ASSIGNMENT DUE TMR PLEASEEEE HELP!!!

ASSIGNMENT DUE TMR PLEASEEEE HELP!!!

Answers

Answer 1

Answer: (x+2)² + (y-2)²= 25

Step-by-step explanation:

remember that the standard equation of a circle is (x-a)²+(y-b)²=r²


Related Questions

Select all sequences of translations, rotations, and reflections below that would take polygon P to polygon Q.

Answers

Answer:

poligon p

poligon q stop

The first pair is rotated and translated, the second pair is reflected and translated, and the third pair is reflected and translated.

What is a transformation of a shape?

A point, line, or geometric figure can be transformed in one of four ways, each of which affects the structure and/or location of the object.

Some of the transformations are given as,

Rotation does not change the shape and size of the geometry. But changes the orientation of the geometry.The translation does not change the shape and size of the geometry. But changes the location.The reflection does not change the shape and size of the geometry. But flipped the image.

The first pair is rotated and translated, the second pair is reflected and translated, and the third pair is reflected and translated.

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Select all sequences of translations, rotations, and reflections below that would take polygon P to polygon

Consider a relation R(A,B,C,D,E) with the following dependencies: {AB-> C, CD -> E, DE -> B} Is AB a candidate key of this relation? If not, is ABD? Explain your answer. No. The closure of AB does not give you all of the attributes of the relation.
If not, is ABD? Explain your answer. A -> A
B -> B
C -> C
D -> D
E -> E
AB -> ABC
AC -> AC
AD -> AD
AE -> AE
BC -> BC
BD -> BD
BE -> BE
CD -> BCDE
CE -> CE
DE -> BDE
ABD -> ABCDE
Yes, ABD is a candidate key. No subset of its attributes is a key.

Answers

The answer is B. You get this answer by multiplying 10 by 87.

The height of AXYZ is the
distance from point Y to XZ
Find the area of the
triangle. Round your answer
to the nearest tenth, if
necessary. y(4,5) a(2,1) x(0,2) z(4,0)
pls help quick:(

The height of AXYZ is thedistance from point Y to XZFind the area of thetriangle. Round your answerto

Answers

Answer:

areas of AXYZ=10 square units1

Answer:

10 units squared

Step-by-step explanation:

Area of a triangle = 1/2 × base × height

Use the distance between two points formula to find the measure of the height and base of ΔXYZ.

Distance between two points

\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

\(\textsf{where }(x_1,y_1) \textsf{ and }(x_2,y_2)\:\textsf{are the two points}\)

Height of Triangle

Define the points:

\(\textsf{Let }(x_1,y_1)=(2,1)\)\(\textsf{Let }(x_2,y_2)=(4,5)\)

Substitute the points into the distance formula to find the height of the triangle:

\(\begin{aligned}\implies \sf height & =\sqrt{(4-2)^2+(5-1)^2}\\& = \sqrt{2^2+4^2}\\& = \sqrt{20}\end{aligned}\)

Base of Triangle

Define the points:

\(\textsf{Let }(x_1,y_1)=(0,2)\)\(\textsf{Let }(x_2,y_2)=(4,0)\)

Substitute the points into the distance formula to find the height of the triangle:

\(\begin{aligned}\implies \sf base & =\sqrt{(4-0)^2+(0-2)^2}\\& = \sqrt{4^2+(-2)^2}\\& = \sqrt{20}\end{aligned}\)

Area of Triangle

\(\begin{aligned}\implies \sf Area & = \dfrac{1}{2} \times \sf base \times height\\& = \dfrac{1}{2}\sqrt{20}\sqrt{20}\\& = \dfrac{1}{2}(20)\\& = 10 \sf \:\:units^2 \end{aligned}\)

Whoever helps me with this I will vemo or cashapp or paypal you $10 but if I get everything right ill paypal $20

Whoever helps me with this I will vemo or cashapp or paypal you $10 but if I get everything right ill

Answers

Answer:

15. 2/6 or 1/3

18. 3/6 or 1/2

21. 25/26

29.

a- 1/12

50/600

b- 22/100

132/600

Step-by-step explanation:

The first ones are self explanatory lol.

to find the ones for 29, i just multiplied divided 12 by 600 and got 50 so that was my numerator and on the 2nd one i just multiplied 22 by 6 to get 132 because the denomenator is 600.

a random sample of 200 voters in a town is selected, and 114 are found to support annexation suit. find the 96% confidence interval for the fraction of the voting population favoring the suit.

Answers

The 96% confidence interval for the fraction of the voting population favoring the suit is 0.498< p< 0.642.

Here we have to find the confidence interval.

Given

Sample, n = 200 voters

Let x represent those that support the annexation suit.

x = 114

First, we have to calculate the probability of supporting the annexation suit.

Let p represent the probability of supporting the annexation suit.

p = x/n

p = 114/200

p = 0.57

From elementary probability;

p + q = 1 where q represents the probability of failure.

In this case, q represent the probability of not supporting the annexation suit

Substitute 0.57 for p

0.57 + q = 1

q = 1 - 0.57

q = 0.43

To find the 96% confidence interval for the fraction of the voting population favoring the suit;

The confidence interval is bounded by the following;

^p - z(α/2) √(pq/n) < p < ^p + z(α/2) √(pq/n)

At this point, we have values for p,q, and n.

Next is to solve z(α/2)

First, we'll find the value of α/2 using

C.I = 100%(1 - α) where C.I = 96%

96% = 100%(1 - α)

1 - α = 96%

1 - α = 0.96

α = 1 - 0.96

α = 0.04

So,

α/2 = 0.04/2

α/2 = 0.02

So, z(α/2) = z(0.02)

Using a normal probability table

z0.02 = 2.055 --- This is the closest value which leaves an area of 0.02 to the right and 0.98 to the left

Recalling our formula to solve 96% interval;

^p - z(α/2) √(pq/n) < p < ^p + z(α/2) √(pq/n)

By substitution, we have

0.57 - 2.055 * √(0.57*0.43/200) < p < 0.57 + 2.055 * √(0.57*0.43/200)

0.57 - 0.071939677073920 < p < 0.57 + 0.071939677073920

0.498060322926079 < p < 0.641939677073920 ---- Approximate

0.498 < p < 0.642

Therefore the confidence interval is 0.498<p<0.642.

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Describe the transformation of g(x) as it relates to the parent function f(x). f(x)=5^x g(x)=-5^x

Answers

The solution is,  the applicable choices are:

2.  reflection in the x-axis,

4. horizontal shift of 4 units right.

c) See the attached with all transformations f(x) → h(x) → g(x).

What is function?

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.

here, we have,

Given function:

g(x) = - (x - 4)³

This is a transformation of the

a) parent cubic function f(x) = x³

b) It was first reflected in the x-axis:

f (x) → h(x)  ⇒ x³ → - x³

Then translated right by 4 units:

h(x) → g(x)  ⇒ - x³ → - (x - 4)³

So the applicable choices are:

2. reflection in the x-axis,

4. horizontal shift of 4 units right.

c) See the attached with all transformations f(x) → h(x) → g(x).

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Describe the transformation of g(x) as it relates to the parent function f(x). f(x)=5^x g(x)=-5^x

Find the surface area of revolution about the x-axis of the graph of y=(4-x^2/3) 3/2, 0≤x≤8.

Answers

To find the surface area of revolution about the x-axis for the graph of y = (4 - x^(2/3))^(3/2), we can use the formula for surface area of revolution:

Surface Area = 2π ∫[a,b] y * √(1 + (dy/dx)^2) dx


First, let's find the derivative of y with respect to x to get dy/dx:

dy/dx = d/dx (4 - x^(2/3))^(3/2)
      = (3/2)(4 - x^(2/3))^(1/2) * d/dx (4 - x^(2/3))
      = (3/2)(4 - x^(2/3))^(1/2) * (-2/3)x^(-1/3)
      = (-3/2)(4 - x^(2/3))^(1/2) * x^(-1/3)

Next, let's simplify the expression inside the square root:

1 + (dy/dx)^2 = 1 + [(-3/2)(4 - x^(2/3))^(1/2) * x^(-1/3)]^2
             = 1 + [(-3/2)^2 * (4 - x^(2/3))] * [x^(-2/3)]
             = 1 + (9/4) * (4 - x^(2/3)) * [x^(-2/3)]
             = 1 + (9/4) * (4x^(-2/3) - x^(-2/3 + 2/3))
             = 1 + (9/4) * (4x^(-2/3) - x^0)
             = 1 + (9/4) * (4x^(-2/3) - 1)
             = 1 + (9/4) * (4/x^(2/3) - 1)

Now, we can substitute y and √(1 + (dy/dx)^2) into the surface area formula:
Surface Area = 2π ∫[0,8] (4 - x^(2/3))^(3/2) * √(1 + (dy/dx)^2) dx
                    = 2π ∫[0,8] (4 - x^(2/3))^(3/2) * √(1 + (9/4) * (4/x^(2/3) - 1)) dx

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The equation of the line of best fit is y = 0.755x + 11.339. Based on the line of best fit, approximately how many customers are predicted to use a card to pay on a day when 90 customers pay by cash?

Answers

Answer:

79

Step-by-step explanation:

Answer:

79

Step-by-step explanation:

Can anyone help? can’t figure this one out

Can anyone help? cant figure this one out

Answers

Answer:

d = 47°

Step-by-step explanation:

Image is attached

Sum of angles drawn along a straight line = 180°:

The angle drawn in pink : 180° - 112° = 68°

The angle drawn in green: 180° - 66° = 114°

The angle drawn in yellowish orange: 180° - 23° = 157°

The angle marked in purple: 180° - 102° = 78°

The figure drawn in the center is a 5-sided irregular pentagon

∴ Sum of interior angles in a 5-sided polygon = (n - 2) ×180°

                                                                            = (5 - 2) × 180°

                                                                            = 540°

The four angles calculated above will assist in calculating the fifth remaining angle of the pentagon (which is marked in light blue):

   68° + 114° + 78° + 157° + angle in light blue = 540°

Angle in light blue = 540° - 68° - 114° - 78° - 157°

                               = 123°

Sum of angles drawn along a straight line = 180°:

Angle marked in dark blue = 180° - 123°

                                                = 57°

Now focus on the angles present inside the triangle

Sum of interior angles of a triangle = 180°

= d + 76° + 57° = 180°

d has to be isolated and made the subject of the equation:

d = 180° - 76° - 57°

d = 47°

Can anyone help? cant figure this one out

PLEASE ANSWER QUICK!!!!! 25 POINTS
Find the probability of exactly one successes in five trials of a binomial experiment in which the probability of success is 5%
round to the nearest tenth

Answers

The probability of exactly one successes in five trials  is 0.20

Finding the probability of exactly one successes in five trials

From the question, we have the following parameters that can be used in our computation:

Binomial experiment Probability of success is 5%Number of trials = 5

The probability is calculated as

P(x) = nCx * p^x * (1 - p)^(n -x)

Where

n = 5

p = 5%

x = 1

Substitute the known values in the above equation, so, we have the following representation

P(1) = 5C1 * (5%)^1 * (1 - 5%)^(5 -1)

Evaluate

P(1) = 0.20

HEnce, the probability value is 0.20

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compare numbers.
Performs arithmetic operations (+, –, *, /) as well as comparison or relational operations (<, >, =); the latter are used to compare numbers.

Answers

Comparison or relational operations, such as <, >, =, allow for comparing numbers. They determine if a number is less than, greater than, equal to, or not equal to another number, enabling decision-making and comparisons within arithmetic and programming operations.

When comparing numbers, you can use various relational operators to determine the relationship between them. Here are the commonly used comparison operators:

Less than (<): This operator compares two numbers and returns true if the first number is smaller than the second number. For example, 5 < 10 is true.

Greater than (>): This operator compares two numbers and returns true if the first number is larger than the second number. For example, 10 > 5 is true.

Less than or equal to (<=): This operator compares two numbers and returns true if the first number is smaller than or equal to the second number. For example, 5 <= 5 is true.

Greater than or equal to (>=): This operator compares two numbers and returns true if the first number is larger than or equal to the second number. For example, 10 >= 10 is true.

Equal to (==): This operator compares two numbers and returns true if they are equal. For example, 5 == 5 is true.

Not equal to (!=): This operator compares two numbers and returns true if they are not equal. For example, 5 != 10 is true.

These comparison operators allow you to compare numbers and make decisions based on their relationships within arithmetic or programming operations.

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Find the surface area of a 6*6 h cube

Answers

Answer:

Surface Area = 6×62 = 216 inches2

Step-by-step explanation:

Formula for surface area of cube is 6 into side²

EXPLANATION

Cube has all sides of equal dimensions and like that it has 6 faces.

So total surface area of cube is equal to the combined area of all the six faces.

Now each face has equal dimensions so same area

Therefore,

area of a single face =side²

Therefore, area of entire cube = 6 times the area of a single face

= 6 * side²

n a large population, 51% of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Answers

The probability that at least one of the randomly selected 5 people has been vaccinated is 0.9502.

The probability of at least one vaccinated person from a population where 51% of people are vaccinated and 5 people are chosen randomly is calculated as follows:Since the probability of a person who has been vaccinated is 51%, and the probability of a person who has not been vaccinated is 49%, the probability of at least one person being vaccinated from 5 people who are selected randomly is given by the formula:P(at least one vaccinated person) = 1 - P(no vaccinated person)The probability of no vaccinated person is calculated as follows: P(no vaccinated person) = 0.49 × 0.49 × 0.49 × 0.49 × 0.49 = 0.0498

The probability of at least one vaccinated person is calculated as follows:P(at least one vaccinated person) = 1 - P(no vaccinated person)P(at least one vaccinated person) = 1 - 0.0498P(at least one vaccinated person) = 0.9502Therefore, the probability that at least one of the randomly selected 5 people has been vaccinated is 0.9502.

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Solve the matrix equation for the unknown matrix X. (If not possible, enter IMPOSSIBLE in any cell of the matrix.) A = [7 3 4 3] B = [7 2 7 2] C = [3 3 4 0 0 8] D = [10 30 30 30 30 0] 5(X - C) = D x = []

Answers

After  solving the equation 5(X - C)  = D , for the given matrices , we get , the unknown matrix X is = \(\left[\begin{array}{ccc}5&9\\10&6\\6&8\end{array}\right]\) .

In the question ,

it is given that , the matrices are

A = \(\left[\begin{array}{ccc}7&3\\4&3\end{array}\right]\)   B = \(\left[\begin{array}{ccc}7&2\\7&2\end{array}\right]\)   C = \(\left[\begin{array}{ccc}3&3\\4&0\\0&8\end{array}\right]\)   and D = \(\left[\begin{array}{ccc}10&30\\30&30\\30&0\end{array}\right]\) .

and the equation is given as 5(X - C)  = D ,

and we have to find the value of unknown matrix X .

let the matrix X be = \(\left[\begin{array}{ccc}a&b\\c&d\\e&f\end{array}\right]\)

Substituting , the given matrices A , B , C and D in the equation 5(X - C)  = D ,

we get ,

5(X - C)  = D ,

5X - 5C = D ,

5\(\left[\begin{array}{ccc}a&b\\c&d\\e&f\end{array}\right]\) - 5\(\left[\begin{array}{ccc}3&3\\4&0\\0&8\end{array}\right]\) = \(\left[\begin{array}{ccc}10&30\\30&30\\30&0\end{array}\right]\)

5\(\left[\begin{array}{ccc}a&b\\c&d\\e&f\end{array}\right]\) = \(\left[\begin{array}{ccc}10&30\\30&30\\30&0\end{array}\right]\) - \(\left[\begin{array}{ccc}15&15\\20&0\\0&40\end{array}\right]\)

Simplifying further ,

we get ,

5\(\left[\begin{array}{ccc}a&b\\c&d\\e&f\end{array}\right]\) = \(\left[\begin{array}{ccc}25&45\\50&30\\30&40\end{array}\right]\)

Dividing both sides by 5 ,

we get ,

X = \(\left[\begin{array}{ccc}a&b\\c&d\\e&f\end{array}\right]\) = \(\left[\begin{array}{ccc}5&9\\10&6\\6&8\end{array}\right]\)

Therefore , the matrix X is = \(\left[\begin{array}{ccc}5&9\\10&6\\6&8\end{array}\right]\) .

The given question is incomplete , the complete question is

Solve the matrix equation for the unknown matrix X.

A = \(\left[\begin{array}{ccc}7&3\\4&3\end{array}\right]\)   B = \(\left[\begin{array}{ccc}7&2\\7&2\end{array}\right]\)   C = \(\left[\begin{array}{ccc}3&3\\4&0\\0&8\end{array}\right]\)   and D = \(\left[\begin{array}{ccc}10&30\\30&30\\30&0\end{array}\right]\) . the equation is

5(X - C)  = D .

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Help pls I really need help

Help pls I really need help

Answers

Answer:

It's B or the second one that says yes that you have not picked yet

Step-by-step explanation:

B the secound answer

Step-by-step explanation:

because -1/4 is greater than -3/4

Look at the shapes and decide wheater to use (1) sine law(2) cosine law, or (3) SOHCAHTOA

Look at the shapes and decide wheater to use (1) sine law(2) cosine law, or (3) SOHCAHTOA

Answers

The triangle ABC, the angle x is 36°.

Given that,

In the picture, we have triangles ABC.

We have to find the angle x.

The sides of the triangle are AB=3cm and AC=5cm.

We have the angle B is 90°.

We use the sine law.

In trigonometry, there are six fundamental ratios that are used to establish a connection between a right triangle's side-to-angle ratio and its angle. If the angle created by the base and hypotenuse of a right-angled triangle is, then

Perpendicular/hypotenuse = sin θ

base/hypotenuse = cos θ

tan θ = Base/Perpendicular

The three additional functions' values, cot, sec, and cosec, are dependent on the values of tan, cos, and sin, as shown below.

cot θ= perpendicular/base

sec θ=hypotenuse/base

cosec θ=hypotenuse/perpendicular

So,

Sin x=opposite side/hypothesis

Sin x=3/5

Sin x= 0.6

x= sin⁻¹(0.6)

x=36.86°

Approximately, x=36°

Therefore, The angle x is 36°.

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Which equation could they use to make a graph that is steeper than the graph of the parent quadratic equation?

Which equation could they use to make a graph that is steeper than the graph of the parent quadratic

Answers

Answer:

first one

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

3. Classify the triangle by its angles and its sides. Explain how you knew which classifications to use. A triangle has sides measuring 8, 11, and 12 and angles measuring 45 degrees, 65 degrees, and 70 degrees.

Answers

Based on the given side lengths of 8, 11, and 12, none of them are equal. Therefore, we can classify the triangle as a Scalene Triangle.

To classify a triangle by its angles and sides, we can use the properties and definitions of different types of triangles. Let's analyze the given triangle with sides measuring 8, 11, and 12 and angles measuring 45 degrees, 65 degrees, and 70 degrees.

Classification by angles:

Acute Triangle: An acute triangle has all three angles less than 90 degrees.

Obtuse Triangle: An obtuse triangle has one angle greater than 90 degrees.

Right Triangle: A right triangle has one angle exactly 90 degrees.

Based on the given angles of 45 degrees, 65 degrees, and 70 degrees, none of them are greater than 90 degrees, so we can classify the triangle as an Acute Triangle.

Classification by sides:

Equilateral Triangle: An equilateral triangle has all three sides of equal length.

Isosceles Triangle: An isosceles triangle has two sides of equal length.

Scalene Triangle: A scalene triangle has all three sides of different lengths.

Based on the given side lengths of 8, 11, and 12, none of them are equal. Therefore, we can classify the triangle as a Scalene Triangle.

In summary, based on the given measurements, the triangle can be classified as an Acute Scalene Triangle. We determined this by comparing the angles to the definitions of acute, obtuse, and right triangles, and comparing the side lengths to the definitions of equilateral, isosceles, and scalene triangles.

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Please Help me - You will get 60 points for the rapid reply- Use isosceles trapezoid ABCD to determine the following measurements-

Please Help me - You will get 60 points for the rapid reply- Use isosceles trapezoid ABCD to determine

Answers

Answer:

1) AD = 9 in

2) DE = 9.25 in

3) ∠EDC = 36°

4) ∠AEB = 108°

5) 11.5 in

Step-by-step explanation:

1) AD = BC = 9in

2) AC = BD (diagonals are equal)

⇒ BD = 14.25

⇒ BE + DE = 14.25

⇒ 5 + DE = 14.25

DE = 9.25

3) Since AB ║CD,

∠ABE = ∠EDC = 36°

4) ∠ABE = ∠BAE = 36°

Also ∠ABE + ∠BAE + ∠AEB = 180 (traingle ABE)

⇒ 36 + 36 + ∠AEB = 180

∠AEB = 108

5) midsegment = (AB + CD)/2

= (8 + 15)/2

11.5

The cafeteria sold 6 more turkey sandwiches than ham sandwiches. They sold 16 sandwiches in all. How many ham sandwiches did the cafeteria sell?

Answers

Answer:

5 ham sandwiches

Step-by-step explanation:

To find the number of ham sandwiches sold, we should find the pair of numbers that have a difference of 6 and add to 16.

1 ham sandwich, 7 turkey sandwiches = 8 sandwiches2 ham sandwiches, 8 turkey sandwiches = 10 sandwiches3 ham sandwiches, 9 turkey sandwiches = 12 sandwiches4 ham sandwiches, 10 turkey sandwiches = 14 sandwiches5 ham sandwiches, 11 turkey sandwiches = 16 sandwiches

While this method isn't always very efficient, it worked pretty well for this case!

Therefore, the answer is 5 ham sandwiches.

Answer:

5

Step-by-step explanation:

11 and 5 have a 6 number gap between them 5 plus 11 gets you 16

I need help plsss, check all that apply

I need help plsss, check all that apply

Answers

they lie in the same plane is the answer ;)

what is the factor form of r^27 - s^30

Answers

Answer:

(r^9−s^10) (r^18+r^9s^10+s^20)

Find an ordered pair (x,y) that is a solution to the equation. 3x-y=9

Answers

Answer:

3x-y=9

(3,-9)

Step-by-step explanation:

Which of the following is NOT an important factor affecting growth in labor productivity?A) the saving rateB) the speed with which prices fallC) the growth rate of physical capitalD) the growth rate of labor productivity

Answers

Labor productivity is an important measure of economic growth, and there are several factors that can affect it. These include the amount of physical capital available, the education and training of workers, and the use of advanced technology. Overall, while falling prices can be a positive factor for economic growth, it is not directly tied to labor productivity, making option B the correct answer.

Option B, "the speed with which prices fall," is not directly related to labor productivity. While falling prices can indicate a decrease in inflation and potentially stimulate economic growth, they do not directly impact the efficiency of workers or the production process.

In contrast, the other options are all important factors that can affect labor productivity. A higher saving rate can lead to greater investment in physical capital and technological advancements, which can improve productivity. Similarly, increasing the growth rate of physical capital and labor productivity can both lead to increases in overall productivity.

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NEED HELP WITH THIS ASAP PLSSS!!!

NEED HELP WITH THIS ASAP PLSSS!!!

Answers

Answer:

(19)          (9)               (17)

               (5)

(11)                               (7)

               (16)

Step-by-step explanation:

b.5/9

Measure of xyz fast and correct please

Measure of xyz fast and correct please

Answers

Answer:

it will be 36 (B)

Step-by-step explanation:

1/2(72+55)=36

hope this help!

What is the slope of 2/3 on a graph

Answers

Answer:

The slope is 0

Step-by-step explanation:

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

y = mx + b

Using the slope-intercept form, the slope is 0.

m = 0

(See the attachment below)

Hope this helps :)

Pls brainliest...

What is the slope of 2/3 on a graph

17. Who am I? ___ Collection of one or more different types of variables, including arrays and pointers, that have been grouped under a single name for each manipulation.
a) template
b) array
c) structure
d) local variables

Answers

You are c) a structure. A structure is a collection of one or more different types of variables, including arrays and pointers, that have been grouped under a single name for each manipulation.

A structure is a user-defined data type that allows you to group together related data. For example, you could create a structure to store the name, age, and address of a person. The structure would have three variables, each of a different type: a string variable for the name, an integer variable for the age, and a string variable for the address.

The advantage of using a structure is that it allows you to treat the related data as a single unit. This makes it easier to manipulate the data and to pass the data to functions.

The other answer choices are incorrect. A template is a blueprint for creating a generic class or function. An array is a collection of elements of the same type. Local variables are variables that are declared within a function and that are only accessible within the function.

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For each of the following arguments, prove it using Natural Deduction. (a) ¬r∨¬p,¬q→r,¬s→p⊢N​q∨s (b) ¬(q∧¬p)⊢N​¬p→¬q

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(a) Using natural deduction, we can prove that ¬r ∨ ¬p, ¬q → r, ¬s → p logically entails q ∨ s. (b) Similarly, we can prove that ¬(q ∧ ¬p) logically entails ¬p → ¬q using natural deduction.

(a) Proof of ¬r ∨ ¬p, ¬q → r, ¬s → p ⊢ q ∨ s using Natural Deduction:

¬r ∨ ¬p (Premise)

¬q → r (Premise)

¬s → p (Premise)

Assume ¬(q ∨ s) (Assumption)

¬q ∧ ¬s (De Morgan's Law, 4)

¬q (Conjunction Elimination, 5)

¬s (Conjunction Elimination, 5)

¬s → p (Reiteration, 3)

p (Modus Ponens, 7, 8)

¬q → r (Reiteration, 2)

r (Modus Ponens, 6, 10)

¬r ∨ ¬p (Reiteration, 1)

¬r (Disjunction Elimination, 12)

¬r ∨ ¬s (Disjunction Introduction, 13)

¬s ∨ ¬r (Commutation, 14)

¬s (Disjunction Elimination, 15)

⊥ (Contradiction, 9, 16)

q ∨ s (Negation Introduction, 4-17)

Therefore, ¬r ∨ ¬p, ¬q → r, ¬s → p ⊢ q ∨ s.

(b) Proof of ¬(q ∧ ¬p) ⊢ ¬p → ¬q using Natural Deduction:

¬(q ∧ ¬p) (Premise)

Assume p (Assumption)

¬q (Negation Introduction, 1, 2)

p → ¬q (Conditional Introduction, 2, 3)

Assume ¬p (Assumption)

q (Negation Introduction, 1, 5)

¬p → q (Conditional Introduction, 5, 6)

¬p → ¬q (Contrapositive, 4, 7)

Therefore, ¬(q ∧ ¬p) ⊢ ¬p → ¬q.

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For an upcoming election, 500 registered voters were surveyed on a tax increase. 43% were for a tax increase and 51% were against a tax increase. The margin of error was 6%. What is an accurate conclusion from the poll?
a. Against a tax increase will win because it is above 50%.
b. The nal result is undecided because of the margin of error.
c. The nal result is undecided because percentages do not add up to 100%.
d. Against a tax increase will win because the margin of error will increase their percentage.

Answers

Against a tax increase because either way the margin of error may fall, against will still have more votes
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