MATHEMATICS COLLEGE

Find an explicit solution to the Bernoulli equation. y'-1/3 y = 1/3 xe^xln(x)y^-2

Answers

Answer 1

Answer:

$y'-\frac13y=\frac13xe^x\ln x\,y^(-2)$

Divide both sides by $\frac13y^(-2)(x)$ :

$3y^2y'-y^3=xe^x\ln x$

Substitute $v(x)=y(x)^3$ , so that $v'(x)=3y(x)^2y'(x)$ .

$v'-v=xe^x\ln x$

Multiply both sides by $e^(-x)$ :

$e^(-x)v'-e^(-x)v=x\ln x$

The left side can be condensed into the derivative of a product.

$(e^(-x)v)'=x\ln x$

Integrate both sides to get

$e^(-x)v=\frac12x^2\ln x-\frac14x^2+C$

Solve for $v(x)$ :

$v=\frac12x^2e^x\ln x-\frac14x^2e^x+Ce^x$

Solve for $y(x)$ :

$y^3=\frac12x^2e^x\ln x-\frac14x^2e^x+Ce^x$

$\implies\boxed{y(x)=\sqrt[3]{\frac14x^2e^x(2\ln x-1)+Ce^x}}$

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The upcoming championship high school football game is a big deal in your little town. The problem is, it is being played in the next biggest town, which is two hours away! To get as many people as you can to attend the game, you decide to come up with a ride-sharing app, but you want to be sure it will be used before you put all the time in to creating it. You determine that if more than three students share a ride, on average, you will create the app.You conduct simple random sampling of 20 students in a school with a population of 300 students to determine how many students are in each ride-share (carpool) on the way to school every day to get a good idea of who would use the app. The following data are collected:6 5 5 5 3 2 3 6 2 25 4 3 3 4 2 5 3 4 5Construct a 95% confidence interval for the mean number of students who share a ride to school, and interpret the results.Part A: State the parameter and check the conditions.Part B: Construct the confidence interval. Be sure to show all your work, including the degrees of freedom, critical value, sample statistics, and an explanation of your process.Part C: Interpret the meaning of the confidence interval.Part D: Use your findings to explain whether you should develop the ride-share app for the football game.

Answers

The parameter used in the probability is the average number of students represented by u.

How to calculate the probability?

The confidence interval based on the information will be:

= 3.85 - 2.09(1.348 / ✓20)

= 3.22

Also, 3.85 + 2.09(1.348 / ✓20) = 4.48

The confidence interval simply means that one is 95% confident that the true mean is between 3.22 and 4.48.

Learn more about probability on:

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Answer:

a) The parameter of interest on this case is $\mu$ who represent the average number of students in order to create an app.

b) $3.85 - 2.09 (1.348)/(√(20))=3.22$

$3.85 + 2.09 (1.348)/(√(20))=4.48$

The 95% confidence interval is given by (3.22;4.48)

c) For this case we have 95% of confidence that the true mean for the average number of students in order to create an app is between 3,22 and 4.48.

d) We have the following criteria in order to decide: "You determine that if more than three students share a ride, on average, you will create the app"

So then since the lower limti for the confidence interval is higher than 3 we can conclude that at 5% of significance we have more than 3 students share a ride so then makes sense create the app.

Step-by-step explanation:

Part a

The parameter of interest on this case is $\mu$ who represent the average number of students in order to create an app.

Part b

Data: 6 5 5 5 3 2 3 6 2 2 5 4 3 3 4 2 5 3 4 5

n=20 represent the sample size

$\bar X$ represent the sample mean

$s$ represent the sample standard deviation

m represent the margin of error

Confidence =95% or 0.95

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".

Calculate the mean and standard deviation for the sample

On this case we need to find the sample standard deviation with the following formula:

$s=sqrt{(\sum_(i=1)^20 (x_i -\bar x)^2)/(n-1)}$

And in order to find the sample mean we just need to use this formula:

$\bar x =(\sum_(i=1)^(20) x_i)/(n)$

The sample mean obtained on this case is $\bar x= 3.85$ and the deviation s=1.348

Calculate the critical value tc

In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 95% of confidence, our significance level would be given by $\alpha=1-0.95=0.05$ and $\alpha/2 =0.025$ . The degrees of freedom are given by:

$df=n-1=20-1=19$

We can find the critical values in excel using the following formulas:

"=T.INV(0.025,19)" for $t_(\alpha/2)=-2.09$

"=T.INV(1-0.025,19)" for $t_(1-\alpha/2)=2.09$

The critical value $tc=\pm 2.09$

Calculate the confidence interval

The interval for the mean is given by this formula:

$\bar X \pm t_(c) (s)/(√(n))$

And calculating the limits we got:

$3.85 - 2.09 (1.348)/(√(20))=3.22$

$3.85 + 2.09 (1.348)/(√(20))=4.48$

The 95% confidence interval is given by (3.22;4.48)

Part c

For this case we have 95% of confidence that the true mean for the average number of students in order to create an app is between 3.22 and 4.48.

Part d

We have the following criteria in order to decide: "You determine that if more than three students share a ride, on average, you will create the app"

So then since the lower limti for the confidence interval is higher than 3 we can conclude that at 5% of significance we have more than 3 students share a ride so then makes sense create the app.

What is the value of 8x + 2x when x = 6?

Answers

The value of 8x + 2x when x = 6 is 60

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be A = 8x + 2x

The value of x = 6

Substituting the value of x in the equation , we get

A = 8x + 2x

A = 8 ( 6 ) + 2 ( 6 )

A = 48 + 12

A = 60

Therefore , the value of A = 60

Hence , the value of 8x + 2x when x = 6 is 60

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#SPJ2

8x + 2x

Plug the variable in.

8×6 + 2×6

48+12 = 60

If $11000 yields $1100 over 5 years, what is the interest rate?

Answers

Answer:

1.92449%

Step-by-step explanation:

Simple Interest Rate Formula: A = P(1 + r)ⁿ

Step 1: Define variables

Principle amount P = 11000

Total amount A = 11000 + 1100 = 12100

Years n = 5

Step 2: Substitute and Evaluate for r

12100 = 11000(1 + r)⁵

11/10 = (1 + r)⁵

1.01924 = 1 + r

r = 0.019245

Step 3: Convert to percentage

0.019245 × 100 = 1.92449%

A university financial aid office polled a random sample of 528 male undergraduate students and 651 female undergraduate students. Each of the students was asked whether or not they were employed during the previous summer. 295 of the male students and 463 of the female students said that they had worked during the previous summer. Give a 80% confidence interval for the difference between the proportions of male and female students who were employed during the summer.A) Find the values of the two sample proportions (round to three decimals)B) Find the critical value that should be used in constructing the confidence interval.C) Find the value of the standard error. Round your answer to three decimal places.D) Construct the 95% confidence interval. Round your answers to three decimal places.

Answers

Answer:

Step-by-step explanation:

A dealer advertises that a certain brand of tire averages 50,000 miles of use before needing to be replaced. Testing this claim, an investigator finds that a sample of 19 randomly selected tires from this dealer had an average lifespan of 48,700 miles with a standard deviation of 4,500 miles. Assume tire life spans are approximately normally distributed. At the α = 0.05 level, find if the investigator can prove the dealer is exaggerating the average lifespan of their tires by performing the hypothesis test.

Answers

Answer:

100k miles

Step-by-step explanation:

Working together, two secretaries can stuff the envelopes for a political fund-raising letter in 4 hours. Working alone, it takes the slower worker 6 hours longer to do the job than the faster worker. How long does it take each to do the job alone

Answers

Answer: The faster one needs 6 hours, the slower one needs 12 hours.

Step-by-step explanation:

Let's define Sa and Sb as the times that each worker needs to stuff the envelopes for a political fundraising letter.

Sa is the faster one

Sb is the slower one.

Let's define 1 as a complete task.

Then:

when they both work together, they need 4 hours:

(1/Sa + 1/Sb)*4h = 1.

The slower one needs 6 more hours than the faster one:

Sb = (Sa + 6h).

We can replace this in the first equation and get:

(1/Sa + 1/(Sa + 6h))*4h = 1.

let's solve this for Sa.

1/Sa + 1/(Sa + 6h) = 1/4h.

(Sa + 6h) + Sa = Sa*(Sa + 6h)/4h.

2*Sa + 6h = Sa^2/4h + Sa*(6/4)

Then we have a quadratic equation:

(1/4h)*Sa^2 - (2/4)*Sa - 6h = 0h

(0.25*1/h)*Sa^2 - 0.5*Sa - 6h = 0h

The solutions come from the Bhaskara equation:

$Sa = (0.5 +- √((0.5)^2 - 4*0.25h*(-6)) )/(2*0.25* 1/h) = (0.5 +- 2.5)/(0.5) h$

Then we have two solutions:

Sa = ((0.5 + 2.5)/0.5 )h = 6h.

Sb = ( (0.5 - 2.5)/0.5) = -4h

The one that makes sense is the positive option (the negative one has no physical meaning in this situation)

Then the faster worker needs 6 hours to stuff all the envelopes.

And the slower one needs 6h + 6h = 12hours to stuff all the envelopes.

So when they work together, the combined rate is:

(1/6h + 1/12h) = (2/12h + 1/12h) = (3/12h) = (1/4h)

So working together they need 4 hours to stuff all the envelopes.

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