Which table shows a proportional relationship between x and y?

The confidence interval in the form of ^p-E < p < ^p+E is 0.013 < p < 0.089.

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. In this case, the population parameter is the true proportion or probability of success, represented by the symbol "p". The interval is defined by a lower bound and an upper bound, represented by "^p-E" and "^p+E" respectively.

E is the margin of error.

In this case, the lower bound of the interval is 0.013, and the upper bound is 0.089. So, we can say that there is a certain level of confidence that the true proportion of success falls between 0.013 and 0.089.

It is important to note that confidence intervals are not a measure of how well a model or estimate predicts future observations. It is only a measure of how uncertain we are about the true population parameter.

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The required solution to the given equation is x = 106.04, which is the correct answer that would be option (A).

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

The equation is given in the question following:

x - 69.80 = 36.24

We have to determine the solution to the given equation

As per the question, we have

x - 69.80 = 36.24

Add 69.80 on both sides of the equation, and we get

x - 69.80 + 69.80 = 36.24 + 69.80

x = 106.04

Hence, the correct answer would be an option (A).

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the centroid of the region bounded by the given curves is (x, y) = (15, 225).

To show that f is continuous on (−∞,∞), we need to show that the limit of f(x) as x approaches any point from the left and from the right is equal.

First, let's consider the limit of f(x) as x approaches 1 from the left.

As x approaches 1 from the left, f(x) = 1-x^2, so the limit of f(x) as x approaches 1 from the left is 1.

Now, let's consider the limit of f(x) as x approaches 1 from the right.

As x approaches 1 from the right, f(x) = ln x, so the limit of f(x) as x approaches 1 from the right is 0.

Since the limit of f(x) as x approaches 1 from the left is 1 and the limit of f(x) as x approaches 1 from the right is 0, we can conclude that f is continuous on (−∞,∞).

To find the centroid of the region bounded by the given curves, we need to calculate the area of the region and the area-weighted centroid of the region.

The area of the region is given by the integral of x3 from 0 to 10, which is equal to 1125/4.

The area-weighted centroid of the region is given by the integral of (x3*x)/(1125/4) from 0 to 10, which is equal to 15.

Therefore, the centroid of the region bounded by the given curves is (x, y) = (15, 225).

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On solving the equations x+2y = 14 and -x+3y = 26 , by equal value method , we get that the value of y is  8 .

The System Of Equations is given as : x+2y = 14 and -x+3y=26 ;

In the Equal value method , we equate the value of any one variable and simplify further .

in the first equation x+2y = 14 ; the value of x can be written as :

[tex]x = 14 - 2y[/tex] ; ....equation(1) ;

the second equation , [tex]-x+3y=26[/tex] , can be expressed as :  

[tex]x=3y-26[/tex] ;  ....equation(2) ;

equating both equation(1) and equation(2) ,

we get ;

⇒ 14 - 2y = 3y - 26 ;

⇒ 3y + 2y = 26 + 14 ;

⇒ 5y = 40 ;

⇒ y = 8 ;

Therefore , the value of y is = 8 .

The given question is incomplete , the complete question is

How do you solve for y in the system of equations x+2y = 14 and -x+3y=26  by  equal value method ?

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The largest image of the mascot that will fit on the Web page is 135 pixels wide and 90 pixels high.

A ratio displays the multiplicity of two numbers. For instance, if a dish of fruit contains eight oranges and six lemons, the ratio of oranges to lemons is eight to six. The ratio of oranges to the overall amount of fruit is 8:14, while the ratio of lemons to oranges is 6:8.

The connection between the amounts of two or more items is defined by a ratio. The amounts of the same kind are compared using this method. It is considered to be in proportion when two or more ratios are equal.

Two ratios are said to be in proportion if they are the same. If the four elements are a, b, c, and d in that order, then a/b = c/d. Extremes are the elements a and d, whereas mean terms are b and c. The ratio shows that the product of averages and the product of extremes are equal.

The height in the web page for the mascot is 90 pixels

let the width be x pixels

Therefore,

[tex]\frac{90}{x} = \frac{400}{600}[/tex]

⇒ 400*x = 90*600

⇒x = 135 pixels.

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Answer:

c ≤ -36

Step-by-step explanation:

c/9 ≤ −4

Multiply both sides by 9.

c ≤ -36

Answer: c ≤ −36

Step-by-step explanation:

Step 1: Simplify both sides of the inequality.

1/9c ≤ −4

Step 2: Multiply both sides by 9.

9*(1/9c) ≤ (9)*(−4)

Answer:

c ≤ − 36

The number of union workers is 42

A ratio can be defined in mathematics as an ordered pair of numbers, like  y and z, that is written in the form y / z such that the values of  z is not equal  to zero.

It also shows how many the times a number contains another number in a given proportion.

From the information given, we have that;

Then,

If 3 = 18 workers

Then 7 = x workers

cross multiply

3x = 18(7)

Multiply the values

3x = 126

x = 42 workers

Hence, the number is 42

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Answer: First point 6x -2y and second point 3x 4y

Step-by-step explanation:

Answer:

Not similar

Not similar

Similar

Step-by-step explanation:

Similar Polygons

If two polygons are similar their:

Pair of right triangles

If the given pair of right triangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 5:3=12:4=13:5[/tex]

[tex]\implies \dfrac{5}{3}= \dfrac{12}{4}= \dfrac{13}{5}[/tex]

[tex]\implies 1.67\neq 3 \neq 2.6[/tex]

Therefore, as the corresponding sides are not in the same ratio, the right triangles are not similar.

Pair of isosceles triangles

If the given pair of isosceles triangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 10:5=10:5=6:6[/tex]

[tex]\implies \dfrac{10}{5}=\dfrac{10}{5}=\dfrac{6}{6}[/tex]

[tex]\implies 2 =2 \neq 1[/tex]

Therefore, as the corresponding sides are not in the same ratio, the isosceles triangles are not similar.

Pair of rectangles

If the given pair of rectangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 1:2=5:10[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{5}{10}[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{5 \div 5}{10 \div 5}[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{1}{2}[/tex]

Therefore, as the corresponding sides are in the same ratio, the rectangles are similar.

The equation of the plane through the origin and the points (2, –4, 6) and (5, 1, 3) is - 9x + 12y + 11 x = 0.

The general equation of a plane through (0, 0, 0) is a(x - 0) + b(y - 0) + c(z - 0) = 0

Since the plane passes though the origin, the equation of the plane is given by ( x, y, z ) = 0. Simplify it so that you write the equation of the plane in the form a x + b y + c z = 0

ax + by + cx = 0...(1)

It will pass through B(2, -4, 6) and C(5, 1, 3) if

a(2) + b(-4) + c(6) = 0

2a - 4b + 6c = 0

a - 2b + 3c = 0 ...(2)

a(5) + b(1) + c(3) = 0

5a + b + 3c = 0 ... (3)

Solving (2) and (3) by cross-multiplication, we have

[tex]& \frac{a}{-6-3}=\frac{b}{15-3}=\frac{c}{1+10} \\[/tex]

[tex]& \Rightarrow \frac{a}{-9}=\frac{b}{12}=\frac{c}{11}=\lambda[/tex] (say) }

[tex]& \Rightarrow[/tex] a = - 9 [tex]\lambda[/tex], b = 12 [tex]\lambda[/tex]  and c = 11 [tex]\lambda[/tex]

Substituting the values of a, b and c in (1), we get

- 9 [tex]\lambda[/tex] x + 12 [tex]\lambda[/tex] y + 11 [tex]\lambda[/tex] x = 0

- 9x + 12y + 11 x = 0

which is the required equation of the plane.

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The Intermediate Value Theorem there must be a time t0: 0 < t0 < 12, where (x1 − x2)(t0) = 0 =⇒ x1(t0) = x2(t0).

What is the intermediate value theorem ?

Intermediate value theorem states that if “f” be a continuous function over a closed interval [a, b] with its domain having values f(a) and f(b) at the endpoints of the interval, then the function takes any value between the values f(a) and f(b) at a point inside the interval.

Let d be the distance between the monastery and the top of the hill.

Define x1(t) to be the distance traveled by the monk in t hours on day one and similarly

define x2(t) to be the distance traveled by the monk in t hours on day two.

In the above setup x1(0) = 0, x1(12) = d, x2(0) = d and x2(12) = 0.

Next consider the function (x1 − x2)(t), which is continuous on [−d, d]. Also, note that

(x1 − x2)(0) = −d and (x1 − x2)(12) = d.

The Intermediate Value Theorem there must be a time t0: 0 < t0 < 12, where (x1 − x2)(t0) = 0 =⇒ x1(t0) = x2(t0).

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Fraction data type is a numerical data type that consists of two numbers: a numerator and a denominator. It is used to represent a part of a whole or any other ratio.

Fraction data type is a numerical data type used to represent a part of a whole or any other ratio. It consists of two numbers: a numerator and a denominator. The numerator is the number of parts and the denominator is the number of equal parts that make up the whole. For example, if a pizza is cut into 4 equally sized pieces, each piece is 1/4 of the pizza. To represent this fraction in a fraction data type, the numerator would be 1 and the denominator would be 4. Fraction data type is used in mathematics, engineering, and other fields to represent relationships between different parts of a whole. It is also used to simplify calculations, such as addition and subtraction, by representing the same value in a simpler form. By using fraction data type, complex calculations can be simplified and the accuracy of results can be improved.

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The numerator and the denominator are the two numbers that make up the fraction data type, which is a numerical data type. It is employed to symbolize any ratio or a portion of a whole.

The fraction data type is a numerical data type that can be used to represent any ratio or portion of a whole. There are two figures in it: a numerator and a denominator.

The denominator is the total number of equally sized parts, while the numerator is the total number of parts. For instance, if a pizza is divided into four pieces of identical size, each one equals 1/4 of the entire pie.

The numerator and denominator of this fraction should be 1 and 4, respectively, to represent it in a fraction data type. In mathematics, engineering, and other disciplines, the fraction data type is used to indicate relationships between various components of a whole.

Additionally, it is used to make calculations like addition and subtraction simpler by expressing the same value in a more straightforward manner. Complex computations can be made simpler and results can be more accurate by using the fraction data type.

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The surface area of the triangular prisms are:

1. 153 cm²

2. 87 cm²

3. 12 cm²

4. 123 cm²

5. 118 cm²

To find the surface area of the triangular prism is the sum of all faces of the net of the triangular prism.

Problem 1:  

Area of the rectangle 1 = 9 * 5 = 45 cm²

Area of the rectangle 2 = 9 * 6 = 54 cm²

Area of the rectangle 3 = 9 * 4 = 36 cm²

Area of the two identical triangles = (6 * 3) = 18 cm²

The surface area = 45 + 54 + 36 + 9 = 153 cm²

Problem 2:  

Area of the rectangle 1 = 8 * 3 = 24 cm²

Area of the rectangle 2 = 6 * 3 = 18 cm²

Area of the rectangle 3 = 5 * 3 = 15 cm²

Area of the two identical triangles = 6 * 5 = 30 cm²

The surface area = 24 + 18 + 15 + 30 = 87 cm²

Problem 3:  

Area of the rectangle 1 = 5 * 5 = 25 cm²

Area of the rectangle 2 = 5 * 4 = 20 cm²

Area of the rectangle 3 = 5 * 3 = 15 cm²

Area of the two identical triangles = 4 * 3 = 12 cm²

The surface area = 25 + 20 + 15 + 12 = 72 cm²

Problem 4:  

Area of the rectangle 1 = 5 * 4 = 20 cm²

Area of the rectangle 2 = 7 * 5 = 35 cm²

Area of the rectangle 3 = 8 * 5 = 40 cm²

Area of the two identical triangles = 7 * 4 = 28 cm²

The surface area = 20 + 35 + 40 + 28 = 123 cm²

Problem 5:  

Area of the rectangle 1 = 7 * 5 = 35 cm²

Area of the rectangle 2 = 7 * 4 = 28 cm²

Area of the rectangle 3 = 7 * 5 = 35 cm²

Area of the two identical triangles = 4 * 5 = 20 cm²

The surface area = 35 + 28 + 35 + 20 = 118 cm²

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An angle is a measurement of how much of a "turn" there is between two lines or line segments. Consider two sticks meeting at a point. The place where they intersect is known as the vertex, and the lines or line segments are known as the angle's sides. When we turn one of the sticks so that it is no longer directed against the other, we have established an angle.

For kids, think of it as the space between two rays with the same terminus. It expresses how "open" or "closed" the space between the two rays is. The amount of rotation around the vertex can be measured in degrees. From 0 to 180, or 0 to 360. A complete rotation is 360 degrees or 2 radians.

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The linear regression equation for the model is Y = ( 0.9629 )X - 2.759 where the slope of the equation is m = 0.9629

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of the line be represented as A

Now , the homework grade is represented as x

The values of x = { 71 , 72 , 62 , 80 , 85 , 74 , 90 , 83 , 62 }

And , the test grade is represented as y

The values of y = { 63 , 75 , 58 , 77 , 71 , 65 , 95 , 68 , 57 }

From the linear regression calculator , the equation of line is given as

y = mx + b where m is the slope and b is the y-intercept

Y = ( 0.9629 )X - 2.759   be equation (1)

where the slope is m = 0.9629

And , Y = Test grade ; X = homework grade

Now ,  for a student with a homework grade of 80

Substitute the value of x as 80 , we get

Y = ( 0.9629 )X - 2.759

when X = 80 , we get

Y = ( 0.9629 ) ( 80 ) - 2.759

Y = 77.032 - 2.759

On simplifying the equation , we get

Y = 74.273

Y = 74

Therefore , the test grade Y of the student is 74

Hence , the linear regression equation is Y = ( 0.9629 )X - 2.759

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A polynomial of the form ax^n + bx^(n-1) + cx^(n-2) + ... + z is called an nth-degree polynomial, where n is a positive integer. The highest exponent in the polynomial is n.

The term "nth-degree polynomial" refers to the highest exponent in the polynomial. In other words, a polynomial is an nth-degree polynomial if the highest exponent in the polynomial is n.

For example, the polynomial x^2 + 3x + 5 is a 2nd-degree polynomial because the highest exponent is 2. Similarly, the polynomial 3x^4 + 2x^3 - 5x^2 - 6x + 9 is a 4th-degree polynomial because the highest exponent is 4.

The degree of a polynomial can be helpful in determining the number of roots that the polynomial has. For example, a 2nd-degree polynomial will have at most 2 roots, while a 4th-degree polynomial can have up to 4 roots.

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The option that is the better deal between both deals for apples, is 8 apples for $9.25.

The better deal on the apples would be the deal that gives the lower amount per apple. This means that the deal that leads to each individual apple having a lower price is the deal that should be considered best.

To find the price of an apple, you would need to divide the price of the deal given, by the number of apples in the deal.

The first deal would therefore have a per apple price of :

= Price of deal / Number of apples

= 6. 25 / 5 apples

= $ 1. 25 per apple

The second deal would be :

= Price of deal / Number of apples

= 9. 25 / 8 apples

= $ 1. 16 per apple

The better deal is therefore 8 apples for $9.25.

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Answer: 10

scince x is equal to -4 the equation becomes g(-4)=6- -4

Answer:

3

Step-by-step explanation:

Pothagorean theorem:

a² + b² = c²

c = 8

b = √55

a² + 55 = 64

a² = 9

a = 3

Answer:$48.80

Step-by-step explanation:

The standard form of the given polynomial is x⁶ + 53x⁵ + 2x⁴ - 2 which has a highest degree of 6 while the leading coefficient is 1. This polynomial is called a quadrinomial because it has 4 terms

A polynomial is an equation that only uses addition, subtraction, multiplication, and non-negative integer exponents of variables and consists of variables (also known as indeterminates) and coefficients. Algebra, calculus, and numerical analysis are just a few of the mathematics and scientific disciplines that use polynomials.

In the presented problem, we must determine the terms, variables, coefficients, and other properties of the polynomial f(x).

f(x) = 2x⁴ + 53x⁵ - 2 + x⁶

The standard form of this polynomial is x⁶ + 53x⁵ + 2x⁴ - 2

The degree of the polynomial is 6

The leading coefficient is 1 which is the coefficient of the degree of the polynomial

This is called a Quadrinomial because it has 4 terms

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The compound inequality for the weight of the ball is represented as 14 > w< 16.

When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.

Given that the ball used in a soccer game may not weigh more than 16 ounces or less than 14 ounces at the start of the match. After 1.5 ounces of air was added to a ball, the ball was approved for use in a game.

The compound inequality will be written as,

14 > w< 16.

Here the weight of the ball varies between 14 ounces to 16 ounces means that the minimum weight should be 14 ounces and the maximum is 16 ounces.

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Linear regression is a statistical tool used in predictive analytics to identify the relationship between a dependent variable and one or more independent variables.

Linear regression is a statistical technique used in predictive analytics to identify the relationship between a dependent variable and one or more independent variables. It is used to predict future outcomes by analyzing data from the past. It works by fitting a linear equation to the data, which is then used to estimate the value of the dependent variable for any given combination of values of the independent variables. The linear equation is constructed by finding the line of best fit through the data points. Linear regression can be used to identify trends and patterns in the data, and to make predictions about future outcomes. It is a powerful tool for predicting the future, and can be used to make informed decisions about how to best use resources.

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The unknown side marked with a question mark (?) is of length 5.0 approximately to one decimal place, using trigonometric ratio of sine for the angle 24.3°

The trigonometric ratios involves the relationship of an angle of a right-angled triangle to ratios of two side lengths. Basic trigonometric ratios includes; sine cosine and tangent.

sine of angle = opposite side/ hypotenuse

sin 24.3° = 2.06/?

? = 2.06/sin 24.3°

? = 5.0059

Therefore, by trigonometric ratio of sine for the angle 24.4°, the unknown side labelled with a question mark (?) has length equal to 5 rounded to one decimal place.

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The function that correctly depicts the scenario is represented as; f(x) = 18.50 (1.06)^x

The cost of the buffet in 10 years is; $33.13

The cost of the buffet 15 years ago is ; $7.71

Function is a type of relation, or rule, that maps one input to specific single output.

Here we have the following parameters that can be used in our computation:

Value of buffet = $18.50

Rate of increment = 6%

The function that correctly depicts the scenario is;

f(x) = Value of buffet  (1 + Rate of increment)^x

Substitute the values in the above equation, so;

f(x) = 18.50 (1 + 6%)^x

So, we have;

f(x) = 18.50 (1.06)^x

The buffet cost 10 years from now means that

x = 10

Substitute the values in the above equation,

f(10) = 18.50 (1.06)^10

Evaluate;

f(10) = 33.13

x = -15

Substitute the values in the above equation, so, we have

f(-15) = 18.50 (1.06)^(-15)

Evaluate;

f(-15) = 7.71

Hence, the amount 15 years ago is $7.71

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The probability:  P(A, A) if two cards are randomly selected with replacement is one ninth. Option B

You should be aware that probability is the chance of occurance of an event.  the probability of an event is written thus

P(E) = Number of required outcome divided by the total number of possible outcomes

The possible outcomes are the spelling the word ADD,

The probabilities are 1/3, 1/3, 1/3 respectively.

So, P(A, A) if two cards are randomly selected with replacement will be

P(A, A) = 1/3 * 1/3

Therefore the probability of the event is 1/9

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Answer: one ninth i took the test

Step-by-step explanation:

Answer:

To solve the equation 2(4x - 1) = -10(x - 3) + 4, we can follow these steps:

Step 1: Distribute the 2 on the left side of the equation:

8x - 2 = -10x + 6 + 4

Explanation: When we distribute the 2, we get 2 * 4x - 2 * 1, which simplifies to 8x - 2.

Step 2: Combine like terms on both sides of the equation:

8x - 10x = 6 + 4 - 2

-2x = 8

Explanation: On the left side of the equation, we have 8x - 10x, which simplifies to -2x. On the right side of the equation, we have 6 + 4 - 2, which simplifies to 8.

Step 3: Divide both sides of the equation by -2 to solve for x:

x = -4

Explanation: When we divide both sides of the equation by -2, we get -2x / -2 = 8 / -2, which simplifies to x = -4.

Therefore, the solution to the equation is x = -4.

To solve the equation 2(4x - 1) = -10(x - 3) + 4,we can follow these steps:

Distribute the 2 on the left side of the equation:

8x - 2 = -10x + 6 + 4

When we distribute the 2, we get 2 * 4x - 2 * 1, which simplifies to 8x - 2.

Combine like terms on both sides of the equation:

8x - 10x = 6 + 4 - 2

-2x = 8

On the left side of the equation, we have 8x - 10x, which simplifies to -2x. On the right side of the equation, we have 6 + 4 - 2, which simplifies to 8.

Divide both sides of the equation by -2 to solve for x:

x = -4

When we divide both sides of the equation by -2, we get -2x / -2 = 8 / -2, which simplifies to x = -4.

Therefore, the solution to the equation is x = -4.

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The measure of angle are:

m∠1 = 35°; m∠2= 125°; m∠3= 55°; m∠4 =125° ; m∠5 =55°.

What are similar triangles?

Similar triangles are triangles that have the same shape but differ in size. Similar objects include all equilateral triangles and squares with any side length. In other words, if two triangles are similar, their corresponding angles and sides are congruent and in equal proportion.

The △ABC is a right triangle.

Given that, 2AE = AC, 2CD = BD

AE/AC = 1/2; CD/BD = 1/2

∠ACD = ∠ECD = right angle

According to SAS rule, △ABC ≅ △ECD.

Thus the corresponding angles of △ABC and △ECD are congruent.

∠A = ∠E; ∠B = ∠D; ∠C = ∠C

Consider  △ABC:

∠A = 35°, ∠C =90°

The sum of all interior angles of a triangle is 180°.

∠A + ∠B + ∠C = 180°

35° +  ∠B + 90° = 180°

∠B + 125° = 180°

∠B  = 180° -  125°

∠B  = 55°

Therefore ∠1 = 55°; ∠5 = 55°; ∠3 = 55°.

The sum of the exterior angle and corresponding to the interior angle is 180°.

∠1  + ∠2 = 180°

55° + ∠2 = 180°

∠2 = 125°

Again:

∠5  + ∠4 = 180°

55° + ∠4 = 180°

∠4 = 125°

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The Probability of receiving 7 three times is 0.0025566, but the likelihood of getting 7 just once is 3.7528.

Simply put, the probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or probability of several outcomes. Statistics is the study of events that follow a probability distribution.

The given question is related to the binomial distribution.

Given n = 10 trials. The probability of one successful trial is p = 1/8. You want k=3 successes and n − k = 7 failures. The probability is:

[tex](\frac{n}{k}) p^k (1-p)^{n-k}[/tex]

One way to understand this formula: You want k successes (probability: pk) and n−k failures (probability: (1−p)n−k). The successes can occur anywhere in the trials, and there are (n/k) to arrange k successes in n trials.

substituting the values in the given formula.

[tex](\frac{10}{3}) 1/8^3 (1-1/8)^{10-3}[/tex]

Probability of getting 7, 3 times = [tex](\frac{10}{3}) 1/8^3 (7/8)^7[/tex]

Probability of getting 7, 3 times = [tex](\frac{10}{3}) 1/216 (7/8)^7[/tex]

Probability of getting 7, 3 times = 0.0025566

The Same goes for the probability of getting 7 only one times

the probability of getting 7 only one time = [tex](\frac{10}{1}) 1/8^1 (1-1/8)^{10-1}[/tex]

the probability of getting 7 only one time = 3.7528

Therefore, The Probability of getting 7, 3 times is 0.0025566 and the probability of getting 7 only one time is 3.7528.

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The required center of the circle's equation is (h, k) = (-4, 1), and the radius, r = 7.

The equation of a circle with center (h,k) and radius r is :

r² = (x−h)² + (y−k)²

To convert the equation to standard form, we complete the square for both x and y. To do this, we add and subtract the square of half the coefficient of the x term and the y term, respectively:

x² + 8x + (8/2)² - 2y + 2(8/2)(1/2)y - 2y + (2/2)² = 32

Simplifying, we get:

x² + 8x + 16 - 2y + y + 1 = 32

Now we can combine like terms:

x² + 8x + 16 + y - 2y + 1 = 32

x² + 8x + y + 17 = 32 + 17

x² + 8x + y = 49

Now we have the equation in standard form:

(x + 4)² + (y - 1)² = 49

The center of the equation is (h, k) = (-4, 1), and the radius, r = 7.

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