dave sold popcorn and hot dogs at the game. he sold a total of $336 worth of both. he sold popcorn for $2.50 and hot dogs for $2 each. he sold twice as many bags of popcorn than hot dogs. how many bags of popcorn did he sell

The diameter decrease rate is approximately 0.041 cm/min when the diameter is 11 cm.

The surface of a sphere of diameter d is given by

[tex]A = 4πr^2 = πd^2[/tex]

where r is the radius of the sphere.

the rate of decrease in diameter (d) when the diameter is 11 cm.

We can start by deriving the formula for the surface area over time.

[tex]dA/dt = d/dt(πd^2)[/tex]

With chain rules it looks like this:

dA/dt = 2πd (dd/dt)

where dd/dt is the rate at which the diameter decreases.

Knowing that dA/dt is given as [tex]-9 cm^2/min[/tex] (unfavorable due to reduced surface area), look for dd/dt at d = 11 cm.

Substituting these values ​​into the above formula yields:

-9 = 2π(11) (dd/dt)

Simplified, it looks like this:

dd/dt = -9/(22π)

dd/dt ≈ -0.041 cm/min (rounded to three decimal places)

So for a diameter of 11 cm, the diameter decrease rate is approximately 0.041 cm/min. 

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We have to solve all of the parts of the given problem using our understanding of a sample and a population. It is given below.

A sample is taken when you want to take data from a specific set and then make observations based on that. A population is, on the other hand, the complete set that you want to make your observations from.

In the given problem -

A. To find how many books are published in one week by a famous publishing company - We are going to use a sample because we need to observe under s set category - one week.

B. To test a new drug produced by a biotech company - We are going to use a population because we need to monitor all individuals who take this drug.

C. To find the number of men and women working in an IT company with 600 people - We use a population here because we are going to use the data of all the people in the company.

D. To estimate the average salary of doctors in California - We use a sample here because the set category is California.

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Kim must receive 75 stickers if Jim gets 60 stickers.

1. Since Kim receives 4 stickers more than Jim, and Kim receives 20 stickers,

we can determine the number of stickers Jim receives:
Jim's stickers = Kim's stickers - 4
Jim's stickers = 20 - 4
Jim's stickers = 16
2. The ratio of Kim's stickers to Jim's stickers is:
Kim's stickers : Jim's stickers
20 : 16
3. To simplify the ratio, divide both numbers by their greatest common divisor (4):
(20/4) : (16/4)
5 : 4
So, the ratio of Kim's stickers to Jim's stickers is 5:4.
4. If Jim receives 60 stickers, we can determine how many stickers Kim must receive using the ratio 5:4:
4 units = 60 stickers (Jim's)
1 unit = 60 stickers / 4
1 unit = 15 stickers
Since Kim has 5 units in the ratio:
Kim's stickers = 5 units * 15 stickers/unit
Kim's stickers = 75 stickers.

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Answer:

Anderson bought 3 posters for his bedroom? What's the actual question.

To test if p - 1 is a factor of p^2 + p + p - k, we can use synthetic division.

Place the coefficients of p^2 + p + p - k in the top row of the division table, and write (p - 1) to the left, outside of the table:

       | 1   1   1   -k

    1  |__________

       |  

Bring down the first coefficient of 1:

       | 1   1   1   -k

    1  |__________

       |      1  

Multiply the 1 by (p - 1) to get p - 1:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

Add the entries diagonally (1 and 1) to get 2 and write the result under the next coefficient of 1:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

             2  

Multiply the 2 by (p - 1) to get 2p - 2:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

             2  

       -2p+2      

Add the entries diagonally (2 and 1) to get 3 and write the result under the next coefficient of 1:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

             2  

       -2p+2      

       ________

            3  

Multiply the 3 by (p - 1) to get 3p - 3:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

             2  

       -2p+2      

       ________

            3  

       -3p+3    

Add the entries diagonally (3 and 1) to get 4 and write the result under the next coefficient of -k:

       | 1   1   1   -k

    1  |__________

       |      1  

       |__________

             2  

       -2p+2      

       ________

            3  

       -3p+3    

       ________

                 4

The remainder is 4, which means that p - 1 is a factor of p^2 + p + p - k if k = 4.

Therefore, the value of k is 4.

Answer:

[tex]54 \: {ft}^{2} [/tex]

Step-by-step explanation:

The surface area is the sum of the lateral and base areas

Since the given figure is a cube (all sides and walls of a cube are equal), instead of finding its surface area the harder way, we can find the area of one side and multiply it by the number of sides a cube has (which is 6):

[tex]a(surface) = 3 \times 3 \times 6 = 54 \: {ft}^{2} [/tex]

Answer: 54 ft²

Step-by-step explanation: The side of a cube is given to be 3ft

We know the formula of the surface area of a cube is: 6A²

putting the value of side in the formula gives

6x(3)²=54 ft²

The height of the triangular window is 8 feet.

To find the height of Amy's triangular window, we need to use the formula for the area of a triangle:

Area = (1/2) × base × height.
From the student question, we know the area of the window is 24 ft², and the base of the window measures 6 ft.

We can plug these values into the formula:
24 = (1/2) × 6 × height
Now, let's solve for the height:
First, simplify the equation: 24 = 3 × height
Next, divide both sides by 3: height = 24 / 3
Finally, calculate the height: height = 8 ft.

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The sum of the given algebraic expression is =6g²-9g +4

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added before a variable and then multiplied by it.

The given expression is

=3g²-g+(3g²-8g+4)

=3g²-g + 3g² -8g +4

=6g²-9g +4

The sum of the given algebraic expression is =6g²-9g +4

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If basketball team sold total of 23, t-shirts and hats as a fund-raiser, then the number of hats sold is 8 and number of T-Shirts sold is 15.

Let us denote the number of t-shirts sold = "x',  and

let the number of hats sold be = "y".

So, a system of two equations to represent the information is written as:

The total-items sold at fund-raiser is = 23,

So,

⇒ x + y = 23      ...equation(1),

The total profit is = $246,

So,

⇒   10x + 12y = 246       ...equation(2)

Solving equation(1) for x,

We get,

⇒ x = 23 - y

Substituting value of "x" into equation 2,

We get,

⇒ 10 × (23 - y) + 12y = 246,

⇒ 230 - 10y + 12y = 246

⇒ 2y = 16

⇒ y = 8

So, the basketball team sold 8 hats.

Substituting this value into equation(1);

We get,

⇒ x + 8 = 23

⇒ x = 15

Therefore, the basketball team sold 15 t-shirts and 8 hats.

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The given question is incomplete, the complete question is

The basketball team sold t-shirts and hats as a fund-raiser. they sold a total of 23 items and made a profit of $246. they made a profit of $10 for every t-shirt they sold and $12 for every hat they sold.
Determine the number of t-shirts and the number of hats the basketball team sold.

The surface area of the right rectangular prism is 87.88 square units.

The surface area of ​​a rectangle is the total area or area of ​​six faces. A prism is a solid with straight parallelogram sides and a base of the same polygon. There are many types of prisms such as triangular prism, square prism, cartesian prism, penta prism, and hexagonal prism.

Given that:

from the set of information :

Length(l) = 5.2

Breadth (b) = 2.6

Height(h) = 3.9

Area of the rectangular prism = 2(lb + bh + lh)

                                                 = 2(5.2×2.6 + 2.6×3.9 + 3.9×5.2)

                                                 = 2(43.94)

                                                 = 87.88

Therefore, the surface area of the right rectangular prism is 87.88 square units.

Complete Question:

The net of a right rectangular prism is shown 2.6 3.9 5.2

What is the surface area of the right rectangular prism? Enter the answer as a decimal in the box unit².

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Hence, After solving this quadratic equation  D is the right response x = -9 or x = 1,

Three fundamental approaches can be used to resolve quadratic equations:

Factoring, first

2. Applying the quadratic equation

3. Finish the square.

To factor an equation with a quadratic term:

1. Place the equal sign with zero on the other side and all terms on one side.

2. Aspect.

3. Define each factor as zero.

4. Handle the equations in order.

5. Verify by entering your solution into the initial equation.

In order to use the quadratic formula to solve a quadratic equation:

1. Type the quadratic formula into your calculator: x = (-b (b2 - 4ac)) / (2a).

2. Determine the quadratic equation's values for the variables a, b, and c.

3. Enter these numbers as replacements in the quadratic formula, then simplify.

(x + 4)² = 25 is the quadratic version of the given equation.

By taking the square root of each side of the equation and then figuring out x,

we can solve this equation.

(x + 4)² = 25

Square roots of both sides result in x + 4 = 5 and x = -4 5.

Because of this, x = -9 or x = 1.

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Answer:

x=-9 and x=1

Step-by-step explanation:

It was correct for me

Therefore , the solution of the given problem of expressions comes out to be 4/x + 10/y can be expressed as 2(2y + 5x)/xy.

It is preferable to make use of shifting integer values that can be increasing, reducing, or blocking rather than approximations generated at random. They were only able to assist one another by exchanging resources, knowledge, or answers to problems. A declaration of truth equation may include the justifications, components, and mathematical.

Here,

We need to determine a common denominator for the two fractions in order to combine the rational expressions 4/x + 10/y. Here, xy is the common factor.

As a result, 4/x can be rewritten as (4y)/xy and 10/y as (10x)/xy.

We can now combine the two fractions:

=> (10x)/xy + (4y)/xy

=> (4y + 10x)/xy

=> 2(2y + 5x)/xy

As a result, 4/x + 10/y can be expressed as 2(2y + 5x)/xy.

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Geometry is the branch of mathematics that deals with shapes, sizes, positions, and properties of space. It involves concepts such as points, lines, angles, and shapes such as triangles, circles, and polygons.

To excel in geometry, it is important to have a good grasp of the basic concepts and formulas. You should understand the properties of different shapes and know how to calculate their areas and volumes.

One important formula in geometry is the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Another important concept is trigonometry, which deals with the relationships between the sides and angles of triangles.

Overall, geometry is a fascinating subject that is essential for a wide range of applications, from architecture and engineering to computer graphics and art. If you need more specific help or have any questions, feel free to ask!

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Answer: 0=0

Step-by-step explanation:

A statistical question will pose a problem with a potential range of solutions. 'How many hours does an average member of the club at Musa's school spend participating in club events each month?' is a statistical question that Musa might use to help him gather data about the club.

Only once varied data have been gathered can a statistical question be addressed. A statistical question is one that may be answered by gathering information, and there will always be variation in the information. Since there is no variability in the data used to answer the question, questions that can be answered by a single data point are not statistical questions. After data has been gathered and examined, the solution can give an interpretation of the data. Although there will be statistics for a non-statistical question, the outcomes won't change. There will be only one solution.

The following qualities should be included in a statistical question:

Indefiniteness

Indeterminism

Answers that are plural

Variability.

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Answer: The transformation is a reflection.

Step-by-step explanation:

The normally distributed with mean 2.4 and standard deviation 0.8:

a) fraction of the students will possess a GPS is 0.4648.

b) probability that all three will possess a gpa in excess of 3.0 is 0.2266.

c) percentage of the students will be dropped is 19.08%.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means.

Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence. With it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can vary from 0 to 1, with 0 being an impossibility and 1 denoting a certainty. For pupils in Class 10, probability is a crucial subject since it teaches all the fundamental ideas of the subject. One is the probability of every event in a sample space.

we have, μ = 2.4, σ = 0.8

(a) z1 = (x1 - μ)/σ = (2 - 2.4)/0.8 = -0.5

z2 = (x2 - μ)/σ = (3 - 2.4)/0.8 = 0.75

P(2 < x < 3) = P(-0.5 < z < 0.75) = 0.4648

(b) P(x > 3) = P(z > 0.75) = 0.2266

(c) z = (1.7 - 2.4)/0.8 = -0.875

P(x < 1.7) = P(z < -0.875) = 0.1908, which is 19.08%.

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Complete question:

The grade point averages (GPAs) of a large population of college students are approximately normally distributed with mean 2.4 and standard deviation 0.8. What fraction of the students will have GPAs between 2.0 and 3.0? What fraction of students will possess a GPA in excess of 3.0? If students possessing a GPA less than 1.7 are dropped from college, what percentage of the students will be dropped?

The measure of angle L is approximately 48.59 degrees.

What is Pythagoras' Theorem?

Pythagoras' theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

We can use trigonometry to solve for angle L. Using the Pythagorean theorem, we can find that the length of side MN is:

MN = sqrt(LN² - LM²) = √(60² - 18²) = √(3444) ≈ 58.66 inches

Now, using the definition of sine, we have:

sin(L) = MN / LN = 58.66 / 60

Taking the inverse sine (sin⁻¹) of both sides, we get:

L = sin⁻¹(58.66/60) ≈ 48.59 degrees

Therefore, the measure of angle L is approximately 48.59 degrees.

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Parabola 1 :

Vertex: (-1,2)

Min: -2

AOS: x=-1

DoO: downwards

y-Int.: -4

parabola 2:

Vertex; (0,0)

Min: 0

AOS: x=0

DoO:upwards

Parabola 3:

Vertex: (2,1)

Min: 1

AOS: x=2

DoO.:upwards

i left some stuff out because I do not know it k think it’s pretty correct

Therefore, the expression that represents the area of the rectangle is A(x) = mnx² + (mc + bn)x + bc.

In mathematics, a polynomial is a mathematical expression consisting of variables (also known as indeterminates) and coefficients, which are combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Here,

Part A:

Let's assume that the length of the rectangle is represented by the linear polynomial L(x) = mx + b, where m is the slope of the line and b is the y-intercept. Similarly, let's represent the width of the rectangle by the linear polynomial W(x) = nx + c, where n is the slope of the line and c is the y-intercept.

The area of the rectangle is given by multiplying the length and width of the rectangle, so we can express the area A(x) as:

A(x) = L(x) * W(x)

= (mx + b) * (nx + c)

= mnx² + (mc + bn)x + bc

Part B:

The degree of the polynomial A(x) is 2, which makes it a quadratic polynomial. The classification of the polynomial A(x) is that of a concave upward parabola. The degree of the polynomial is 2 because of the presence of the quadratic term mnx² in the expression for the area. The classification of the polynomial as a concave upward parabola is because the coefficient of the x² term is positive (i.e., mn > 0), and this causes the graph of the polynomial to curve upwards.

Part C:

Closure under multiplication means that if we multiply two numbers in a certain set, the result is always within that same set. In the context of the dimensions and area of a rectangle, closure under multiplication applies because if we multiply the length and width (which are both represented by linear polynomials), the resulting area (which is represented by a quadratic polynomial) is still within the same set of polynomial functions. Specifically, multiplying two linear polynomials results in a quadratic polynomial, and since polynomial functions are closed under multiplication, the resulting area polynomial is still a polynomial function. This means that the area of the rectangle can be represented by a single polynomial function, which is useful for various mathematical computations and applications.

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Answer:

D. y = 1.5[tex](4)^{x}[/tex]

Step-by-step explanation:

There are a lot of ways to solve this, but here we will just use the simplest way is to put the number in for x and y to see if it is true or not, and that will be the correct answer.

Let's use the point (4, 384)

A. y = 4[tex](2)^{x}[/tex]

  384 = 4[tex](2)^{4}[/tex]

  384 = 64

This is not true, so A is wrong.

B. y = 3[tex](2)^{x}[/tex]

  384 = 3[tex](2)^{4}[/tex]

  384 = 48

This is not true, so B is wrong.

C. y = 4[tex](1.5)^{x}[/tex]

  384 = 4[tex](1.5)^{4}[/tex]

  384 = 20.25

This is not true, so C is wrong.

D. y = 1.5[tex](4)^{x}[/tex]

  384 = 1.5[tex](4)^{4}[/tex]

  384 = 384

This is true, so D is the answer.

11. The relation is given by the set of ordered pairs: {(0.5, 2), (3, 2), (1, 3), (5, -5)}. To graph this relation, plot each point on a coordinate plane and connect them with a smooth curve.

12. The relation is given by the set of ordered pairs: {(-1, -4), (0, -1), (1, 2), (2, 5)}. To graph this relation, plot each point on a coordinate plane and connect them with a smooth curve.

13. The relation graphed below is a straight line with slope -2 and y-intercept 0.

14. This is a linear function with slope -2.

15.

x y

-5 -8

3 0

7 4

16.

x y

-10 16

-5 11

0 6

2 2

17.

x y

-3 9

-1 7

3 3

18.

x y

-3 1

0 -5

3 -11

In mathematics, a function is a relationship between two variables in which one variable (the independent variable) determines the value of the other variable (the dependent variable). It assigns a unique output value to each input value. Functions are commonly denoted as f(x), where x is the independent variable and f(x) is the dependent variable.

Here,

The relation ((0.5), (3-1), (1,3), (5,-5)) has three pairs of values: (0.5, 3-1), (1, 3), and (5, -5).

To plot this relation, you can plot each point on a coordinate plane. The first value in each pair represents the x-coordinate, and the second value represents the y-coordinate.

The relation ((-1,-4), (0, -1), (1, 2), (2, 5)) has four pairs of values: (-1, -4), (0, -1), (1, 2), and (2, 5).

To plot this relation, you can plot each point on a coordinate plane. The first value in each pair represents the x-coordinate, and the second value represents the y-coordinate.

The equation y = -2x represents a linear function with a slope of -2 and a y-intercept of 0.

To graph this function, you can plot any two points that satisfy the equation and draw a line passing through those points. For example, you can use the points (0, 0) and (1, -2).

To complete the function table for y = x - 3, you can substitute each value of x in the equation and solve for y.

For example, when x = -5, y = -5 - 3 = -8. Repeat the process for the other values of x.

To complete the function table for y = 4x + 4, you can substitute each value of x in the equation and solve for y.

For example, when x = -2, y = 4(-2) + 4 = -4. Repeat the process for the other values of x.

To complete the function table for y = -x + 6, you can substitute each value of x in the equation and solve for y.

For example, when x = -10, y = -(-10) + 6 = 16. Repeat the process for the other values of x.

To complete the function table for y = -2x - 5, you can substitute each value of x in the equation and solve for y.

For example, when x = -3, y = -2(-3) - 5 = 1. Repeat the process for the other values of x.

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To find the number that 495 is 55% of, we can use the percent proportion or percent equation.

Using the percent proportion:

Part/Whole = Percent/100

Let x be the whole (or total) number we are looking for. Then:

495/x = 55/100

We can cross-multiply to solve for x:

49500 = 55x

Dividing both sides by 55:

x = 900

Therefore, 495 is 55% of 900.

Using the percent equation:

Percent * Whole = Part

Let x be the whole (or total) number we are looking for. Then:

55% * x = 495

We can divide both sides by 0.55 to solve for x:

x = 495 / 0.55

Simplifying:

x = 900

Therefore, 495 is 55% of 900.

Answer: 55 %

Step-by-step explanation:

To find the number that $495 is 55% of, we can use the following proportion:

part/whole = percent/100

Let x be the number we are trying to find, then we can write:

495/x = 55/100

To solve for x, we can cross-multiply:

55x = 49500

Dividing both sides by 55, we get:

x = 900

Therefore, $495 is 55% of 900.

Answer: 5387.05 [tex]in^3[/tex]

Step-by-step explanation:

9.5 / 2 = 4.75 (Radius of 1 Basketball)

V = [tex]\frac{4}{3} \pi r^3[/tex]

V = (4 x π x 4.75^3) / 3

V = 1346.761 / 3

V = 448.92

Thus we need 448.92 [tex]in^3[/tex] to fill 1 basketball

12 Basketballs = 449.92 * 12

12 Basketballs = 5387.046 ≈ 5387.05

(a) (i) 0.3174 = 31.74% probability of a defect

(ii) The expected number of defects for a 1,000-unit production run is 317.

(b) (i) 0.0026 = 0.26% probability of a defect

(ii) The expected number of defects for a 1,000-unit production run is 3.

(C) Reduces the process standard deviation and causes no change in the number of defects.

When the distribution is normal, we use the z-score formula.

In a set with mean  and standard deviation, the score of a measure X is given by:

         Z = X-μ /σ

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Question a:

We have that: μ = 10 and σ = 0.12

(i). Calculate the probability of a defect.

Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.

Probability of less than 9.88:

This is the p-value of Z when X = 9.88. Thus,

  Z = X-μ /σ

⇒ Z = 9.88 - 10/0.12

⇒ Z = -1, has a p-value of 0.1587

⇒ 2× 0.1587 = 0.3174

This means 0.3174 = 31.74% probability of a defect

(ii)  Calculate the expected number of defects for a 1,000-unit production run.

The expected number of defects is 31.74% of 1000. So

0.3174*1000 = 317.4

Rounding to the nearest integer

The expected number of defects for a 1,000-unit production run is 317.

Question (b):

The mean remains the same, but the standard deviation is now

(i) Calculate the probability of a defect.

Less than 9.88 or greater than 10.12. These probabilities are equal, so we find one and multiply by 2.

Probability of less than 9.88:

This is the p-value of Z when X = 9.88. Thus,

   Z = X-μ /σ

⇒ Z = 9.88 -10/0.04

⇒ Z = -3

⇒ Z = -3, has a p-value of 0.0013

which means 2× 0.0013 = 0.0026

from which 0.0026 = 0.26% probability of a defect

(ii) Calculate the expected number of defects for a 1,000-unit production run. The expected number of defects is 31.74% of 1000. Thus,

⇒ 0.0026*1000 = 2.6

Rounding to the nearest integer

The expected number of defects for a 1,000-unit production run is 3.

(C) The advantage of reducing process variation, thereby causing problem limits to be at a greater number of standard deviations from the mean Reduces the process standard deviation and causes no change in the number of defects.

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The statement that is not correct is in marketing research, the null hypothesis is formulated in such a way that rejection leads to the acceptance of the desired conclusion.

The null hypothesis (often denoted H0)[1] is the claim that no relationship exists between two sets of data or variables

being analyzed. The null hypothesis is that any experimentally observed difference is due to chance alone, and an

underlying causative relationship does not exist, hence the term "null". In addition to the null hypothesis, an alternative

hypothesis is also developed, which claims that a relationship does exist between two variables.

The null hypothesis is the actual hypothesis being tested and it can never be accepted based on a single statistical

test. Failing to reject the null hypothesis will not lead to any changes in opinions or action.

The statement that is not correct is in marketing research, the null hypothesis is formulated in such a way that rejection

leads to the acceptance of the desired conclusion.

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So, the largest angle in quadrilateral PQRS is < 120 degrees.

A cyclic quadrilateral is what?

A quadrilateral that may be encircled by a circle is known as a cyclic quadrilateral.

Considering that,<P=3x , <Q=4x and <R=5x

∠P + ∠R = 180º            ∠Q + ∠S = 180º

Then we can write

3x+4x+5x+<S=360

12x+<S=360

<S=360-12x

We are aware that in a cyclic quadrilateral, the largest angle lies opposite the smallest angle.

Hence, in order to determine the largest angle, we must first determine the smallest angle.

P being the smallest angle leads us to believe that it is equal to x.

Therefore:

x+4x+5x+<s=360

10x+<s=360

<S=360-10x

therefore, Largest angle

- 10x < 360 - (10 * 24) = 120

So, the largest angle in quadrilateral PQRS is less than 120 degrees.

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Answer:

You're right -- it's B!

Step-by-step explanation:

When raising a power to a power, multiply exponents. The answer would be D if it were 4^6 times 4^3, but it's (4^6)^3, so I'm not sure what is happening to the computer.

Solving the question with the laws of combinatorics, specifically permutation problems.(a) 5,040 (b) 1,440. (c) 4,320.

(a) The word "mailbox" has 7 distinct letters. Therefore, the number of ways the letters can be arranged in a row is 7! = 5,040.

(b) If m and a must remain together (in order) as a unit, we can think of them as a single letter "ma". Now we have 6 letters to arrange in a row, which can be done in 6! ways. However, within the "ma" unit, the letters can be arranged in 2! ways. Therefore, the total number of arrangements is 6! × 2! = 1,440.

(c) If the letters ilb must remain together (in order) as a unit, we can think of them as a single letter "ilb". Now we have 5 letters to arrange in a row, which can be done in 5! ways. However, within the "ilb" unit, the letters can be arranged in 3! ways. Therefore, the total number of probable arrangements is 5! × 3! = 720 × 6 = 4,320.

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