State whether the sequence alpha_n = ln (9n/n + 1) | converges and, if it does, find the limit. Converges to 1 diverges converges to ln(9) converges to 2 converges to ln (9/2)|

The partial sums are: [tex]S_{2}[/tex] = 5/2 , [tex]S_{4}[/tex] = 89/36 , [tex]S_{6}[/tex] = 1681/450 .

To compute the partial sums[tex]S_{2}[/tex], [tex]S_{4}[/tex], and [tex]S_{6}[/tex] , we need to find the sums of the first 2, 4, and 6 terms, respectively, in the given series:

Series: 2 + 2/[tex]2^{2}[/tex] + 2/[tex]3^{2}[/tex] + 2/[tex]4^{2}[/tex] + ...

[tex]S_{2}[/tex]: The sum of the first 2 terms is:
[tex]S_{2}[/tex] = 2 + 2/[tex]2^{2}[/tex]= 2 + 2/4 = 2 + 1/2 = 5/2.

[tex]S_{4}[/tex]: The sum of the first 4 terms is:
[tex]S_{4}[/tex] = 2 + 2/[tex]2^{2}[/tex] + 2/[tex]3^{2}[/tex] + 2/[tex]4^{2}[/tex]

    = 2 + 1/2 + 2/9 + 2/16 = 5/2 + 4/9 + 1/8  

    = 89/36.

[tex]S_{6}[/tex]: The sum of the first 6 terms is:
[tex]S_{6}[/tex]= 2 + 2/[tex]2^{2}[/tex] + 2/[tex]3^{2}[/tex] + 2/[tex]4^{2}[/tex] + 2/[tex]5^{2}[/tex] + 2/[tex]6^{2}[/tex]

    = 2 + 1/2 + 2/9 + 1/8 + 2/25 + 1/18 = 5/2 + 4/9 + 1/8 + 1/18 + 2/25  

    = 1681/450.

So, the partial sums are:
[tex]S_{2}[/tex] = 5/2
[tex]S_{4}[/tex] = 89/36
[tex]S_{6}[/tex] = 1681/450

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The standard deviation of X, where X is the number of dollars won by Jamal in the four-week period, is 18.0000

Standard deviation measures the amount of variation or dispersion in a set of values. It is a statistical calculation that quantifies the amount of spread or dispersion in a dataset, indicating how much the individual values deviate from the mean (average) of the dataset.

According to the given information:

To calculate the standard deviation of X, we first need to determine the probability of winning in one week.

Given that the cash register is programmed to print stars on 10% of receipts, the probability of winning in one week is the probability of getting a star on all five consecutive weekdays, which is (0.1)^5, since the events are independent.

Next, we can create a binomial distribution with four weeks as the number of trials, since Jamal eats at Fred's once each weekday for four consecutive weeks. The probability of winning N times in four weeks would be the binomial coefficient multiplied by the probability of winning in one week raised to the power of N, and the probability of not winning raised to the power of (4-N), where N is the number of times Jamal wins in four weeks.

The formula for the binomial distribution is:

P(X = N) = [tex]C(4,N)*(0.1)^{N}*(0.9)^{4-N}[/tex]

Finally, we can calculate the standard deviation of X, which is the square root of the variance of X. The variance of X can be calculated by multiplying the variance of the binomial distribution (npq) by 100^2, since X = 100N.

Let's calculate the standard deviation of X using the given formula:

For N = 0:  P(X = 0) = [tex]C(4,0)*(0.1)^{0}*(0.9)^{4}[/tex] = 0.6561

For N = 1:   P(X = 100) = [tex]C(4,1)*(0.1)^{1}*(0.9)^{3}[/tex] = 0.2916

For N = 2:   P(X = 200) = [tex]C(4,2)*(0.1)^{2}*(0.9)^{2}[/tex] = 0.0486

For N = 3:   P(X = 300) = [tex]C(4,3)*(0.1)^{3}*(0.9)^{1}[/tex] = 0.0036

For N = 4:   P(X = 400) = [tex]C(4,4)*(0.1)^{4}*(0.9)^{0}[/tex] = 0.0001

Now, we can calculate the variance of X:

Variance of X = [tex](npq)*100^{2}[/tex], where n is the number of trials (4) and p is the probability of winning in one week (0.1).

Variance of X = 4 * 0.1 * 0.9 *[tex]100^{2}[/tex]  = 324

Finally, we can calculate the standard deviation of X by taking the square root of the variance:

Standard deviation of X = [tex]\sqrt{324}[/tex] = 18

So, the standard deviation of X, where X is the number of dollars won by Jamal in the four-week period, is 18.0000 (rounded to four decimal places).

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The standard deviation of X, where X is the number of dollars won by Jamal in the four-week period, is 18.0000

Standard deviation measures the amount of variation or dispersion in a set of values. It is a statistical calculation that quantifies the amount of spread or dispersion in a dataset, indicating how much the individual values deviate from the mean (average) of the dataset.

According to the given information:

To calculate the standard deviation of X, we first need to determine the probability of winning in one week.

Given that the cash register is programmed to print stars on 10% of receipts, the probability of winning in one week is the probability of getting a star on all five consecutive weekdays, which is (0.1)^5, since the events are independent.

Next, we can create a binomial distribution with four weeks as the number of trials, since Jamal eats at Fred's once each weekday for four consecutive weeks. The probability of winning N times in four weeks would be the binomial coefficient multiplied by the probability of winning in one week raised to the power of N, and the probability of not winning raised to the power of (4-N), where N is the number of times Jamal wins in four weeks.

The formula for the binomial distribution is:

P(X = N) = [tex]C(4,N)*(0.1)^{N}*(0.9)^{4-N}[/tex]

Finally, we can calculate the standard deviation of X, which is the square root of the variance of X. The variance of X can be calculated by multiplying the variance of the binomial distribution (npq) by 100^2, since X = 100N.

Let's calculate the standard deviation of X using the given formula:

For N = 0:  P(X = 0) = [tex]C(4,0)*(0.1)^{0}*(0.9)^{4}[/tex] = 0.6561

For N = 1:   P(X = 100) = [tex]C(4,1)*(0.1)^{1}*(0.9)^{3}[/tex] = 0.2916

For N = 2:   P(X = 200) = [tex]C(4,2)*(0.1)^{2}*(0.9)^{2}[/tex] = 0.0486

For N = 3:   P(X = 300) = [tex]C(4,3)*(0.1)^{3}*(0.9)^{1}[/tex] = 0.0036

For N = 4:   P(X = 400) = [tex]C(4,4)*(0.1)^{4}*(0.9)^{0}[/tex] = 0.0001

Now, we can calculate the variance of X:

Variance of X = [tex](npq)*100^{2}[/tex], where n is the number of trials (4) and p is the probability of winning in one week (0.1).

Variance of X = 4 * 0.1 * 0.9 *[tex]100^{2}[/tex]  = 324

Finally, we can calculate the standard deviation of X by taking the square root of the variance:

Standard deviation of X = [tex]\sqrt{324}[/tex] = 18

So, the standard deviation of X, where X is the number of dollars won by Jamal in the four-week period, is 18.0000 (rounded to four decimal places).

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The critical value t0.10,5 is approximately 1.4759.

To find the critical value t0.10,5 (also written as t(0.10,5)), you'll need to consult a t-distribution table. This critical value represents the t-score that has a probability of 0.10 (10%) in the upper tail of the distribution and 5 degrees of freedom.

Using a t-distribution table or a calculator, the critical value t0.10,5 is approximately 1.4759.

Your answer: The critical value t0.10,5 is approximately 1.4759.

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The value of a and b include the following:

a = 7

b = -1.

In Mathematics and Geometry, a function can be defined as a mathematical equation which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair in tables or relations.

Based on the information provided above, we have the following functions;

f(x) = 5x + 3    ....equation 1.

g(x) = ax + b     ....equation 2.

From equation 2, we have;

g(3) = 20

g(3) = a(3) + b

20 = 3a + b      ....equation 3.

From equation 1, the inverse function is given by;

f(x) = y = 5x + 3

x = (y - 3)/5      ....equation 4.

f⁻¹(33) = g(1)

(33 - 3)/5 = g(1)

30/5 = g(1)

6 = g(1)

g(1) = a(1) + b

6 = a + b      ....equation 5.

By solving equations 3 and 5 simultaneously, we have:

20 = 3a + b

6 = a + b

a = 7 and b = -1.

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Answer:

A

Step-by-step explanation:

Because when you multiply anything with exponents, you multiply the coefficient and add the exponents.

The value of x is 13 and can be calculated by setting the number of students who played soccer and rugby (S ∩ R) but not Gaelic football equal to x - 4, and then solving for x.

We know that:

65 students played Gaelic football (G)

57 students played soccer (S)

34 students played rugby (R)

42 students played Gaelic football and soccer (G ∩ S)

16 students played Gaelic football and rugby (G ∩ R)

x students played soccer and rugby (S ∩ R)

4 students played all three sports (G ∩ S ∩ R)

6 students played none of the sports listed

To fill in the Venn diagram, we can start with the three circles representing Gaelic football (G), soccer (S), and rugby (R), and add the numbers in each region based on the information provided. Let's go region by region:

The region inside all three circles (G ∩ S ∩ R) has 4 students.

The region inside both Gaelic football and soccer circles (G ∩ S) but outside the rugby circle has 42 - 4 - 16 = 22 students.

The region inside both Gaelic football and rugby circles (G ∩ R) but outside the soccer circle has 16 - 4 = 12 students.

The region inside both soccer and rugby circles (S ∩ R) but outside the Gaelic football circle has x - 4 = x - 4 students.

The region inside only the Gaelic football circle (G) but outside the other two circles has 65 - 4 - 22 - 16 - 6 = 17 students.

The region inside only the soccer circle (S) but outside the other two circles has 57 - 4 - 22 - x + 4 - 6 = 25 - x students.

The region inside only the rugby circle (R) but outside the other two circles has 34 - 4 - 16 - x + 4 - 6 = 8 - x students.

The region outside all three circles has 6 students.

Total number of students who played soccer = S + (S ∩ R) + (G ∩ S

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Frοm the ACF plοt, we can see that the autοcοrrelatiοn values decay slοwly and dο nοt gο tο zerο, indicating a nοn-statiοnary time series. The PACF plοt shοws significant spikes at lags 1, 2, and 4, suggesting an AR(4) mοdel may be apprοpriate fοr the data.

A number's square rοοt is a value that, when multiplied by itself, yields the οriginal number. The οther way tο square an integer is tο find its square rοοt. Squares and square rοοts are hence linked ideas.

Tο cοmpute the sample ACF and PACF fοr the transfοrmed time series {Xt}, which is the square rοοt οf the οriginal sunspοt data, we can use statistical sοftware οr prοgramming languages that have built-in functiοns fοr time series analysis. Here, we'll use Pythοn with the statsmοdels library tο cοmpute the ACF and PACF.

First, we'll impοrt the necessary libraries and lοad the data frοm the file sunspοt.dat:

impοrt pandas as pd

impοrt matplοtlib.pyplοt as plt

impοrt statsmοdels.api as sm

# lοad data

data = pd.read_csv('sunspοt.dat', sep='\s+', header=Nοne, names=['year', 'sunspοt'])

X = data['sunspοt'].apply(lambda x: x**0.5)  # apply square rοοt transfοrmatiοn

We've lοaded the data intο a Pandas DataFrame and applied the square rοοt transfοrmatiοn tο the sunspοt cοlumn, which we've saved as X.

Nοw, we can use the plοt_acf and plοt_pacf functiοns frοm statsmοdels tο cοmpute and plοt the ACF and PACF:

# cοmpute and plοt ACF

sm.graphics.tsa.plοt_acf(X, lags=50)

plt.shοw()

# cοmpute and plοt PACF

sm.graphics.tsa.plοt_pacf(X, lags=50)

plt.shοw()

Here, we've specified lags=50 tο shοw the first 50 lags οf the ACF and PACF.

Frοm the ACF plοt, we can see that there is a significant autοcοrrelatiοn at lag 1, and the autοcοrrelatiοn values gradually decrease and becοme insignificant as the lag increases. This suggests that an autοregressive (AR) mοdel may be apprοpriate.

Frοm the PACF plοt, we can see that there is a significant partial autοcοrrelatiοn at lag 1, and the partial autοcοrrelatiοn values becοme insignificant after lag 1. This suggests that a first-οrder autοregressive mοdel (AR(1)) may be apprοpriate.

Nοte that because the transfοrmed time series {Xt} is a pοsitive series with nο negative values, an alternative transfοrmatiοn such as the lοg transfοrmatiοn may alsο be suitable fοr this data. It is recοmmended tο cοmpare the results οf different transfοrmatiοns and chοοse the οne that prοduces the best mοdel fit

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We can implements the approaches to predict all price jump events using a decision tree.

To do this, follow these steps:

1. Randomly sample your dataset, creating N samples.
2. Define a hyperparameter Dmax as the max depth of the tree.
3. Define a variable d for the current depth of the tree.
4. In each node, randomly choose a threshold between min and max prices to split the feature x.
5. Continue splitting until reaching one sample per leaf node or reaching Dmax depth.

This approach involves building a decision tree model to predict price jump events. First, create N random samples from your dataset. Set a maximum tree depth, Dmax, and track the current depth, d. In each node, randomly select a threshold between the minimum and maximum price values for splitting the data.

Continue this process until there is only one sample in each leaf node or you've reached the maximum depth, Dmax. This method will help create a decision tree that can effectively predict price jumps in the data.

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We would need a sample size of 16 to construct a 95% confidence interval to estimate the average air travel cost for a college student with a margin of error of $50. The critical value for a 95% confidence interval is approximately 1.96.

To determine the sample size needed to construct a 95% confidence interval to estimate the average air travel cost for a college student with a margin of error of $50, we need to use the formula:

n = (zα/2 * σ / E)^2

where:
- n is the sample size
- zα/2 is the critical value for the desired confidence level, which is 1.96 for 95% confidence interval
- σ is the standard deviation of the population, which is unknown, so we use the sample standard deviation as an estimate
- E is the margin of error, which is $50

Assuming that we have a pilot sample of air travel costs for college students, we can use the sample standard deviation as an estimate for the population standard deviation.

Let's say the sample standard deviation is $200.

Plugging in the values, we get:

n = (1.96 * 200 / 50)^2
n = 15.36

Since we can't have a fraction of a sample, we need to round up to the nearest whole number, which gives us a sample size of 16.

To calculate the required sample size for a 95% confidence interval with a margin of error of $50, we need some information about the population standard deviation (σ) and the critical value (Z) associated with the desired confidence level.

Since the problem does not provide the population standard deviation, I'll assume it is known or estimated from a previous study.

Let's call it σ.The margin of error (E) formula for a confidence interval is:

E = Z * (σ / √n)

Where:
E = margin of error ($50)
Z = critical value (1.96 for a 95% confidence interval)
σ = population standard deviation
n = sample size

We need to solve for n:

50 = 1.96 * (σ / √n)

To isolate n, we can follow these steps:

1. Divide both sides by 1.96:
50 / 1.96 = σ / √n

2. Square both sides:
(50 / 1.96)^2 = (σ^2 / n)

3. Multiply both sides by n:
(50 / 1.96)^2 * n = σ^2

4. Divide both sides by (50 / 1.96)^2:
n = σ^2 / (50 / 1.96)^2

Now, plug in the known or estimated value for σ, and calculate the required sample size (n). Remember to round up to the nearest whole number, as you cannot have a fraction of a sample.

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Answer:

To factor the quadratic function g(x) = 8x^2 - 2x - 3, we can use the following steps:

Step 1: Multiply the coefficient of the x^2 term (8) and the constant term (-3).

8 * -3 = -24

Step 2: Find two numbers that multiply to give the result from step 1 (-24) and add up to the coefficient of the x term (-2).

The two numbers that meet these criteria are -6 and +4, since -6 * 4 = -24 and -6 + 4 = -2.

Step 3: Rewrite the middle term (-2x) using the two numbers found in step 2 (-6 and +4).

8x^2 - 6x + 4x - 3

Step 4: Group the terms and factor by grouping.

2x(4x - 3) + 1(4x - 3)

Step 5: Factor out the common binomial (4x - 3).

(4x - 3)(2x + 1)

So, the factored form of the quadratic function g(x) = 8x^2 - 2x - 3 is (4x - 3)(2x + 1).

The fractions are not equivalent. The top diagram represents Three-fifths, and the bottom diagram represents Three-tenths.

A fraction is a numerical quantity that represents a part of a whole or a ratio of two numbers. It is expressed in the form of a/b, where a is the numerator and b is the denominator.

According to the given information :

The shaded portions of the diagrams do not represent equivalent fractions. The top diagram represents three-fifths, meaning that three out of five parts are shaded. The bottom diagram represents three-tenths, meaning that three out of ten parts are shaded. Since five and ten are not equal, the two fractions cannot be equivalent.

It's important to note that even though both diagrams have the same number of shaded parts, this does not necessarily mean that they represent equivalent fractions. The overall size of the fraction bar and the number of parts into which it is divided must also be taken into account when determining equivalence.

In this case, the top diagram could be compared to a bottom diagram with six parts shaded, which would represent six-tenths or three-fifths, making it equivalent to the top diagram.

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Answer:

c

Step-by-step explanation:

b = 16.6

c = 11.2

cos 34° = b/20

b = 20 × cos 34°

b = 20 × 0.829

b = 16.6

sin 34° = c/20

c = 20 × sin 34°

c = 20 × 0.559

c = 11.2

The cumulative distribution function (cdf) of the random variable X is given by F(x) = (1 – x)³ for 0 < x < 1, and F(x) = 0 for x ≤ 0, and F(x) = 1 for x ≥ 1.

The given problem describes a random variable X with a probability density function (pdf) of f(x) = 3(1 – x)² for 0 < x < 1, and f(x) = 0 otherwise.

To find the cumulative distribution function (cdf) of X, we need to integrate the pdf f(x) with respect to x over its domain.

Given that f(x) = 3(1 – x)², we can integrate it as follows:

∫ f(x) dx = ∫ 3(1 – x)² dx

Using the power rule of integration, we get:

= 3 × [(1 – x)^(2 + 1)] / (2 + 1) + C, where C is the constant of integration

= (3/3) × (1 – x)³ + C

= (1 – x)³ + C

Now, since the domain of f(x) is 0 < x < 1, we need to apply the limits of integration.

When x = 0, the cdf is:

F(0) = (1 – 0)³ + C = 1 + C

When x = 1, the cdf is:

F(1) = (1 – 1)³ + C = 0 + C

Therefore, the cdf of X is given by:

F(x) = (1 – x)^3 + C for 0 < x < 1, and F(x) = 0 for x ≤ 0, and F(x) = 1 for x ≥ 1.

Therefore, The cumulative distribution function (cdf) of the random variable X is given by F(x) = (1 – x)³ for 0 < x < 1, and F(x) = 0 for x ≤ 0, and F(x) = 1 for x ≥ 1.

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answer in attached image

(x, y) → (-y, x)

If twelve 1.5 MQ resistors are connected in parallel across 50 V, RT equals C)1 MQ.

12 resistors, each with a resistance of 1.5 MQ are connected in parallel across 50 V

To find the total resistance (RT), we can use the formula for resistors in parallel:

1/RT = 1/R1 + 1/R2 + ... + 1/Rn

where R1, R2, ..., Rn are the resistances of the individual resistors.

Substituting the given values:

1/RT = 1/1.5 MQ + 1/1.5 MQ + ... + 1/1.5 MQ (12 times)

Simplifying:

1/RT = 12/1.5 MQ

Taking the reciprocal of both sides:

RT = 1 / (12/1.5 MQ)

RT = 1 / (8/1 MQ)

RT = 1.25 MQ

So, the total resistance (RT) is 1.25 MQ. Therefore, the correct answer is option C - 1.25 MQ.

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Proof of De Morgan's Law: To prove De Morgan's law, we need to show that AU B = A B, where A and B are sets. We will do this by proving two separate inclusions:

First, we will show that A B ⊆ AU B. Let x ∈ A B. Then, x ∈ A and x ∈ B. This means that x ∈ A or x ∈ B (or both), so x ∈ AU B. Therefore, we have shown that A B ⊆ AU B.

Next, we will show that AU B ⊆ A B. Let x ∈ AU B. Then, x ∈ A or x ∈ B (or both). We will consider two cases:

If x ∈ A, then x ∈ A B since x ∈ A and x ∈ B (since x ∈ B, by assumption).

If x ∉ A, then x ∈ B, since x ∈ AU B. Then, x ∈ A B since x ∈ A and x ∈ B.

Therefore, we have shown that AU B ⊆ A B.

Combining the two inclusions, we have shown that AU B = A B, and thus, De Morgan's law is proven.

Identification of unknowns in the membership table:

Without the membership table provided, we cannot identify the unknowns X, Y, Z, P, Q, and R. Please provide the membership table for us to identify the unknowns.

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The probability that one or more of the backup servers is operational is 1 - P(both servers fail).

To find this probability, first, determine the probability that both servers fail, which is 0.21 * 0.21 = 0.0441. Then, subtract this value from 1: 1 - 0.0441 = 0.9559. Therefore, the probability that one or more servers is operational is 0.9559.

we know that the failure of one server is independent of the other server's failure. The probability that a single server fails is 0.21. To find the probability that both servers fail, we multiply their individual failure probabilities: 0.21 * 0.21 = 0.0441.

However, the question asks for the probability that at least one server is operational, which is the opposite of both servers failing.

So, we subtract the probability of both servers failing from 1 (the total probability of all possible outcomes): 1 - 0.0441 = 0.9559. This means there's a 95.59% chance that at least one server will be operational.

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7k+1−1 is divisible by 6 by using inductive hypothesis by putting different values on k.

To prove 7k+1-1 is divisible by 6 for all non-negative integers k we need to follow these steps

By using mathematical induction we need to proof the base case is true. When k=0, we have

7k+1-1 = 7^0+1-1 = 1

1 is divisible by 6 as = 6*0 + 1. Therefore, the base case is true.

Now, lets assume that 7k+1-1 is divisible by 6 for some non-negative integer k.

We will use the assumption to prove that 7(k+1)+1-1 is also divisible by 6.

We have:

7(k+1)+1-1 = 7k+7+1-1 = 7(7k+1)-6

By the inductive hypothesis, 7k+1-1 is divisible by 6, so we can write:

7k+1-1 = 6m

where m is an integer.

Putting these values into the previous equation, we get:

7(k+1)+1-1 = 7(6m+1)-6 = 42m+1

42m+1 is  divisible by 6, as 42m+1 = 6(7m)+1.

Therefore,  7k+1-1 is divisible by 6 for some non-negative integer k, then 7(k+1)+1-1 is also divisible by 6.

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The option that is the  difference of cubes is option A) 125x²¹- 64y³  

125x²¹ - 64y³, can be written as the difference of cubes due to:

a³ - b³ = (a - b) (a² + ab + b²)

Hence 125x²¹ - 64y³ = (5x⁷ - 4y) (25x¹⁴ + 20x⁷y + 16y²)

Note that:

x⁶ + 27y⁹ = (x²)³ + (3y³)³  - sum of cubes

3x⁹ - 64y³ - the first term is not a cube

27x¹⁵ - 9y³ - the second term is not a cube

125x²¹- 64y³ = (5x⁷)³ - (4y)³ - difference of cubes

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Answer:

2.75

Step-by-step explanation:

9.25-6.5=2.75

The number using ones and thousandths is 7 ten, 8 units, 2 tenths, 6 hundredth and 3 thousandth

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

78.263

The place values of the digits in the number are

7 = Ten

8 = Units

2 = Tenth

6 = Hundredth

3 = Thousandth

When the number is expressed using ones and thousandths, we have

7 ten, 8 units, 2 tenths, 6 hundredth and 3 thousandth

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The standard error of the sampling distribution of the sample proportion is given by:

The standard error for the sampling distribution of the sample proportion when n = 50 is approximately 0.059.

SE = sqrt[p(1-p)/n]

where p is the population proportion and n is the sample size.

For n = 20 and p = 0.12, we have:

SE = sqrt[(0.12)(1-0.12)/20] ≈ 0.083

Therefore, the standard error for the sampling distribution of the sample proportion when n = 20 is approximately 0.083.

For n = 50 and p = 0.12, we have:

SE = sqrt[(0.12)(1-0.12)/50] ≈ 0.059

Therefore, the standard error for the sampling distribution of the sample proportion when n = 50 is approximately 0.059.

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The solution of the differential equation of the  initial value problem is:

y(x) = (-4/sqrt(3))cos(x) + (2/sqrt(3))sin(x)

The given differential equation is:

d²y/dx² + y = 0

The characteristic equation is:

r² + 1 = 0

Solving for r, we get:

r = ±i

The general solution of the differential equation is:

y(x) = c1 cos(x) + c2 sin(x)

To find the values of the constants c1 and c2, we use the initial conditions:

y(pi/3) = 0

y'(pi/3) = 2

Substituting x = pi/3, we get:

c1 cos(pi/3) + c2 sin(pi/3) = 0

-c1 sin(pi/3) + c2 cos(pi/3) = 2

Simplifying, we get:

c1/2 + c2(sqrt(3)/2) = 0

-c1(sqrt(3)/2) + c2/2 = 2

Solving this system of equations, we get:

c1 = -4/sqrt(3)

c2 = 4/2sqrt(3)

Therefore, the solution of the initial value problem is:

y(x) = (-4/sqrt(3))cos(x) + (2/sqrt(3))sin(x)

So, the solution satisfies the differential equation and the initial conditions.

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The final expression shows that:

(a) -9 - 7x - 15x2 = (5/6)P1 - (11/6)P2 - (5/6)P3

(b) 6 + 11x + 6x2 = (7/2)P1 - (5/2)P2 + 2P3

(c) 0 = (1/3)P1 - (1/3)P3

(d) 7 + 8x + 9x2 = (-1/2)P1 + (5/2)P2 + (3/2)P3

To express the given vectors as a linear combination of P1, P2, and P3, we need to solve a system of equations.

Let's set up the augmented matrix for each vector and row reduce to find the coefficients:

(a) -9 - 7x - 15x2 = c1(2 + x + 4x2) + c2(1 - x + 3x2) + c3(3 + 2x + 5x2)

The augmented matrix for this system is:

[2 1 3 -9]

[1 -1 2 -7]

[4 3 5 -15]

Row reducing this matrix using elementary row operations, we get:

[1 0 0 -3]

[0 1 0 2]

[0 0 1 -1]

So the coefficients for the linear combination are

c1 = -3, c2 = 2, and c3 = -1:

-9 - 7x - 15x2 = -3(2 + x + 4x2) + 2(1 - x + 3x2) - (3 + 2x + 5x2)

Therefore, -9 - 7x - 15x2 = -7 - 7x + 5x2.

(b) 6 + 11x + 6x2 = c1(2 + x + 4x2) + c2(1 - x + 3x2) + c3(3 + 2x + 5x2)

The augmented matrix for this system is:

[2 1 3 6]

[1 -1 2 11]

[4 3 5 6]

Row reducing this matrix using elementary row operations, we get:

[1 0 0 3]

[0 1 0 2]

[0 0 1 -1]

So the coefficients for the linear combination are

c1 = 3, c2 = 2, and c3 = -1:

6 + 11x + 6x2 = 3(2 + x + 4x2) + 2(1 - x + 3x2) - (3 + 2x + 5x2)

Therefore, 6 + 11x + 6x2 = 7 + 2x + 13x2.

(c) 0 = c1(2 + x + 4x2) + c2(1 - x + 3x2) + c3(3 + 2x + 5x2)

The augmented matrix for this system is:

[2 1 3 0]

[1 -1 2 0]

[4 3 5 0]

Row reducing this matrix using elementary row operations, we get:

[1 0 0 0]

[0 1 0 0]

[0 0 1 0]

So the coefficients for the linear combination are

c1 = 0, c2 = 0, and c3 = 0:

0 = 0(2 + x + 4x2) + 0(1 - x + 3x2) + 0(3 + 2x + 5x2)

Therefore, 0 = 0.

(d) 7 + 8x + 9x2 = c1(2 + x + 4x2) + c2(1 - x + 3x2) + c3(3 + 2x + 5x2)

The augmented matrix for this system is:

[

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This is an algebraic word problem and it has a solution of $120 which is Jonathan's pocket money for each month.

In algebraic word problems, we can represent an unknown number using letters and then carry out basic mathematics operations to get the value of the unknown number.

We shall represent Jonathan's pocket money for each month with the letter x so that;

In July he saved: x - $80 and in August he saved x - $72

Since his savings increased by 20%, then;

x - $72 + x - $80 = (20/100)(x - $80)

2x - $252 = (1/5)(x - $80)

5(2x - $252) = x - $80 {cross multiplication}

10x - $1260 = x - $80

10x - x = $1260 - $80 {collect like terms}

9x = $1080

x = $1080/9 {divide through by 9}

x = $120.

Therefore, the agebraic word problem have a solution of $120 which is Jonathan's pocket money for each month.

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Answer:

8 centimeters : 2 kilometers =

1 centimeter : 1/4 kilometer

After the addition of the given expression (x + 9) + (2x + 3), the resultant answer is 3x + 12.

A finite collection of symbols that are properly created in line with context-dependent criteria is referred to as an expression, sometimes known as a mathematical expression.

An example is the expression x + y, which combines the terms x and y with an addition operator.

In mathematics, there are two different types of expressions: algebraic expressions, which also include variables, and numerical expressions, which solely comprise numbers.

So, we have the expression:

(x + 9) + (2x + 3)

Now, perform the addition as follows:

(x + 9) + (2x + 3)

x + 9 + 2x + 3

3x + 12

Therefore, after the addition of the given expression (x + 9) + (2x + 3), the resultant answer is 3x + 12.

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Complete question:

Find the sum of the given expression.

(x + 9) + (2x + 3)

Answer:

-3

Step-by-step explanation:

 f(x)=2x^2 - 3x - 2

if x = 1

f(1) = 2(1)^2 - 3(1) - 2

= 4 - 3 - 2

= -3

Hope this helps :)

Pls brainliest...

Since angles CAD and CDE are both right angles, and angle CAE is equal to angle CDE, we can conclude that angle CAE is also a right angle. Therefore, AE is tangent to circle C at point A, as required.

A line that touches ellipses or circles only once is said to be tangential. Assuming a line contacts the curve at P, "P" is referred to be the point of tangency.

To prove that AE is tangent to circle C, we need to show that the angle CAE is a right angle.

First, we can use the fact that CD = DE to show that triangle CDE is isosceles, and therefore, angles CED and CDE are equal.

Next, since CA is a radius of circle C, we know that angle CAD is a right angle. Therefore, angle CAE is equal to the sum of angles CAD and DAE.

Using the fact that angles CED and CDE are equal, we can write:

angle DAE = angle CED = angle CDE

Substituting this into the expression for angle CAE, we get:

angle CAE = angle CAD + angle CED + angle CDE

= 90 degrees + angle CED + angle CED

= 90 degrees + 2 angle CED

Since triangle CDE is isosceles, angles CED and CDE are equal. Therefore, we can substitute either one of them for angle CED, and we get:

angle CAE = 90 degrees + 2 angle CED

= 90 degrees + 2 angle CDE

But the sum of angles in a triangle is 180 degrees. Therefore, we can write:

angle CED + angle CDE + angle DCE = 180 degrees

Substituting angle CED for angle CDE, we get:

2 angle CED + angle DCE = 180 degrees

Solving for angle CED, we get:

angle CED = (180 degrees - angle DCE) / 2

Substituting this into our expression for angle CAE, we get:

angle CAE = 90 degrees + 2 angle CED

= 90 degrees + 2 [(180 degrees - angle DCE) / 2]

= 180 degrees - angle DCE

Therefore, angle CAE is equal to the supplement of angle DCE. But since CD = DE, angles CDE and DCE are equal, and therefore, angle CAE is equal to angle CDE.

Since angles CAD and CDE are both right angles, and angle CAE is equal to angle CDE, we can conclude that angle CAE is also a right angle. Therefore, AE is tangent to circle C at point A, as required.

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The probability that X is less than 75 is approximately 0.133, rounded to three decimal places.

To find the probability that a random variable X is less than 75, given that X is normally distributed with µ = 80 and

σ = 4.5, you can follow these steps:

1. Standardize the random variable X using the z-score formula:
  z = (X - µ) / σ
  Here, X = 75, µ = 80, and σ = 4.5.

2. Calculate the z-score:
  z = (75 - 80) / 4.5 = -5 / 4.5 ≈ -1.111

3. Use a standard normal distribution table or calculator to find the probability corresponding to the z-score:
  P(Z < -1.111) ≈ 0.133

So, the probability that X is less than 75 is approximately 0.133, rounded to three decimal places.

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Answer:

We can standardize the normal distribution with µ = 80 and σ = 4.5 by using the z-score formula:

z = (x - µ) / σ

Substituting the values given in the problem, we get:

z = (75 - 80) / 4.5 = -1.1111

Using a standard normal distribution table or calculator, we can find the probability that a standard normal random variable is less than -1.1111, which is approximately 0.132.

Therefore, the probability that x is less than 75 is approximately 0.132, rounded to three decimal places.

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