Answer:
96 ways
Step-by-step explanation:
Let´s call flags of countries from Africa as A₁, A₂, A₃, A₄, and from Europe as E₁, E₂, E₃ and E₄
We have to display the flags always beginning with one flag from Africa let´s take A₁
For that flag, we have 4! possibilities Then 4! is:
4*3*2*1 = 24
As we have 4 different Flags from Africa, we will have
4*24 = 96 possibilities
a box is filled with chocolates and its mass is 480g. The same box is now filled with mints and its mass is 350g. The chocolates weigh twice as much as the mints. what is the mass of the box
Answer:
The box weighs 220 grams.
Step-by-step explanation:
Since the box full of chocolates weighs 480 grams, and the same box full of mints weighs 350, the weight difference between them is 130 grams. According to the statement, the quantity of chocolate weighs twice that of mint, while the weight of the box does not vary.
Therefore, since chocolate weighs twice as much as mints, and the weight is reduced by 130 grams, that is the difference in weight between the two, with which chocolate weighs 260 grams and mints 130 grams.
Therefore, the box weighs 220 grams: 220 + 130 = 350, and 220 + 260 = 480.
Phoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of 26 miles. The second and third days she averaged 12 miles per day. The last two days she hiked a total of 28 miles. The total hike for the first and third days was 22 miles. How many miles long was the trail?
Answer:
50 miles
Step-by-step explanation:
let he hiked a,b,c and d miles on each of the four days respectively.
then, according to the question.
a+b=26...i
b+c= 24...ii
c+d=28...iii
a+c=22...iv
now, adding i,ii,iii,iv we get
2(a+b+c+d) = 100
a+b+c+d= 50 miles.
Hence, he traveled in all 50 miles.
You and 3 of your friends decide to sell lemonade around town, and then split the money you make evenly. You decide to sell each cup of lemonade for 50 cents. In total, you all sell 120 cups of lemonade. How much money will each of you earn? Write an expression for the problem too.
Expression:
Answer:
$15
Step-by-step explanation:
Each cup is 50 cents which is basically $0.50
Multiply $0.50 by 120= $60
Because you and your three friends equal 4 total people,
divide 60 by 4 to get your own profit:
60/4=15
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
If this procedure is repeated 100 times, what is the probability that the number of times that the coin lands tails will be less than 40
Answer:
1.79% probability that the number of times that the coin lands tails will be less than 40
Step-by-step explanation:
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 standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
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.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
Fair coin:
Equally as likely to be heads or tails, so [tex]p = 0.5[/tex]
100 times
[tex]n = 100[/tex]
Then
[tex]\mu = E(X) = np = 100*0.5 = 50[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.5*0.5} = 5[/tex]
What is the probability that the number of times that the coin lands tails will be less than 40
Using continuity correction, this is [tex]P(X < 40 - 0.5) = P(X < 39.5)[/tex], which is the pvalue of Z when X = 39.5.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{39.5 - 50}{5}[/tex]
[tex]Z = -2.1[/tex]
[tex]Z = -2.1[/tex] has a pvalue of 0.0179
1.79% probability that the number of times that the coin lands tails will be less than 40
Suppose a basketball team had a season of games with the following characteristics: Of all the games, 60% were at-home games. Denote this by H (the remaining were away games). Of all the games, 25% were wins. Denote this by W (the remaining were losses). Of all the games, 20% were at-home wins. Of the at-home games, we are interested in finding what proportion were wins. Which of the following probabilities do you need to find in order to determine the proportion of at-home games that were wins?A. P(H)B. P(W)C. P(H and W)D. P(H | W)E. P(W | H)
Answer:
E. P(W | H)
Step-by-step explanation:
What each of these probabilities mean:
P(H): Probability of the game being at home
P(W): Probability of the game being a win.
P(H and W): Probability of the game being at home and being a win.
P(H|W): Probability of a win being at home.
P(W|H): Probability of winning a home game.
Which of the following probabilities do you need to find in order to determine the proportion of at-home games that were wins?
This is the probability of winning a home game. So the answer is:
E. P(W | H)
what is the length of ac? a)96 b)132 c)72 d)136
Answer:
show a picture
Step-by-step explanation:
Let g be the function defined by g(x) = − 1 2 x + 5 if x < 6 x − 6 if x ≥ 6. Find g(−6), g(0), g(6), and g(12). g(−6) = g(0) = g(6) = g(12) =
Answer:
g(-6) = 8; g(0) = 5; g(6) = 0; g(12) = 6
Step-by-step explanation:
We assume your function definition is ...
[tex]g(x)=\left\{\begin{array}{ccc}-\dfrac{1}{2}x+5&\text{for}&x<6\\x-6&\text{for}&x\ge 6\end{array}\right.[/tex]
For each given value of x, determine which segment applies, then evaluate.
For x = -6 and for x = 0, the first segment applies:
g(-6) = (-1/2)(-6) +5 = 3 +5 = 8
g(0) = (-1/2)(0) +5 = 5
For x = 6 and x = 12, the second segment applies:
g(6) = (6) -6 = 0
g(12) = (12) -6 = 6
In summary, ...
g(-6) = 8; g(0) = 5; g(6) = 0; g(12) = 6
A car’s value varies inversely with its age. Jackie bought a 10-year-old car for $2,400. Write the equation that relates the car’s value, v, to its age, a. What will be the value of Jackie’s car when it is 15 years old ?
Answer:
$1,600
Step-by-step explanation:
Inverse relation:
v = k/a
where v = value of car, and a = age in years.
To find k, we use a known value
2400 = k/10
k = 24000
The inverse relation is
v = 24,000/a
At 15 years, a = 15.
v = 24,000/15 = 1,600
The value of Jackie’s car when it is 15 years old will be $1,600.
What is the equation?
A mathematical statement consisting of an equal symbol between two algebraic expressions with the same value is known as an equation.
Given data;
Let y be the car’s value and x is the age then, if there is an inverse relation between them;
y = k/x
Substitute the given values;
k=xy
k=2400 × 10
k = 24000
Substitute the value of k;
y = 24000/x
Condition 2;
at x = 15 and y = ?
y = 24,000/15
y= 1,600
Hence, the value of Jackie’s car when it is 15 years old will be 1600.
To learn more, about equations, refer;
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Teresa is investigating if grade level has any effect on time spent studying. What is the response variable?
Answer:
The time spent studying is the response variable.
Step-by-step explanation:
The response variable, also known as the dependent variable is the main question which the experiment wants to provide an answer for. Usually, the predictors determine or affect the response variable. In the study where Teresa investigates the effect of grade level on time spent studying, the response variable is the time spent studying, while the predictor which is the grade level provides an explanation as to the time spent studying.
The changes or variations on time spent studying depends on the grade level. This means that the grade level provides an explanation of the length of time dedicated to studying.
PLEASE HELP AND SHOW WORK
Answer:
7.5
Step-by-step explanation:
If we look at the 4 by 4 square around the triangle we can just do the area of the square minus the area of the 3 little triangles which is:
4 * 4 - 4 * 1 / 2 - 3 * 3 / 2 - 4 * 1 / 2
= 16 - 2 - 4.5 - 2
= 16 - 8.5
= 7.5
A right triangle has two shorter sides that differ in length by 7cm. The length of the
hypotenuse is 8 cm longer than the shortest side. Find the lengths of the three sides.
Show all of your steps.
Pls help!!! 75 points
Answer:
a = 5
b = 12
c = 13
Step-by-step explanation:
a^2+b^2=c^2
b-a=7(b=a+7)
c=a+8
Then, substitute
a^2+((a+7)*(a+7))=c^2
a^2+a^2+7a+7a+49=c^2
2a^2+14a+49=c^2
Because c = a+8
2a^2+14a+49=(a+8)(a+8)
2a^2+14a+49=a^2+16a+64
a^2-2a=15
a^2-2a-15=0
(a-5)(a+3)=0
a = 5,-3
a = 5(a side cannot be negative)
Plug in a=5 to the other equations to get
a = 5, b = 12, c = 13
Hope it helps <3
Answer:
The lengths of the sides are 5, 12, 13.
Step-by-step explanation:
In a right triangle, the two shorter sides are the legs. The longest side is the hypotenuse.
Let the shorter leg = x.
The longer leg is 7 cm longer, so its length is x + 7.
The length of the hypotenuse is 8 cm longer than the shorter leg, so its length is x + 8.
The lengths are:
x, x + 7, x + 8
Since the triangle is a right triangle, we can use the Pythagorean theorem.
a^2 + b^2 = c^2
x^2 + (x + 7)^2 = (x + 8)^2
Square the trinomials.
x^2 + x^2 + 14x + 49 = x^2 + 16x + 64
Combine like terms and place them all on the left side equaling zero.
x^2 - 2x - 15 = 0
Factor the left side.
(x - 5)(x + 3) = 0
x - 5 = 0 or x + 3 = 0
x = 5 or x = -3
Since the length of a side of a triangle cannot be negative, we discard the solution x = -3.
x = 5
x + 7 = 5 + 7 = 12
x + 8 = 5 + 8 = 13
Answer: The lengths of the sides are 5, 12, 13.
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.
Draw a graph of f(x) =3^-x+3
Answer:
Use a graphing calc or desmos
Step-by-step explanation:
What is the solution to the system of equations below? HELP!!!! y = negative one-fourth x + 2 and 3 y = negative three-fourths x minus 6 no solution infinitely many solutions (–16, 6) (–16, –2)
Answer:
No solution
Step-by-step explanation:
Step 1: Write out equations
y = -1/4x + 2
3y = -3/4x - 6
Step 2: Substitution
3(-1/4x + 2) = -3/4x - 6
Step 3: Distribute
-3/4x + 6 = -3/4x - 6
From here, we can see that we have the same slope but different y-intercept. This means that the 2 lines are parallel and therefore never intersect.
Alternatively, you could graph the equations and see that the 2 lines are parallel and never intersect.
Answer:
No solution
Step-by-step explanation:
y = -1/4x + 2
3y = -3/4x - 6
Plug y as -1/4x + 2 in the second equation.
3(-1/4x + 2) = -3/4x - 6
-3/4x + 6 = -3/4x - 6
-3/4x + 3/4x = -6 -6
0 = -12
No solution.
Follow the directions to solve the system of equations by elimination. 8x + 7y = 39 4x – 14y = –68 Multiply the first equation to enable the elimination of the y-term. Add the equations to eliminate the y-terms. Solve the new equation for the x-value. Substitute the x-value back into either original equation to find the y-value. Check the solution.
Answer:
x=½
y=5
Step-by-step explanation:
(8x+7y=39)2
16x+14y=78
4x-14y=-68 add the two equations
20x=10.
divide both sides by 20
x=½
8x+7y=39
4+7y=39
7y=39-4
7y=35
y=5
The value of x and y in the system of equation using elimination method is 1 / 2and 5 respectively.
8x + 7y = 39
4x – 14y = –68
Multiply the first equation to enable the elimination of the y-term:Multiply by 2
16x + 14y = 78
Add the equations to eliminate the y-terms:-14y + 14y = 0
4x + 16x = 20x
-68 + 78 = 10
Solve the new equation for the x-value20x = 10
x = 1 / 2
Substitute the x-value back into either original equation to find the y-value8(1 / 2) + 7y = 39
4 + 7y = 39
7y = 35
y = 35 / 7
y = 5
learn more on system of equation here: https://brainly.com/question/3861421?referrer=searchResults
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.
What is the equation of the line ( -4,8 ) ( 0,0 )
Answer:
Step-by-step explanation:
First you need to find the slope of the line that contains those 2 points.
[tex]m=\frac{0-8}{0-(-4)}=\frac{-8}{4}=-2[/tex]
So the slope is -2. Now we can pick one of those points and sub it into the point-slope formula to find the equation:
y - 0 = -2(x - 0) gives us an equation of
y = -2x
You can retry this question below
A6 inch personal pizza has 610 calories, with 240 of those from fat. A 12 inch pizza is cut into 8 slices.
Estimate the number of calories in one slice of a 12 inch pizza.
Answer:
305 calories, 120 from fat
Step-by-step explanation:
The ratio of the area of the larger pizza to that of the smaller pizza is the square of the ratio of the diameters. So, the larger pizza has an area that is ...
(12/6)² = 4
times that of the smaller pizza. When that area is divided into 8 parts, each part has an area that is 4/8 = 1/2 the area of the smaller pizza.
We expect a slice of the larger pizza to have 1/2 the calories of a smaller pizza, so 305 calories, 120 from fat.
__
610/2 = 305; 240/2 = 120.
PLEASE HELP!!!!!! Find common difference
Answer:
d = [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
The n th term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference , thus
a₇ = a₁ + 6d
a₄ = a₁ + 3d
Given a₇ - 2a₄ = 1 , then
a₁ + 6d - 2(a₁ + 3d) = 1, that is
a₁ + 6d - 2a₁ - 6d = 1
- a₁ = 1 ( multiply both sides by - 1 )
a₁ = - 1
Given a₃ = 0 , then
a₁ + 2d = 0 , thus
- 1 + 2d = 0 ( add 1 to both sides )
2d = 1 ( divide both sides by 2 )
d = [tex]\frac{1}{2}[/tex]
Find the radius of the cylinder when volume is 304 cm^3 and height is 10 cm
Answer:
3.11 cmsolution,
Volume of cylinder=304 cm^3
height=10 cm
Radius=?
Now,
[tex]volume = \pi {r}^{2} h \\ or \: 304 = 3.14 \times {r}^{2} \times 10 \\ or \: 304 = 31.4 \times {r}^{2} \\ or \: {r}^{2} = \frac{304}{31.4} \\ or \: {r}^{2} = 9.68 \\ or \: r = \sqrt{9.68} \\ or \: r = \sqrt{ {(3.11)}^{2} } \\ r = 3.11 \: cm[/tex]
Hope this helps..
Good luck on your assignment..
3. What is the explicit formula for the arithmetic sequence 2, 7, 12, 17, ...?
Step-by-step explanation:
The given sequences are;
2,7,12,17......
difference =5
by using formula,
we get,
tn=a+(n-1)d
tn= 2+(n-1)5
Therefore, tn is 5n-3 is required formula for this arithmetic sequences.
Hope it helps....
rounded to the nearest whole, what is the radius length if minor arcYZ = 12 and angleYXZ is one-third of a full circle? (i guessed it idk if it’s right)
Answer:
Option (1)
Step-by-step explanation:
Since the length of arc YZ = 12 units
m∠YXZ = one third of the full circle = [tex]\frac{360}{3}[/tex] = 120°
From the formula of arc length,
Length of arc = [tex]\frac{\theta}{360}(2\pi r)[/tex]
Where θ = Central angle subtended by the arc
r = radius of the circle
By substituting these values in the formula,
12 = [tex]\frac{120}{360}(2\pi r)[/tex]
12 = [tex]\frac{2}{3}\pi r[/tex]
[tex]18=\pi r[/tex]
r = [tex]\frac{18}{\pi }[/tex]
r = 5.73
r ≈ 6 units
Therefore, Option (1) will be the answer.
Pleaseeeeeee helppppppp
Answer:
[tex]\boxed{Option \ D}[/tex]
Step-by-step explanation:
The combination of a rational number (3) and an irrational no. ([tex]4i[/tex]) is called a complex number.
So,
[tex]3+4i[/tex] is a complex no.
Answer:
D. Complex number.
Step-by-step explanation:
This number is not irrational, since 3 is rational.
The number is not entirely rational, since 4i is irrational.
The number is not real because i is not real.
So, the number is a Complex number, since it includes both real and nonreal numbers.
Hope this helps!
Solve 56000(1+1.8%)^5
Answer:
The solution to this expression is 61,224.74
Step-by-step explanation:
To solve we initially have to convert the percentage to a decimal:
[tex]1.8\% = \frac{1.8}{100} = 0.018[/tex]
So
56000*(1+1.8%)^5 = 56000(1+0.018)^5 = 56000(1.018)^5 = 61,224.74
The solution to this expression is 61,224.74
In a 2-card hand, what is the probability of holding only face cards?
Answer:
12
Step-by-step explanation:
( J , Q, K 4 each)
so prob that for 2 cards, both cards are face
= C(12,2)/C(52,2) = 66/1326 = 11/221
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.
Select all numbers that are in the range.
-3
-2
-1
0
1
2
-2
0
2
Answered on edge
Answer:
-2, 0, 2
Step-by-step explanation:
edge 2020
What is the measure of C?
Answer:
C.) 60°
Step-by-step explanation:
The triangle is an equilateral triangle. So that means that all the angles measure the same. And we have to remember that a triangle always equals 180°
So, to find out the measure of an angle. We must divide 180 by 3. Which is 60
=60°
Hope this helps you out! : )
Elif is arranging 28 chairs in rows in a room. Each row must be the same
length. The room is wide enough to make a row of 9 chairs, but no more
The room is deep enough to make 8 rows, but no more. What are the
possible numbers of rows and chairs in each row that Elif can make?
Answer:
Hey there!
Elif can only arrange the chairs like: 4 by 7, and 7 by 4.
Hope this helps :)