Derive the statements as corollaries of Theorems 4.3.1, 4.3.2.
*Theorem 4.3.1 = Every integer is a rational number.
**Theorem 4.3.2 = The sum of any two rational numbers is rational.
For any rational numbers r and s, 2r + 3s is rational.

The facts about the quotient expression p(x) / (2x+5)

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

The expression p(-5/2) = 0 means that

The divisor is a factor of the polynomial

This statement represents the first fact

The second fact in the expression p(x) / (2x+5) is that p(x) has a zero at x = -5/2

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The value of the random variable x0 such that P (x > x0) = 0.025 is approximately 55.88.

To find the value of x0 such that P(x > x0) = 0.025, we need to use the standard normal distribution table or a calculator to find the corresponding z-score.

First, we need to standardize x using the formula:

z = (x - µ) / σ

where µ = 50 and σ = 3.

So,

z = (x - 50) / 3

Now, we want to find the z-score that corresponds to a probability of 0.025 in the right tail of the standard normal distribution. From the standard normal distribution table, we can see that the z-score for this probability is approximately 1.96.

Therefore,

1.96 = (x0 - 50) / 3

Multiplying both sides by 3, we get:

x0 - 50 = 1.96 * 3

x0 - 50 = 5.88

Adding 50 to both sides, we get:

x0 = 55.88

So, the value of the random variable x0 such that P(x > x0) = 0.025 is approximately 55.88.

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The value of the angle in degrees and radian is 45° and π/4 respectively.

Arc length formula is used to calculate the measure of the distance along the curved line making up the arc.

length of an arc is calculated as!

l =( tetha)/360 × 2πr

16π = x/360 × 2 × π × 64

16 π = 128xπ/360

x = 16 × 360/128

x = 5760/128

x = 45°

therefore in degrees the angle is 45°

π in radian is 180°

1° = π/180

45° = π/180 × 45

45° = π/4

therefore the angle in radians is π/4

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The area of Kimberly's wall hanging will be 32 square units.

Given that:

Points, A(1, 5), B(1, 9), C(7, 9), D(7, 5), E(5, 3), and F(3, 3)

The area of a two-dimensional figure is the area that its perimeter encloses. The quantity of unit squares that occupy a closed figure's surface is its region.

The polygon is shown in the graph below.

The area of the polygon is given as,

A = Area of rectangle + Area of trapezium

A = 4 x 6 + 1/2 x (6 + 2) x 2

A = 24 + 8

A = 32 square units

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The value of x + 6 is: x + 6 = [(TO - 14) / 2] + 6

To find the value of x, we need to use the given information about the lengths of line segments AB, BC, and AC.

We know that AB = x + 6, BC = x + 8, and AC = TO.

Since AC is the sum of AB and BC, we can write the equation:

AC = AB + BC

TO = (x + 6) + (x + 8)

TO = 2x + 14

Now, we need to solve this equation for x.

Given that AC = TO, we can substitute the value of AC with TO:

TO = 2x + 14

Since the question asks for the value of x + 6, we can subtract 14 from both sides of the equation:

TO - 14 = 2x

Now, simplify the equation:

2x = TO - 14

Finally, divide both sides of the equation by 2 to solve for x:

x = (TO - 14) / 2

Simplifying further will give the final expression for x + 6.

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Answer:

1 gram is the equivalent of 1000 mg to g

Answer:

0,001

Step-by-step explanatio}

The equivalent expressions are;

1. ((x² -1)/x-5. Option B

2. (x+2)(x-5)/x² + 14. Option D

Algebraic expressions are described as expressions composed of terms, coefficients, variables, constants and factors.

These expressions are also made up of mathematical operations, such as;

From the information given, we have that;

x-3/x - 5  + x² - x +2/x - 5

find the LCM

1/x-5(x-3 + x² - x +2)

add the like terms

1/x-5(x² -1)

2. x+2/x + 8 ÷ x² + 14/(x + 8)(x-5)

Take the inverse of the divisor and multiply

x+2/x+ 8 × (x + 8)(x-5)/x² + 14

(x+2)(x-5)/x² + 14

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We should use the one mean z-test for large samples (size 30 or more) when dealing with quantitative data, the sample size is at least 30, and the population standard deviation is known.

When to use the one mean z-test for large samples (size 30 or more).
we should use the one mean z-test for large samples in the following situations:
Sample size (n) is greater than or equal to 30:

The one mean z-test is appropriate for large samples, typically when the sample size is 30 or more.

This is because the Central Limit Theorem states that for a sufficiently large sample size, the sampling distribution of the mean is approximately normally distributed.

Population standard deviation (σ) is known:

The one mean z-test requires knowledge of the population standard deviation.

If the population standard deviation is unknown, you should use the t-test instead.
The variable of interest is quantitative:

The one mean z-test is designed for quantitative data (interval or ratio scale).

If you are dealing with categorical or ordinal data,  would need to use different tests, such as the chi-square test or Mann-Whitney U test.

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Answer:

We can use proportion to estimate the number of households that have a dog in the town of Nowaday.

From the data given in the random sample, we know that 24 out of 60 households had a dog. This can be written as:

24/60 = x/3000

where x is the number of households in the town that have a dog.

To solve for x, we can cross-multiply and simplify:

24 x 3000 = 60 x x

x = (24 x 3000) / 60

x = 1,200

Therefore, we can estimate that approximately 1,200 households in the town of Nowaday have a dog.

The statement as a division problem is Sharing 21 apples among 5 friends

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

21/5

The above expression is a quotient expression

A quotient expression is represented as

Quotient expression = dividend/divisor

Using the above as a guide, we have the following:

Dividend = 21

Divisor = 5

A division problem that represents this statement is

Sharing 21 apples among 5 friends

There are several other division problems that can be generated from the expression

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According to the data, it can be established that the game of Scrabble is popular and successful around the world.

To conclude on a subject we must analyze the data that exists and establish a statement that summarizes these data including our opinion. In this case we can state that the game of Scrabble has been very popular all over the world which has caused it to sell billions of units.

Additionally, we can affirm that it is very successful and popular all over the world because it has been translated into 29 languages, which indicates that this game is suitable for different cultures.

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The normal approximation to the sampling distribution of a proportion can be used when both np and n(1-p) are greater than or equal to 10.

In this problem, we are selecting an SRS of size n=100 from a large population with a proportion p of successes. We need to find the value of p that would make it safe.

To use the normal approximation to the sampling distribution of p.

For each given value of p, we can calculate np and n(1-p), and then determine whether both of these values are greater than or equal to 10.

a) p = 0.85

np = 85, n(1-p) = 15

Both np and n(1-p) are greater than or equal to 10, so the normal approximation is safe.

b) p = 0.975

np = 97.5, n(1-p) = 2.5

Only np is greater than or equal to 10, so the normal approximation is safe.

c) p = 0.999

np = 99.9, n(1-p) = 0.1

Only np is greater than or equal to 10, so the normal approximation is safe.

d) p = 0.01

np = 1, n(1-p) = 99

Only n(1-p) is greater than or equal to 10, so the normal approximation is safe.

e) p = 0.09

np = 9, n(1-p) = 91

Only n(1-p) is greater than or equal to 10, so the normal approximation is safe.

Therefore, for all given values of p, it is safe to use the normal approximation to the sampling distribution of p.

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A quadrilateral with exactly one pair of opposite sides always parallel is an isosceles trapezoid. An isosceles trapezoid is a quadrilateral with exactly one pair of opposite sides parallel, which distinguishes it from other quadrilaterals like squares, rhombuses, rectangles, parallelograms, and kites.

1. Square: A square has all sides equal in length and all angles measuring 90 degrees. It has two pairs of parallel sides.
2. Rhombus: A rhombus has all sides equal in length and two pairs of parallel sides.
3. Rectangle: A rectangle has two pairs of parallel sides and all angles measuring 90 degrees.
4. Parallelogram: A parallelogram has two pairs of parallel sides, but not necessarily right angles.

5. Kite: A kite has two pairs of adjacent sides equal in length, but not parallel.
6. Trapezoid: A trapezoid has only one pair of parallel sides, but its non-parallel sides are not necessarily equal in length.The isosceles trapezoid stands out because it has one pair of parallel sides, and its non-parallel sides are equal in length.

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Answer:

y-5=3(x+4)

Step-by-step explanation:

point slope form: y-y1=m(x-x1)

plug in 5 for y1, -4 for x1

y-5=3(x+4)

Assuming that the sample size and standard error remain constant, the confidence interval widens as the confidence level increases. The correct answer is option a.

This means that a 99% confidence interval will be wider than a 95% confidence interval, which will be wider than a 90% confidence interval, and so on.

The reason for this is that as we increase the confidence level, we want to be more certain that our interval contains the true population parameter. This requires us to widen the interval to allow for more potential values of the parameter that could be within the range of our interval.

Therefore, option A, a 99% confidence interval, will be the widest of the four options given.

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The probability would depend on the specific numbers of Car Riders, Bus users, and people who use both.

Let's assume there are 'n' people in the group, out of which 'm' are Car Riders, 'p' use the Bus, and 'q' use both Car and Bus. Then, the probability of randomly selecting a person who is either a Car Rider or uses the Bus can be calculated using the following formula:

P(Car Rider OR Bus user) = (m + p - q) / n

Here, we subtract the number of people who use both Car and Bus (i.e., 'q') to avoid counting them twice.

For example, if there are 100 people in the group, out of which 40 are Car Riders, 30 use the Bus, and 10 use both Car and Bus, then the probability of randomly selecting a person who is either a Car Rider or uses the Bus would be:

P(Car Rider OR Bus user) = (40 + 30 - 10) / 100 = 0.6 or 60%

So, the probability would depend on the specific numbers of Car Riders, Bus users, and people who use both.

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The amount of candy remaining in the dispenser is 10.35 cups. There are 10.05 cups of candy remaining in the dispenser.

The candy dispenser can hold 15 cups of candy. If the dispenser is 75% full, then it contains 75% of 15 cups, which is:

0.75 x 15 = 11.25 cups

After a few hours, 8% of the candy has been sold. This means that 8% of 15 cups has been sold, which is:

0.08 x 15 = 1.2 cups

To find out how much candy remains in the dispenser, we need to subtract the amount sold from the amount that was initially in the dispenser. So, the amount of candy remaining in the dispenser is:

11.25 - 1.2 = 10.05 cups

Therefore, there are 10.05 cups of candy remaining in the dispenser.

Alternatively, we can calculate the amount of candy remaining in the dispenser directly as follows:

The dispenser is 75% full, which means that 25% of its capacity is empty. Therefore, the empty space in the dispenser is:

0.25 x 15 = 3.75 cups

If 8% of the candy has been sold, then 92% of the candy remains in the dispenser. Therefore, the amount of candy remaining in the dispenser is:

0.92 x (15 - 3.75) = 0.92 x 11.25 = 10.35 cups

This gives us the same answer as before, which is that there are 10.05 cups of candy remaining in the dispenser.

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None  of the given pairs of lines are perpendicular. The answer is none of the above (E).

To determine which two lines are perpendicular, we need to check whether the slopes of the lines are negative reciprocals of each other. If they are, then the lines are perpendicular.

The slope of a line can be determined by examining the coefficient of x in the line's equation.

For Equation A, the slope is -3 (the coefficient of x is 3, and the slope is the negative of the coefficient).
For Equation B, the slope is 3 (the coefficient of x is 6, and the slope is the coefficient divided by -2).
For Equation C, the slope is 3 (the coefficient of x is 3).
For Equation D, the slope is 13.

Now we can check which two lines have slopes that are negative reciprocals of each other. The negative reciprocal of 3 is -1/3, and the negative reciprocal of -3 is 1/3. None of the pairs of lines have slopes that are negative reciprocals of each other. Therefore, none of the given pairs of lines are perpendicular. The answer is none of the above (E).

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The number 25% represents the proportion of the entire population of students at the school who have period 4 lunch. Therefore, it is a option D- parameter.

A statistic is a numerical summary of a sample, while a parameter is a numerical summary of a population. In this case, the proportion of students who have period 4 lunch in the sample of 40 students is 30%, which is a statistic. The proportion of students who have period 4 lunch in the entire population of 380 students is 25%, which is a parameter.

It's important to distinguish between a sample and a population because statistical conclusions drawn from a sample cannot be automatically generalized to the population from which the sample was drawn. In other words, the characteristics of a sample may not perfectly reflect the characteristics of the larger population, due to random sampling variation. Therefore, when making statistical inferences or drawing conclusions about a population, it's important to use appropriate sampling methods and to carefully consider the limitations and assumptions of statistical analyses.

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The symbol for a hectare is "ha." A hectare is a unit of area measurement, primarily used in the field of agriculture and land management. It is equal to 10,000 square meters or approximately 2.47 acres.

The term "hectare" is derived from the metric prefix "hecto," which means one hundred, and the unit of area "are," which equals 100 square meters. Therefore, a hectare is equivalent to 100 ares (100 x 100 square meters).

The hectare is widely used in many countries around the world to describe the size of agricultural land, forests, parks, and other large areas. Its use is more common in countries that have adopted the metric system, such as in Europe and Asia. In contrast, countries like the United States often use acres as their primary unit of land measurement.

In summary, "ha" is the symbol used to represent a hectare, a unit of area measurement often utilized in agriculture and land management. It is equivalent to 10,000 square meters or about 2.47 acres and is commonly used in countries that follow the metric system.

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After analysing the given data we conclude that the measure of <ABD is 117 degrees, under the condition that <ABC has a measure of 40°.

Now in order to evaluate the angle we have to relie on the principles of angle to derive a formula
The formula is
180 - ( remaining angles) = ( unknown angle)
Now, to find the measure of <ABD, we can apply the fact that the sum of the angles in a triangle is 180 degrees. It is known that <ABC has a measure of 40 degrees and <CBD has a measure of 23 degrees. Since <ABC and <CBD share a ray, we can add their measures to get the measure of <ABD.
Placing the values in the formula
Therefore,
<ABD = 180 - 40 - 23
= 117 degrees.
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Answer:

  No, it is not reasonable.

Step-by-step explanation:

Given an 8-inch pizza costs $14.35, you want to know if an estimated cost of $28.70 is reasonable for a 16-inch pizza.

The scale factor for area is the square of the scale factor for linear dimensions such as diameter.

The ratio of diameters is ...

  (16 in)/(8 in) = 2

Then the ratio of areas will be 2² = 4. The larger pizza will have 4 times the area of the smaller one.

If cost is based on area, then a better estimate of the cost of a 16-inch pizza is ...

  4 × $14.35 = $57.40

Marco's estimate of $28.70 is not reasonable. A better estimate is $57.40.

__

Additional comment

The area is given by the formula ...

  A = (π/4)d²

When the diameter is doubled, the area becomes ...

  A' = (π/4)(2d)² = 4×(π/4)d² = 4A

You can see that the scale factor 2 was squared when we expanded (2d)² to 2²·d² = 4d².

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The mean of the Poisson distribution is 0.1.

The mean of a Poisson distribution is given by µ, which is the expected number of occurrences in the specified interval. In this case, µ = 0.1, which means we expect to have 0.1 occurrences in the specified interval.

The mean can also be calculated by using the formula µ = ΣxP(x), where ΣxP(x) is the sum of the product of each value of x and its corresponding probability.

µ = (0 × 0.9048) + (1 × 0.0905) + (2 × 0.0045) + (3 × 0.0002)

µ = 0 + 0.0905 + 0.009 + 0.0006

µ = 0.1

Therefore, the mean of the Poisson distribution is 0.1.

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The volume of the cone is 441π cubic meters.

The formula for the volume of a cone is:

V = (1/3)πr²h

where r is the radius of the base of the cone, h is the height of the cone, and π is the mathematical constant pi.

Substituting the given values, we get:

V = (1/3)π(7²)(27)

Simplifying, we get:

V = (1/3)π(49)(27)

V = (1/3)π(1323)

V = 441π

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we can conclude that there is significant evidence to support the claim that μ is less than 10.

We need to compare the test statistic with the critical value to determine whether to reject or fail to reject the null hypothesis. Since the test statistic (-2.50) is less than the critical value (-1.645), we can reject the null hypothesis at the 5% level of significance.

The conclusion is that there is sufficient evidence to support the alternative hypothesis Ha: μ < 10. This means that the sample data suggests that the true population mean is less than 10, and we can reject the possibility that the population mean is greater than or equal to 10.

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We can simplify [tex](16x^4)^(-1/2) to 1/(4x^2)[/tex] using the properties of rational exponents.

Certainly! Here's an example:

Simplify the expression: [tex](16x^4)^(-1/2)[/tex]

We can apply the property of rational exponents which states that [tex](a^m)^n = a^(m*n)[/tex]. Using this property, we get:

[tex](16x^4)^(-1/2) = 16^(-1/2) * (x^4)^(-1/2)[/tex]

Next, we can simplify [tex]16^(-1/2)[/tex] using the rule that [tex]a^(-n) = 1/a^n[/tex]:

[tex]16^(-1/2) = 1/16^(1/2) = 1/4[/tex]

Similarly, we can simplify [tex](x^4)^(-1/2)[/tex] using the rule that [tex](a^m)^n = a^(m*n)[/tex]:

[tex](x^4)^(-1/2) = x^(4*(-1/2)) = x^(-2)[/tex]

Substituting these simplifications back into the original expression, we get:

[tex](16x^4)^(-1/2) = 1/4 * x^(-2) = 1/(4x^2)[/tex]

Therefore, the simplified expression is [tex]1/(4x^2).[/tex]

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Answer:

33 - 4i

Step-by-step explanation:

note that i² = - 1

(7 - 6i)(3 + 2i) ← expand using FOIL

= 21 + 14i - 18i - 12i²

= 21 - 4i - 12(- 1)

= 21 - 4i + 12 ← collect like terms

= 33 - 4i