[tex]f(x) = 2x^{3} - 5x^{2} - 14x + 8[/tex] synthetic division

possible zeros:
Zeros:
Linear Factors:

The following parts can be answered by the concept of standard deviation.

a.  The population mean is $55.
b.  The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

(a) The mean of the sampling distribution of the mean amount spent per sale for samples of size 50 is equal to the population mean. In this case, the population mean is $55.

(b) To find the standard deviation of the sampling distribution, we use the formula:

Standard deviation of the sampling distribution = (Population standard deviation) / sqrt(sample size)

In this case, the population standard deviation is $19 and the sample size is 50. Plugging these values into the formula, we get:

Standard deviation of the sampling distribution = 19 / sqrt(50) ≈ 19 / 7.071 ≈ 2.688

So, the standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

Therefore,

a.  The population mean is $55.
b.  The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

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The given set F = {f(x) in C[-1,1] : ∫ f(x)dx = 0 from -1 to 1} is a vector space.

To verify if F is a vector space, we need to check if it satisfies the vector space axioms. Let f(x) and g(x) be elements of F, and c be a scalar.

1. Closure under addition: ∫ (f(x) + g(x))dx = ∫ f(x)dx + ∫ g(x)dx = 0 + 0 = 0. So, (f(x) + g(x)) is in F.
2. Closure under scalar multiplication: ∫ (cf(x))dx = c∫ f(x)dx = c(0) = 0. So, (cf(x)) is in F.
3. Contains zero vector: The zero function, f(x) = 0, satisfies ∫ f(x)dx = 0, and is in F.

Since F satisfies these axioms, it is a vector space.

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The given set F = {f(x) in C[-1,1] : ∫ f(x)dx = 0 from -1 to 1} is a vector space.

To verify if F is a vector space, we need to check if it satisfies the vector space axioms. Let f(x) and g(x) be elements of F, and c be a scalar.

1. Closure under addition: ∫ (f(x) + g(x))dx = ∫ f(x)dx + ∫ g(x)dx = 0 + 0 = 0. So, (f(x) + g(x)) is in F.
2. Closure under scalar multiplication: ∫ (cf(x))dx = c∫ f(x)dx = c(0) = 0. So, (cf(x)) is in F.
3. Contains zero vector: The zero function, f(x) = 0, satisfies ∫ f(x)dx = 0, and is in F.

Since F satisfies these axioms, it is a vector space.

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The exact value of the cos 105 degrees by the using the half angle formula is -0.2588.

To find the exact value of cos(105°) using the half-angle formula, we first need to express 105° as half of another angle. Since 105° is equal to 210°/2,

we can use the half-angle formula for cosine:

cos(x/2) = ±√[(1 + cos(x))/2]

In our case, x = 210°. Now, we need to find the value of cos(210°):

210° lies in the third quadrant, where both sine and cosine are negative. To find the reference angle, subtract 180°:

210° - 180° = 30°

So, cos(210°) = -cos(30°) = -√3/2.

Now, let's plug the value of cos(210°) into the half-angle formula:

cos(105°) = ±√[(1 - √3/2)/2]

Since 105° lies in the second quadrant, where cosine is negative, we choose the negative root:

cos(105°) = -√[(1 - √3/2)/2]

               =-0.2588

Explanation:- Here to find the value of the cosine 105 degree by using the half angle formula first we write the half angle formula of the cosine and substituted x=210 degree and simplify.

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Answer: (-8, -6, -3, 7, 9, 10)

Step-by-step explanation:

The domain of a function is the set of all possible x-values that correspond to the given ordered pairs.

Therefore, the domain of the function is:  (-8, -6, -3, 7, 9, 10)

Having drawn and attached the scatter plot for the given data, the equation for the line of best fit is y = 0.083x + 97.55

After plotting the scatter point, we choose two points on the line of best fit that are far apart.

The points chosen are:

(150, 110) And (1240, 200)

Using the slope-intercept form we proceed to find the equation

y = mx + b

y =  cost of plane tickets

x = distance

m = slope

m = (y2-y1)/(x2-x1)
m = (200-110) / (1240 - 150)
m = 0.083

So, we can state

y = mx + b

110 = 0.083 * 150 + b
b = 97.55

hence,

the line of best fit is:

y = 0.083x + 97.55


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Answer:

32

Step-by-step explanation:

43-11=32

Answer:

6/8, 9/12, 12/16, 15/20

Step-by-step explanation:

If you simplify the fractions it will all equal 3/4.

Simplify by finding the Greatest Common Factor and dividing both numbers by the GCF.

Raymond gets a discount of $27.

Let the marked price of the radio be 'x'.

According to the problem, a discount of 33% is given, which means that the selling price is 67% of the marked price.

So, the selling price of the radio is 67% of x, which is given as $54 in the problem.

Hence, 67% of x = $54

Solving for x, we get x = $80

The discount amount is the difference between the marked price and selling price, which is $80 - $54 = $26.

Therefore, Raymond gets a discount of $26.

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if the bias is calculated as 2.234, you would round it to 2.23 since the third digit, 4, is less than 5.

I can explain how to represent a number with two decimal places.
When a number is represented with two decimal places, it means that it has two digits after the decimal point. For example, the number 2.23 has two decimal places, with the digits 2 and 3 after the decimal point.

Once you have calculated the bias or have the number you want to represent with two decimal places, you can round the number accordingly. To round a number to two decimal places:

1. Identify the second digit after the decimal point.
2. Look at the third digit after the decimal point.
3. If the third digit is 5 or greater, add 1 to the second digit. If it is less than 5, leave the second digit unchanged.
4. Remove all digits after the second decimal place.

For example, if the bias is calculated as 2.234, you would round it to 2.23 (since the third digit, 4, is less than 5).

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N(e ∪ f) = 194.

Using the inclusion-exclusion principle, we have:

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

We are given n(e ∩ f) = 73 and n(e ∪ f) = 194, so we can rearrange to solve for n(f):

n(f) = n(e ∪ f) - n(e) + n(e ∩ f)

n(f) = 194 - 106 + 73

n(f) = 161

Finally, to find n(e ∪ f), we can substitute the values we have found into the first equation:

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

n(e ∪ f) = 106 + 161 - 73

n(e ∪ f) = 194

Therefore, n(e ∪ f) = 194.

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This would calculate the perimeter of a square beam with an area of 25 square units, oriented horizontally.

```
function [perimeter] = geometry(across_section, area, orientation)
   % Calculate the perimeter of a beam given the desired cross section

   % Define constants for the shape of the cross section
   switch across_section
       case 'square'
           side_length = sqrt(area);
           num_sides = 4;
       case 'circle'
           radius = sqrt(area / pi);
           num_sides = 0; % Circles have no sides
       case 'rectangle'
           aspect_ratio = 2; % Set this to whatever you need for your application
           width = sqrt(area / aspect_ratio);
           height = width * aspect_ratio;
           num_sides = 4;
       otherwise
           error('Unknown cross section type');
   end

   % Calculate the perimeter based on the number of sides and shape
   switch across_section
       case 'circle'
           perimeter = 2 * pi * radius;
       otherwise
           perimeter = num_sides * (width + height);
   end

   % Adjust the perimeter based on the orientation
   switch orientation
       case 'horizontal'
           % No adjustment necessary
       case 'vertical'
           % Swap the width and height
           temp = width;
           width = height;
           height = temp;
       otherwise
           error('Unknown orientation type');
   end
end
```

To use this function, you would call it with the desired values for `across_section`, `area`, and `orientation`. For example:

```
perimeter = geometry('square', 25, 'horizontal');
```

This would calculate the perimeter of a square beam with an area of 25 square units, oriented horizontally.

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Answer:

[tex]26m^{2}[/tex]

Step-by-step explanation:

To solve this problem you find the total area of the entire rectangle and subtract the area of the sand from it. That will give you the area of the grass.

To find the total area you need to do [tex]b*h[/tex], in this case, the base is [tex]2+3+2[/tex] or 7. The height is 5. So to find the area, you have to multiply  [tex]7*5[/tex] to get[tex]35m^{2}[/tex].

To find the area of the grass you multiply the [tex]b*h[/tex] or [tex]3*3[/tex] to get the area of 9.

Now the last step is to subtract [tex]35 - 9[/tex], doing so gives you your answer of 26 m

I would love to get Brainliest. I'm trying to get to the Genius status on Brainly.

The area of the shaded area is 0.314m²

First let's analyze the shape.

It comprises:

A square whose side length is 1m

Then there is the circle that passes through a corner of the square and is tangent to the opposite two sides of the square. So the region shaded in blue is the region contained by the circle and the square but not common to both shapes.

To get the area of this portion shaded in blue, here is what we must do:

So the Area of the region shaded blue is:

Area of the circle + area of the square - 2 x Area of the overlapping area.

To further help our analysis, we must deconstruct this shape into familiar ones by creating a composite right triangle as shown in the attached image.


Since we have a right triangle, the arc that it subtends will be  a 180° arc thus, the hypotenuse of the triangle is the diameter of the circle.

Thus, when we created a right triangle from the shape, we also carved out a semi-circle.

This means that the overlapping areas are:

Area of the composite right triangle + the Area of the Semi Circle.

Since we now have two semi circles, we can cancel out the area of the circle above.

Our new equation for the area shaded in blue becomes:

The area of the square  - the area of 2 Right Triangles

Now we can being to compute for the area of the shapes:


Taking the right triangle firs, create a line from the radius bisecting the triangle at 90 degrees. This gives us two equal smaller right triaangles.

Since the ∠90 is bisected, this means that the sides will be r√2 and the height of both right triangles = r (radius)

extending the radius into the opposite direcitn, we now have the diagonal of the square.

Diagonal = r + r√2 = √2
Solving for r we say:

r(1+√2) = √2
r = √2/(1+√2)

To further simplify the above, we can multiply it by the conjugate:

r = √2/(1+√2) x (1-√2)/(1-√2)

Thus,

r = 2-√2


So since the length of the square is 1m, then the area = 1x1 = 1m

The area of the composite triangle =( r√2 x r√2)/ 2 = r²

Since the area of the shaded region as given above is
The area of the square  - the area of 2 Right Triangles

Then it's area = 1 - 2r²

We can simplify this further to get:

1-2(2-√2)²

= 8 √2 - 11m²
area of the shaded region ≈ 0.314m²

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The value of f ⁻¹ (x) would be,

⇒ f⁻¹(x) = - x/2 - 1

We have to given that;

Function is,

⇒ f (x) = - 2x - 2

Now, We can find the inverse of function as;

⇒ f (x) = - 2x - 2

⇒ y = - 2x - 2

Solve for x;

⇒ y + 2 = - 2x

⇒ x = - (y + 2)/2

⇒ x = - y/2 - 1

⇒ f⁻¹(x) = - x/2 - 1

Thus, The value of f ⁻¹ (x) would be,

⇒ f⁻¹(x) = - x/2 - 1

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7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution, hence it is not a good description of the center of this data set.

What is a data set?

A data set is a group of related data. In the case of tabular data, a data set relates to one or more database tables, where each row refers to a specific record in the corresponding data set and each column to a specific variable.

Here, we have

Given: Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution. A rating of 0 means “not at all important” and a rating of 10 means “very important.”

From the plot given in this exercise, it is found that only 7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution.

Hence it is not a good description of the center of this data set.

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For the relation "≤", the correct options: a. Linear order, d. Partial order, e. Well-ordering order

Hi! I'd be happy to help you with this question. Let's go through each option and determine if it applies to the relation "≤" (less than or equal to).

a. Linear order: A linear order is a partial order in which every pair of elements is comparable. Since "≤" allows us to compare any two elements in a set, it is a linear order.

b. Strict partial order: A strict partial order is a binary relation that is irreflexive (no element is related to itself) and transitive. "≤" is not a strict partial order because it is not irreflexive (an element can be related to itself, e.g., a≤a).

c. (non-strict): This term is incomplete and cannot be properly evaluated. Please provide more context or a complete term.

d. Partial order: A partial order is a binary relation that is reflexive, antisymmetric, and transitive. "≤" satisfies these conditions, so it is a partial order.

e. Well-ordering order: A well-ordering is a linear order in which every non-empty subset has a least element. "≤" satisfies this condition, so it is a well-ordering order.

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We would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.

To estimate the true proportion of children under age 6 living in poverty in Washington with 95% confidence and a margin of error of 2%, we can use the formula:

n = (Z² * p * q) / E²

where:

Z = the Z-score corresponding to the desired confidence level (1.96 for 95% confidence)

p = the estimated proportion (0.17 based on the federal report)

q = 1 - p

E = the desired margin of error (0.02)

Plugging in these values, we get:

n = (1.96² * 0.17 * 0.83) / 0.02²

n = 1072.45

Rounding up to the next whole number, we would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.

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Answer: 108 cups of cranberry juice. Brainliest?

Step-by-step explanation:

mixed juice. How many cups of cranberry juice did Jetia use?

Let's start by assuming that Jetia used x cups of cranberry juice to make the mixed juice. Then, since the ratio of cranberry juice to apple juice is 5:8, she must have used (5/8)x cups of apple juice.

We know that the total amount of mixed juice is 177 cups, so we can set up an equation based on the total amount of juice:

x + (5/8)x = 177

Simplifying this equation, we get:

(13/8)x = 177

Multiplying both sides by 8/13, we get:

x = 108

Therefore, Jetia used 108 cups of cranberry juice to make the mixed juice.

The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.

The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.

When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.

When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.

When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.

When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.

When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.

When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.

Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

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The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.

The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.

When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.

When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.

When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.

When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.

When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.

When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.

Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

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Th solution of the given seperable differential equation is :

u = (1/5)ln(25t^2 + e^75)

To solve the separable differential equation for u, we need to separate the variables and integrate both sides.

First, we can write the equation as:
(1/e^5u)du = 5t dt

Now we can integrate both sides:
∫(1/e^5u)du = ∫5t dt

To integrate the left side, we can use u-substitution:
Let u = 5u
Then du = 5e^5u du

Substituting into the integral, we get:
(1/5)∫e^5u du = ∫5t dt
(1/5)e^5u = 5t^2/2 + C
Where C is the constant of integration.

Now we can solve for u:
e^5u = 25t^2 + 2C

Taking the natural logarithm of both sides:
5u = ln(25t^2 + 2C)
u = (1/5)ln(25t^2 + 2C)

Using the initial condition u(0) = 15, we can solve for C:
15 = (1/5)ln(2C)
ln(2C) = 75
2C = e^75
C = (1/2)e^75

Substituting this value of C into our solution for u, we get:
u = (1/5)ln(25t^2 + e^75)

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From the list of options the equation with roots of ±3 is: (d) (x + 0)^2 = 3^2

The equation that has roots of ±3 is:

(x - 3)(x + 3) = 0

Expanding the left side of the equation using FOIL method, we get:

x^2 - 9 = 0

Therefore, the equation with roots of ±3 is:

x^2 - 9 = 0

Add 9 to both sides

x^2 = 9

Express 9 as 3^2

x^2 = 3^2

So, we have

(x + 0)^2 = 3^2

Therefore, the equation with roots of ±3 is: (d) (x + 0)^2 = 3^2

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If the person filling the tank realizes something is wrong with the hose. The time  it take to completely fill the tank by using the second hose is: 5.56 minutes.

Using this formula find  long does it take to completely fill the tank by using the second hose

Let Assume  the tank has a capacity of 100 gallons.

Time =Amount of water ÷ rate

Let plug in the formula

Time = 100 gallons ÷ 18 gallons/minute

Time = 5.56 minutes (Approximately)

Therefore  it would take 5.56 minutes to fill a 100-gallon tank.

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If the person filling the tank realizes something is wrong with the hose. The time  it take to completely fill the tank by using the second hose is: 5.56 minutes.

Using this formula find  long does it take to completely fill the tank by using the second hose

Let Assume  the tank has a capacity of 100 gallons.

Time =Amount of water ÷ rate

Let plug in the formula

Time = 100 gallons ÷ 18 gallons/minute

Time = 5.56 minutes (Approximately)

Therefore  it would take 5.56 minutes to fill a 100-gallon tank.

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The most general antiderivative of f(x) = 1/2 + 5/6x² − 4/5x³ is F(x) = 1/2x + 5/18x³ − 1/5x⁴ + C, where C is the constant of the antiderivative.

To check this answer, we can differentiate F(x) and see if it gives us back f(x). Taking the derivative of F(x), we get f(x) = d/dx (1/2x + 5/18x³ − 1/5x⁴ + C) = 1/2 + 5/6x² − 4/5x³, which matches the original function f(x). Therefore, F(x) is the most general antiderivative of f(x).

The constant of integration, denoted by C, is added because when taking the derivative of a constant, it is equal to zero. Thus, the constant of integration can be any real number, and it is included in the antiderivative to account for all possible functions that have f(x) as their derivative.

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Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:

1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.

So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

The smallest possible set that includes both  {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}  subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

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Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:

1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.

So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

The smallest possible set that includes both  {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}  subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

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Explanation: A linear function would be a diagonal line from left to right either moving up or down in direction. An exponential function would be a straight line on the x axis from left or right until it reaches y axis, then a sharp curve up or down. A quadratic function would be a U shape either facing up or down.

Linear equation: y = mx + b
Exponential equation: y = abˣ
Quadratic function: y = axˣ + bx + c

The graph listed in the picture would be a quadratic function according to the explanation.

The Euclidean algorithm gives gcd(a,b) = gcd(b,r) and gcd(b,r) = gcd(r,s).

The Euclidean algorithm is a recursive algorithm to find the greatest common divisor (gcd) of two integers a and b. It works by repeatedly finding the remainder of the division of the larger number by the smaller number, until the remainder is 0, at which point the gcd is the last non-zero remainder.

When applying the algorithm to a and b with a > b, we can write a = qb + r, where q is the quotient and r is the remainder. Then, we apply the algorithm to b and r to find the gcd(b,r), and so on until we reach 0. At each step, the gcd of the current pair of numbers is equal to the gcd of the previous pair, since any common divisor of the current pair must also divide the previous pair. Therefore, we have gcd(a,b) = gcd(b,r) = gcd(r,s), where s is the final non-zero remainder.

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The best description of the data on the line plot is: "D. The data is symmetric, with a range of 4. This might happen because the most popular price of a book at this store is $4."

A symmetric data on a line plot means that the data is evenly distributed around the center. In other words, the data points on one side of the center are mirror images of the data points on the other side of the center.

For example, the set of data values displayed on the line plot shows that the data points on one side of the line (the center) are balanced by the data points on the other side of the line. This indicates that the data is evenly distributed and has no significant skewness or bias towards one side.

The range of the data = 6 - 2 = $4.

The mode is also $4

Therefore, the correct option is: option D.

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The compound statement is not entirely clear as it contains several separate expressions. However, I can break down and analyze each of the parts:

7 + 1 2 7: This is a mathematical expression that represents the sum of 7, 1, and 27, which evaluates to 35.

7,1,7: This is a list of three individual numbers.

7+1: This is a mathematical expression that represents the sum of 7 and 1, which evaluates to 8.

74157: This is a five-digit number with no apparent mathematical relationship to the other expressions.

7+1=7: This is a mathematical equation that tests the equality of the expressions 7+1 and 7. Since 7+1 is not equal to 7, this equation is false.

27: This is a single number that is not obviously related to the other expressions.

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Answer:

V = 4188.97

Step-by-step explanation:

Formula for volume of a sphere is 4/3(pi)(r^3)

Using the formula for volume of a sphere, we plug in (4/3 * pi * 10^3).

1) The sample space consists of all possible outcomes of coin tosses until the first "H" occurs

2) Probability of each outcome given by (1/2)^(n+1) where n is the number of tails before the first head

The sample space for this experiment is the set of all possible outcomes of the coin tosses until the first "H" occurs. This includes all possible sequences of "T" (tails) and "H" (heads), with the restriction that the first "H" must be the last element in the sequence. For example, some possible outcomes are:

"H" (the first toss is heads)

"TH" (the first heads is on the second toss)

"TTTH" (the first heads is on the fourth toss)

To find the probability law for this experiment, we can use a tree diagram to represent all possible outcomes and their probabilities. At each node in the tree, we branch to represent the two possible outcomes of the next coin toss (heads or tails). The probability of each branch is 1/2, since the coin is fair.

Here is the first level of the tree:

H (probability 1/2)

T (probability 1/2)

If the first toss is heads, we have reached the desired outcome and the experiment ends. If the first toss is tails, we continue branching:

T - H (probability 1/2 * 1/2 = 1/4)

T - T (probability 1/2 * 1/2 = 1/4)

If the second toss is heads, the experiment ends with a total of two tosses. If the second toss is tails, we continue branching:

T - T - H (probability 1/2 * 1/2 * 1/2 = 1/8)

T - T - T (probability 1/2 * 1/2 * 1/2 = 1/8)

We can continue this process to generate the full tree, which has an infinite number of levels (since the experiment could theoretically go on forever). However, we can see that each outcome corresponds to a unique path through the tree, and the probability of that outcome is the product of the probabilities along that path. For example, the outcome "TH" has probability 1/2 * 1/2 = 1/4, while the outcome "TTTH" has probability 1/2 * 1/2 * 1/2 * 1/2 = 1/16.

Therefore, the probability law for this experiment is:

P("H") = 1/2

P("TH") = 1/4

P("TTH") = 1/8

P("TTTH") = 1/16

In general, the probability of the outcome "T^nH" (where there are n tails before the first heads) is (1/2)^{n+1}. The probability of the experiment going on forever (i.e. never getting heads) is 0, since the probability of this outcome is the limit of (1/2)^{n+1} as n approaches infinity, which is 0.

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