What is the measure of angle R, given the largest triangle is a right triangle?

33°

12°

45°

78°

The equations that are true for x = -2 and x = 2 are x² – 4 = 0 and 4x² = 16. So, correct options are A and D.

The solutions for the equations when x = -2 and x = 2 can be found by substituting these values of x into the equations and simplifying.

For x² – 4 = 0, substituting -2 and 2 for x gives (-2)² – 4 = 0 and (2)² – 4 = 0, which are both true. Therefore, this equation is true for x = -2 and x = 2.

For x² = –4, substituting -2 and 2 for x gives (-2)² = –4 and (2)² = –4, which are both false. Therefore, this equation is not true for x = -2 and x = 2.

For 3x² + 12 = 0, substituting -2 and 2 for x gives 3(-2)² + 12 = 0 and 3(2)² + 12 = 0, which are both false. Therefore, this equation is not true for x = -2 and x = 2.

For 4x² = 16, substituting -2 and 2 for x gives 4(-2)² = 16 and 4(2)² = 16, which are both true. Therefore, this equation is true for x = -2 and x = 2.

For 2(x – 2)² = 0, substituting -2 and 2 for x gives 2(-2 – 2)² = 0 and 2(2 – 2)² = 0, which are both true. Therefore, this equation is true for x = -2 and x = 2.

Therefore, correct options are A and D.

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Answer:

Step-by-step explanation:

you can solve with a proportion between the real measurements and those on the map

52 : 2 = x : 3.5

x = 52 x 3.5 : 2

x = 182 : 2

x = 91 miles

------------------------- check

52 : 2 = 91 : 3.5

26 = 26

the answer is good

The situation that describes the use of an ATM is option C, where Abby entered a PIN to begin a transaction and received an amount of cash.

An ATM, or Automated Teller Machine, is a self-service banking machine that allows users to perform various financial transactions, including withdrawing cash, depositing money, checking account balances, and transferring funds. To use an ATM, the user typically needs to have a debit card or ATM card linked to their bank account, and they need to enter a unique Personal Identification Number (PIN) to access their account.

Once the user enters the correct PIN, they can choose the type of transaction they want to perform, such as withdrawing cash, and the machine dispenses the requested amount of cash.

Therefore, option C is the correct answer that describes the use of an ATM.

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A statement that compares two expressions that are not equal is called an inequality. An inequality is a mathematical expression that shows that two quantities are not the same, unlike an equation which shows that two quantities are equal. Inequalities use symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to) to compare two expressions.

For example, 4 < 7 is an inequality because 4 is less than 7. Inequalities are commonly used in algebra and real-world scenarios to represent situations where there are limits, ranges or restrictions. Solving inequalities involve finding the values of the unknown variable that make the inequality true. There are several methods to solve inequalities, including adding or subtracting the same value to both sides of the inequality, multiplying or dividing by a positive or negative number, and graphing the inequality on a number line.

Understanding inequalities is crucial for various fields such as economics, engineering, physics, and computer science.

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Answer:

E=Mc squared

Step-by-step explanation:

To summarize:

- The relation is not reflexive.

- The relation is not symmetric.

- The relation is transitive.

A homogeneous relation R over the set A, which comprises the elements x, y, and z, is known as a transitive relation. If R relates x to y and y to z, then R likewise relates x to z.

To determine whether each relation is reflexive, symmetric, or transitive, we need to examine the properties individually. Let's analyze each property for the given relation R on Z, where x, y ∈ Z and xRy if and only if (x + y)² ≡ ±1.

(a) Reflexive: A relation R is reflexive if every element in the set is related to itself. In this case, we need to check if (x + x)² ≡ ±1 for all x ∈ Z.

If we simplify (x + x)², we get (2x)² = 4x². Since we are looking for the relation (x + y)² ≡ ±1, this relation is not reflexive because 4x² is not equivalent to ±1 for all integers x.

(b) Symmetric: A relation R is symmetric if whenever x is related to y, then y is also related to x. In this case, we need to check if (x + y)² ≡ ±1 implies (y + x)² ≡ ±1 for all x, y ∈ Z.

Let's consider a counterexample to show that it is not symmetric. Suppose we have x = 1 and y = 2. (1 + 2)² = 9, which is not equivalent to ±1. However, (2 + 1)² = 9, which is also not equivalent to ±1. Since the relation is not symmetric for these values, we can conclude that it is not symmetric for all values.

(c) Transitive: A relation R is transitive if whenever x is related to y and y is related to z, then x is related to z. In this case, we need to check if (x + y)² ≡ ±1 and (y + z)² ≡ ±1 imply (x + z)² ≡ ±1 for all x, y, z ∈ Z.

To show that this relation is transitive, we need to verify that if (x + y)² ≡ ±1 and (y + z)² ≡ ±1, then (x + z)^2 ≡ ±1.

Expanding (x + y)², we have (x + y)² = x² + 2xy + y². Similarly, expanding (y + z)², we have (y + z)² = y² + 2yz + z².

Now, if we add these two equations, we get:

(x + y)² + (y + z)² = x² + 2xy + y² + y² + 2yz + z² = x² + 2xy + 2yz + z² + 2y².

We want this expression to be equivalent to ±1, so we need to consider the cases when it is equal to ±1:

Case 1: (x² + 2xy + 2yz + z² + 2y²) ≡ 1

In this case, (x + z)² ≡ 1, which satisfies the transitive property.

Case 2: (x² + 2xy + 2yz + z² + 2y²) ≡ -1

In this case, (x + z)² ≡ -1, which also satisfies the transitive property.

Therefore, the given relation is transitive.

To summarize:

- The relation is not reflexive.

- The relation is not symmetric.

- The relation is transitive.

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Yes, the conditions for inference are met. The teacher conducts 50 trials, which is large enough to meet the large counts condition (np ≥ 10 and n(1-p) ≥ 10).

The teacher's attempt to make the number cube unfair by inserting lead weights raises the question of whether the proportion of rolls that will land on a 1 has changed. To determine if she was successful, the teacher rolls the cube 50 times and keeps track of the number of times she rolls a 1. The data shows that she rolled a 1 15 times.

To determine if the data provides convincing evidence that the proportion of rolls that will land on a 1 is greater than one-sixth, we need to check if the conditions for inference are met.

The 10% condition requires that the sample size is less than 10% of the population size. Since we do not know the population size, we cannot determine if this condition is met.

The large counts condition requires that both the number of successes (15) and the number of failures (35) are greater than or equal to 10. This condition is met.

The randomness condition requires that the sample is random. Since the teacher rolled the cube, it is assumed that the rolls were random.

Therefore, the conditions for inference are met. We can use a hypothesis test to determine if the data provides convincing evidence that the proportion of rolls that will land on a 1 is greater than one-sixth.

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Answer:no there not because they don't make a 90 degree angle                      

so your answer is no

The standard deviation of the sampling distribution of sample means is 0.054 pounds.

Now, let's consider the standard deviation of the sampling distribution of sample means. This is a measure of how much the means of different samples of the same size vary from each other. It is also referred to as the standard error of the mean (SEM). The formula for calculating the SEM is:

SEM = standard deviation of population / square root of sample size

In this case, the population is all people on this diet, and the standard deviation of the population is given as 4.9 pounds. The sample size is 8282 people. Therefore, we can calculate the SEM as:

SEM = 4.9 / square root of 8282

= 0.054 pounds

This means that the means of different samples of 8282 people on this diet are expected to deviate from each other by an average of 0.054 pounds.

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Divide the number of percent changes that fall within the range by the total number of percent changes to get the percentage.

What is percentage?

Percentage is a way of expressing a number or proportion as a fraction of 100. It is represented by the symbol "%".

To find the percentage of percent changes that is 1 standard deviation away from the mean in Excel, you can follow these steps:

1. Enter the percent changes in a column in Excel.

Calculate the mean of the percent changes using the AVERAGE function.

2. Calculate the standard deviation of the percent changes using the STDEV.S function.

3. Calculate the lower and upper bounds of the range that is 1 standard deviation away from the mean. To do this, subtract the standard deviation from the mean to get the lower bound, and add the standard deviation to the mean to get the upper bound.

4. Use the COUNTIF function to count the number of percent changes that fall within the range of 1 standard deviation away from the mean.

Therefore,  Divide the number of percent changes that fall within the range by the total number of percent changes to get the percentage.

Here's an example formula:

=COUNTIF(A2:A10,">="&B1-STDEV.S(A2:A10)," <="&B1+STDEV.S(A2:A10))/COUNT(A2:A10)*100

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The definition of continuity on an interval allows using one-sided limits at the endpoints.

Continuity of a function at a point means that the value of the function at that point can be determined by its nearby values. For an interval, the continuity requires that the function is continuous at every point within that interval. However, the boundaries of the interval might not have values on both sides. Therefore, the definition of continuity on an interval allows using one-sided limits to evaluate the continuity of the function at the boundaries of that interval. This means that the function is considered continuous at a boundary point if its limit exists from the left or right side. Continuity on an interval uses one-sided limits at boundaries.

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Option D is the correct response, and it is false to say that it is always necessary to use the Pythagorean Theorem when finding the distance between two points on a coordinate plane.

What is Pythagorean theorem?

The Pythagorean theorem is a fundamental concept in mathematics that relates to the lengths of the sides of a right triangle. It 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 other two sides.

The correct response is D) false; If the two points create a horizontal or vertical line segment, the Pythagorean Theorem is not needed.

When finding the distance between two points on a coordinate plane, the Pythagorean Theorem is used to calculate the distance only when the two points do not create a horizontal or vertical line segment.

If the two points create a horizontal line segment, then the distance between the two points is simply the difference between their x-coordinates. If the two points create a vertical line segment, then the distance between the two points is simply the difference between their y-coordinates. In both cases, the Pythagorean Theorem is not needed.

Therefore, option D is the correct response, and it is false to say that it is always necessary to use the Pythagorean Theorem when finding the distance between two points on a coordinate plane.

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The function S(t) = -15t^(5/3) + C which we get by integration of S'(t) and the company  will continue to manufacture this computer for approximately 5.47 months.

To find S(t), we need to integrate the given rate of decline, S'(t), with respect to time (t). We have:

S'(t) = -25t^(2/3)

Integrating both sides with respect to t, we get:

S(t) = ∫(-25t^(2/3) dt)

Using the power rule for integration, we obtain:

S(t) = (-25 * (3/5) * t^(5/3)) + C
S(t) = -15t^(5/3) + C

Now, we're given that at t = 0, S(0) = 2,050 computers. We can use this information to find the constant of integration, C:

S(0) = -15(0)^(5/3) + C
2,050 = C

Thus, the function for the monthly sales is:

S(t) = -15t^(5/3) + 2,050

The company will stop manufacturing when S(t) = 1,000 computers. To find when this occurs, we'll set S(t) equal to 1,000 and solve for t:

1,000 = -15t^(5/3) + 2,050
-1,050 = -15t^(5/3)

Now, isolate t^(5/3) and solve for t:

t^(5/3) = 1,050 / 15
t^(5/3) = 70

Take the cube root of both sides:

t^(5/3)^(3/5) = 70^(3/5)
t = 70^(3/5)

Calculating this value, we get approximately:

t ≈ 5.47

So, the company will continue to manufacture this computer for approximately 5.47 months.

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Required statistic options are option a and option c and required parameter options are b and d respectively.

a. In a survey of 360 credit card holders in the U.S., the mean service fee charged last year was $42.75. This is a statistic because it is a numerical value that describes a characteristic of a sample.

b. In the 2014-2015 academic year, 78.2% of public high school seniors from the state of Washington graduated. This is a parameter because it is a numerical value that describes a characteristic of an entire population.

c. A recent study of 1,625 elementary school children in the state of Washington found that 18.3% were living in poverty as classified by the federal government. This is a statistic because it is a numerical value that describes a characteristic of a sample.

d. The mean salary of all public college presidents in the U.S. was $428,000 in 2014. This is a parameter because it is a numerical value that describes a characteristic of an entire population.

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In the given function T(x) = [tex]0.29(1.14)^{x}[/tex],

1. The given function is T(x) = [tex]0.29(1.14)^{x}[/tex], which is an exponential function with the variable X is an exponent. So, the function is representing the Exponential Growth of the tumor. So, the correct option is D.

2. The value 0.29 represents the original size of the tumor, in  [tex]cm^{3}[/tex]. Because it is a constant value that has been multiplied by growth rate of the tumor. So, the correct option is D.

3. To find the average rate of change of the tumor from day 0 to day 7, we need to find the difference from day 0 to day 7.

Tumor at day 0 = T(0) = [tex]0.29(1.14)^{0}[/tex] = 0.29 x 1 = 0.29.

Tumor at day 7 = T(7) =  [tex]0.29(1.14)^{7}[/tex] = 0.473

So, the difference is 0.473 - 0.29 = 0.183

The average rate of change from day 0 to day 7 which is 7 days = 0.183/7

which is equal to 0.027  [tex]cm^{3}[/tex]/day.

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Answer:

To find the coordinates of point B on AC such that the ratio of AB to BC is 2:1, we can use the section formula:

Let the coordinates of point B be (x, y).

Then, we know that AB is twice as long as BC. Let's call the length of BC "d".

So, the length of AB is 2d.

Using the distance formula, we can find the distances between the points:

AB = √[(x - 2)² + (y + 8)²]

BC = √[(x + 4)² + (y - 7)²]

Since the ratio of AB to BC is 2:1, we have:

AB/BC = 2/1

(√[(x - 2)² + (y + 8)²]) / (√[(x + 4)² + (y - 7)²]) = 2/1

(√[(x - 2)² + (y + 8)²])² / (√[(x + 4)² + (y - 7)²])² = (2/1)²

[(x - 2)² + (y + 8)²] / [(x + 4)² + (y - 7)²] = 4

Multiplying both sides by [(x + 4)² + (y - 7)²], we get:

4[(x + 4)² + (y - 7)²] = (x - 2)² + (y + 8)²

Expanding both sides, we get:

4x² + 16x + 4y² - 56y + 129 = x² - 4x + 4y² + 16y + 68

Simplifying, we get:

3x² + 20x - 3y² - 72y + 61 = 0

To solve for x and y, we need another equation. We know that point B lies on the line AC, so we can use the slope formula to find the equation of the line:

m = (y₂ - y₁) / (x₂ - x₁)

m = (7 - (-8)) / (-4 - 2)

m = 15 / (-6)

m = -2.5

Using point-slope form, we can find the equation of the line:

y - 7 = -2.5(x + 4)

Simplifying, we get:

y = -2.5x - 2

Now we have two equations:

3x² + 20x - 3y² - 72y + 61 = 0

y = -2.5x - 2

We can substitute the second equation into the first equation to get an equation in terms of x:

3x² + 20x - 3(-2.5x - 2)² - 72(-2.5x - 2) + 61 = 0

Simplifying and solving for x, we get:

x = -3

Substituting x = -3 into y = -2.5x - 2, we get:

y = -9.5

Therefore, the coordinates of point B are (-3, -9.5).

Step-by-step explanation:

The value of P(not yellow) if the spinner is spun once is 75%

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

Spinner with repeated colors numbered from 1 to 8

The sample space of a eight-sided number cube is

S = {8 colors}

Where, the sections that are not yellow are

not yellow = 6 sections

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

P(not yellow) = n(not yellow)/n(S)

So, we have

P(not yellow) = 6/8

Evaluate

P(not yellow) = 75%

Hence, the value of P(not yellow) is 75%

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a) We would expect about 4 people in the sample to name chocolate as their favorite flavor of ice cream.

b) The probability of exactly six people naming chocolate is 0.20190.

c) The probability of six or more people naming chocolate is 0.1118.

a) We can use the expected value formula for a binomial distribution, where the probability of success is p = 0.3 and the number of trials is n = 14:

E(X) = np = 14 × 0.3 = 4.2

Rounding this to the nearest whole number, we would expect about 4 people in the sample to name chocolate as their favorite flavor of ice cream.

b) To find the probability that exactly six people name chocolate, we can use the binomial probability formula:

P(X = 6) = (14 choose 6) ÷ 0.3⁶ × 0.7⁸

Using a calculator, we can evaluate this expression as:

P(X = 6) = 0.2019

Rounding this to 5 decimal places, the probability of exactly six people naming chocolate is 0.20190.

c) To find the probability that six or more people name chocolate, we can use the cumulative binomial probability formula:

P(X ≥ 6) = 1 - P(X < 6) = 1 - P(X ≤ 5)

We can calculate P(X ≤ 5) using the binomial cumulative distribution function or by summing the probabilities for X = 0, 1, 2, 3, 4, and 5. Using the latter method, we get:

P(X ≤ 5) = (14 choose 0) × 0.3⁰ × 0.7¹⁴ + (14 choose 1) × 0.3¹ × 0.7¹³ + (14 choose 2) × 0.3² × 0.7¹² + (14 choose 3) × 0.3³ × 0.7¹¹ + (14 choose 4) × 0.3⁴ × 0.7¹⁰ + (14 choose 5) × 0.3⁵ × 0.7⁹

Using a calculator, we can evaluate this expression as:

P(X ≤ 5) = 0.8882

Therefore, the probability that six or more people name chocolate is:

P(X ≥ 6) = 1 - P(X ≤ 5) = 1 - 0.8882 = 0.1118

Rounding this to 4 decimal places, the probability of six or more people naming chocolate is 0.1118.

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If in  triangle ABC, EG = x inches and BG = (5x - 12) inches, the length of EB is 7x - 12 inches.

In triangle ABC, the centroid G is the point of intersection of the three medians AD, BE, and CF. We are given that EG = x inches and BG = (5x - 12) inches. We need to find the length of EB.

We can use the fact that the centroid G divides each median in the ratio of 2:1. Let M be the midpoint of AC, and let N be the midpoint of BC. Then, we have:

BE/EN = 2/1

Using this proportion, we can solve for EB as follows:

BE/EN = 2/1

BE/(BN + EN) = 2/1

BE/((5x - 12)/2 + x) = 2/1 (substituting BN = (5x - 12)/2 and EN = x)

BE/(7x/2 - 6) = 2/1

BE = 2(7x/2 - 6)

BE = 7x - 12

In conclusion, we can use the fact that the centroid divides each median in the ratio of 2:1 to find the length of EB in triangle ABC. We can solve for the unknown using a proportion and the lengths of the medians that pass through it.

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Complete question is:

Answer:

12

Step-by-step explanation:

i got it right

The amount of filling she would need for a dozen pastries is 37.704 cubic inches.

In Mathematics and Geometry, the volume of a cone can be determined by using this formula:

V = 1/3 × πr²h

Where:

Note: a dozen pastries is equal to 12 pastries.

By substituting the given parameters into the formula for the volume of a cone, we have the following;

Volume of cone, V = 1/3 × 3.142 × 1² × 3 × 12

Volume of cone, V = × 3.142 × 12

Volume of cone, V = 37.704 cubic inches.

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Standardized scores, also known as z-scores, allow us to compare and analyze data from different sources, scales, and units of measurement. They provide a common metric for comparing individual scores to the distribution of scores for a given population, making it easier to interpret and compare data.

Z-scores indicate the number of standard deviations that a particular score falls above or below the mean of a distribution.

A z-score of 0 represents a score that is exactly at the mean of the distribution, while a z-score of +1 represents a score that is one standard deviation above the mean, and a z-score of -1 represents a score that is one standard deviation below the mean.

Standardized scores allow us to:

Compare scores from different populations:

Z-scores make it possible to compare scores from different populations with different means and standard deviations.

This makes it easier to analyze data from multiple sources and to draw meaningful conclusions.

Identify outliers:

By using z-scores, we can identify scores that are significantly higher or lower than the rest of the distribution.

This can help to identify potential outliers or anomalies in the data.

Calculate percentiles:

Z-scores can be used to calculate percentiles, which indicate the percentage of scores that fall below a particular score.

A z-score of +1.5 represents a score that is higher than 93.32% of the scores in the distribution.

Standardize data:

By converting raw scores to z-scores, we can standardize the data, making it easier to compare scores and draw meaningful conclusions.

Standardized scores (z-scores) provide a powerful tool for analyzing and interpreting data in a way that is consistent and meaningful.

To compare scores from different populations, identify outliers, calculate percentiles, and standardize data.

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in the sample of 100 respondents from Vermont, 65.3% of them reported exercising for at least 30 minutes a day for three days out of the week.

The value of the sample proportion of people from Vermont who exercised for at least 30 minutes a day 3 days a week can be calculated as:

sample proportion = 65.3% = 0.653

what is minute?

In this context, "minute" likely refers to the amount of time spent exercising per day. Specifically, the survey asked if participants had exercised "more than 30 minutes a day for three days out of the week."

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The sound pressure level for input sound intensity using nonlinear regression is SPL = a x log10(sound intensity) + b

To estimate the SPL for a given sound intensity using nonlinear regression, we need to first understand the relationship between sound intensity and SPL. The formula that describes this relationship is:

SPL = 20 * log10 (pressure / reference Pressure)

where SPL is the sound pressure level in dB, pressure is the sound pressure in pascals (Pa), and reference Pressure is a standard reference pressure of 20 µPa.

In the context of the problem at hand, we have a set of sound intensity values and their corresponding SPL measurements. We can use this data to fit a nonlinear regression model that estimates the SPL for a given sound intensity value.

To perform nonlinear regression, we need to choose a functional form that models the relationship between the independent variable (sound intensity) and the dependent variable (SPL). One commonly used functional form for this type of problem is the power law:

SPL = a x log10(sound intensity) + b

where a and b are the parameters that we want to estimate from the data.

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Complete Question:

Calculating sound pressure level using nonlinear regression. Humans perceive sound intensity logarithmically. If the sound intensity increases by a factor of 100, then a human perceives the sound twice as loud. The sound intensity  and corresponding sound pressure level data is provided for a fixed number of samples.

Estimate the sound pressure level for input sound intensity using nonlinear regression.

At a sales level of $15,000, the two options yield the same salary.

To find the sales level at which the two options yield the same salary, we need to set the two salaries equal to each other and solve for sales:

2500 + 0.09x = 3100 + 0.05x

where x is the sales level.

Simplifying this equation, we get:

0.04x = 600

x = 15000

If the sales are less than $15,000, the first option with a lower base salary but a higher commission rate would be more favorable, while if sales exceed $15,000, the second option with a higher base salary but a lower commission rate would be more favorable.

It's worth noting that the decision between the two options depends on more than just the sales level. Other factors, such as the type of work, job security, and benefits, should also be considered when making a decision.

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(17) The length of apothem of the given regular hexagon is 6.9 units approximately and area of the same is 166.3 square units.

(18) The length of each side of the given regular hexagon is 15 units and area of the same is 585 square units.

(17) The figure is given below with required construction.

Side of the given regular hexagon = 8 units.

We know that the line joining midpoint of a side and center is perpendicular to the side.

So triangle APO is a right angled triangle.

So, P is midpoint of FA.

AP = 8/2 = 4 units.

Each angle of hexagon = 180 - 360/6 = 120

Then angle OAP = 120/2 = 60 degree

Now the length of apothem = AP tan 60 = 4√3 = 6.9 units (rounding to nearest tenth)

So the area of the hexagon = (1/2)*side*apothem*6 = (6.9*8*6)/2 = 166.3 square units.

(18) The figure is given below with required construction.

Apothem (OI) = 13 units

Similarly triangle OYI is right angled triangle.

Angle OYI = 60 degrees

So, the length of YI = OI tan 60 = 13/tan 60 = 13/√3 = 7.5 units (rounding off to nearest tenth)

So the side = 2*7.5 = 15 units.

So the area = (1/2)*15*6*13 = 585 square units.

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a)  There is not enough evidence to support the claim that the mean battery life exceeds 4 hours.

(b) The power of the test is 0.001, or 0.1%.

(c) We would need a sample size of at least 119 to achieve a power level of at least 0.90.

From the data,

The artificial heart will last and operate almost indefinitely once it is implanted in the patient's body, but the battery pack needs to be recharged about every four hours.

A random sample of 50 battery packs is selected and subjected to a life test. the average life of these batteries is 4.05 hours. assume that battery life is normally distributed with a standard deviation hour.

(a) We can test the null hypothesis that the mean battery life is equal to 4 hours against the alternative hypothesis that the mean battery life exceeds 4 hours.

The test statistic is:

t = ([tex]\overline x[/tex] - μ) / (s / √n)

Where [tex]\overline x[/tex] is the sample mean, μ is the hypothesized population mean (4 hours), s is the sample standard deviation (1 hour), and n is the sample size (50).

Substituting in the values, we get:

t = (4.05 - 4) / (1 / √50) = 1.58

Using a t-distribution table with 49 degrees of freedom (df = n - 1), the p-value for a one-tailed test with a test statistic of 1.58 is 0.0643.

Since the p-value is greater than the commonly used significance level of 0.05, we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the mean battery life exceeds 4 hours.

(b) To compute the power of the test, we need to specify the alternative hypothesis and the significance level.

The alternative hypothesis is that the true mean battery life is 4.5 hours, which means the null hypothesis is that the true mean battery life is 4 hours.

The significance level is 0.05, which means we will reject the null hypothesis if the p-value is less than 0.05.

We can use the formula for the power of a t-test:

power = P(t > tα/2 + (μ - μ0) / (s / √n))

where tα/2 is the critical value of the t-distribution with n-1 degrees of freedom and a significance level of α/2 (0.025 in this case), μ is the true population means (4.5 hours), μ0 is the hypothesized population mean (4 hours), s is the population standard deviation (1 hour), and n is the sample size.

Substitute in the numbers, we get:

power = P(t > 2.01 + (4.5 - 4) / (1 / √50)) = P(t > 3.31)

Using a t-distribution table with 49 degrees of freedom, the probability of t being greater than 3.31 is approximately 0.001. Therefore, the power of the test is 0.001, or 0.1%.

(c) We can use the formula for the sample size required to achieve the desired power level:

n = [(zβ + zα/2)σ / (μ - μ0)]²

where zβ is the critical value of the standard normal distribution corresponding to the desired power level (0.90 in this case, which gives zβ = 1.28), zα/2 is the critical value of the standard normal distribution corresponding to the desired significance level (0.05/2 = 0.025 in this case, which gives zα/2 = 1.96), σ is the population standard deviation (1 hour), and μ and μ0 are the true and hypothesized population means, respectively (4.3 hours and 4 hours, respectively).

Substituting in the numbers, we get:

n = [(1.28 + 1.96) x 1 / (4.3 - 4)]² = 118.7

Rounding up to the nearest integer, we would need a sample size of at least 119 to achieve a power level of at least 0.90.

Therefore,

a)  There is not enough evidence to support the claim that the mean battery life exceeds 4 hours.

(b) The power of the test is 0.001, or 0.1%.

(c) We would need a sample size of at least 119 to achieve a power level of at least 0.90.

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If Mr. Mathewson increased the weight he lifted from 80 pounds to 90 pounds, then the percentage increase weight he lifted is 12.5%.

To find the percentage increase in weight lifted by Mr. Mathewson, we can use the formula:

⇒ Percentage Increase = (New Weight - Old Weight)/(Old Weight) × 100%,

The old-weight that Mr. Mathewson lifted is = 80 pounds,

The new-weight that Mr. Mathewson lifted is = 90 pounds,

⇒ Percentage Increase = (90 - 80)/80 × 100 = 12.5%,

Therefore, there is an increase of 12.5% in the amount of weight lifted.

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Answer:

f(x) = (x - 2)(x - 5) = x^2 - 7x + 10

At the 0.05 level, there is not enough evidence to support the hospital director's claim that the percentage of insured patients is less than the expected percentage.

To test the claim that the percentage of insured patients is less than the expected percentage, the hospital director can use a one-tailed hypothesis test with a significance level of 0.05.

The null hypothesis (H0) would be that the percentage of insured patients is equal to or greater than the expected percentage, while the alternative hypothesis (Ha) would be that the percentage of insured patients is less than the expected percentage.

Using the given information, we know that 54% of emergency room visitors are insured, which means the expected percentage of insured patients is 0.54. The sample size is 300 patients, and 150 of them were insured.

To determine if there is enough evidence to support the director's claim, we can calculate the test statistic using the formula:

Z = (p - P) / sqrt[P(1-P)/n]

where p is the sample proportion, P is the expected proportion, and n is the sample size.

Plugging in the values, we get:

Z = (0.5 - 0.54) / sqrt[0.54(1-0.54)/300]

Z = -1.33

Using a standard normal distribution table, we can find that the probability of getting a Z-score of -1.33 or lower is approximately 0.0918.

Since this p-value (0.0918) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to support the director's claim that the percentage of insured patients is less than the expected percentage.

In conclusion, at the 0.05 level, there is not enough evidence to support the hospital director's claim that the percentage of insured patients is less than the expected percentage.

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