iq scores in a certain population are normally distributed. it is given that: (1) 13% of people have iq that does not exceed 80. (2) mensa is an organization whose members have iqs in the top 3% of the population. the minimum iq score required for admission to mensa is 128. 9(a) find the mean and the standard deviation of iq scores. round your answers to the nearest whole numbers. hint. find z-scores.

Answer:

A≈615.75cm²

Step-by-step explanation:

A=πr2=π·142≈615.75216cm²

A polynomial having integer coefficients and the lowest degree: p(x) =  x² - (2 + √2)x + 2√2. Here, integer coefficients for x² is 1: x is - (2 + √2) and 2√2 is the constant.

Sums (including differences) of polynomial "contexts" are polynomials. Any variables there in expression has to have whole-number powers for it to be a polynomial term.

A polynomial term can also be a simple number. In particular, an expression must not contain any square roots of variables, any fractional and negative powers mostly on variables, and any variables in any fractions' denominators in order to qualify as a polynomial term.

The given zeros for the polynomial are-

√2 and 2

There are two zeros so, the smallest degree of polynomial will be 2.

Thus, Let p(x) be the 2 degree polynomial.

Then, general equation will be:

p(x) = (x - √2)(x - 2)

On expansion:

p(x) = x² -2x -√2x +2√2

On simplification:

p(x) =  x² - (2 + √2)x + 2√2

Here, integer coefficients for x² is 1: x is - (2 + √2) and 2√2 is the constant.

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x - numero primero

y - numero segundo

x = y + 12

x - 4 = 2(y - 4)

x - y = 12

x - 4 = 2y - 8

- x + y = - 12

x - 2y = - 4

_______________

- y = - 16

y = 16

x = 16 + 12

x = 28

Answer:

312.5

Step-by-step explanation:

125g sultans for every 40g sugar

40 times 2 is 80, 125 times two is 250, then divide 125 by two to get how many for 20g of sugar

250+62.5=312.5

The solution for the equation given would be x = 2 and x = -1.

To solve the equation 2[x² + 2x + 1] - x² - 4x + 4 = x² - 3x, we can simplify and rearrange the terms as follows:

2x² + 4x + 2 - x² - 4x + 4 = x² - 3x

x² - 3x + 6 = x² - 3x

2x² - 2x - 6 = 0

Dividing both sides by 2, we get:

x² - x - 3 = 0

We can then solve for x by factoring or using the quadratic formula. Factoring gives:

(x - 2)(x + 1) = 0

So, the solutions are x = 2 and x = -1.

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To answer your question about the negative, in the expression (a + b) (x²-5) - (a + b) (3x + 5), the negative sign in front of the second term indicates that you should subtract the second term from the first term.

how to solve quadratic equation ?

There are different methods to solve quadratic equations, but one of the most common methods is the quadratic formula:

Given a quadratic equation in the form of ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero, the quadratic formula is:

x = (-b ± sqrt(b² - 4ac)) / 2a

To solve the quadratic equation using the quadratic formula, follow these steps:

Write the quadratic equation in the standard form ax² + bx + c = 0.

Identify the values of a, b, and c in the equation.

Substitute the values of a, b, and c into the quadratic formula.

Simplify the expression under the square root sign.

Apply the plus-minus sign and simplify the numerator.

Divide the simplified numerator by the denominator.

Write the solution(s) in the form of x = value.

To simplify the expression (a + b) (x²-5) - (a + b) (3x + 5), you can factor out the common factor of (a + b) from both terms:

(a + b) (x² - 5 - 3x - 5)

Simplifying the expression within the parentheses:

(a + b) (x² - 3x - 10)

Now, you can factor the trinomial inside the parentheses by finding two numbers that multiply to -10 and add up to -3. These numbers are -5 and 2:

(a + b) (x - 5) (x + 2)

So the final simplified expression is (a + b) (x - 5) (x + 2).

To answer your question about the negative, in the expression (a + b) (x²-5) - (a + b) (3x + 5), the negative sign in front of the second term indicates that you should subtract the second term from the first term.

And regarding the x^2 term, in the simplified expression (a + b) (x - 5) (x + 2), the x² term is represented by the (x - 5)(x + 2) part, which expands to x² - 3x - 10.

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The P(Z > 2.90) = 1 - P(Z ≤ 2.90)= 1 - 0.9986= 0.0014Thus, the proportion of overweight children with left atrial diameters greater than the mean of healthy children is 0.0014, rounded to four decimal places.Answer: 0.0014

According to the given question, we have to find what proportion of overweight children have left atrial diameters greater than the mean for healthy children. To calculate the answer, we need to use the z-score of the left atrial diameter. We have to find the z-score value corresponding to the mean of healthy children and the standard deviation of the left atrial diameter of healthy children. Then, we will have to use the z-score formula.

Let’s solve it.Step-by-step explanation:Given that we need to calculate the proportion of overweight children with left atrial diameters greater than the mean of healthy children. Here, we can use the standard normal distribution to solve this problem. To solve this problem, we need to calculate the z-score for the mean of healthy children and the left atrial diameter of healthy children.1. Find the z-score for the mean of healthy children.z = (x - μ) / σz = (0 - 0) / 1z = 0 Thus, the z-score for the mean of healthy children is 0.2.

Find the z-score for the left atrial diameter of healthy children.z = (x - μ) / σz = (3.95 - 2.5) / 0.5z = 2.90Thus, the z-score for the left atrial diameter of healthy children is 2.90.Using the z-score formula, we can find the proportion of overweight children with left atrial diameters greater than the mean of healthy children. Let’s use the formula for it.P(Z > 2.90) = 1 - P(Z ≤ 2.90)From the z-score table, we can find that the value of P(Z ≤ 2.90) is 0.9986.

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Using the idea of the angle of elevation as it could be applied to the problem, the height of the statue is 182 ft

When gazing up at an object or point, the angle of elevation is the angle formed between the horizontal plane and the observer's line of sight. It is, in other words, the angle at which the line of sight of an observer is tilted upward from the horizontal plane.

The angle of elevation is an important concept in geometry and trigonometry and is used to calculate the height of objects, such as buildings, towers, or trees, or to measure the distance between two points.

We know that;

Tan 34 = x/270

x = 270 tan 34

x = 182 ft

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Sonya is approximately 47 feet away from Thomas. And

height of the kite above the ground is approximately 43 feet.

How to find the height?

To find the height of the kite above the ground, we can use trigonometry. Let's draw a diagram:

       /|

      / |

     /  | height (h)

    /   |

   /    |

/θ___|___

 distance (d)

We know that the angle θ is 25°, the hand height is 3 feet, and the distance from the hand to the kite is 100 feet. We want to find the height of the kite above the ground, which we'll call h.

Using trigonometry, we can write:

tan(θ) = h / d

where tan(θ) is the tangent of the angle θ, and d is the distance from the hand to the kite. Solving for h, we get:

h = d * tan(θ)

Substituting the known values, we get:

h = 100 * tan(25°) ≈ 43.1 feet

So the height of the kite above the ground is approximately 43 feet.

2.Let's draw a diagram:

        Sonya

          |

          |

 66°      |       48°

          |    

          |

-----------o----------- Balloon

          |

          |

        Thomas

We want to find the distance between Sonya and Thomas, which we'll call x. We know that the height of the balloon above the ground is 126 feet. We also know the angles of elevation from Sonya and Thomas to the balloon.

Let's first find the distance from Sonya to the balloon. We can use trigonometry again:

tan(66°) = 126 / d

where d is the distance from Sonya to the balloon. Solving for d, we get:

d = 126 / tan(66°) ≈ 50.5 feet

Now let's find the distance from Thomas to the balloon:

tan(48°) = 126 / (x + d)

where x + d is the total distance from Thomas to the balloon (the sum of the distances from Thomas to the point directly below the balloon, and from that point to the balloon). Solving for x + d, we get:

x + d = 126 / tan(48°) ≈ 97.5 feet

Finally, we can solve for x:

x = (x + d) - d ≈ 97.5 - 50.5 ≈ 47 feet

So Sonya is approximately 47 feet away from Thomas.

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Answer:

Step-by-step explanation:

Answer: A. You cannot use any theorem along with option A. For B, you can use SAS. For option C, you can use SAS. For option D, you can use SSS. For option D, you can use ASA. So, the answer is A.

Lines AB and CD are parallel. If 6 measures (4x - 31)°, and 5 measures 111°, the value of x will be 21 if both angles are supplementary.

If lines AB and CD are parallel, then we can assume that m<5 and m<6 are supplementary that is the sum of both angles is 180 degrees

Therefore we can say that;

then 3x - 31 + 148 = 180

then 3x + 117 = 180

we need to then subtract 117 from both sides

therefore, 3x+117-117 = 180 - 117

then 3x = 180 - 117

then 3x = 63

then x = 63/3

therefore, x = 21

Hence the value of x will be 21 if both angles are supplementary.

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The slope of the line AB is 1.5 and the ratio of the sides is 2

The slope, m of line AB is solved using the formula

m = y2 - y1  / x2 - x1

using points 'A (-9, -5) and B (3, 3)

m = -9 - 3 / -5 - 3

m = -12 / -8

m = 1.5

For a similar triangle the ratio of side lengths are equal

AB / BC = AE / BD = BE / CD

The length of AB and BC is solved as follows

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points we want to find the distance between.

For AB:

x1 = -9, y1 = -5 (coordinates of A)

x2 = 3, y2 = 3 (coordinates of B)

distance_AB = sqrt((3 - (-9))^2 + (3 - (-5))^2)

distance_AB = sqrt(144 + 64)

distance_AB = sqrt(208)

distance_AB ≈ 14.42

For BC:

x1 = 3, y1 = 3 (coordinates of B)

x2 = 9, y2 = 7 (coordinates of C)

distance_BC = sqrt((9 - 3)^2 + (7 - 3)^2)

distance_BC = sqrt(36 + 16)

distance_BC = sqrt(52)

distance_BC ≈ 7.21

From the graph,

AE = 12

BD = 6

BE = 8

CD = 4

AB / BC = 14.42 / 7.21 = 2

AE / BD = 12 / 6 = 2

BE / CD = 8 / 4 = 2

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The shape of the cross section is a triangle.

To find the shape of the cross section after slicing through the triangular prism with piano wire, follow these :

Determine the original shape of the triangular prism.

Consider the orientation of the dotted lines indicating where the piano wire sliced through the prism.

Analyze the resulting shape after the cut is made.

In this case, Christina has a triangular prism, and she sliced it with piano wire.

Based on your description, the cross section would be a triangle since the wire cut through the triangular sides of the

prism.

an emotional relationship involving a couple and a third person with whom one of them is also involved.

Therefore, the shape of the cross section is a triangle.

a plane figure with three straight sides and three angles

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Coefficients and indeterminates are both parts of polynomials. The polynomial  [tex]x^{2}[/tex] - 15x + 36 has the components (x-3) (x-12).

To factor the polynomial  [tex]x^{2}[/tex]- 15x + 36, we need to find two numbers whose product is 36 and whose sum is -15. These numbers are -3 and -12,

=[tex]x^{2}[/tex] -12x -3x + 36

now, we'll use x as the common term between the first two terms and 3 as the common term between the next two terms.

=x(x-12)-3(x-12)

=(x-12)(x-3)

so we can write the polynomial as (x - 3)(x - 12). This is the factored form of the polynomial.

To verify this, we can expand the expression (x - 3)(x - 12) using the distributive property, which gives us  [tex]x^{2}[/tex]- 15x + 36. This confirms that the factored form is correct.

Factoring polynomials is an important skill in algebra as it helps simplify expressions and solve equations more easily. By factoring a polynomial, we can often find its roots, which are the values of x that make the polynomial equal to zero. This is useful in solving various types of problems in mathematics and science.

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The difference between the two expressions is:

(k³-7k+2)-(k²-12) = k³ - k² - 7k + 14

What is an expression?

We can begin by simplifying the left side of the equation:

(k³ - 7k + 2) - (k² - 12)

= k³ - 7k + 2 - k² + 12 // Distribution of the negative sign

= k³ - k² - 7k + 14 // Combining like terms

So the difference between the two expressions is:

k³ - k² - 7k + 14

An expression is a combination of one or more values, variables, and operators that can be evaluated to produce a result. Expressions can be as simple as a single number or variable, or they can be complex, combining multiple operators and functions.

What are variables?

A variable is a symbol or a named memory location that can hold a value. It is used to store and manipulate data during the execution of a program. Variables can be assigned different types of values, such as numbers, text, or Boolean (true/false) values. The value of a variable can change during the execution of a program, and it can be used in expressions and statements to perform calculations, make decisions, or control the flow of the program.

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Complete question is: (k³-7k+2)-(k²-12) = k³ - k² - 7k + 14

A unit square is a square with sides of length 1 unit. When we talk about the area of a unit square, we mean the amount of space that the square covers, which is equal to 1 square unit.

To demonstrate that 0.1 of 1 is the same as 0.1 x 1 using a unit square area model, we can imagine dividing the unit square into 10 equal parts vertically and horizontally. This creates a grid of 100 smaller squares, each with an area of 0.01 square units.

Now, let's consider the expression 0.1 of 1. This means taking 0.1 times the area of a unit square. Since the area of the unit square is 1 square unit, 0.1 of 1 is equal to 0.1 times 1, which is 0.1.

Similarly, we can represent the expression 0.1 x 1 visually on the unit square area model by shading in 0.1 of the unit square, as before. Since 0.1 x 1 is also equal to 0.1, the shaded area will once again cover 0.1 square units, confirming that these two expressions are equal.

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We may conclude after answering the presented question that As a expression result, the huge container is at 87.5% of its full capacity.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the depiction of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.

The big container is 8 ounces less than its maximum capacity of 64 ounces, which indicates it is 64 - 8 = 56 ounces full.

We may use the following formula to calculate the proportion of the container's capacity that is filled:

(Amount filled / Maximum capacity) x 100% = Percentage filled

Substituting the values yields:

% filled = (56 / 64) x 100%

87.5% of the spaces were filled.

As a result, the huge container is at 87.5% of its full capacity.

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Answer:

x ≤ 40

Step-by-step explanation:

-75-6x+13x ≤ 200

-75 is the cost of the booth a day

-6 is the intial cost of the paintings for Chris

x is the amount of paintings

13 is the price Chris sells the paintings for

200 is the profit needed

First thing you do is add like terms -75-6x+13x ≤ 200

add -6x and 13x

-75+7x ≤ 200

Now add 75 to both sides

7x ≤ 275

Now divide both sides by 7

x≤ 275

x≤ 39.2857142857

round up to x≤ 40   (because 39 paintings is not enough, 40 gives him $205 in profit while 39 only gives him $198 in profit)

Therefore, Jozef had 15 dimes and 27 quarters in his piggy bank.

An equation is a mathematical statement that shows that two expressions are equal. It typically consists of variables, constants, and mathematical operations. Equations can be solved by manipulating the expressions to find the value of the variables that satisfy the equation. Equations can be used to model real-world situations, and they are an important tool in many fields, including mathematics, physics, engineering, and economics.

Here,

Let's use the following variables to represent the number of dimes and quarters in Jozef's piggy bank:

d: the number of dimes

q: the number of quarters

We know that Jozef has a total of 42 dimes and quarters, so we can write the equation:

d + q = 42

We also know that the total value of the coins is $8.25. We can express this value in cents as:

10d + 25q = 825

We can simplify this equation by dividing both sides by 5:

2d + 5q = 165

Now we have two equations with two variables:

d + q = 42

2d + 5q = 165

We can use the first equation to solve for one of the variables in terms of the other. For example, we can solve for q:

q = 42 - d

Now we can substitute this expression for q into the second equation:

2d + 5(42 - d) = 165

Simplifying and solving for d, we get:

2d + 210 - 5d = 165

-3d = -45

d = 15

So Jozef had 15 dimes in his piggy bank. We can use the first equation to find the number of quarters:

15 + q = 42

q = 27

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Answer:

A = 18 cm²

Step-by-step explanation:

the area (A) of a trapezoid is calculated as

A = [tex]\frac{1}{2}[/tex] h (b₁ + b₂ )

where h is the perpendicular height between the bases b₁ and b₂

here h = 3 , b₁ = 8 , b₂ = 4 , then

A = [tex]\frac{1}{2}[/tex] × 3 × (8 + 4) = 1.5 × 12 = 18 cm²

Answer:

10 / 11  bananas in one week, I believe.

I'm sorry, but I need more information to answer your question.

The equation y = -10x + 216 represents a linear relationship between two variables, where y represents the number of students who drop out and x represents the year. However, I don't have any information about the values of x for previous years or any other data that can be used to make a prediction for the year 2012.

If you have additional information, such as the number of dropouts for previous years, you can substitute those values into the equation and solve for the value of y in 2012.

your messageConvert to number of atoms 294 grams AuTo convert 294 grams of Au to the number of atoms, we need to use the Avogadro's number, which is 6.022 x 10^23 atoms/mole. First, we need to find the number of moles of Au in 294 grams: 294 grams Au / 196.97 g/mole = 1.49 moles Au Next, we can calculate the number of atoms: 1.49 moles Au x 6.022 x 10^23 atoms/mole = 8.97 x 10^23 atoms Au Therefore, there are approximately 8.97 x 10^23 atoms of gold in 294 grams.PLEASE HELP ME!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!This figure is a rectangle with a semicircle on the shorter side.What is the perimeter of this figure?Use 3.14 for pi.A. 20.28 ftB. 30.28 ftC. 46.28 ftD. 74.24 ftTo find the perimeter of the figure, we need to add up the lengths of all the sides. Let's call the length of the rectangle "L" and the width "W". The rectangle has two sides of length L and two sides of length W, so the perimeter of the rectangle is: 2L + 2W The semicircle has a diameter equal to the width of the rectangle (W), so the circumference of the semicircle is: 1/2 (pi) W To get the total perimeter, we need to add the circumference of the semicircle to the perimeter of the rectangle. Since the semicircle only covers half of the width of the rectangle, we only need to add one width (W) to the perimeter of the rectangle. So the total perimeter is: 2L + 3W + 1/2 (pi) Wevaluate C(4,2)C(4,2) represents the number of ways to choose 2 items from a set of 4 distinct items. The formula for C(n,r) is n! / (r! * (n-r)!), where n is the total number of items and r is the number of items being chosen. Plugging in the values for C(4,2), we get: C(4,2) = 4! / (2! * (4-2)!) = 24 / (2 * 2) = 6 Therefore, there are 6 ways to choose 2 items from a set of 4 distinct items.Let f(x)=\sqrt(x) and g(x)=5\sqrt(x). Find (f-g)(x).(f-g)(x) represents the difference between f(x) and g(x). So we can write: (f-g)(x) = f(x) - g(x) Substituting the given expressions for f(x) and g(x), we get: (f-g)(x) = sqrt(x) - 5sqrt(x) To simplify this expression, we can factor out sqrt(x) as a common factor: (f-g)(x) = sqrt(x) * (1 - 5) Simplifying the expression in the parentheses, we get: (f-g)(x) = -4sqrt(x) Therefore, (f-g)(x) = -4sqrt(x)

Answer:

[tex] \begin{gathered} \mathfrak{\large\bold \bf \mapsto {\underline{ \boxed{ \sf{\ \: p \: = 1 }}}}} \: \purple \: \bigstar \\ \end{gathered}[/tex]

Step-by-step explanation:

[tex]\begin{gathered} \large\bold \bf \mapsto { { \sf{\ 5 ( 5p - 1) = 20 }}} \: \\ \end{gathered}[/tex]

[tex]\begin{gathered} \large\bold \bf \mapsto { { \sf{\ 5p - 1 = \frac{20}{5} }}} \: \\ \end{gathered}[/tex]

[tex]\begin{gathered} \large\bold \bf \mapsto { { \sf{\ 5p - 1= 4 }}} \: \\ \end{gathered}[/tex]

[tex]\begin{gathered} \large\bold \bf \mapsto { { \sf{\ 5p = 4 + 1 }}} \: \\ \end{gathered} \\ \\ \begin{gathered} \large\bold \bf \mapsto { { \sf{\ 5p = 5 }}} \: \\ \end{gathered}[/tex]

[tex]\begin{gathered} \large\bold \bf \mapsto { { \sf{\ p \: = \frac{5}{5} }}} \: \\ \end{gathered}[/tex]

[tex]\begin{gathered} \large\bold \bf \mapsto { \pink{ \bf{\ p \: = 1 }}} \: \\ \end{gathered}[/tex]

[tex] \underline{ \rule{250pt}{7pt}}[/tex]

The solution to the equation 5(5p-1) = 20 is p = 1.

An equation contains one or more terms with variables connected by an equal sign.

Example:

2x + 4y = 9 is an equation.

2x = 8 is an equation.

We have,

To solve this equation, we can follow these steps:

Simplify the left side of the equation by using the distributive property of multiplication:

5(5p-1) = 25p - 5

Solve for the unknown variable by isolating it on one side of the equation. In this case, we want to get p by itself, so we'll add 5 to both sides of the equation:

25p - 5 + 5 = 20 + 5

25p = 25

Finally, divide both sides of the equation by 25 to get the value of p:

25p/25 = 25/25

p = 1

Therefore,

The solution to the equation 5(5p-1) = 20 is p = 1.

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Answer:

x² + y² = 16

Step-by-step explanation:

the circle has its centre at the origin (0, 0 )

the equation of a circle centred at the origin is

x² + y² = r² ( r is the radius )

the radius r of the circle is 4 , then

x² + y² = 4² , that is

x² + y² = 16

The answer of the given question based on the area of Aaron’s bedroom is 8,448 inches².

Perimeter is total distance around  boundary of  two-dimensional shape. It is  sum of the lengths of all sides of the shape. For example, the perimeter of a rectangle is found by adding the length and width of the rectangle and multiplying the sum by 2, while the perimeter of a circle is found by multiplying the diameter by π. Perimeter is usually measured in units like inches, feet, meters, or centimeters.

We can begin by converting the width of the room from feet to inches, since the given perimeter is in inches:

8 ft = 8x12 inch =96 inch

Let the length of the room be L. Then, the perimeter P is given by the formula:

P = 2L + 2W

Substituting the given values, we have:

368 = 2L + 2(96)

368 = 2L + 192

2L = 176

L = 88 inches

So, the length of the room is 88 inches. The area A of the room is given by the formula:

A = L x W = 88 x 96 = 8,448  inches²

Therefore, the area of Aaron's bedroom is 8,448  inches².

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Option C : The slope of Function A is 6 and the slope of the line graphed in Function B, going through the ordered pairs (1, 4) and (-1, -2), is 3.

To compare the slopes of Function A and Function B, we need to determine the slopes of the line graphed in Function B and the slope of Function A, and then compare them using the answer choices provided.

The slope of the line going through the ordered pairs (1, 4) and (-1, -2) can be found using the slope formula:

slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(-1 - 1) = -6/-2 = 3

So the slope of Function B is 3.

The equation for Function A is f(x) = 6x - 1, which means its slope is 6.

Now we can compare the slopes using the answer choices provided:

Slope of Function B = 2 x Slope of Function A: 3 = 2 x 6 is not true

Slope of Function A = Slope of Function B: 6 = 3 is not true

Slope of Function A = 2 x Slope of Function B: 6 = 2 x 3 is true

Slope of Function B = − Slope of Function A: 3 = -6 is not true

Therefore, the equation that best compares the slopes of the two functions is "Slope of Function A = 2 x Slope of Function B".

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The equation represents Function A, and the graph represents Function B: Function A f(x) = 6x - 1 Function B graph of line going through ordered pairs 1, 4 and negative 1, negative 2 and negative 2, negative 5 Which equation best compares the slopes of the two functions? (1 point)

A. Slope of Function B = 2 x Slope of Function A

B. Slope of Function A = Slope of Function B

C. Slope of Function A = 2 x Slope of Function B

D. Slope of Function B = − Slope of Function A

On solving the provided question we can say that Therefore, the simplified equivalent expression is -6 - 12p.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the depiction of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.

[tex]2(-2-4p) + 2(-2p-1) = -4 - 8p - 4p - 2\\-4 - 8p - 4p - 2 = -6 - 12p\\[/tex]

Therefore, the simplified equivalent expression is -6 - 12p.

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5) Triangles QRN and MNL are not similar.

6) Triangle ECD is similar to triangle MNL

What is similarity of triangle?

Similarity of triangles is a concept in geometry that describes the relationship between two triangles that have the same shape but possibly different sizes. Two triangles are considered similar if their corresponding angles are congruent (equal in measure) and their corresponding sides are proportional (having the same ratio).

This can be expressed using the following notation: if triangle ABC is similar to triangle DEF, we can write it as:

∆ABC ~ ∆DEF

The symbol "~" means "is similar to."

5) To determine whether the two triangles QRN and MNL are similar, we can use the SSS~ (side-side-side) similarity criterion or the SAS~ (side-angle-side) similarity criterion.

Let's first check if the triangles satisfy the SSS~ criterion:

SSS~ criterion: If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

The sides of triangles QRN and MNL are:

QR = QN + RN = 56 + 48 = 104

RN = 48

QN = 56

MN = ML + NL = 60 + 70 = 130

NL = 70

ML = 60

We can see that the ratios of the corresponding sides are not equal:

QR/MN = 104/130 = 0.8

RN/NL = 48/70 = 0.686

QN/ML = 56/60 = 0.933

Since the ratios of the corresponding sides are not equal, the triangles QRN and MNL are not similar by the SSS~ criterion.

Now let's check if the triangles satisfy the SAS~ criterion:

SAS~ criterion: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

We can see that the included angles are not congruent, since we don't have any angle measures given. Therefore, we cannot use the SAS~ criterion to determine similarity.

Since the triangles do not satisfy either the SSS~ or SAS~ similarity criterion, we can conclude that they are not similar by either criterion.

6) In triangle ECD, we have:

angle ECD = 180 - angle CED - angle CDE

= 180 - (angle MNL + angle LNM) - angle CDE (since MNL is given as similar to ECD)

= 180 - (angle MNL + angle LNM) - angle CDM (since triangle CDM is similar to triangle LNM)

= 180 - 45 - 36 = 99 degrees

Similarly, in triangle MNL, we have:

angle MNL = 180 - angle LNM - angle MLN

= 180 - angle LNM - (angle ECD + angle CDE) (since MNL is given as similar to ECD)

= 180 - angle LNM - (angle CDM + angle CDE) (since triangle CDM is similar to triangle LNM)

= 180 - 36 - 45 = 99 degrees

Therefore, the triangles have two congruent angles: angle ECD is congruent to angle MNL, and angle CED is congruent to angle MLN.

Next, we need to check if the corresponding sides are proportional. We can do this by finding the ratios of the corresponding sides:

EC/MN = 96/36 = 8/3

CD/NL = 64/45

ED/ML = 80/54 = 40/27

If we simplify these ratios, we get:

EC/MN = 8/3

CD/NL = 64/45 = 16/9

ED/ML = 40/27

Since the ratios of the corresponding sides are not equal, the triangles are not similar by SSS~ or SAS~.

However, we can see that the triangles are similar by AA~, since they have two congruent angles. Therefore, a valid similarity statement would be:

Triangle ECD is similar to triangle MNL by AA~.

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Answer:

[tex]{ \tt{ - 1 > - 1 \frac{1}{4} }} \\ [/tex]