Now suppose that out of the 10 dishes that the restaurant offers, only 3 of them are vegetarian. If Fiona must select a vegetarian option on Friday, how many ways are there for her to select her lunches?

Answer:

11.69% probability that it will weigh between 523 grams and 534 grams

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 551, \sigma = 20[/tex]

If you pick one fruit at random, what is the probability that it will weigh between 523 grams and 534 grams

This is the pvalue of Z when X = 534 subtracted by the pvalue of Z when X = 523. So

X = 534

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{534 - 551}{20}[/tex]

[tex]Z = -0.85[/tex]

[tex]Z = -0.85[/tex] has a pvalue of 0.1977

X = 523

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{523 - 551}{20}[/tex]

[tex]Z = -1.4[/tex]

[tex]Z = -1.4[/tex] has a pvalue of 0.0808

0.1977 - 0.0808 = 0.1169

11.69% probability that it will weigh between 523 grams and 534 grams

Answer:

[tex]P(523<X<534)=P(\frac{523-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{534-\mu}{\sigma})=P(\frac{523-551}{20}<Z<\frac{534-551}{20})=P(-1.4<z<-0.85)[/tex]

And we can find this probability with this difference:

[tex]P(-1.4<z<-0.85)=P(z<-0.85)-P(z<-1.4)[/tex]

And in order to find these probabilities we can use the table for the normal standard distribution, excel or a calculator.  

[tex]P(-1.4<z<-0.85)=P(z<-0.85)-P(z<-1.4)=0.198-0.0808=0.1172[/tex]

Step-by-step explanation:

Let X the random variable that represent the weigths of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(551,20)[/tex]  

Where [tex]\mu=551[/tex] and [tex]\sigma=20[/tex]

We are interested on this probability

[tex]P(523<X<534)[/tex]

We can solve the problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(523<X<534)=P(\frac{523-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{534-\mu}{\sigma})=P(\frac{523-551}{20}<Z<\frac{534-551}{20})=P(-1.4<z<-0.85)[/tex]

And we can find this probability with this difference:

[tex]P(-1.4<z<-0.85)=P(z<-0.85)-P(z<-1.4)[/tex]

And in order to find these probabilities we can use the table for the normal standard distribution, excel or a calculator.  

[tex]P(-1.4<z<-0.85)=P(z<-0.85)-P(z<-1.4)=0.198-0.0808=0.1172[/tex]

Answer:

x = 2.298

Step-by-step explanation:

we have to use the sin formula which is sin = opp / hyp

sin 50 = x / 3

x = 3 sin 50

x = 2.298

The given options are:

Answer:

Step-by-step explanation:

If Elizabeth has a combined total of 20 apps and movies.

Where:

Number of apps=x

Number of Movies =y

Then:

Their total,

If we subtract x from both sides

x+y-x=-x+20

In Option E

8 apps and 12 movies  add up to 20. Therefore, this could also apply.

Answer:

base : ¼

three points : (¼,1), (1,0), (4,–1)

domain : x>0

range : all real number

asymptote : x=0

Answer:

your answer would be x=0

Step-by-step explanation:

I nee to write a 5-paragraph eassy, so please help me it is base on an article name "Schools in Maryland Allow Elementary Students to Carry Cellphones, by Amanda Lenhart, The Washington Post" here are some pic. I just need help writing two paragraph. I already have my Introduction, Body Paragraph #1 and my Body paragraph #2 just need my Body paragraph #3 and my Conclusion I will give brainlis and 30 pnt.

Prompt:

Write an argumentative essay answering the questions: Should students be allowed to carry cellphones on campus? You must support your claim with evidence from the text. You may also use relevant examples from your own experience, observations, and other readings.

Directions:

Before you begin, read the text below, which presents information about the advantages and disadvantages of carrying a cell phone at school. Use the Student Writing Checklist on the back of this page to plan and write a multi-paragraph essay that addresses the prompt. Use your own words, except when quoting directly from the text.

PLEASE DONT WAST THEM

Answer:

  612

Step-by-step explanation:

Both x and y are linear functions of t, so for each increment of t, the x- and y-coordinates will increment by 8 and 15, respectively. The length of a line segment joining points 8 units in the horizontal direction and 15 units in the vertical direction is given by the Pythagorean theorem as ...

  d = √(8² +15²) = 17

From t=4 to t=40, there are 36 increments in t, so the length of the line segment defined by the given functions is ...

  36×17 = 612 . . . units

Answer:

8-4x= 0 has only 1 solution

Step-by-step explanation:

This can only have one solution because there will only be one value that can set the equation equal to 0 that number is 2. -2 on the other hand will not work because you will get 8+8=0 which does not make sense so you can only have 1 solution.

Hope this helps

Answer:

Probability that the sample will have a mean that is greater than $52,000 is 0.0057.

Step-by-step explanation:

We are given that the population mean for income is $50,000, while the population standard deviation is 25,000.

We select a random sample of 1,000 people.

Let [tex]\bar X[/tex] = sample mean

The z-score probability distribution for sample mean is given by;

               Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]  ~ N(0,1)

where, [tex]\mu[/tex] = population mean = $50,000

            [tex]\sigma[/tex] = population standard deviation = $25,000

            n = sample of people = 1,000

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

So, probability that the sample will have a mean that is greater than $52,000 is given by = P([tex]\bar X[/tex] > $52,000)

  P([tex]\bar X[/tex] > $52,000) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] > [tex]\frac{52,000-50,000}{\frac{25,000}{\sqrt{1,000} } }[/tex] ) = P(Z > 2.53) = 1 - P(Z [tex]\leq[/tex] 2.53)

                                                                    = 1 - 0.9943 = 0.0057

Now, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 2.53 in the z table which has an area of 0.9943.

Therefore, probability that the sample will have a mean that is greater than $52,000 is 0.0057.

The probability that a sample of 1,000 people will have an average income greater than $52,000 is approximately 0.57%, calculated using the standard error and the z-score in a normal distribution.

To calculate the probability that a sample will have a mean that is greater than $52,000, we would use the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough, even if the population distribution itself is not normal. Since we have a large sample size of 1,000 in this case, we can assume that the sampling distribution of the sample mean is approximately normal.

The first step is to determine the standard error of the mean, which is calculated as the population standard deviation divided by the square root of the sample size. In this case:

Standard Error = 25,000 / √(1,000) = 25,000 / 31.62 ≈ 790.57

Next, we calculate the z-score for the sample mean of $52,000:

Z = (Sample Mean - Population Mean) / Standard Error

Z = (52,000 - 50,000) / 790.57 ≈ 2.5262

Using the z-score value, we find the corresponding probability in a standard normal distribution table or use a calculator with a normal distribution function to find the probability that Z is greater than 2.5262. The area to the right of this z-score represents the probability we're seeking.

Since in standard normal tables we typically find the area to the left, we subtract this value from 1:

P(Z > 2.5262) = 1 - P(Z < 2.5262)

If we assume P(Z < 2.5262) = 0.9943 (from standard normal tables), then:

P(Z > 2.5262) = 1 - 0.9943 = 0.0057

Thus, the probability that the sample mean is greater than $52,000 is approximately 0.57%.

Answer:i think is [tex]30r2[/tex]

Step-by-step explanation:

Answer:

30r^3

Step-by-step explanation:

The answer is 30r^3 because first you multiply 5r and 6r and then you add the exponents of them together.

5r * 6r=30r

the exponent of 5r is two and the exponent of 6r is one so you add those together.

hope this helped

Answer:

Step-by-step explanation:

Given that the dresser is in shape of a rectangular prism

Note that, a rectangular prism has a shape of a cuboid

So, area of the rectangular prism is same as area of a cuboid

A = (2LB + 2LH + 2BH)

Where

L is length

B is breadth

H is height

Then, given the dimension of the rectangular prism to be

2ft by 2ft by 6ft

Then, you can assume that,

Length L = 2ft

Breadth B = 2ft

Height H = 6ft.

NOTE: you can take you assumption anyhow, there is no standard, you will get the same answer.

Then,

A = (2LB + 2LH + 2BH)

A = (2×2×2 + 2×2×6 + 2×2×6)

A = (8 + 24 + 24)

A = 56 ft²

The total surface area of the dresser is 56ft²

Answer:

Surface area of the dresser is 56ft²

Step-by-step explanation:

Surface area of a rectangular prism is expressed as S = 2(LW+LH+WH)

L is the length the prism

W is the width

H is the height

If the rectangular prism measures 2 feet by 2 feet by 6 feet

L = 2feet, W = 2feet, height = 6feet

S = 2{(2)(2)+2(6)+2(6)}

S = 2{4+12+12}

S = 2×28

S = 56ft²

Answer:

a) "This is an upper-tail test because the company wants to show the stands will hold 525 pounds (or more) easily."

b) "They will decide the stands are safe when they're not."

c) "They will decide the stands are unsafe when they are in fact safe."

Step-by-step explanation:

a) As the alternative hypothesis Ha is μ>575, the rejection region lays in the upper tail. Then, it is an upper-tail test.

They are testing the claim that the stand supports an average of 575 pounds or more, and are looking for statistical evidence for that claim.

"This is an upper-tail test because the company wants to show the stands will hold 525 pounds (or more) easily."

b)  A Type I error happens when a true null hypothesis is rejected.

This would mean that a stands that doesn't really hold 575 pounds or more has passed the test. The stand would appear more safe than it is.

"They will decide the stands are safe when they're not."

c) A Type II error happens when a false null hypothesis failed to be rejected. In this case, a safe stand, that suports 575 pounds or more, does not pass the test.

"They will decide the stands are unsafe when they are in fact safe."

Final answer:

The hypothesis test in question is an upper-tail test aiming to prove that the stands can support more than 575 pounds. A Type I error would misclassify safe stands as unsafe, while a Type II error would incorrectly certify unsafe stands as safe.

Explanation:

a) This is an upper-tail test because the goal is to show that the metal stand can support more than the standard set weight (>575 pounds). This is vital to ensure that the product holds significantly more weight than the heaviest TVs, thereby providing a safety margin.

b) A Type I error occurs if the inspectors reject the null hypothesis when it is actually true. In this context, it means they will incorrectly deem the stands to be unsafe when they are actually safe.

c) A Type II error occurs if the inspectors fail to reject the null hypothesis when it is false. In this scenario, it means that the inspectors will wrongfully certify the stands as safe when they are, in fact, not capable of holding the standard weight.

Answer:

Null hypothesis:[tex]p_{1} \leq p_{2}[/tex]    

Alternative hypothesis:[tex]p_{1}> p_{2}[/tex]  

[tex]z=\frac{0.117-0.08}{\sqrt{0.0979(1-0.0979)(\frac{1}{230}+\frac{1}{250})}}=1.363[/tex]    

Since is a right tailed test the p value would be:    

[tex]p_v =P(Z>1.363)= 0.0864[/tex]    

Comparing the p value and the significance level given we see that [tex]p_v <\alpha=0.1[/tex] so then we have enough evidence to reject the null hypothesis and then the proportion of defectives for the retailer is significantly higher than the proportion of defectives for the competitor at a 10% of significance level used.

Step-by-step explanation:

Data given

[tex]X_{1}=27[/tex] represent the number of defectives from the retailer

[tex]X_{2}=20[/tex] represent the number of defectives from the competitor

[tex]n_{1}=230[/tex] sample for the retailer

[tex]n_{2}=3377[/tex] sample for the competitor

[tex]p_{1}=\frac{27}{230}=0.117[/tex] represent the proportion of defectives for the retailer

[tex]p_{2}=\frac{20}{250}=0.08[/tex] represent the proportion of defectives for the competitor

[tex]\hat p[/tex] represent the pooled estimate of p

z would represent the statistic (variable of interest)    

[tex]p_v[/tex] represent the value for the test (variable of interest)  

[tex]\alpha=0.1[/tex] significance level given  

System of hypothesis

We need to conduct a hypothesis in order to check if the percentage of defective cellular phones found among his products, ( p1 ), will be no higher than the percentage of defectives found in a competitor's line, ( p2 ), the system of hypothesis would be:    

Null hypothesis:[tex]p_{1} \leq p_{2}[/tex]    

Alternative hypothesis:[tex]p_{1}> p_{2}[/tex]    

The statistic is given by:

[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)  

Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{27+20}{230+250}=0.0979[/tex]  

Calculate the statistic  

Replacing in formula (1) the values obtained we got this:    

[tex]z=\frac{0.117-0.08}{\sqrt{0.0979(1-0.0979)(\frac{1}{230}+\frac{1}{250})}}=1.363[/tex]    

P value

Since is a right tailed test the p value would be:    

[tex]p_v =P(Z>1.363)= 0.0864[/tex]    

Comparing the p value and the significance level given we see that [tex]p_v <\alpha=0.1[/tex] so then we have enough evidence to reject the null hypothesis and then the proportion of defectives for the retailer is significantly higher than the proportion of defectives for the competitor at a 10% of significance level used.

Answer:20%

Step-by-step explanation:

Answer:Triangle A

Step-by-step explanation: It has one right angle.

Answer:

THE ANSWERS A

Step-by-step explanation:

OMG GOOD LUCK!!! BECAUSE IT HAS THE AREA LAYED OUT HAVE A BLESSED DAY!!!

Answer:

[tex]7.3-1.440\frac{1.2}{\sqrt{830}}=7.240[/tex]    

[tex]7.3+1.440\frac{1.2}{\sqrt{830}}=7.360[/tex]    

So on this case the 85% confidence interval would be given by (7.2;7.4)  

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X=7.3[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma =1.2[/tex] represent the population standard deviation

n=830 represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

Since the Confidence is 0.85 or 85%, the value of [tex]\alpha=0.15[/tex] and [tex]\alpha/2 =0.075[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.075,0,1)".And we see that [tex]z_{\alpha/2}=1.440[/tex]

Now we have everything in order to replace into formula (1):

[tex]7.3-1.440\frac{1.2}{\sqrt{830}}=7.240[/tex]    

[tex]7.3+1.440\frac{1.2}{\sqrt{830}}=7.360[/tex]    

So on this case the 85% confidence interval would be given by (7.2;7.4)    

Answer:

10%

Step-by-step explanation:

Percentage increase for any change is calculated by formula

{(Final value - initial value)/initial value} * 100

Given

population in 2010 = 2 million ----------->initial value

population in 2015 = 2.2 million ----------->Final value

[tex]Percentage \ \ increase = {(2.2 - 2)/2} *100= (0.2/2)*100 = 10%[/tex]

Answer:

10%

Step-by-step explanation:the population in 2010 - 2015 is grown though 10%

as a percent incease the number willl incease in increase will increase and increase until the number can increase to 2.2 million or 2 millions and also the population would probably decrease

Answer:

209.467mm³

Step-by-step explanation:

the explanation is in the picture

hope this helps<3

please like and Mark as brainliest

Answer:

73.90%

Step-by-step explanation:

Let Event D=Defective,  D' = Non Defective

Let Event N=New Machine, N' = Old Machine

From the given information:

[tex]P(D|N')=0.25\\P(D|N)=0.09\\P(N)=0.7\\P(N')=0.3[/tex]

We are required to calculate the probability that a widget was manufactured by the new machine given that it is non defective.  

i.e. [tex]P(N|D')[/tex]

[tex]P(D'|N')=1-P(D|N')=1-0.25=0.75\\P(D'|N)=1-P(D|N)=1-0.09=0.91[/tex]

Using Baye's Law of conditional Probability

[tex]P(N|D')=\dfrac{P(D'|N)P(N)}{P(D'|N)P(N)+P(D'|N')P(N')} \\=\dfrac{0.91*0.7}{0.91*0.7+0.75*0.3}\\ =0.73897\\\approx 0.7390[/tex]

Therefore given that a selected widget is non-defective, the probability that it was manufactured by the new machine is 73.9%.

Answer:

A, B and F

Step-by-step explanation:

Area of 2 triangles:

2(½ × 6 × 8)

48 in²

Area of 3 rectangles:

3(18 × 6)

324 in²

Total surface area:

324 + 48

372 in²

Answer:

See answer below

Step-by-step explanation:

Hi there,

To get started, recall the circumference formula.

[tex]C = \pi r^{2}[/tex] where π is the irrational number 3.14159... and r is the radius of the circle. Circumference is like perimeter but for a circle; it is the distance around the boundary.

[tex]C = \pi (4 \ yd)^{2}= 16\pi \ yd^{2} = 50.27 \ yd^{2}[/tex] (approximately)

thanks,

Answer:

C≈25.13yd

Step-by-step explanation:

The circumference of a circle can be found by multiplying pi ( π = 3.14 ) by the diameter of the circle. If a circle has a diameter of 4, its circumference is 3.14*4=12.56. If you know the radius, the diameter is twice as large.

Hope that was helpful.Thank you!!!

Answer:

x=22

Step-by-step explanation:

The value of x is 14.

Arrangement of reference lines or curves used to identify the location of points in space.

Given that Parallel lines q and r are cut by transversals s and t.

The angles formed by the intersection of lines q, s, and t, clockwise from top left, are blank, 53 degrees, blank, 57 degrees,

We need to find the value of x.

The unknown angle between 53 and 57 be u

53+57+u=180

110+u=180

u=70

Now this angle is corresponding to 5x

5x=70

Divide both sides by 5

x=14

Hence, the value of x is 14.

To learn more on Coordinate System click:

https://brainly.com/question/29762404

#SPJ6

Answer:

Check the explanation

Step-by-step explanation:

Scatter plot

x---square feet

y---rent

SolutionA:

Kindly check the attached image below:

THERE EXISTS A WEAK POSITIVE RELATIONSHIP BETWEEN SQUARE FEET AND RENT .

AS SQUARE FEET INCREASES RENT INCREASES

SolutionB:

Rent=0.4932(squarefeet)+992.99

slope=m=0.4932

Y intercept=992.99

SolutionC:

Interpretation of slope:

slope =m=change in rent/change in square foot

For a unit increase in square feet rent increases by 0.4932

Interpretation of y intercept:

when square feet=0 rent is 992.99

Solutiond:

when squarefeet=800

Rent=0.4932(squarefeet)+992.99

=Rent=0.4932(800)+992.99

=1387.55

Answer:

c

Step-by-step explanation:

Answer:

Do your classmates prefer warm or cool places to travel for vacation?

Step-by-step explanation:

Answer: 4 outcomes

Step-by-step explanation:

For two number cubes, the total possible outcomes are:

6 events for the first and 6 events for the second, then the total number of combinations is 6*6 = 36

If the dice are different, the possible outcomes are:

Dice 1 = 3, Dice 2 = 2

Dice 1 = 2, Dice 2 = 3

Dice 1 = 4, Dice 2 = 1

Dice 1 = 1, Dice 2 = 4

Then we have 4 outcomes in the collection for event E.

Answer:

4

Step-by-step explanation:

I did it on college board

Answer:

the first one is 5,2 and the second one is -5,-2 on edge 2020!

Answer:

5,2 and -5,-2

Step-by-step explanation:

Answer:

measure each side of the triangle and make sure it is right and then graph is out.

Step-by-step explanation:

Answer:

[tex]A(30)=40000(1.05)^{30}[/tex]

Step-by-step explanation:

Given that the land was purchased  for $40,000 in 1990, the initial amount/principal =$40,000

Since its value increases by approximately 5% per year, we can model this growth using the compound interest formula:

[tex]A=P(1+r)^n[/tex]

P=$40,000, r=5%=0.05, n=2020-1990=30 Years

Therefore, we have the value of the land in 30 years time to be:

[tex]A=40000(1+0.05)^{30}\\A(30)=40000(1.05)^{30}[/tex]

Since the options are not available, the relationship which represents the value of the land in the year 2020 is:

[tex]A(30)=40000(1.05)^{30}[/tex]

Answer:

We need to conduct a hypothesis in order to check if the true mean for sales is significantly higher than 8000, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 8000[/tex]  

Alternative hypothesis:[tex]\mu > 8000[/tex]  

[tex]z=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]    

[tex]p_v =P(z>2)=0.0228[/tex]  

Step-by-step explanation:

Data given  

[tex]\bar X=8300[/tex] represent the sample mean  for the sales

[tex]\sigma=1200[/tex] represent the population standard deviation

[tex]n=64[/tex] sample size  

[tex]\mu_o =8000[/tex] represent the value that we want to test

z would represent the statistic (variable of interest)  

System of hypothesis

We need to conduct a hypothesis in order to check if the true mean for sales is significantly higher than 8000, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 8000[/tex]  

Alternative hypothesis:[tex]\mu > 8000[/tex]  

The statistic to check this hypothesis is given by:

[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex]  (1)  

Calculate the statistic

[tex]t=\frac{8300-8000}{\frac{1200}{\sqrt{64}}}=2[/tex]    

P-value

Since is a one right tailed test the p value would be:  

[tex]p_v =P(z>2)=0.0228[/tex]  

Answer:

Option C) is the correct answer.

Step-by-step explanation:

We are given the following in the question:

Mean = 192

Sample mean, [tex]\bar{x}[/tex] = 212

Sample size, n = 40

Alpha, α = 0.05

Population standard deviation, σ = 56.5

95% Confidence interval:

[tex]\mu \pm z_{critical}\dfrac{\sigma}{\sqrt{n}}[/tex]

Putting the values, we get,

[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]

[tex]192 \pm 1.96(\dfrac{56.5}{\sqrt{40}} ) = 192 \pm 17.5 = (174.5,209.5)[/tex]

Thus, the correction answer is

Option C)

"The interval that contains 95% of the sample means is 174.5 and 209.5 visits. Because the sample mean is not between these two values, we have support for the results of the May 2011 study."

Answer:

y_g(x) = C1*x^2 + C2*x^-2 +  x^4 / 12

Step-by-step explanation:

Given:-

- The following second order ODE :

                                  x^2y''+xy'-4y=x*(x+x^3)  

Find:-

Find a particular solution of the nonhomogeneous equation    

Solution:-

- First note that the ODE given is a Cauchy Euler ODE. The order of derivative of independent and dependent variables are similar. The general form of Cauchy Euler ODE is:

                           a*x^n y^(n) + b*x^n-1 y^(n-1) + c*x^n-2 y^(n-2) + ... + d*y = f(x)

- We will use the following Auxiliary Equation to find the complementary solutions - Solving Homogeneous part of ODE.

                           am*(m-1) + bm + c = 0

Where, a,b,c are constants such that:

                            x^2y'' + xy' - 4y = 0

                            a = 1 , b = 1 , c = -4

- Solve the Auxiliary equation for (m) as follows:

                            m*(m-1) + m - 4 = 0

                            m^2 - 4 = 0

                            m = +/- 2  ...... ( Real and distinct roots )

- The complementary solutions to the Real and distinct roots from Auxiliary Equation is:

                            yc(x) = y1(x) + y2(x)

                            yc(x) = C1*x^2 + C2*x^-2   .... ( Complementary Solution ).

- Now for the non-homogeneous part of ODE. The function f(x) is defined as:

                            f(x) = x*( x + x^3 ) = x^2 + x^4

- We see that (x^2) term is common to both f(x) and complementary solution yc(x). So when we develop a particular solution, we have to make sure that the solution is independent from complementary solution. If not we multiply the particular solution with (x^n). Where n is the smallest possible integer for which the solution is independent. So in our case ( Using undetermined Coefficient method ) :

                           y_p (x) = A*x^4 + B*x^3 + C*x^2 + D*x + E

- To make the solution independent we multiply y_p by (x^3) where n = 3.

                          y_p (x) = A*x^7 + B*x^6 + C*x^5 + D*x^4 + E*x^3

- Take first and second derivatives of the y_p(x) as follows:

                          y'_p(x) = 7A*x^6 + 6B*x^5 + 5C*x^4 + 4D*x^3 + 3E*x^2

                          y''_p(x) = 42Ax^5 + 30Bx^4 + 20Cx^3 + 12Dx^2 + 6Ex

- Substitute y_p(x) , y'_p(x) and y''_p(x) into the ODE given:

                     42Ax^7 + 30Bx^6 + 20Cx^5 + 12Dx^4 + 6Ex^3

              +       7Ax^7  + 6B*x^6  + 5C*x^5 + 4D*x^4 + 3E*x^3

              -      ( 4Ax^7 +  4B*x^6 +  4C*x^5 +  4D*x^4 + 4E*x^3 )

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

                       45Ax^7  + 32Bx^6 + 21Cx^5 + 12Dx^4 + 5Ex^3  

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

            45Ax^7  + 32Bx^6 + 21Cx^5 + 12Dx^4 + 5Ex^3 = x^2 + x^4

- Compare the coefficients:

                                           A = B = C = E = 0

                                           D = 1 / 12.

The particular solution is:

                                          y_p(x) = x^4 / 12

- The general solution is as follows:

                                     y_g(x) = yc(x) + y_p(x)

                                     y_g(x) = C1*x^2 + C2*x^-2 +  x^4 / 12

Answer:

The particular solution to the differential equation

x²y'' + xy' - 4y = x(x + x³)

is

y_p = (1/12)x^4 - x²/2 - x/3

Step-by-step explanation:

Given the differential equation:

x²y'' + xy' - 4y = x(x + x³)...............(1)

First, we solve the homogeneous part of (1)

x²y'' + xy' - 4y = 0...........................(2)

Let x = e^z

=>z = lnx

Let D = d/dz

dz/dx = (1/x)

dy/dx = (dy/dz).(dz/dx)

= (1/x)(dy/dz)

dy/dz = xdy/dx = xy' = Dy

d²y/dx² = (-1/x²)(dy/dz) + (1/x)(d²y/dx²)(dz/dx)

= (1/x²)(d²y/dx² - dy/dz) = (1/x²)(D² - D)y

Using these, (2) becomes

(D² - D)y + Dy - 4y = 0

(D² - 4)y = 0

The auxiliary equation is

m² - 4 = 0

(m - 2)(m + 2) = 0

m1 = 2, m2 = -2

The complementary function is

y = C1e^(2z) + C2e^(-2z)

But z = lnx

y_c = C1x² + C2/x² ...........................(3)

Now we solve (1) using the method of undetermined coefficients.

The nonhomogeneous part is

x(x + x³) = x² + x^4

So, we assume a particular solution of the form

y_p = Ax^4 + Bx³ + Cx² + Dx + E

y'_p = 4Ax³ + 3Bx² + 2Cx + D

y''_p = 12Ax² + 6Bx + 2C

Using these in (1)

x²y''_p + xy'_p - 4y_p = x²(12Ax² + 6Bx + 2C) + x(4Ax³ + 3Bx² + 2Cx + D) - 4(Ax^4 + Bx³ + Cx² + Dx + E)

= x² + x^4

12Ax^4 + 6Bx³ + 2Cx + 4Ax^4 + 3Bx³ + 2Cx² + Dx - 4Ax^4 - 4Bx³ - 4Cx² - 4Dx - 4E = x² + x^4

Comparing the coefficients of various powers of x, we have

12A + 4A - 4A = 1

12A = 1

=> A = 1/12

6B + 3B - 4B = 0

5B = 0

=> B = 0

2C - 4C = 1

-2C = 1

=> C = -1/2

2C + D - 4D = 0

2C - 3D = 0

2C = 3D

2(-1/2) = 3D

=> D = -1/3

-4E = 0

=> E = 0

(A, B, C, D, E) = (1/12, 0, -1/2, -1/3, 0)

y_p = Ax^4 + Bx³ + Cx² + Dx + E

= (1/12)x^4 - (1/2)x² - (1/3)x

The general solution is

y = y_c + y_p

= C1x² + C2/x² + (1/12)x^4 - x²/2 - x/3

Answer:

0.95

Step-by-step explanation:

Let's convert this into a fraction that has a denominator of 100. In order to do so, we need to multiply the numerator and denominator both by 5 (because 20 * 5 = 100):

[tex]\frac{19}{20}* \frac{5}{5} =\frac{95}{100}[/tex]

So, we have 95/100, which is technically 95 divided by 100. Whenever we divide something by a power of 10, like [tex]10^n[/tex], where n is any integer, we move the decimal point of the dividend n places to the left.

Here, we have 100, which is [tex]10^2[/tex], so we want to move the decimal point of 95 (which equals 95.0) 2 places to the left:

0.95

Hope this helps!

Answer:

confidence interval using a  two sample t test between percents

Step-by-step explanation:

confidence interval using a  two sample t test between percents This can be used to compare percentages drawn from two independent samples in this case employees. It is used to compare two sub groups from a single sample example the population on the planet