A marketing firm obtained random samples of 20 people in five regions of the country to investigate the level of interest in a new product. People in the sample were asked to rate their level of interest on a scale from 1 to 10, with 1 being the least amount of interest and 10 being the greatest. The histograms show the results for each region. The graph for which region displays data for level of interest with the least standard deviation?.

Answer: I think C

Step-by-step explanation:

Okay, here is the frequency table for the given data:

Number of books Frequency

0 3

1 4

2 4

3 3

5 1

Let me walk you through how I created this frequency table:

   The data gives the number of books checked out by 15 students

   I listed the possible numbers of books (0, 1, 2, 3, 5) in the first column

   For each number, I counted how many times it appeared in the data.

   So 0 appears 3 times, so under frequency I wrote 3

   1 appears 4 times, so frequency is 4

   2 appears 4 times, so frequency is 4

   3 appears 3 times, so frequency is 3

   5 appears only once, so frequency is 1

Answer:

Last table

Step-by-step explanation:

Given-

0,3,2,3,2,1,1,0,1,2,5,1,2,3,1

As we can see based on the given, there are;

Two ⇒ 0

Five ⇒ 1

Four ⇒ 2

Three ⇒ 3

Zero ⇒ 4

One ⇒ 5

                          ---------------------------------------------------------------    

Number of Books ║   0     ║   1     ║   2     ║   3     ║   4      ║  5   ║

 Tally                      ║ ||        ║|||||     ║ ||||      ║ |||       ║           ║ |      ║

Frequency              ║ 2       ║  5      ║ 4       ║3        ║    0       ║ 1      ║

According to the tables given we can see that [D] also known as the last table matches the data given.

Xkavinsky-

Step-by-step explanation:

You need to show us the function. To solve it you need to take f(x)=0 and solve for x, if x=-1 then -1 is a zero of a function.

The dimensions of the rectangle that minimize the perimeter are 18 feet by 18 feet.

To find the dimensions of a rectangle with an area of 324 square feet and the smallest possible perimeter, we need to minimize the perimeter while maintaining the given area. The area of a rectangle is given by the formula:

Area = length × width

Now, let's find the dimensions that result in the smallest perimeter.


The dimensions of the rectangle that minimize the perimeter are 18 feet by 18 feet.

1. We know the area is 324 square feet, so we can write the equation:

  324 = length × width

2. To minimize the perimeter, the rectangle should be as close to a square as possible because a square has the smallest possible perimeter for a given area.

3. Find the factors of 324 to determine which pair of numbers is closest to being equal. The factors are:

  1×324, 2×162, 3×108, 4×81, 6×54, 9×36, 12×27, and 18×18

4. Among these factor pairs, 18×18 is the closest to being equal, so the dimensions are 18 feet by 18 feet.

5. The smallest possible perimeter is achieved when the rectangle is a square with side lengths of 18 feet. The perimeter can be calculated as:

  Perimeter = 2 × (length + width) = 2 × (18 + 18) = 2 × 36 = 72 feet

So, the dimensions of the rectangle with an area of 324 square feet and the smallest possible perimeter are 18 feet by 18 feet, and the perimeter is 72 feet.

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The minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 4.123 and that about the ordinary point x = 1 is 4 units.

What is radius of convergence?

In mathematics, the radius of convergence of a power series is  stated as the radius of the largest disk at the center of the series in that the series converges. Radius of convergence is either a non-negative real number or infinity.

The given differential equation is

(x² -2x+17) y" +xy' -4y =0

We have to find the minimum radius of convergence R of power series solutions about the ordinary point x = 0 and x = 1.

The minimum radius of convergence can be defined as the distance between the ordinary point and the singularity of the differential equation.

Singularity point is the root of the polynomial attached with the second derivative. So, the singularity points can be calculated as,

(x² -2x+17)=0 Comparing this with ax²+bx+c=0 we get,

x=[tex]\frac{-b+\sqrt{b^{2} -4ac } }{2a}[/tex]  and x= [tex]\frac{-b-\sqrt{b^{2} -4ac } }{2a}[/tex]

x= [tex]\frac{2+\sqrt{4 -68 } }{2}[/tex]    and x= [tex]\frac{2-\sqrt{4 -68 } }{2}[/tex]

x= (2±8i)/2

x= 1±4i

So, the singularity points are x₁= 1+4i  and x₂ = 1-4i

Now, the ordinary points can be written as  z₁= 0+0i and z₂ = 1+0i

The minimum radius of convergence can be calculated as,

r₁ = | z₁ - x₁ |

   = | 0+0i-1-4i |

   = | -1-4i |

   = √17

   = 4.123

r₂ =  | z₂ - x₂ |

   = | 1+0i-1+4i |

   = | 4i |

   = √16

   = 4

Hence, the minimum radius of convergence R of power series solutions about the ordinary point x = 0 is 4.123 and that about the ordinary point x = 1 is 4 units.

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The circumcenter of an obtuse triangle will be located in the exterior of the triangle.

Recall that the circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle, and it is the center of the circumcircle that passes through all three vertices of the triangle.

In an acute triangle, the circumcenter lies inside the triangle, and in a right triangle, the circumcenter coincides with the midpoint of the hypotenuse. However, in an obtuse triangle, one of the angles measures more than 90 degrees, and the perpendicular bisectors of the two shorter sides intersect outside the triangle, while the perpendicular bisector of the longest side intersects inside the triangle.

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Answer: 8b^2 + 9b + 3y + 5 =

Step-by-step explanation:

8b^2 + 9b + 3y + 5 =

Answer:

24 coins

Step-by-step explanation:

We give the dimes a unknown number like y since the nickels are twice as much we give it 2y we multiply the numbers with their value and form an equation like 2y×5+y×10=160 find the value of y then get value of 2y+y to get the number of coins

The expression that gives the volume of the pyramid is (1/6) Bh. So, correct option is D.

The volume of a prism is given by the formula V = Bh, where B is the base area and h is the height of the prism. Similarly, the volume of a pyramid is given by the formula V = 1/3 Bh, where B is the base area and h is the height of the pyramid.

In this case, the pyramid has the same base area B as the prism, but its height is half of the prism's height. Therefore, the height of the pyramid is h/2.

Using the formula for the volume of a pyramid, we get:

V = 1/3 Bh = 1/3 B(h/2) = (1/6) Bh

Thus, the expression that gives the volume of the pyramid is (1/6) Bh, which is equivalent to option D.

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Complete question is:

A pyramid is placed inside a prism as shown. The pyramid has the same base area, B, as the prism but half the height, h, of the prism. Which expression gives the volume of the pyramid?

A) V = 2/3bh

B) V = 1/4bh

C) V = 2bh

D) V = 1/6bh

The p-value of 0.0316 indicates that if the null hypothesis were true, there would be a 3.16% chance of observing a test statistic as extreme or more extreme than 2.16.

Step 1: State the null and alternative hypotheses.

The null hypothesis (H0) is that the population parameter equals 6, and the alternative hypothesis (Ha) is that the population parameter is not equal to 6.

H0: μ = 6

Ha: μ ≠ 6

Step 2: Determine the level of significance (α).

The level of significance (α) represents the maximum acceptable probability of making a type I error (rejecting the null hypothesis when it is actually true). Let's assume a significance level of 0.05 (or 5%).

α = 0.05

Step 3: Calculate the test statistic.

The test statistic for this problem is given as z = 2.16.

z = 2.16

Step 4: Determine the critical value(s).

Since this is a two-tailed test with a 0.05 level of significance, we need to find the critical values that would capture 95% of the area under the standard normal distribution. For a two-tailed test with α = 0.05, the critical values are ±1.96.

Step 5: Calculate the p-value.

To calculate the p-value, we need to find the probability of getting a test statistic as extreme or more extreme than 2.16 assuming that the null hypothesis is true. Since this is a two-tailed test, we need to find the probability in both tails of the standard normal distribution.

Using a standard normal distribution table, we can find that the area to the right of 2.16 is 0.0158, and the area to the left of -2.16 is also 0.0158. Therefore, the total area in both tails is:

P-value = 2 × 0.0158 = 0.0316

Step 6: Compare the p-value to the level of significance.

The p-value of 0.0316 is less than the level of significance of 0.05. Therefore, we can reject the null hypothesis at the 0.05 level of significance.

Step 7: Interpret the result.

The p-value of 0.0316 indicates that if the null hypothesis were true, there would be a 3.16% chance of observing a test statistic as extreme or more extreme than 2.16.

This is relatively small, which suggests that the observed test statistic is unlikely to have occurred by chance alone, and provides evidence against the null hypothesis.

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Jenna is actually correct. The square root of a rational number can be an irrational number.

A rational number is defined as a number that can be expressed as a ratio of two integers (i.e., a fraction), where the denominator is not zero. An irrational number, on the other hand, is a number that cannot be expressed as a ratio of two integers.

For example, the number 2 is a rational number because it can be expressed as the ratio 2/1. The square root of 2, however, is an irrational number because it cannot be expressed as a ratio of two integers.

Therefore, it is possible for the square root of a rational number to be irrational. Jenna's argument is correct.

While Sam's example of the square root of 121 is a valid point, it is not a general proof that all square roots of rational numbers are rational.

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Considering the expression of a line, the equation of the line that passes through the pair of points (-8,5) and (2,0) is y=-1/2x +1.

A linear equation o line can be expressed in the form y = mx + b

where

Knowing two points (x₁, y₁) and (x₂, y₂) of a line, the slope can be calculated using:

m= (y₂ - y₁)÷ (x₂ - x₁)

Substituting the value of the slope m and the value of one of the points, the value of the ordinate to the origin b can be obtained.

In this case, being (x₁, y₁)= (-8,5) and (x₂, y₂)= (2,0), the slope m can be calculated as:

m= (0 -5)÷ (2 - (-8))

m= (-5)÷ (2 +8)

m= (-5)÷ 10

m= -1/2

Considering point 2 and the slope m, you obtain:

0= (-1/2)×2 + b

0= -1 +b

0 + 1= b

1= b

Finally, the equation of the line is y=-1/2x +1.

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We have proved that f o g is uniformly continuous on R, as desired.

F(g(x)) or (f g)(x) denotes the combination of the functions f(x) and g(x), where g(x) acts first. It brings together two or more functions to produce a new function.

To prove that the composite function f o g is uniformly continuous on R, we need to show that for any ε > 0, there exists a δ > 0 such that for any x, y ∈ R,

| x - y | < δ implies | (f o g)(x) - (f o g)(y) | < ε

We can start by using the uniform continuity of g to choose a δ1 > 0 such that for any x, y ∈ R,

| x - y | < δ1 implies | g(x) - g(y) | < ε

Now, we can use the uniform continuity of f with ε replaced by δ₁ to choose a δ₂ > 0 such that for any u, v ∈ R,

| u - v | < δ₂ implies | f(u) - f(v) | < δ₁

Finally, we can choose δ = δ₂ to show that for any x, y ∈ R,

| x - y | < δ implies | (f o g)(x) - (f o g)(y) | < ε

To see why this is true, let's assume that | x - y | < δ. Then, by the definition of the composite function,

(f o g)(x) - (f o g)(y) = f(g(x)) - f(g(y))

Now, since | g(x) - g(y) | < δ₁, we know that | f(g(x)) - f(g(y)) | < ε, by the choice of δ₁. And since | x - y | < δ₂ implies | g(x) - g(y) | < δ₁, we have shown that

| x - y | < δ₂ implies | (f o g)(x) - (f o g)(y) | < ε

Therefore, we have proved that f o g is uniformly continuous on R, as desired.

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The language that Pattie and Jairo speak to each other upon meeting was Spanish.

Holly Goldberg Sloan's novel "Counting by Sevens" centers around Willow Chance, a math and science prodigy who finds difficulty in connecting with others emotionally.

Following an unforeseen incident that drastically alters her life, Willow meets various individuals whose guidance enables her to understand love, friendship, and family. When Pattie and Jairo first encounter each other, they communicate solely in Spanish. Overall, the book is an inspiring and reflective read, skillfully examining topics such as resilience, loss, grief, and community.

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The first step to conducting a statistical study is to clearly define the research question or problem.

This involves identifying the population of interest, the variables to be measured, and the hypothesis or objective of the study. Once the research question is clearly defined, the next step is to design the study, including selecting an appropriate sample size, choosing a sampling method, and determining the data collection method.

It is also important to consider potential sources of bias and confounding factors that may impact the study results. Overall, a well-designed statistical study should be able to produce reliable and accurate results that can be used to draw meaningful conclusions about the population of interest.

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The  answer  is that since the company uses the allowance method for uncollectible accounts, it needs to record an allowance for doubtful accounts at the end of each accounting period. This allowance is an estimated amount that represents the portion of accounts receivable that is expected to be uncollectible.

In this specific situation, the company has determined that it will not be able to collect $1,200 owed by one of its customers. Therefore, it needs to adjust the allowance for doubtful accounts to reflect this uncollectible amount. This adjustment will decrease the accounts receivable balance and the net income for the period.

The allowance method is a way of accounting for uncollectible accounts that estimates the amount of accounts receivable that is likely to be uncollectible. This method is used because it is not possible to know exactly which accounts will not be collected. The estimated amount is based on historical experience and other relevant factors, such as economic conditions and the creditworthiness of customers.

When a company determines that a specific account is uncollectible, it needs to adjust the allowance for doubtful accounts to reflect this. This adjustment reduces the accounts receivable balance and the net income for the period. The adjustment is made by debiting the allowance for doubtful accounts and crediting the accounts receivable.

In this situation, the company has determined that it will not be able to collect $1,200 owed by one of its customers. Therefore, it needs to adjust the allowance for doubtful accounts to reflect this uncollectible amount. This adjustment will decrease the accounts receivable balance and the net income for the period. The adjustment is made by debiting the allowance for doubtful accounts and crediting the accounts receivable.

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This alternative hypothesis is either greater than or less than 0.87, but not exactly equal to it.

The alternative hypothesis (H1) is the hypothesis that is tested when the null hypothesis (H0) is rejected.

The null hypothesis is that the population proportion using the new drug that have a quick recovery is 0.87.

The alternative hypothesis would be that the population proportion using the new drug that have a quick recovery is different from 0.87.

This can be expressed as:

H1: p ≠ 0.87

The alternative hypothesis in this scenario would be that the new drug being tested is effective in increasing the chance of a quick recovery in people who have come down with the flu in the past week.

The alternative hypothesis would state that the population proportion of those who use the new drug and experience a quick recovery is greater than 0.87.
The alternative hypothesis is necessary because it allows us to determine whether the results of the study are statistically significant.

If the null hypothesis is accepted, it means that there is no significant difference between the rate of quick recovery with or without the new drug.

The alternative hypothesis is accepted, it means that the new drug has a significant effect on increasing the rate of quick recovery.
To test this hypothesis, a statistical analysis will need to be performed using the data collected from the study.

This analysis will allow us to determine whether the results are statistically significant and whether we can reject the null hypothesis in favor of the alternative hypothesis.

Ultimately, this will provide valuable information on the effectiveness of the new drug and whether it should be recommended for use in treating the flu.

The "≠" symbol means "not equal to".

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Using the following pattern and inductive reasoning, predicted answer to 9 x 7,654,321 - 1 is 73,888,889.

The pattern:

When you subtract 1 from a number that ends with a sequence of n consecutive digits (all equal to d), the result is a number that ends with the same n digits followed by ([tex]10^{n}[/tex] - 1) -d.

For example:

If you subtract 1 from a number that ends with three 7's, the result is a number that ends with three 6's, i.e., (777-1=776).

If you subtract 1 from a number that ends with four 2's, the result is a number that ends with four 1's, i.e., (2222-1=2221).

Applying this pattern to 9 x 7,654,321 - 1:

The number ends with one 9, so n=1 and d=9.

Therefore, the result will end with one 8 (one less than 9), followed by ([tex]10^{1}[/tex] - 1) - 9 = 0, i.e., it will end with 8.

So, the predicted answer is 73,888,889.

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A recursive function is a function f(N) that is defined in terms of the same function operating on a value < N.

In a recursive function, the function calls itself repeatedly until it reaches a base case, which is a condition that stops the recursion. The function then returns the result of the base case to the previous call, which uses it to compute the final result.

Recursive functions are commonly used in programming for solving problems that can be broken down into smaller subproblems of the same type. They are especially useful for tasks such as traversing trees or searching through graphs.

The process of recursion involves breaking down a problem into smaller, similar subproblems that can be solved using the same approach. This leads to a more elegant and concise solution, as compared to iterative approaches. However, recursive functions can also be less efficient than their iterative counterparts and may have issues with a stack overflow if the recursion depth becomes too large.

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The probability that a Jordanian couple is both sterile and infertile is 0.0384 or about 3.84%, given that 6% of Jordanian couples are sterile and 64% of sterile couples in Jordan are infertile.

Let S be the event that a couple is sterile and I be the event that the couple is infertile. We are given that P(S) = 0.06 and P(I|S) = 0.64.

We want to find P(S and I), the probability that a couple is both sterile and infertile. Using the conditional probability formula, we have

P(S and I) = P(I|S) * P(S)

Substituting the given values, we get

P(S and I) = 0.64 * 0.06

= 0.0384

Therefore, the probability that a Jordanian couple is both sterile and infertile is 0.0384 or about 3.84%.

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The surface area of the new, larger ball is 160 square inches.

What is surface area?

The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area.

The surface area of a sphere with radius r is given by:

A = 4πr²

If the radius is doubled to 2r, the surface area of the new sphere is:

A' = 4π(2r)² = 4π(4r²) = 16πr²

So the new surface area is 16 times the original surface area.

Since the original surface area is 10 square inches, the new surface area is:

A' = 16(10) = 160 square inches

Therefore, the surface area of the new, larger ball is 160 square inches.

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The scale factor that does not belong with the other three is 2.

The scale factor that does not belong with the other three is determined as follows;

5/4 ------->  interpreted as a ratio of the new size to the original size, where the new size is 1.25 times larger than the original size.

60% ------->  interpreted as a percentage increase in size or quantity, where the new size is 1.6 times larger than the original size.

115% ------>  interpreted as a percentage increase in size or quantity, where the new size is 2.15 times larger than the original size.

2 is not a scale factor because it does not provide any information about the relationship between the new and original size or quantity.

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Yes, the wait time significantly decreased after implementing the new system, based on hypothesis testing.

To determine if the wait time has significantly decreased, we can conduct a hypothesis test using the one-sample t-test. The null hypothesis, denoted as H0, is that the population mean waiting time is still 84.3 seconds, while the alternative hypothesis, denoted as Ha, is that the population mean waiting time has decreased to less than 84.3 seconds.

Let's set the significance level at α = 0.05. Since the sample size is large (n = 45), we can use the t-distribution to conduct the hypothesis test. The test statistic can be calculated as:

t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

t = (79.1 - 84.3) / (17.3 / sqrt(45))

t = -2.16

Using a t-distribution table or calculator with 44 degrees of freedom

(df = n - 1), the p-value for this test is found to be approximately 0.018.

Since the p-value (0.018) is less than the significance level (0.05), we reject the null hypothesis. This means we have evidence to suggest that the wait time has significantly decreased after implementing the new drive-through system.

Therefore, based on the given data, we can conclude that the wait time has significantly decreased at the fast-food restaurant after implementing the new drive-through system.

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The average study time of students is greater than 6.5 hours per week, at a significance level of 0.05.

What is the average?

This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.

To test the claim that students on average spend more than 6.5 hours studying, we can use a one-sample t-test.

The null hypothesis is that the mean study time is equal to 6.5 hours:

H0: μ = 6.5

The alternative hypothesis is that the mean study time is greater than 6.5 hours:

Ha: μ > 6.5

We can calculate the test statistic using the formula:

t = (x - μ) / (s / √n)

where x is the sample mean, μ is the hypothesized population mean (6.5), s is the sample standard deviation, and n is the sample size.

Plugging in the values given in the problem, we get:

t = (7.2 - 6.5) / (2.8 / √120) = 3.18

Using a t-table with 119 degrees of freedom (df = n - 1), and a significance level of 0.05, the critical t-value for a one-tailed test (since we are testing for the mean being greater than 6.5) is 1.66.

Since our calculated t-value (3.18) is greater than the critical t-value (1.66), we can reject the null hypothesis.

Therefore, we have sufficient evidence to conclude that the average study time of students is greater than 6.5 hours per week, at a significance level of 0.05.

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Answer:

√(20^2 + 21^2) = √(400 + 441) = √841 = 29

The diameter of the circle is 29 inches, so the radius is 14.5 inches.

Area of shaded region:

π(14.5^2) - (1/2)(20)(21) = 450.52 in^2

The edges of the square may be linearly parameterized as follows:- The bottom edge from (0,0) to (5,0) can be parameterized as (t, 0) for t in [0, 5].

- The right edge from (5,0) to (5,5) can be parameterized as (5, t) for t in [0, 5], - The top edge from (5,5) to (0,5) can be parameterized as (t, 5) for t in [5, 0]., - The left edge from (0,5) to (0,0) can be parameterized as (0, t) for t in [5, 0].
To linearly parameterize the edges of square C with corners (0,0), (5,0), (5,5), and (0,5), oriented counterclockwise,

we can use the following parameterizations: 1. Bottom edge: from (0,0) to (5,0)
Parameterization: (x, y) = (t, 0), where 0 ≤ t ≤ 5

2. Right edge: from (5,0) to (5,5)
Parameterization: (x, y) = (5, t), where 0 ≤ t ≤ 5

3. Top edge: from (5,5) to (0,5)
Parameterization: (x, y) = (t, 5), where 0 ≤ t ≤ 5

4. Left edge: from (0,5) to (0,0)
Parameterization: (x, y) = (0, t), where 0 ≤ t ≤ 5, These linear parameterizations represent each edge of the square and follow the counterclockwise orientation.

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The answer is that the resulting function is called a composite function.

A composite function is formed by combining two or more functions, where the output of one function becomes the input of another. In this case, when two functions are combined so that the range of one becomes the domain of the other, the resulting function is a composite function.

This can be given by considering two functions f(x) and g(x). If we want to create a composite function, we need to use the output of one function as the input for the other. So, if we let g(x) be the first function, then the output of g(x) becomes the input for f(x). This can be written as f(g(x)).

Now, if we want the range of g(x) to become the domain of f(x), we need to make sure that the output of g(x) is a valid input for f(x). In other words, the range of g(x) should be a subset of the domain of f(x).

For example, let's say g(x) = x² and f(x) =√(x). The domain of g(x) is all real numbers, and its range is all non-negative real numbers. The domain of f(x) is also all non-negative real numbers. So, we can create a composite function f(g(x)) by using the output of g(x) (which is always non-negative) as the input for f(x).

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The value of x when central angle is 122 degrees and inscribed angle is (3x + 5) degrees is given by, x = 39.

So the blank will be '39'

Central angle is angle made at center by the end points of an arc of a circle.

Here the central angle in variable is = (3x + 5) degrees

The central angle is = 122 degrees

So by the figure we can see that,

3x + 5 = 122

3x = 122 - 5

3x = 117

x = 117/3

x = 39

So the value of the x is given by, x = 39.

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Based on the Gallup poll conducted on November 2, 2014, it can be inferred that the majority of Americans are in favor of legalizing marijuana.

The poll reported that 51 percent of Americans support legalization while 47 percent oppose it.

However, it is important to note that the reported margin of sampling error was +/- 4 percent, which means that the actual percentage of Americans who support or oppose legalization could be slightly different from the reported percentages.

The margin of sampling error is a measure of the accuracy of the poll results and indicates the amount of random sampling error that is expected. In this case, the margin of sampling error is relatively small, at +/- 4 percent, which suggests that the poll results are fairly reliable.

Overall, the poll indicates that attitudes towards marijuana legalization have shifted in recent years, with a majority of Americans now supporting legalization. This could have implications for future policy decisions related to marijuana legalization at both the state and federal levels.

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The value of the integral ∫[tex]e^{cos(t)[/tex] sin(2t) dt over the interval [0, π] is 0.

The summing of discrete data is indicated by the integration. To determine the functions that will characterise the area, displacement, and volume that result from a combination of small data that cannot be measured separately, integrals are calculated.

To evaluate the integral ∫[tex]e^{cos(t) sin(2t)} dt[/tex] over the interval [0, π], we can use integration by parts with the substitution u = sin(t) and dv = [tex]e^{cos(t)cos(t)}dt[/tex]. Then, du = cos(t)dt and [tex]v = e^{cos(t)[/tex].

Using this substitution and the formula for integration by parts:

∫[tex]e^{cos(t) sin(2t)} dt[/tex] = [tex]-e^{cos(t) sin(t)}[/tex] + ∫[tex]e^{cos(t) cos(t)} dt[/tex]

We can use another substitution, z = cos(t), to simplify the integral on the right-hand side:

∫[tex]e^{cos(t) cos(t)} dt = \int e^{z} dz = e^z + C[/tex]

Substituting back to the original integral, we get:

∫[tex]e^{cos(t) sin(2t)} dt = -e^{cos(t) sin(t)} + e^{cos(t)} + C[/tex]

Evaluating the definite integral over the interval [0, π], we get:

∫[0,π] [tex]e^{cos(t) sin(2t)} dt = -e^{cos(\pi ) sin(\pi )} + e^{cos(\pi )} - (-e^{cos(0) sin(0)} + e^{cos(0)})[/tex]

Simplifying the trigonometric terms sin(π) = 0 and sin(0) = 0, and using the fact that cos(π) = -1 and cos(0) = 1, we get:

∫[0,π] [tex]e^{cos(t)[/tex] sin(2t) dt = [tex]-e^{-1} + e^{-1} = 0[/tex]

Therefore, the value of the integral ∫[tex]e^{cos(t)[/tex] sin(2t) dt over the interval [0, π] is 0.

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There are 396 different ways to pair up a team from league a with a team from league b and 153 different ways to pair up two teams from league a.

(a) To pair up a team from league a with a team from league b, we can use the fundamental counting principle which states that if we have m ways to do one thing and n ways to do another, then there are m x n ways to do both. Therefore, the number of ways to pair up a team from league a with a team from league b is:

18 x 22 = 396

(b) To pair up two teams from league a, we can use the combination formula which states that the number of ways to choose r items from a set of n items is nCr = n! / (r! (n-r)!). Since we want to choose 2 teams from a set of 18 teams in league a, we can use the formula:

18C2 = 18! / (2! (18-2)!) = 153

Therefore, there are 153 different ways topair up two teams from league a.

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