.3. Find P(A ∪ B ∪ C).

P(A ∪ B ∪ C) = ?

Answer:

C) ∠1 ≅ ∠7

Step-by-step explanation:

If ∠1 = ∠3, then ∠3 = ∠7, behind there is written ∠1 = ∠3, so it's A) ∠1 = ∠7.

Answer:

Step-by-step explanation:

We know that the time it takes to wash the dishes is uniformly distributed between 6 and 15 minutes. Let X be the time it takes to wash the dishes. Then, X ~ U(6, 15), where U represents the uniform distribution.

To find the probability that washing dishes tonight will take between 13 and 14 minutes, we need to find P(13 ≤ X ≤ 14). Since the distribution is uniform, the probability density function is f(x) = 1/(15-6) = 1/9 for 6 ≤ x ≤ 15.

The probability of a continuous random variable falling between two values is given by the integral of its probability density function between those values. So, we need to find:

P(13 ≤ X ≤ 14) = ∫13^14 f(x) dx

Substituting f(x) = 1/9, we have:

P(13 ≤ X ≤ 14) = (1/9) ∫13^14 dx

P(13 ≤ X ≤ 14) = (1/9) [x]13^14

P(13 ≤ X ≤ 14) = (1/9) [14 - 13]

P(13 ≤ X ≤ 14) = 1/9

Therefore, the probability that washing dishes tonight will take between 13 and 14 minutes is 1/9, or approximately 0.11 rounded to two decimal places.

We can explain to the boss that the advertising suggestion of "all bags have equally likely flavors" would be inaccurate based on the sample bag of candy.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

Based on the sample bag of Dream Pop candy, we can calculate the probability of getting each flavor.

If all bags have equally likely flavors, then each flavor should have the same probability of being selected.

However, we can see from the sample that this is not the case.

To communicate this to the boss, we can calculate the probability of getting two different flavors and compare them.

For example:

The probability of getting a grape flavor is 16/40 or 0.4

The probability of getting a lemon flavor is 6/40 or 0.15

These probabilities are not equal, indicating that the flavors are not equally likely.

We can also compare the probabilities of getting two other flavors, such as:

The probability of getting a strawberry flavor is 12/40 or 0.3

The probability of getting a blueberry flavor is 6/40 or 0.15

Again, these probabilities are not equal, further indicating that the flavors are not equally likely.

Therefore,

We can explain to the boss that the advertising suggestion of "all bags have equally likely flavors" would be inaccurate based on the sample bag of candy.

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The value of b is 73° as opposite angles of congruent sides are equal in an isosceles triangle.

An angle is created by cοmbining twο rays (half-lines) that have a cοmmοn terminal. The angle's vertex is the latter, while the rays are alternately referred tο as the angle's legs and its arms.

An angle within a shape that has the shape's base as οne οf its sides is knοwn as the base angle οf a shape in geοmetry. Cοnsider the triangle in the image as an example. We can οbserve that the triangle's base side is made up οf an angle B side and an angle C side. As a result, the triangle's base angles are angles B and C.

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Answer:

  "permutations that preserve distances"

Step-by-step explanation:

You want to know the meaning of "isometries" in the given discussion of dihedral groups.

The wording of the paragraph tells you the meaning:

  "isometries ... are permutations that preserve distances."

__

Additional comment

Often, writing that introduces an unfamiliar word will describe the meaning of that word. Here, the meaning is described, along with several examples (translations, rotations, reflections).

The word isometry has its origin in ancient Greek. The prefix "iso-" means "equal", and "-metry" comes from metron, meaning "measure." Effectively, an isometry is a transformation that preserves measures.

Step-by-step explanation:

You probably have a graphing tool to do this:

The two equations of a system that can be used to find the ages of both girls are y - x = 6 and 3x - y = 0.

A group of two or more equations that must be solved all at once is known as a system of equations. The objective is to determine the values of the variables that make both equations true in a system of two equations with two variables. These numbers represent the system of equations' answer. There are several ways to solve a system of equations, including substitution, elimination, and graphing. They are used to simulate and resolve issues in many branches of mathematics, science, engineering, and other disciplines.

Let us suppose the age of Celine = x.

Let us suppose the age of Cindy = y.

Given that, Celine is 6 years younger than her sister Cindy.

x = y - 6

She is also one third as old as Cindy.

x = 1/3y or y = 3x.

Substitute the value of y in equation 1:

x = 3x - 6

6 = 2x

x = 3

Substitute the value of x:

y = 3(3) = 9

Hence, the two equations that can be used to find the ages of  both girls are y - x = 6 and 3x - y = 0.

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The value of yp is equal to negative two divided by the expression 27x raised to the power of 27/4 and multiplied by e raised to the power of 3x.. This could be found using the Method of Undetermined Coefficients.

To find the particular solution using the Method of Undetermined Coefficients, we assume that the particular solution is of the form:

yp = Axⁱe⁽³ˣ⁾

where A is a constant to be determined and i is the smallest positive integer that makes yp linearly independent from the complementary function.

First, we find the complementary function by solving the characteristic equation:

r² - (4/3)r + 1 = 0

Using the quadratic formula, we get:

r = (4/3 ± √(16/9 - 4))/(-9/4) = -3/4 or -1

So the complementary function is:

yc = c₁e⁻ˣ + c₂e⁽-(3/4)x⁾

To determine the value of A, we substitute yp into the differential equation and equate coefficients of like terms.

yp' = A(xⁱe⁽³ˣ⁾)' = A(xⁱe⁽³ˣ⁾)(3e⁽³ˣ⁾ + i)

yp" = A(xⁱe⁽³ˣ⁾)" = A(xⁱe⁽³ˣ⁾)(9e⁽³ˣ⁾ + 6ie⁽³ˣ⁾ + i(i+3))

Substituting these into the differential equation and simplifying, we get:

(81/4)A(xⁱe⁽³ˣ⁾) + 4A(xⁱe⁽³ˣ⁾)(3e⁽³ˣ⁾ + i) + Axⁱe⁽³ˣ⁾ = 2xe⁽³ˣ⁾

Simplifying further, we get:

A(81/4 + 12i)e⁽³ˣ⁾xⁱ = 2x

To satisfy this equation for all x, we must have:

A(81/4 + 12i) = 0

and

A = 2/(xⁱe⁽³ˣ⁾)

Since A cannot be zero, we must have:

81/4 + 12i = 0

i = -27/4

Therefore, the particular solution is:

yp = -2/(27x⁽27/4⁾e⁽³ˣ⁾)

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Segment C has a length of √10, which is the closest to √10 compared to the other segments.

Segment is a customer data platform (CDP) that enables companies to collect, store, and analyze customer data from multiple sources. It helps companies build customer profiles and create personalized experiences for their customers. Segment allows businesses to track website visits, user actions, and other events in real-time, as well as to create custom events and store customer data in a secure and unified data warehouse. With Segment, companies can create powerful customer segmentation, which allows them to target customers with personalized messages and offers. Segment also integrates with various marketing, analytics, and CRM tools to provide a complete picture of customer behavior. It enables companies to build cohesive customer journeys, run campaigns, and optimize their customer experience.

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Complete Question.

The probability of at least three out of the next four songs being correctly identified by the application is 0.63 or approximately 63%.

What is probability?

Probability theory is a branch of mathematics that deals with the study of random events and their probabilities. It has applications in various fields, such as statistics, finance, engineering, and computer science, among others.

We want to find the probability that at least three out of the next four songs are correctly identified by the application. We can break down this event into four mutually exclusive cases:

The probability of all four songs being correctly identified is [tex](2/3)^4[/tex] or approximately 0.2.

The probability of exactly three out of four songs being correctly identified can be calculated by using the binomial probability formula:

[tex]P(X = 3) = (4 choose 3) * (2/3)^3 * (1/3)^1 = 4/27[/tex]

The probability of exactly two out of four songs being correctly identified can also be calculated using the binomial probability formula:

[tex]P(X = 2) = (4 choose 2) * (2/3)^2 * (1/3)^2 = 8/81[/tex]

The probability of one or none of the four songs being correctly identified is the complement of the sum of the probabilities of the first three cases:

[tex]P(X < = 1) = 1 - P(X = 4) - P(X = 3) - P(X = 2) = 44/81[/tex]

Therefore, the probability of at least three out of the next four songs being correctly identified is the sum of the probabilities of the first three cases:

[tex]P(X > = 3) = P(X = 4) + P(X = 3) + P(X = 2) = 0.2 + 4/27 + 8/81 = 0.63[/tex]

So the probability of at least three out of the next four songs being correctly identified by the application is 0.63 or approximately 63%.

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The second term of a geometric sequence of real numbers is -2 and the fifth term is 16, then the fourteenth term of real number is -8192.

In mathematics, a geometric series, also called a geometric sequence, is a sequence of non-zero numbers in which each term after the first is found by multiplying the previous term by a non-zero fixed number giving a common calling relation of ratio.

Given that:

In geometric sequence of real number is is -2

And The fifth term is 16.

According to the Question:

Let r  be  the common ratio

So

   (-2)r⁽⁵⁻²⁾  = 16

⇒ (-2)r³  = 16

⇒ r³ =  -16/2

⇒ r³ =  - 8

⇒ r   = -2

So the 14th term is

   (16)(-2)⁽¹⁴⁻⁵⁾

⇒ 16 (-2)⁹

⇒ 16 × (-512)  = -8192

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The correct option is D, The grades on Mr. Mueller's math exam were displayed in a histogram is I and II grades.

A histogram is a graphical representation of a distribution of data that shows how often each value or range of values occurs in a dataset. The horizontal axis of the histogram represents the range of values, while the vertical axis represents the frequency of their occurrence.

Histograms are commonly used to display and analyze large sets of numerical data. They are especially useful when dealing with continuous data, such as heights or weights, where the data can be grouped into intervals or bins. The height of each bar in the histogram indicates the number of observations in the dataset that fall within each interval. Histograms can reveal patterns, such as the shape of the distribution, the presence of outliers, and the presence of gaps or clusters in the data.

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Complete Question: -

The grades on Mr. Mueller's math exam were displayed in a histogram.

Based on this histogram, which of the following could be a list of the exam grades?

I. 55, 55, 60, 60, 65, 65, 70, 70, 70, 70, 75, 75, 80, 85, 85, 85, 85, 85, 90, 90, 95

II. 50, 55, 60, 60, 60, 65, 70, 70, 70, 75, 75, 75, 80, 80, 80, 80, 80, 85, 90, 95, 95

III. 50, 55, 65, 65, 65, 65, 70, 70, 70, 70, 75, 75, 75, 80, 80, 85, 85, 85, 85, 95, 95

a. III only   b. I,II, and III   c. I only   d. I and II

Therefore, the scale factor of the dilation of the shape is 2.

A scale factor is a ratio that describes the proportional relationship between two similar figures. It represents how much larger or smaller one figure is compared to the other, and it is calculated by dividing a corresponding measurement (such as side length, perimeter, or area) of the larger figure by the corresponding measurement of the smaller figure. Scale factor is used in mathematics, particularly in geometry and measurement, to describe the transformation of one figure into another through dilation or resizing. It is represented by a number or a ratio, such as 2:1 or 1/2, which indicates how many times larger or smaller the new figure is compared to the original.

Here,

In the given picture, it appears that the distance from the center of dilation (the origin) to the pre-image (the original figure) is 4 units, and the distance from the center of dilation to the image (the transformed figure) is 8 units. The scale factor of the dilation is equal to the ratio of the distance from the center of dilation to the image and the distance from the center of dilation to the pre-image.

So, the scale factor of the dilation is:

8 units ÷ 4 units = 2

Therefore, the scale factor of the dilation is 2.

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The graph that shows the electricity usage on a record-breaking summer day is Sacramento, California is a function.

The domain is 24 hours of a day.

The number of megawatts used at 8 am is 1, 200 megawatts.

The time with the most electricity used was 4 pm to 6 pm and least used was 4 am.

f ( 12 ) would be 1, 900 megawatts.

Usage is increasing from 4 am to 5 pm and decreasing from 5 pm to 4 am.

The graph is a function because each point on the graph represents a distinct megawatt usage. The domain would be 24 hours of a day as this graph of electricity usage shows the usage per day.

The megawatts used at 8 am is:

= 1, 300 - ( 200 / 2 )

= 1, 200 megawatts

From 4 am to 5 pm, we see that electricity usage is increasing as people are getting ready for work and going to work, but from 5 pm to 4 am, electricity usage decreases.

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In the abοve diagram rays are EC and Eb while lines are EL and PA.

A line is a straight path οn a plane that extends fοrever in bοth directiοns with nο endpοints while a ray is a line segment that extends indefinitely in οne directiοn.

1. Two rays EC and Eb

2. Two lines are EL and PA

Line

The line is a cοllectiοn οf pοints in a straight path.  It has nο endpοint and nο fixed length. A Line is οne-dimensiοnal. The line has length but nο width. A line is made up οf a set οf pοints that extend infinitely in οppοsite directiοns. Fοr example, a square is made up οf fοur lines οf equal length, while a triangle is made up οf three lines jοined at the ends.              

Ray

Ray is a part οf the line. It has οne endpοint and nο fixed length. A ray can be extended infinitely in any οne directiοn. Sunlight is an example οf a ray. A ray is alsο called a half-line. We dο nοt measure the length οf a ray.

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Answer:

Step-by-step explanation:

The line y = x is the line of symmetry in the reflection, which means that the x-coordinate and y-coordinate of each point get swapped. Therefore, the x-coordinate of A becomes its y-coordinate in its reflection A', and the y-coordinate of A becomes its x-coordinate in A'. Thus, the coordinates of A' are (n, m).

Therefore, the correct answer is J. (n, m).

The vertex is frequently identified by the letter that appears in the middle, like the letter V. Angles are often measured in degrees, and degrees are used to characterize angles. Thus, option C is correct.

Angle 7 is perpendicular to angle 3, and angle 3 is supplementary to angle 8, therefore they both fall along a straight line. Hence, angle 8 is complementary to angle 7.

Since they form a straight line and are opposite angles, angles 5 and 7 are supplementary. Hence, angle 5 is complementary to angle 7.

The only angle in the figure that is not a support for angle 7 is angle 6. If angles 6 and 7 do not overlap but share a vertex and side, then they are considered neighbouring.

Because they do not go to a straight line, angles 6 and 7 are not extra in this case. The answer is therefore Angle 6.

Therefore, They are not always complementary, even if two nearby angles come together to create a straight line.

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The value of PF1 + PF2 equals to 10 for any point P on curve ellipse of equation 16x^2 + 25y^2 = 400. So, the correct answer is B).

We can start by finding the coordinates of the point P on curve of the ellipse. We can write the equation of the ellipse as:

16x^2 + 25y^2 = 400

Dividing both sides by 400, we get:

x^2/25 + y^2/16 = 1

So, the center of the ellipse is at the origin (0,0) and the semi-axes are a=5 and b=4.

Let the coordinates of point P be (x,y). Then, we can use the distance formula to find the distances PF1 and PF2:

PF1 = sqrt((x-3)^2 + y^2)

PF2 = sqrt((x+3)^2 + y^2)

Therefore, PF1 + PF2 = sqrt((x-3)^2 + y^2) + sqrt((x+3)^2 + y^2)

We can use the property that the sum of the distances from any point on an ellipse to its two foci is constant, and is equal to 2a, where a is the semi-major axis. So, we have:

PF1 + PF2 = 2a = 2(5) = 10

Therefore, PF1 + PF2 equals to 10 for any point P on the ellipse 16x^2 + 25y^2 = 400. So, the correct option is B).

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Therefore , the solution of the given problem of probability comes out to be   z-score formula z =  2.59

The primary goal of the systems inside a procedure known as criteria is to determine the average possibility that a statement is true or that a specific occurrence will occur. Any value from 0 to 1, when 0 is frequently utilized to denote that something is likely and 1 is typically used to denote a level of confidence, can be used to represent chance. The chance that a specific event will occur is displayed in a probability diagram.

Here,

a.) The sample proportion's sampling distribution's mean is:

=> mu = p = 0.48

The sample proportion's sampling distribution's standard deviation is:

=> p(1-p)/n = √[α] = √[(0.48)(0.52)/400] = 0.027

We can standardize the sample percentage to a standard normal distribution using the z-score formula:

=> z = (0.55 - 0.48) / 0.027 = 2.59

=> z = (0.65 - 0.48) / 0.027 = 6.30

Therefore, the likelihood of making a Type I error is less than 0.001 if the real percentage of voters who support the increased tax is 48%.

b.) In the case of the alternate theory, the mean of the sampling distribution of the sample proportion is:

=>  mu = p = 0.48

Under the alternative theory, the sampling distribution of the sample proportion has a standard deviation of:

=> p(1-p)/n = √[α] = √[(0.48)(0.52)/400] = 0.027

When calculating z-scores,

We can standardize the sample percentage to a standard normal distribution using the z-score formula:

=>  z = (0.55 - 0.48) / 0.027 = 2.59

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The complete question is " A Random Sample Of 400 Voters In A Certain City Are Asked If They Favor An Additional 4% Gasoline Tax To Provide Badly Needed Revenues For Street Repairs. If More Than 220 But Fewer Than 260 Favor The Sales Tax, We Shall Conclude That 60% Of The Voters Are For It. A.) Find The Probability Of Committing A Type I Error If 60% Of The Voters Favor The Increased

A random sample of 400 voters in a certain city are asked if they favor an additional 4% gasoline tax to provide badly needed revenues for street repairs. If more than 220 but fewer than 260 favor the sales tax, we shall conclude that 60% of the voters are for it.

a.) Find the probability of committing a type I error if 60% of the voters favor the increased tax.

b.) What is the probability of committing a type II error using this test procedure if actually 48% of the voters are in favor of the additional gasoline tax? "

Answer:

Step-by-step explanation:

An Isosceles triangle is a triangle that has two equal sides. Also, the two angles opposite the two equal sides are equal.

the sum of the interior angles is 180° (triangles only)

so we have two angles of 62° and an unknown angle, we remove the known angles from 180° and we will have the angle x

180 - 62 - 62 =

56°

The statement " the continuity correction must be used because the normal distribution assumes variables whereas the binomial distribution uses discrete variables" is true because  continuity correction is used to adjust for the discrepancy between continuous and discrete variables when approximating a discrete distribution

The continuity correction is used when approximating a discrete distribution, such as the binomial distribution, with a continuous distribution, such as the normal distribution. The normal distribution assumes continuous variables, while the binomial distribution uses discrete variables.

The continuity correction helps to account for the fact that the normal distribution is continuous, whereas the binomial distribution is not. It adjusts the boundaries of the intervals used in the approximation, to better reflect the underlying discrete nature of the data.

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The problem asks for the time, we know that the blue whale will reach a depth of 190.65 meters below sea level after 26.7 minutes.

If a function is linear or not, its graph can be examined to determine. A straight line results from the graphing of a linear function. A nonlinear function has some sort of curve in contrast to a linear function.

The slope and y-intercept must first be determined in order to build a linear function. The slope can be determined using the two provided locations (4, -33.7) and (13, -99.4):

slope is the product of (y-change) and (change in x)

slope = (-99.4 - (-33.7)) / (13 - 4)

slope = -65.7 / 9

slope = -7.3

We can now use the point-slope form of a linear equation to obtain the equation of the line since we know its slope:

y - y1 = m(x - x1)

y - (-33.7) = -7.3(x - 4)

y + 33.7 = -7.3x + 29.2

y = -7.3x - 4.5

The following is the linear function that plots the blue whale's depth below sea level against time (x):

y = -7.3x - 4.5

We can plug in this number for y and solve for x to get how long it will take the blue whale to descend 190.65 metres below sea level:

190.65 = -7.3x - 4.5

195.15 = -7.3x \sx = -26.7

We know that the blue whale will descend to a depth of 190.65 metres below sea level in 26.7 minutes because the problem specifically asks for the time.

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to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{-6}{3 +3} \implies \cfrac{ -6 }{ 6 } \implies - 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{- 1}(x-\stackrel{x_1}{(-3)}) \implies y -2 = - 1 ( x +3) \\\\\\ y-2=-x-3\implies {\Large \begin{array}{llll} y=-x-1 \end{array}}[/tex]

Answer:

B) Group 2

Step-by-step explanation:

Group 1 wrote it correctly because there are 3 groups of 4x and 3 groups of 3

Group 2 wrote it incorrectly because that would mean 9 groups of 4x, which is not the case here

Group 3 wrote it correctly because 3(4x) + 3(3) = 3(4x+3) due to the Distributive Property

Group 4 wrote it correctly because it is an expansion of Group 3's expression

Answer:

Approximately 13.89 inches

Step-by-step explanation:

To find the hypotenuse, we use the Pythagorean theorem, in which c is the hypotenuse and a and b are the other two sides of the triangle:

a² + b² = c²

Plug in the values:

c² = 12² + 7²

c² = 144 + 49

c² = 193

c = √193

c ≈ 13.89 inches

As per the given velocity, the torque T that must be applied to the tube to maintain the motion is zero.

To determine the torque required to maintain the motion of the tube, we can use the formula for torque:

T = I x α

where T is the torque, I is the moment of inertia, and α is the angular acceleration.

The angular acceleration is given by:

α = (d² θ) / (dt²)

where θ is the angle of rotation and t is time.

Since the velocity profile is linear, we know that the velocity of the oil can be described by:

v = (δ v / δ y) x y

where v is the velocity of the oil, δ v is the change in velocity across the oil layer, δ y is the thickness of the oil layer, and y is the distance from the bottom of the oil layer.

We can use this equation to determine the velocity of the oil at the top of the oil layer, which is where the tube is in contact with the oil.

v = (δ v / δ y) x y

v = (δ v / 1.5 x 10^⁻³) x 1.5 x 10⁻³

v = δ v

Therefore, the velocity of the oil at the top of the oil layer is equal to the change in velocity across the oil layer.

Now, we can use the velocity of the oil to determine the angular acceleration of the tube.

α = (d² θ) / (dt²)

α = (d/dt) (σ)

α = 0

Since the angular velocity is constant at omega = 4.5 rad/s, the angular acceleration is zero.

Therefore, the torque required to maintain the motion of the tube is:

T = I x α

T = 0

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