Answer:
a) [tex]P(E) = \frac{6}{8}[/tex]
b) [tex]P(F|E) = \frac{5}{7}[/tex]
c) [tex]P(E \cap F) = \frac{15}{28}[/tex]
Step-by-step explanation:
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
We have that:
8 vehicles, of which 6 are vans.
a.) Let E denote the event that the first vehicle assigned is a van. What is P(E) ?
8 vehicles, of which 6 are vans.
So
[tex]P(E) = \frac{6}{8}[/tex]
b.) Let F denote the probability that the second vehicle assigned is a van. What is P(F|E)?
P(F|E) is the probability that the second vehicle assigned is a van, given that the first one was.
In this case, there are 7 vehicles, of which 5 are vans. So
[tex]P(F|E) = \frac{5}{7}[/tex]
c.) Use the results of parts(a) and (b) to calculate P(E and F)
[tex]P(F|E) = \frac{P(E \cap F)}{P(E)}[/tex]
[tex]P(E \cap F) = P(F|E)P(E)[/tex]
[tex]P(E \cap F) = \frac{6}{8}\frac{5}{7}[/tex]
[tex]P(E \cap F) = \frac{15}{28}[/tex]
The probability of the first vehicle assigned being a van is 3/4, and the conditional probability of the second vehicle being a van given the first was a van is 5/7. The probability that both vehicles assigned are vans is 15/28.
Explanation:The subject of this question is probability, a topic in Mathematics. We are asked to find the probability of a certain event occurring under certain conditions.
P(E), the probability that the first vehicle assigned is a van. Since there are 6 vans out of 8 vehicles, the probability is 6/8 or 3/4.P(F|E), the conditional probability that the second vehicle assigned is a van given that the first vehicle assigned was a van. After the first van has been assigned, there are now 5 vans left out of 7 vehicles. Therefore, the probability is 5/7.Finally, to find P(E and F), which is the probability that both vehicles assigned are vans, we multiply the probabilities we found in parts (a) and (b), so (3/4) * (5/7) = 15/28.Learn more about Probability here:https://brainly.com/question/22962752
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A market surveyor wishes to know how many energy drinks teenagers drink each week. They want to construct a 85% confidence interval for the mean and are assuming that the population standard deviation for the number of energy drinks consumed each week is 1.2. The study found that for a sample of 830 teenagers the mean number of energy drinks consumed per week is 7.3. Construct the desired confidence interval. Round your answers to one decimal place.
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)
You have inherited land that was purchased for $40,000 in 1990. The value of the land
increased by
approximately 5% per year. Which relationship would represent the value of the land in the
year 2020
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]
Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 pounds per square inch (psi).
Write the equation that relates pressure, P, to volume, V.
If the pressure increases to 330 psi, how much air is held in Braydon’s tank?
Answer: 4 litres of air is held in Braydon’s tank.
Step-by-step explanation:
The law relating pressure to volume is the Boyle's law. It states that the volume of a given mass of gas is inversely proportional to its pressure as long as temperature remains constant. It is expressed as
P1V1 = P2V2
Where
P1 and P2 are the initial and final pressures of the gas.
V1 and V2 are the initial and final volumes of the gas.
From the information given,
V1 = 6 litres
P = 220 psi
P2 = 330 psi
Therefore,
6 × 220 = 330V2
V2 = 1320/330 = 4 litres
Answer:
V=1320/p
the tank holds 4 liters
Step-by-step explanation:
The radius of a circle is 4 yards. What is its circumference?
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!!!
Lines q and r are parallel.
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, blank, blank; formed by s and r are blank, 5 x degrees, blank, blank.
What is the value of x?
14
22
53
70
Answer:
x=22
Step-by-step explanation:
The value of x is 14.
What is Coordinate System?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.
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Can someone please answer these for me?
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
Choose all that are correct. Choosing the brainliest.
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
A recent study reported that 18- to 24-year-olds average 192 restaurant visits per year. Assume that the standard deviation for number of visits per year for this age group is 56.5. To validate these findings, a random sample of forty 18- to 24-year-olds was selected and found to average 212 restaurant visits per year. Which of the following statements is correct
A.)The interval that contains 95% of the sample means is 170.3 and 213.7 visits. Because the sample mean is between these two values, we have support for the results of the May 2011 study.
B.)The interval that contains 95% of the sample means is 170.3 and 213.7 visits. Because the sample mean is between these two values, we do not have support for the results of the May 2011 study.
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.
D.)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 do not have support for the results of the May 2011 study.
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."
Suppose that the population mean for income is $50,000, while the population standard deviation is 25,000. If we select a random sample of 1,000 people, what is the probability that sample will have a mean that is greater than $52,000?
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%.
B is located at (−5, 2).
If B is reflected across the y-axis, the coordinates will be .
If B is reflected across the x-axis, the coordinates will be .
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:
Paulo works at the United Nations. He researched what percent of the world's population lives on each continent. He surveys a sample of employees at the United Nations about their continent of origin to see if the distribution in the sample agrees with the percentages he researched.
Which of these inference procedures is most appropriate?
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
the equation 8- 4X=0 has blank real solutions
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
Directions for questions 4 & 5: We selected a random sample of 100 StatCrunchU students, 67 females and 33 males, and analyzed their responses to the question, "What is the total amount (in dollars) of credit card debt you have accrued to date?" With more than 30 in each random and independent sample, conditions are met for modeling the distribution of differences in sample means using a T-model. Therefore, we will proceed with finding a confidence interval to estimate the gender difference in credit card debt for StatCrunchU students. Summary statistics for CC Debt: Group by: Gender Gender Mean Std. dev. n Female 2577.75 1916.29 67 Male 3809.42 2379.47 33 Use StatCrunch to find the 95% confidence interval estimating the difference µ1 – µ2, where µ1 is the mean credit card debt for all female StatCrunchU students and µ2 is the mean credit card debt for all male StatCrunchU students. (directions) Since the numbers are dollars, round to two decimal places when you enter your answer. Flag this Question Question 42 pts The lower limit on the confidence interval is
The lower limit of the 95% confidence interval for the difference in mean credit card debt between female and male students can be calculated by formula using sample means, standard deviations, sample sizes, and accounting for the t-value associated with 95% confidence.
Explanation:To calculate the 95% confidence interval for the difference between the mean credit card debt of female and male StatCrunchU students, we use the given information: µ1 (mean credit card debt of females) = 2577.75, µ2 (mean credit card debt of males) = 3809.42, std. dev. of females = 1916.29, std. dev. of males = 2379.47, number of females = 67, number of males = 33.
To calculate the confidence interval, we will use the t-model formula for confidence intervals for difference in means, which is:
(µ1-µ2) ± t*(sqrt([std. dev.1/sqrt(n1)] + [std. dev.2/sqrt(n2)]))
After plugging in the objective values, we would get the confidence range. The lower limit will be (µ1-µ2) - t*(sqrt([std. dev.1/sqrt(n1)] + [std. dev.2/sqrt(n2)])).
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The population in the city of Millstone was approximately 2 million in 2010 and 2.2 million in 2015. What is the percent increase from 2010 to 2015
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
A particular fruit's weights are normally distributed, with a mean of 551 grams and a standard deviation of 20 grams. If you pick one fruit at random, what is the probability that it will weigh between 523 grams and 534 grams
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]
For problems 13-17 find a particular solution of the nonhomogeneous equation, given that the functions y1(x) and y2(x) are linearly independent solutions of the corresponding homogeneous equation. x^2y''+xy'-4y=x(x+x^3)
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
Based on these tables, what can you determine about the students in this class? Check all that apply.
There are 35 students in the class.
11 of the students are boys who have summer birthdays.
19 of the students are boys.
There is not enough information shown to determine how many girls have summer birthdays.
Answer:
Step-by-step explanation:
It’s 1 3,and 4
Answer:
1,3,4
Step-by-step explanation:
I NEED HELP DUE IN 2 MINS!!
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!!!
Consider rolling two number cubes, each of which has its faces numbered
from 1 to 6. The cubes will be rolled and the sum of the numbers landing
face up will be recorded. Let the event E represent the event of rolling a sum
of 5. How many outcomes are in the collection for event E?
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
A grocery store has an average sales of $8000 per day. The store introduced several advertising campaigns in order to increase sales. To determine whether or not the advertising campaigns have been effective in increasing sales, a sample of 64 days of sales was selected. It was found that the average was $8300 per day. From past information, it is known that the standard deviation of the population is $1200. The correct null hypothesis for this problem is
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]
Find the arc length of the following curve on the given interval. x equals 8 t minus 7 comma y equals 15 t minus 6x=8t−7, y=15t−6, 0 less than or equals t less than or equals 40≤t≤4 The length of the curve is nothing. (Type an integer or a fraction.)
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
part of a $3,600 bonus was invested at 9% annual simple interest. The rest was invested at %8 annual simple interest. The total interest at the end of one year was $312. How much was invested in the %9 account?
Answer:
$2400
Step-by-step explanation:
Let x represent the amount invested at 9%. (3600-x) will be the amount invested at 8%. The total interest earned is then ...
312 = 0.09x +0.08(3600 -x)
24 = .01x . . . . subtract 288, simplify
2400 = x . . . . divide by .01
$2400 was invested in the 9% account.
Item 16
A dresser in the shape of a rectangular prism measures 2 feet by 2 feet by 6 feet. What is the surface area of the dresser?
square feet
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
what is 81,007-26,318?
Answer:
54689
Step-by-step explanation:
In a certain region, 15% of people over the age of 50 didn’t graduate from high school. We would like to know if this percentage is the same among the 25-50 year age group. What is the minimum number of 25-50 year old people who must be surveyed in order to estimate the proportion of non-grads to within 6% of the true parameter with 99% confidence?
Answer:
235 people
Step-by-step explanation:
Given:
P' = 15% = 0.15
1 - P' = 1 - 0.15 = 0.85
At 99% confidence leve, Z will be:
[tex] \alpha [/tex] = 1 - 99%
= 1 - 0.99 = 0.01
[tex] \alpha /2 = \frac{0.01}{2} = 0.005 [/tex]
[tex] Z\alpha/2 = 0.005 [/tex]
Z0.005 = 2.576
For the minimum number of 25-50 year old people who must be surveyed in order to estimate the proportion of non-grads to within 6%, we have:
Margin of error, E = 6% = 0.06
sample size = n = [tex] (\frac{Z\alpha /2}{E})^2 * P* (1 - P) [/tex]
[tex] = (\frac{2.576}{0.06}) ^2 * 0.15 * 0.85 [/tex]
= 235.02 ≈ 235
A number of 235 people between 25-30 years should be surveyed .
Answer:
n = 236
Step-by-step explanation:
Solution:-
- The proportion of people over the age of 50 who didn't graduate from high school are, p = 0.15 - ( 15 % )
- We are to evaluate the minimum sample size " n " from the age group of 25-50 year in order to estimate the proportion of non-grads within a standard error E = 6% of the true proportion p within 99% confidence.
- The minimum required sample size " n " for the standard error " E " for the original proportion p relation is given below:
[tex]n = \frac{(Z_\alpha_/_2)^2 * p* ( 1 - p )}{E^2}[/tex]
- The critical value of standard normal is a function of significance level ( α ), evaluated as follows:
significance level ( α ) = ( 1 - CI/100 )
= ( 1 - 99/100 )
= 0.01
- The Z-critical value is defined as such:
P ( Z < Z-critical ) = α / 2
P ( Z < Z-critical ) = 0.01 / 2 = 0.005
Z-critical = Z_α/2 = 2.58
- Therefore the required sample size " n " is computed as follows:
[tex]n = \frac{(2.58)^2 * 0.15* ( 1 - 0.15 )}{0.06^2}\\\\n = \frac{6.6564 * 0.1275}{0.0036}\\\\n = \frac{0.848691}{0.0036}\\\\n = 235.7475\\[/tex]
Answer: The minimum sample size would be next whole number integer, n = 236.
Find the volume of the cone
Answer:
209.467mm³
Step-by-step explanation:
the explanation is in the picture
hope this helps<3
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For which survey is a sample not necessary?
What percentage of Colorado residents support planting more trees in the community?
Which electronic gadget will be the most popular among middle school students this year?
Do your classmates prefer warm or cool places to travel for vacation?
How many car accidents involve air bag malfunctions?
Answer:
c
Step-by-step explanation:
Answer:
Do your classmates prefer warm or cool places to travel for vacation?
Step-by-step explanation:
How do I do a triangles heirarchy
Answer:
measure each side of the triangle and make sure it is right and then graph is out.
Step-by-step explanation:
1 poir
Elizabeth's tablet has a combined total of 20 apps and movies. Let x
represent the number of apps and y represent the number of movies.
Which of the following could represent the number of apps and movies on
Elizabeth's tablet? Select all that apply.
The given options are:
(A)x+y = 20 (B)7 apps and 14 movies (C)x-y= 20 (D)y=-x+ 20 (E)8 apps and 12 movies (F)xy= 20Answer:
(A)x+y = 20 (D)y=-x+ 20 (E)8 apps and 12 moviesStep-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,
x+y=20 (Option A)If we subtract x from both sides
x+y-x=-x+20
y=-x+20 (Option D)In Option E
8 apps and 12 movies add up to 20. Therefore, this could also apply.
An experiment is carried out 400 times the possible outcomes are void fail and success if the frequency of void is 96 and the relative frequency is 0.24 then how much is the frequency of fail and success
the frequency of success is 244.
part b's answer is 240.
A relative frequency distribution shows the proportion of the total number of observations associated with each value or class of values and is related to a probability distribution, which is extensively used in statistics.
Relative frequency can be defined as the number of times an event occurs divided by the total number of events occurring in a given scenario. The relative frequency formula is given as:
Relative Frequency = Subgroup frequency/ Total frequency.
1. Frequency of success
=400 - 96 - 60
=244
relative frequency of fail
=60/400= 0.15
Relative Frequency of success
=1-0.15 - 0.24
=0.61
Learn more about relative frequency here:
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