Answer:
the minimum sample size n = 11.03
Step-by-step explanation:
Given that:
approximate value of the population standard deviation [tex]\sigma[/tex] = 49
level of significance ∝ = 0.01
population mean = 38
the minimum sample size n = ?
The minimum sample size required can be determined by calculating the margin of error which can be re[resented by the equation ;
Margin of error = [tex]Z_{ \alpha /2}} \times \dfrac{\sigma}{\sqrt{n}}[/tex]
[tex]38 = \dfrac{2.576 \times 49}{\sqrt{n}}[/tex]
[tex]\sqrt{n} = \dfrac{2.576 \times 49}{38}[/tex]
[tex]\sqrt{n} = \dfrac{126.224}{38}[/tex]
[tex]\sqrt{n} = 3.321684211[/tex]
[tex]n= (3.321684211)^2[/tex]
n ≅ 11.03
Thus; the minimum sample size n = 11.03
Calculate 15% of 36, to one decimal place
Answer:
5.4
Step-by-step explanation:
15% = 15/100 = 0.15
0.15 × 36 = 5.4
Answer = 5.4
Answer:
5.4
Step-by-step explanation:
→ To find one percentage of another, I first multiply both numbers together then divide by 100
15 × 36 = 540
→ Divide by 100
540 ÷ 100 = 5.4
A football coach compared the yards per game of two of his running backs over the course of 10 games. Based on the data represented in the box plots, which football player had greater success during the 10 games? Nasir was more successful because he had the greatest number of yards in one game. Aaron was more successful because he had the greater total spread. Nasir was more successful because he had the greater measure of center. Aaron was more successful because he had an outlier.
Answer:
C Nasir was more successful because he had the greater measure of center
Step-by-step explanation:
Answer: C
Step-by-step explanation:
What is the area W. Please help geometry
Answer:
C. 9π
Step-by-step explanation:
r=x-1
A=πr^2
--------
(x-1)^2+(2x-4)^2= (x-1+2)^2x^2-2x+1+4x^2- 16x+16= x^2+2x+14x^2 -20x +16=0x^2-5x+4=0(x-1)(x-4)=0x=1 - not possible as radius can't be zerox=4 is the solution--------
A= πr^2= π*(4-1)^2= 9π, option C.
Solve the system of equations for the variables: 5x+2y=13 x+2y=9
Answer:
x = 1
y = 4
Step-by-step explanation:
5x + 2y = 13
x + 2y = 9
Add both equations.
6x + 4y = 22
Solve for x.
6x = 22 - 4y
x = 22/6 - 4/6y
Put x as 22/6 - 4/6y in the second equation and solve for y.
22/6 - 4/6y + 2y = 9
-4/6y + 2y = 9 - 22/6
4/3y = 16/3
y = 16/3 × 3/4
y = 48/12
y = 4
Put y as 4 in the first equation and solve for x.
5x + 2(4) = 13
5x + 8 = 13
5x = 13 - 8
5x = 5
x = 5/5
x = 1
Answer:
x = 1, y = 4
Step-by-step explanation:
5x+2y = 13
x+2y = 9
Subtracting both equations
=> 5x+2y-x-2y = 13-9
=> 4x = 4
=> x = 1
Now, Putting x = 1 in the first equation
=> 5(1)+2y = 13
=> 2y = 13-5
=> 2y = 8
=> y = 4
Assume that you earned an 87 on Exam 1 in this course. The class had an average of 78 (s=8.69). How many people earned a score below your score? (in percentage)
Answer:
85%
Step-by-step explanation:
Given data
Exam score earned by student = 87
class average = 78
s = 8.69
Calculate the percentage of people that earned a score below your score
P ( z < 1.04 ) = 0.8508 = 85%
Note : Z ( z score ) = (exam score - class average) / s
= (87 - 78) / 8.69 = 1.04
Find the measure of each angle: Supplementary angles with measures (2x+3)° and (3x+2)°.
Answer: 73 degrees and 107 degrees.
Step-by-step explanation:
The total of supplementary angles are 180 degrees. So you add 2x+3 and 3x+2. Then you get 5x+5=180.
Subtract 5 from both sides. Now the equation is 5x=175.
Divide 5 on each side. x=35
Replace x with 35 in the equations. The angles are 73 and 107.
They both add up to 180 degrees so it is correct.
6 drinks make a six-pack. Marty has 23 drinks. name the mixed number of 6- packs Marty has.
━━━━━━━☆☆━━━━━━━
▹ Answer
[tex]3\frac{5}{6}[/tex]
▹ Step-by-Step Explanation
6 drinks = 6 Pack (one pack)
23 ÷ 6 = [tex]\frac{23}{6} = 3\frac{5}{6}[/tex]
Mixed number - [tex]3\frac{5}{6}[/tex]
Hope this helps!
- CloutAnswers ❁
Brainliest is greatly appreciated!
━━━━━━━☆☆━━━━━━━
Answer: It's 3 5/6
but isn't this 4th-grade work?
Step-by-step explanation:
The Nielsen Company reported that U.S. residents aged 18 to 24 years spend an average of 32.5 hours per month using the Internet on a computer.13 You wonder if this it true for students at your large university because so many students use their smartphones to access the Internet. You collect an SRS of n=75 students and obtain ¯x=28.5 hours with s=23.1 hours.
Required:
a. Report the 95% confidence interval for μ, the average number of hours per month that students at your university use the Internet on a computer.
b. Use this interval to test whether the average time for students at your university is different from the average reported by Nielsen. Use the 5% significance level. Summarize your results.
Answer:
a) [tex]28.5-1.993\frac{23.1}{\sqrt{75}}=23.18[/tex]
[tex]28.5+1.993\frac{23.1}{\sqrt{75}}=33.82[/tex]
b) For this case since the value 32.5 is in the confidence interval obtained then we can't conclude that the statement by Nielsen is wrong
Step-by-step explanation:
Information given
[tex]\bar X=28.5[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=23.1 represent the sample standard deviation
n=75 represent the sample size
Part a
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)
The degrees of freedom are given by:
[tex]df=n-1=75-1=74[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex] and the critical value would be [tex]t_{\alpha/2}=1.993[/tex]
Now we have everything in order to replace into formula (1):
[tex]28.5-1.993\frac{23.1}{\sqrt{75}}=23.18[/tex]
[tex]28.5+1.993\frac{23.1}{\sqrt{75}}=33.82[/tex]
Part b
For this case since the value 32.5 is in the confidence interval obtained then we can't conclude that the statement by Nielsen is wrong
38â% of women consider themselves fans of professional baseball. You randomly select six women and ask each if she considers herself a fan of professional baseball. Complete partsâ (a) throughâ (d) below.(a) Find the mean of the binomial distribution.
μequals= ( ) (Round to the nearest tenth asâ needed.) â
(b) Find the variance of the binomial distribution.
sigmasquared= ( ) â(Round to the nearest tenth asâ needed.)
â(c) Find the standard deviation of the binomial distribution.
sigma = ( ) (Round to the nearest tenth asâ needed.) â
(d) Interpret the results in the context of theâ real-life situation.
Onâ average ( ) out of 6 women would consider themselves baseball fans. The standard deviation is ( ) âwomen, so in most samples of 6â women, the number of women who consider themselves baseball fans would differ from the mean by no more than ( ).â(Type integers or decimals rounded to the nearest tenth asâneeded.)
Answer:
a) 2.3
b) 1.4
c) 1.2
d) On average, 2.3 out of 6 women would consider themselves baseball fans. The standard deviation is 1.2 women, so in most samples of 6 women, the number of women who consider themselves baseball fans would differ from the mean by no more than 1.2.
Step-by-step explanation:
For each woman, there are only two possible outcoes. Either they are a fan of professional baseball, or they are not. The prbability of a woman being a fan of professional baseball is independent of other woman. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The variance of the binomial distribution is:
[tex]V(X) = np(1-p)[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
38% of women consider themselves fans of professional baseball.
This means that [tex]p = 0.38[/tex]
Six women are sampled:
This means that [tex]n = 6[/tex]
(a) Find the mean of the binomial distribution.
[tex]E(X) = np = 6*0.38 = 2.3[/tex]
(b) Find the variance of the binomial distribution
[tex]V(X) = np(1-p) = 6*0.38*0.62 = 1.4[/tex]
(c) Find the standard deviation of the binomial distribution.
[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{6*0.38*0.62} = 1.2[/tex]
(d) Interpret the results in the context of theâ real-life situation.
On average, 2.3 out of 6 women would consider themselves baseball fans. The standard deviation is 1.2 women, so in most samples of 6 women, the number of women who consider themselves baseball fans would differ from the mean by no more than 1.2.
There is good evidence that moderate alcohol use improves health. Some people think that red wine is better for your health than other alcoholic drinks. You have recruited 300 adults aged 45-65 who are willing to follow your orders about alcohol consumption over the next five years. You want to compare the effects on heart disease of moderate drinking of just red wine, just beer, or just spirits.what experimental design will you use?
Answer:
Independent measures, or repeated measures,
Step-by-step explanation:
I would use those experimental designs because it is used to compare different participants over a length of time.
Draw the reflected image of ABCD over line l.
Answer: The second image, the second image where b' is right above b is the correct answer for this, hope this helped!
<!> Brainliest is appreciated! <!>
Step-by-step explanation:
Answer
i woukldnt know
Step-by-step explanation:
ahahahahahahahahahahahahhahahahahaha
find the probability of being delt 5 clubs and 3 cards with one of each remaining suit in 8 card poker
Answer: 0.003757(approx).
Step-by-step explanation:
Total number of combinations of selecting r things out of n things is given by:-
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]
Total cards in a deck = 52
Total number of ways of choosing 8 cards out of 52 = [tex]^{52}C_8[/tex]
Total number of ways to choose 5 clubs and 3 cards with one of each remaining suit = [tex]^{13}C_5\times^{13}C_1\times^{13}C_1\times^{13}C_1[/tex] [since 1 suit has 13 cards]
The required probability = [tex]=\dfrac{^{13}C_5\times^{13}C_1\times^{13}C_1\times^{13}C_1}{^{52}C_8}[/tex]
[tex]=\dfrac{\dfrac{13!}{5!8!}\times13\times13\times13}{\dfrac{52!}{8!44!}}\\\\=\dfrac{24167}{6431950}\\\\\approx0.003757[/tex]
Hence, the required probability is 0.003757 (approx).
Please answer it now in two minutes
Answer:
3/4
Step-by-step explanation:
Rise over run.
Go up 3 units and 4 units to the right to find the next point
Answer:
Using points ( 8 , 9 ) and ( 4 , 6)
Slope = 6-9/4-8
= -3/-4
= 3/4
Hope this helps
The University of Arkansas recently reported that 43% of college students aged 18-24 would spend their spring break relaxing at home. A sample of 165 college students is selected.
a. Calculate the appropriate standard error calculation for the data.
b. What is probability that more than 50% of the college students from the sample spent their spring breaks relaxing at home?
Answer:
a. 0.0385
b. 3.44% probability that more than 50% of the college students from the sample spent their spring breaks relaxing at home
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this question:
[tex]p = 0.43, n = 165[/tex]
a. Calculate the appropriate standard error calculation for the data.
[tex]s = \sqrt{\frac{0.43*0.57}{165}} = 0.0385[/tex]
b. What is probability that more than 50% of the college students from the sample spent their spring breaks relaxing at home?
This is 1 subtracted by the pvalue of Z when X = 0.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.5 - 0.43}{0.0385}[/tex]
[tex]Z = 1.82[/tex]
[tex]Z = 1.82[/tex] has a pvalue of 0.9656
1 - 0.9656 = 0.0344
3.44% probability that more than 50% of the college students from the sample spent their spring breaks relaxing at home
En una inecuación, al multiplicar o dividir por un número negativo: *
Answer:
hyg
Step-by-step explanation:
The gable end of the roof shown is divided in half by a vertical brace. Find the distance h (in ft) from an eave to the peak. 41 feet (height) 80 feet (width) h = _________ ft.
Answer:
h = 9 ft
Step-by-step explanation:
Assume the diagram is like the one below.
∆ABD is a right triangle.
We can use Pythagoras' Theorem to find the height h.
[tex]\begin{array}{rcl}h^{2} + 40^{2} & = & 41^{2}\\h^{2} + 1600 & = & 1681\\h^{2} & = & 1681 - 1600\\& = &81\\h& = & \sqrt{81}\\&=& \mathbf{9}\end{array}\\\text{The height of the gable is $\large \boxed{\textbf{9 ft}}$}[/tex]
The high temperatures (in degrees Fahrenheit) of a random sample of 6 small towns are: 99 97.5 97.9 99.4 97 97.7 Assume high temperatures are normally distributed. Based on this data, find the 95% confidence interval of the mean high temperature of towns. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).
Consider a student loan of $20,000 at a fixed APR of 12% for 15 years. a. Calculate the monthly payment.(Do not round until the final answer. Then round to the nearest cent as needed.) b. Determine the total amount paid over the term of the loan. (Round to the nearest cent as needed.) c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest. (Round to the nearest tenth as needed.)
Answer:
a. $240.03
b. $43,205.04
c. Principal = 46.29%
Interest = 53.71%
Step-by-step explanation:
The computation is shown below:
a. For computing the monthly payment we need to apply the PMT formula i.e to be shown in the attachment below
Given that,
Present value = $25,000
Future value or Face value = 0
RATE = 12% ÷ 12 months = 1%
NPER = 15 years × 12 months = 180 months
The formula is shown below:
= PMT(RATE;NPER;-PV;FV;type)
The present value come in negative
So, after applying the above formula the monthly payment is $240.03
b. Now the total amount paid is
= Monthly payment × time period
= $240.03 × 180 months
= $43,205.04
c. Now the percentage is
For principal
= $25,000 ÷ $43,205.04 × 100
= 46.29%
And, for the interest, it is
= 100% - 46.29%
= 53.71%
two positive intergers have a product of 50 one interger is twice the other . what are the intergers
Answer:
10 and 5.
Step-by-step explanation:
Let the integers be x and y.
xy = 50
x = 2y
Put x as 2y in the first equation.
(2y)y = 50
2y² = 50
y² = 50/2
y² = 25
y = √25
y = 5
Put y as 5 in the second equation.
x = 2(5)
x = 10
The product of three consecutive integers is 210. What is their sum?
Answer: 69, 70, 71
x + x + 1 + x + 2 = 210
3x + 3 = 210
3x = 210 - 3
x = 207 / 3
x = 69
x + 1 = 70
x + 2 = 71
PLZ NEED A PROFESSIONAL :( Which is the equation of a parabola with a directrix at y = 2 and a focus at (5, 0)
.
y = one fourth(x − 5)2 + 1
y = one fourth(x + 1)2 − 5
y = −one fourth(x + 1)2 − 5
y = −one fourth(x − 5)2 + 1
Answer:
Answer D (the fourth one)
Step-by-step explanation:
First, we need to set up a (y−k)^2=4p(x−h). After plugging in all of your values, you would get (x-5)^2=-4(y-1). Now, we need to solve in terms of y by dividing each side by the factors that don't contain the variable. Your final solution should be [tex]y = -\frac{1}{4}(x - 5^{2}) + 1[/tex]
The equation is y = −one fourth(x − 5)2 + 1.
What type of equation is a parabola?
This form is called the usual shape of a quadratic function. The graph of the quadratic feature is a U-shaped curve is known as a parabola. The graph of the equation y=x2, shown under, is a parabola.
What is a standard form parabola?In trendy shape, the “a” cost represents the identical element because it does in vertex shape. Just as noted earlier than, it dictates the path of the parabola's commencing. If a > zero the parabola opens upwards, resulting in a minimal price. If a < zero, however, the parabola opens downwards, resulting in a maximum price.
Learn more about the equation of a parabola here https://brainly.com/question/4148030
#SPJ2
finding the missing angles 35° and 145°
Answer:
145°Step-by-step explanation:
There are two ways to find the value of X
[tex]x + 35 = 180[/tex] ( sum of co-interior angles)
Move constant to R.H.S and change its sign
[tex]x = 180 - 35[/tex]
Calculate the difference
[tex]x = 145[/tex]°
You can use another way too.
[tex]x = 145[/tex]° ( being vertically opposite angles)
Vertically opposite angles are always equal.
Hope this helps...
Best regards!!
Answer:
x = 145°
Step-by-step explanation:
Vertically opposite, also interior angles always add up to 180° so if you want to double check this, do 145° + 35° you should get 180°
I hope this helped you :)
Ken invests $6,000 into two accounts. One account earns 8% interest and the other earns 12% interest. After one year his total interest earned from the two accounts totals $580. How much did Ken invest in the 8% account?
Answer:
2400
Step-by-step explanation:
using simple ratio
we have Account A : account B as
8:12
2:3 .....the 8% account gets
2/5 × 6000
= 2× 1200 = 2400
The Ship It Anywhere Company bought a truck for $245,000. According to the company’s accounting department, the truck will depreciate $32,500 per year. 1. Find a linear function V(t) of the form V(t) = mt + b that models the value of the truck. V is the value of the truck and t is the number of years after the truck was bought. a. What is the slope of the function? Interpret what the slope means. b. What is the V intercept? Interpret what the V intercept means. c. Give the formula for the function. 2. Usethefunctiontofindthetintercept.Interpretwhatthetinterceptmeans. 3. Graphthefunction. 4. What is the domain and range of V(t)? 5. Find V(8) and explain what it means. Does your answer make sense? 6. When will the truck have a value of $128,000? 7. When will the truck have a value between $62,000 and $140,000?
Answer:
1. [tex]V(t) = -32500t + 245000[/tex]
a. The slope of the function is m = -32500, and it means the change in the value of V(t) for each unitary change in the value of t.
b. The V-intercept is b = 245000, and it means the value of V(t) when t = 0, that is, the inicial value of V(t).
c. The formula is: [tex]V(t) = -32500t + 245000[/tex]
2. t-intercept: [tex]t = 7.5385[/tex]
The t-intercept means when the function V(t) will be zero, that is, the truck has no value anymore.
3. Graph in the image attached.
4. The domain is t = [0, 7.5385] and the range is V(t) = [245000, 0].
5. [tex]V(8) = -15000[/tex]
It means the price the truck will have after 8 years. It does not make sense, because the truck can't have a negative price.
6. After 3.6 years.
7. Between 3.23 years and 5.63 years.
Step-by-step explanation:
1.
The inicial value is 245,000, and each year the value decreases 32,500, so we can write the equation:
[tex]V(t) = -32500t + 245000[/tex]
a. The slope of the function is m = -32500, and it means the change in the value of V(t) for each unitary change in the value of t.
b. The V-intercept is b = 245000, and it means the value of V(t) when t = 0, that is, the inicial value of V(t).
c. The formula is: [tex]V(t) = -32500t + 245000[/tex]
2.
To find the t-intercept we just need to use V(t) = 0 and then find the value of t:
[tex]0 = -32500t + 245000[/tex]
[tex]32500t = 245000[/tex]
[tex]t = 7.5385[/tex]
The t-intercept means when the function V(t) will be zero, that is, the truck has no value anymore.
3.
The graph of the function is in the image attached.
4.
The domain is t = [0, 7.5385] and the range is V(t) = [245000, 0].
5.
[tex]V(8) = -32500*8 + 245000 = -15000[/tex]
It means the price the truck will have after 8 years. It does not make sense, because the truck can't have a negative price.
6.
[tex]128000 = -32500t + 245000[/tex]
[tex]32500t = 117000[/tex]
[tex]t = 3.6[/tex]
After 3.6 years.
7.
[tex]62000 = -32500t + 245000[/tex]
[tex]32500t = 183000[/tex]
[tex]t = 5.6308[/tex]
[tex]140000 = -32500t + 245000[/tex]
[tex]32500t = 105000[/tex]
[tex]t = 3.2308[/tex]
Between 3.23 years and 5.63 years.
Gilda drove 650 miles at an average speed of 50 miles per hour. How many miles per hour faster would she have had to drive in order for the trip to have taken 1 hour less? Express your answer as a common fraction.
Answer:
325/78 = 4 13/78 miles per hour
Step-by-step explanation:
Total distance for Gilda to travel = 650 miles
Speed used for the journey= 50 miles per hour
Time taken = distance/speed
Time taken = 650/50
Time taken = 13 hours
Then how many miles per hour in essence what will be her speed in other for the journey to last less than one hour of her journey , that is 13 -1 = 12 hours
Speed = distance/time
Speed= 650/12
Speed= 54.1667 miles per hour
She would have gone
=650/12 -650/13
=(8450-7800)/156
=650/156
= 325/78
= 4 13/78
AWARDING BRAINLIEST!
What is the first step to solve for x? -4= x+3/2
A: Subtract 2 to both sides
B: multiply 3 to both sides
C: subtract 3 to both sides
D: multiply 2 both sides
Question 2:
If x-5/7=1 then which answer shows the correct steps to solve x?
(Answers listed in photo)
A
B
C
D
Answer:
1. Is (A) Subtract 2 to both sides
2. Is (C)
Step-by-step explanation:
What is the sum of the measures of the interior angles of this heptagon? A 7-sided figure. 720 degrees 900 degrees 1,080 degrees 1,260 degrees
Answer:
900°
Step-by-step explanation:
interior angles of a polygon = (n−2)×180°, where n is number of sides
for heptagon it is: (7-2)×180°= 900°
Suppose you need to graph the function f(x) = x(4x + 7)(x - 1)^7. Are there going to be any repeated zeros?
Answer:
yes
Step-by-step explanation:
hello,
as we have
[tex](x-1)^7[/tex]
the zero 1 is repeated 7 times
hope this helps
Answer:
yes
Step-by-step explanation:
The exponent on each factor tells you the number of times it is repeated. The zero represented by the factor (x -1)⁷ is repeated 7 times.
__
When a zero is repeated, the graph is flattened there. The more times it is repeated, the "flatter" the function is at that point. If the repeat-count is even, the graph touches the axis, but does not cross. Here, the repeat-count is odd, so the graph flattens out at the x-intercept, and crosses the x-axis.
Find the mode of 1, 4, 24, 14, 98, 37
Answer:
There is no mode
Step-by-step explanation:
Mode is the most occurring no. and there's no number which is the most occurring.
Answer:
No mode
Step-by-step explanation:
In a set of numbers, mode is the most repeated number.
1, 4, 24, 14, 98, 37
There are no repeated numbers in this set.
Consider the initial value problem my′'+ c y′+ k y=F(t),y(0)=0,y,(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t),where the unit of force is the Newton (N). Assume that m=2 kilograms, c=8 kilograms per second, k= 80 Newtons per meter, and F(t)=20sin(6t)Newtons.a. Solve the initial value problem.b. Determine the long-term behavior of the system.c. Is limt→[infinity]y(t)=0?If no, enter a function that approximates y(t)for very large positive values of t.
Answer:
A) [tex]y_g = e^-^2^t*\frac{15}{37}cos(6t) + e^-^2^t*\frac{5}{74}sin(6t) + \frac{5}{74}sin(6t) - \frac{15}{37}cos(6t) \\\\y_g =\frac{15}{37}cos(6t)* [ e^-^2^t - 1 ] + \frac{5}{74}sin(6t)* [ e^-^2^t + 1 ][/tex]
B) [tex]\frac{5}{74}sin(6t) - \frac{15}{37}cos(6t) = y_p[/tex]
Step-by-step explanation:
- The following initial value problem is given as follows:
[tex]my'' + cy' + ky = F(t) \\\\y(0) = 0\\y'(0) = 0[/tex]
- The above equation is the Newtonian mathematical model of a spring-mass-dashpot system. The displacement ( y ) and velocity ( y' ) are zeroed at the initial value t = 0.
- The equivalent mass ( m ) , damping constant ( c ) and the equivalent spring stiffness ( k ) are given as follows:
[tex]m = 2 kg\\\\c = 8 \frac{kg}{s} \\\\k = 80 \frac{N}{m} \\\\[/tex]
- The system is subjected to a sinusoidal force F ( t ) given. We will plug in the constants ( m , c, and k ) and applied force F ( t ) into the given second order ODE.
[tex]2y'' + 8y' + 80y = 20sin(6t)[/tex]
- The solution to a second order ODE is comprised of a complementary function ( yc ) and particular function ( yp ).
- To determine the complementary function ( yc ) we will solve the homogeneous part of the given second order ODE. We will assume the independent solution to the homogeneous ODE takes the form:
[tex]y = e^-^a^t[/tex]
Where,
a: The root of the following characteristic equation
- Substitute ( y ) into the given ODE as follows:
[tex]( 2a^2 + 8a + 80 )*e^-^a^t = 0\\\\2a^2 + 8a + 80 = 0[/tex]
- Solve the above characteristic quadratic equation:
[tex]a = 2 +/- 6i[/tex]
- The complementary solution for the complex solution to the characteristic equation is of the form:
[tex]y_c = e^-^\alpha^t * [ Acos (\beta*t) + Bcos (\beta*t) ][/tex]
Where,
a = α ± β
Therefore,
[tex]y_c = e^-^2^t * [ Acos (6t) + Bcos (6t) ][/tex]
- To determine the particular solution we will scrutinized on the non-homogeneous part of the given ODE. The forcing function F ( t ) the applied force governs the form of the particular solution. For sinusoidal wave-form the particular solution takes form as following:
[tex]y_p = Csin (6t ) + Dcos(6t )[/tex]
Where,
C & D are constants to be evaluated.
- Determine the first and second derivatives of the particular solution (yp) as follows:
[tex]y'_p = 6Ccos(6t) - 6Dsin(6t)\\\\y''_p = -36Ccos(6t) - 36Dcos(6t)\\[/tex]
- Plug in the particular solution ( yp ) and its derivatives ( first and second ) into the given ODE.
[tex]-72Csin(6t) - 72Dcos(6t) + 48Ccos(6t) - 48Dsin(6t) + 80Csin(6t) + 80Dcos(6t) = 20sin(6t) \\\\sin(6t)* ( 8C -48D ) + cos(6t)*(8D + 48C ) = 20sin(6t)\\\\D + 6C = 0\\\\C - 6D = 2.5\\\\C = \frac{5}{74} , D = -\frac{15}{37}[/tex]
- The particular solution can be written as follows:
[tex]y_p = \frac{5}{74}sin(6t) - \frac{15}{37}cos(6t)[/tex]
- Now we use the principle of super-position and combine the complementary and particular solution and form a function of general solution as follows:
[tex]y_g = y_c + y_p \\\\y_g = e^-^2^t* [ Acos(6t) + Bsin (6t) ] + \frac{5}{74}sin(6t) - \frac{15}{37}cos(6t)[/tex]
- To determine the complete solution of the given ODE we have to calculate the constants ( A and B ) using the given initial conditions as follows:
[tex]y_g ( 0 ) = 1*[A(1) + 0 ] + 0 - \frac{15}{37}(1) = 0\\\\A = \frac{15}{37}\\\\y'_g = -2e^-^2^t*[Acos(6t) + Bsin(6t) ] +e^-^2^t*[-6Asin(6t) + 6Bcos(6t) ] + \\\\\frac{15}{37}cos(6t) +\frac{90}{37}sin(6t) \\\\y'_g(0) = -2*[A(1) + 0] + 1*[0 + 6B] + \frac{15}{37}(1) +0 = 0\\\\B = \frac{15}{6*37} = \frac{5}{74}[/tex]
- The complete solution to the initial value problem is:
[tex]y_g = e^-^2^t*\frac{15}{37}cos(6t) + e^-^2^t*\frac{5}{74}sin(6t) + \frac{5}{74}sin(6t) - \frac{15}{37}cos(6t) \\\\y_g =\frac{15}{37}cos(6t)* [ e^-^2^t - 1 ] + \frac{5}{74}sin(6t)* [ e^-^2^t + 1 ][/tex]
- To determine the long term behavior of the system we will apply the following limit on our complete solution derived above:
[tex]Lim (t->inf ) [ y_g ] = \frac{15}{37}cos(6t)* [ 0 - 1 ] + \frac{5}{74}sin(6t)* [ 0 + 1 ]\\\\Lim (t->inf ) [ y_g ] = \frac{5}{74}sin(6t) - \frac{15}{37}cos(6t) = y_p[/tex]
- We see that the complementary part of the solution decays as t gets large and the particular solution that models the applied force F ( t ) is still present in the system response when t gets large.