By applying the initial conditions and inverse Laplace transforming, we can obtain the solution x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function. Therefore, the solution to the initial-value problem is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5)
Taking the Laplace transform of the given differential equation x" + 4 = f(t), we obtain the algebraic equation in the Laplace domain: s^2X(s) + 4sX(s) + 4 = F(s), where X(s) is the Laplace transform of x(t) and F(s) is the Laplace transform of f(t).
Next, applying the initial conditions x(0) = 0 and x'(0) = 0, we get X(0) = 0 and sX(0) = 0. Substituting these initial conditions into the Laplace domain equation, we have s^2X(s) + 4sX(s) + 4 = F(s), with X(0) = 0 and sX(0) = 0.
Now, let's consider the Laplace transform of f(t) using the given piecewise function. For [tex]t < 5, f(t) = t^2/5, and for t > 5, f(t) = 3sin(t-5).[/tex]Taking the Laplace transform of f(t) in each interval, we have [tex]F(s) = (1/s^3) + (3/s^2) for t < 5 and F(s) = (3/s^2) * (1/(s^2+1)) for t > 5.[/tex]
Substituting these Laplace transforms into the equation[tex]s^2X(s) + 4sX(s) +[/tex]4 = F(s), we can solve for X(s). Simplifying, we obtain [tex]X(s) = (1/s^3) + (3/s^2) / (s^2 + 4s + 4) + (3/s^2) * (1/(s^2+1)).[/tex]
To find the inverse Laplace transform of X(s), we can split it into partial fractions and apply the inverse Laplace transform formula. The solution is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function.
Therefore, the solution to the initial-value problem is x(t) = (1 - cos(2t))u(t-5) + (3 sin(t-5))u(t-5), where u(t) is the unit step function that ensures the piecewise function is activated at t = 5.
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Question (5 points): Using Laplace transform to solve the IVP: y" + 5y = eat, y(0) = 0, y' (O) = 0, then, we have Select one: y(t) =(-1 1 93 – 4s2 + 5s – 20 O y(t) = 2-1 1 482 33 5s + 20 O None of these. Ο Ο 1 yt 3 – (t) = c{+60-60-20) {{+40 + 5a + 20 O 1 y(t) = 2-1
Given the differential equation is y''+5y = e^(at) with initial conditions y(0) = 0 and y'(0) = 0, the correct answer is: y(t) = (-1/5√(5)) cos (√(5)t)+(1/5) sin (√(5)t).
To solve the given initial value problem using Laplace transform, we need to apply Laplace transform on both sides of the differential equation.
y''+5y=e^(at) L{y''+5y} = L{e^(at)) s^2Y(s)-sy(0)-y'(0)+5Y(s)=1/(s-a) [by Laplace transform formula] s^2Y(s)+5Y(s)=1/(s-a) ... [i]
Applying Laplace transform on both sides, we get:
L{y''+5y}=L{e^(at))
Using the initial conditions, we get Y(s)=1/[(s-a)(s^2+5)] Y(s) = [A/(s-a)] + [(Bs+C)sin(t)+ (Ds+E)cos(t)]/√(5) (i)
To find the values of A, B, C, D, and E, we take the inverse Laplace transform of both sides of equation (i) using partial fraction expansion. Let's solve for A:
Y(s)=A/(s-a)+(Bs+C)sin(t)/√(5)+(Ds+E)cos(t)/√(5)
Multiplying by s-a on both sides: (s-a)Y(s)=A+Bssin(t)/√(5)+Csincos(t)/√(5)+Dscos(t)/√(5)+Esin(t)/√(5)
Taking the inverse Laplace transform: y(t)=Ae^(at)+(B/√(5))sin(√(5)t)+(C/√(5))cos(√(5)t)+(D/√(5))cos(√(5)t)+(E/√(5))sin(√(5)t)
Differentiating y(t) with respect to t, we get:
y'(t)=Aae^(at)+Bcos(√(5)t)-Csin(√(5)t)-Dsin(√(5)t)+Ecos(√(5)t)
Using the initial conditions, y(0)=0 and y'(0)=0 in equation (iii), we get:
0=A+E ...(iv)0=A+B/√(5)+D/√(5) ...(v)
Solving equations (iv) and (v) simultaneously, we get A=0, B=√(5)/5, D=-√(5)/5, and E=0
Substituting these values in equation (iii), we get: y(t)=(√(5)/5)sin(√(5)t)-(√(5)/5)cos(√(5)t)
Therefore, the correct answer is: y(t)= (-1/5√(5))cos(√(5)t)+(1/5)sin(√(5)t).
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Which expression is equivalent
Answer:
the first one
Step-by-step explanation:
A telephone company representative estimates that 40% of its customers have call-waiting service. To test this hypothesis, she selected a sample of 100 customers and found that 37% had call waiting. At α-0.01 , is there enough evidence to reject the claim?
Using the null-hypothesis to calculate the test-statistic at a significance level of 0.01, there is not enough evidence to reject the claim that 40% of the telephone company's customers have call-waiting service based on the sample data.
Is there enough evidence to reject the claim?To test the hypothesis that 40% of the telephone company's customers have call-waiting service, we can conduct a hypothesis test.
Let's set up the null and alternative hypotheses:
Null hypothesis (H₀): The proportion of customers with call-waiting service is 40%.
Alternative hypothesis (H₁): The proportion of customers with call-waiting service is not 40%.
We can use the t-test for proportions to perform the hypothesis test. The test statistic is given by:
z = (p - p₀) / √((p₀ * (1 - p₀)) / n)
where p is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.
In this case, p₀ = 0.40, p = 0.37, and n = 100.
Calculating the test statistic:
z = (0.37 - 0.40) / √((0.40 * (1 - 0.40)) / 100)
z = -0.03 / √(0.24 / 100)
z = -0.03 / 0.049
z = -0.612
To determine if there is enough evidence to reject the null hypothesis, we compare the calculated z-value with the critical z-value at a significance level of α = 0.01.
The critical z-value for a two-tailed test at α = 0.01 is approximately ±2.576.
Since -0.612 falls within the range of -2.576 to 2.576, we fail to reject the null hypothesis.
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Find the distance between the points (7,–9) and (–2,–4).
[tex](2 \div 2 \sqrt{2) ^{2} } [/tex]
what is the answer
Which of the following is equivalent to the expression below?
Answer:
A. 7✓6 - 6x✓6
Explanation:
7✓6 - 5x✓6 - x✓6
= 7✓6 - (5x + x)✓6
= 7✓6 - 6x✓6
Ming le invited 12 people to her chrismas party.(that is,there are total 13 people at her party )Each person have each other person a present.How many presents were given
Answer:
There will be 13 gifts given in the party.
Step-by-step explanation:
Given that Ming invited 12 people to her chrismas party, with a total of 13 people at her party, and each person have each other person a present, to determine how many presents were given the following logical reasoning must be performed:
Given that there are 13 people at the party, and each of them will bring a gift, by mathematical logic there will be 13 gifts distributed in the party.
Find the area of the painted part of the image below. (Units in m)
Answer:
11 meters square.
Step-by-step explanation:
if the painted area is the gray it would be 10m
if its the white area its 9m
please please please help
Answer:
I wanna say 16 but im not sure
Step-by-step explanation:
what weight is shown
Answer:
325 grams is the weight.
5. How far can you get away from your little brother with a squirt gun filled point
with paint if you can travel at 3 m/s and you have 15 s before he sees you?"
45 m
5 m
2 m
20 m
Answer:
45 meters
Step-by-step explanation:
I hope this helps.
Is 0.25 greater or less than 0.03?
Answer:
0.25 is greater than 0.03
Step-by-step explanation:
You can solve this simply by looking for the position of the decimal point.
For example, 0.30 is greater than 0.25, but 0.03 is less than 0.250.25 > 0.03
Answer:
Step-by-step explanation:
please help dont answer with random
Answer:
Step-by-step explanation:
17x + 14 + 4x -2 = 21x + 12
21x + 12 = 180
-12 -12
21x = 168
x = 8 degrees
[tex]\boxed{\large{\bold{\green{ANSWER~:) }}}}[/tex]
Here,
The two lines l and m are parallel,
cut by a transversat line t
So The two angles are supplementary each other.
we know that,The sum of two supplementary angles are 180°
According to the question,(17x+14)°+(4x-2)°=180°
17x+14°+4x-2°=180°
17x+4x+14°-2°=180°
21x+12°=180°
21x=180°-12°
21x=168°
x=168°/21
x=8°
Therefore,The value of x is 8°
Please answer correctly! I will mark you as Brainliest
Answer:
Choice D
Step-by-step explanation:
Choice D, 3072ft^2 is the correct answer.
Here's how! The length AND width are both 24, so we know that 24 * 24 is our start. Then the height of the pyramid is 16, which now makes our equation 24 * 24 * 16. When finding the volume of a pyramid, you must divide by 3, so now the answer is (24 * 24 * 16) / 3. Making the answer 3072ft^2.
Hope this helped!
Answer:
3072
Step-by-step explanation:
volume =(1/3) hight ×width×length
v=(1/3) × 16 × 24 × 24
v=3072
Angle W is shown in the diagram.
w
400
What is the measure of Angle w in degrees?
Answer:
The measure of Angle w in degrees is 50°
Step-by-step explanation:
We know that in a straight line, 180°
So,
90° + 40° + w = 180°
130° + w = 180°
w = 180° - 130°
w = 50°
Thus, The measure of Angle w in degrees is 50°
-TheUnknownScientist
Roger brought some sponges. The number of packages he bought was 14 less than the number of sponges per package. Roger bought 51 sponges in all. How many packages of sponges did roger buy?
Answer:
the number of sponges per package bought be 32.5
Step-by-step explanation:
Let us assume the number of sponges per package be x
So the number of packages he bought be x - 14
And, the overall he bought 51 sponges
So
the number of packages of sponges did roger buy is
x + x - 14 = 51
2x = 51 + 14
2x = 65
x = 32.5
So the number of sponges per package bought be 32.5
Find the area. 14 m 4.5 m 4.5 m 9m 12 m 3 m 3 m
plzzz help meeee
Answer:
52
Step-by-step explanation:
Identify the type of conic that follwoing equation represents and write the standard form of each equation.
25x² + 100y² - 450x - 400y - 75 = 0
The standard form of the equation for the ellipse is ((x - 9)²/376) + ((y - 2)²/94) = 1.
The given equation represents an ellipse. The standard form of the equation is ((x-h)²/a²) + ((y-k)²/b²) = 1, where (h,k) is the center of the ellipse, 'a' is the semi-major axis, and 'b' is the semi-minor axis.
To determine the type of conic represented by the equation, we examine the coefficients of the x² and y² terms. In the given equation, both the x² and y² terms have positive coefficients, indicating that it represents an ellipse.
To write the equation in standard form, we need to complete the square for both the x and y terms. Let's rearrange the equation:
25x² - 450x + 100y² - 400y = 75
Now, we group the x and y terms and complete the square separately:
25(x² - 18x) + 100(y² - 4y) = 75
To complete the square for the x term, we take half of the coefficient of x (-18/2 = -9) and square it to get 81. Similarly, for the y term, we get 4.
25(x² - 18x + 81) + 100(y² - 4y + 4) = 75 + 25*81 + 100*4
Simplifying further:
25(x - 9)² + 100(y - 2)² = 9400
Dividing both sides by 9400, we get:
((x - 9)²/376) + ((y - 2)²/94) = 1
Therefore, the standard form of the equation for the ellipse is ((x - 9)²/376) + ((y - 2)²/94) = 1.
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Let A(x)=∫x0f(t)dtA(x)=∫0xf(t)dt, with f(x)f(x) as in figure.
A(x)A(x) has a local minimum on (0,6)(0,6) at x=x=
A(x)A(x) has a local maximum on (0,6)(0,6) at x=x=
To determine the local minimum and local maximum of the function A(x) = ∫₀ˣ f(t) dt on the interval (0, 6), we need to analyze the behavior of A(x) and its derivative.
Let's denote F(x) as the antiderivative of f(x), which means that F'(x) = f(x).
To find the local minimum and maximum, we need to look for points where the derivative of A(x) changes sign. In other words, we need to find the values of x where A'(x) = 0 or A'(x) is undefined.
Using the Fundamental Theorem of Calculus, we have:
A(x) = ∫₀ˣ f(t) dt = F(x) - F(0)
Taking the derivative of A(x) with respect to x, we get:
A'(x) = (F(x) - F(0))'
Since F(0) is a constant, its derivative is zero, and we are left with:
A'(x) = F'(x) = f(x)
Now, let's analyze the behavior of f(x) based on the given figure to determine the local minimum and maximum of A(x) on the interval (0, 6). Without the specific information about the shape of the graph, it is not possible to determine the exact values of x that correspond to local minimum or maximum points.
To find the local minimum, we need to locate a point where f(x) changes from decreasing to increasing. This point would correspond to x = x_min.
To find the local maximum, we need to locate a point where f(x) changes from increasing to decreasing. This point would correspond to x = x_max.
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If angle A=320∘, what is the radian measure of A? Give your answer as an exact fraction in terms of π.
The radian measure of angle A is (16/9)π.
To convert degrees to radians, we use the conversion factor:
[tex]1\ degree = \pi /180\ radians[/tex]
Given that angle A is 320 degrees, we can calculate its radian measure as follows:
Angle A in radians = (320 degrees) * (π/180 radians/degree)
= (320π)/180 radians
= (16/9)π radians
Therefore, the radian measure of angle A is (16/9)π.
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PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!
Answer:
Infinitely many solutions
Step-by-step explanation:
7x - 11 = 5(x - 2) + 2x - 1
7x - 11 = 10x - 10 + 2x - 1
7x - 11 = 7x - 11
No matter what number you plugin, it will always be equal.
1) 7x-11 = 5(x-2)+2x-1
2) 7x-11=5x-10+2x-1
3)7x-11=7x-11
4)0=0
5)Would cancel each other out.
6) So I guess no solutions?
Which expression is equivalent to 1/4m + 3/4m- 3/8(m+1)?
Answer:
5/8m-3/8
Step-by-step explanation:
thats prolly it
The radius of a circle is 4 feet. What is the area?
r=4ft
Give the exact answer in simplest form.
Answer:
100.571429.
Step-by-step explanation:
2*22/7*(4)^2
=44/7*16
=100.571429
6.18 heart transplant success. the stanford university heart transplant study was conducted to determine whether an experimental heart transplant program increased lifespan. each patient entering the program was officially designated a heart transplant candidate, meaning that he was gravely ill and might benefit from a new heart. patients were randomly assigned into treatment and control groups. patients in the treatment group received a transplant, and those in the control group did not. the table below displays how many patients survived and died in each group.22 control treatment alive 4 24 dead 30 45 suppose we are interested in estimating the difference in survival rate between the control and treatment groups using a confidence interval. explain why we cannot construct such an interval using the normal approximation. what might go wrong if we constructed the confidence interval despite this problem?
We cannot construct a confidence interval using the normal approximation in this case because the sample sizes of the control and treatment groups are relatively small (less than 30), and the data does not meet the assumptions required for the normal approximation.
The normal approximation assumes that the data follows a normal distribution, and for sample sizes less than 30, the distribution of the sample mean may not be well approximated by a normal distribution. In addition, the normal approximation assumes that the observations are independent, which may not hold true in this study due to the nature of the treatment and control groups.
If we constructed a confidence interval despite these problems, the interval may not accurately reflect the true difference in survival rates between the control and treatment groups. The interval could be biased or too wide/narrow, leading to incorrect conclusions about the effectiveness of the heart transplant program. Therefore, it is important to use appropriate statistical methods that account for the specific characteristics of the data and study design in order to obtain reliable estimates and valid inferences.
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Right triangle EFG it is known that the side opposite of acute angle F has a length of 18 inches and the leg adjacent to F has a length of 14 inches. Which of the following is closest to angle F
1. 38° 2. 65° 3. 52° 4. 72°
Answer:
3
Step-by-step explanation:
The closest angle to F in the triangle is 52 degrees.
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
Given that Right triangle EFG it is known that the side opposite of acute angle F has a length of 18 inches and the leg adjacent to F has a length of 14 inches.
We can use the tangent function to find the measure of angle F:
tan(F) = opposite/adjacent = 18/14
we can find the inverse tangent of this value:
F = tan⁻¹(18/14)
= tan⁻¹(1.28)
F= 52.15°
Hence, the closest angle to F in the triangle is 52 degrees.
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The following data repite the resundew of students to short-zule test (out of 10) of cours Alb. X. and Cae). X 7 10 3 8 3 0 9 8 Sum 9 G 8 5 2 9 10 1. Calculate the correlation coeffici
The correlation coefficient between the scores of students in courses Alb. X and Cae is approximately -0.333.
Correlation refers to the strength of the relationship between two variables while coefficient refers to the numerical value that measures the strength of the correlation.
To calculate the correlation coefficient between the scores of students in courses Alb. X and Cae, we need to first organize the data into two separate lists or arrays representing the scores in each course. Let's denote the scores in Alb. X as X_scores and the scores in Cae as C_scores:
X_scores: 7, 10, 3, 8, 3, 0, 9, 8
C_scores: 8, 5, 2, 9, 10, 1
Next, we need to calculate the mean (average) of both sets of scores.
Mean of X_scores (denoted as X_mean):X_mean = (7 + 10 + 3 + 8 + 3 + 0 + 9 + 8) / 8
X_mean = 48 / 8
X_mean = 6
Mean of C_scores (denoted as C_mean):C_mean = (8 + 5 + 2 + 9 + 10 + 1) / 6
C_mean = 35 / 6
C_mean ≈ 5.83
Now, we calculate the covariance between the two sets of scores using the formula:cov(X_scores, C_scores) = Σ((X_i - X_mean) * (C_i - C_mean)) / (n - 1)
where Σ denotes the sum, X_i and C_i are individual scores, X_mean and C_mean are the means calculated above, and n is the number of scores.
Let's calculate the covariance:cov(X_scores, C_scores) = ((7-6)(8-5.83) + (10-6)(5-5.83) + (3-6)(2-5.83) + (8-6)(9-5.83) + (3-6)(10-5.83) + (0-6)(1-5.83) + (9-6)(8-5.83) + (8-6)(0-5.83)) / (8-1)
cov(X_scores, C_scores) ≈ -3.39
Next, we calculate the standard deviations of both sets of scores:
Standard deviation of X_scores (denoted as X_std):X_std = √(Σ(X_i - X_mean)² / (n - 1))
Let's calculate X_std:
X_std = √(((7-6)² + (10-6)² + (3-6)² + (8-6)² + (3-6)² + (0-6)² + (9-6)² + (8-6)²) / (8-1))
X_std ≈ 3.20
Standard deviation of C_scores (denoted as C_std):C_std = √(Σ(C_i - C_mean)² / (n - 1))
Let's calculate C_std:
C_std = √(((8-5.83)² + (5-5.83)² + (2-5.83)² + (9-5.83)² + (10-5.83)² + (1-5.83)²) / (6-1))
C_std ≈ 3.18
Finally, we can calculate the correlation coefficient (r) using the formula:r = cov(X_scores, C_scores) / (X_std * C_std)
Let's calculate r:
r ≈ -3.39 / (3.20 * 3.18)
r ≈ -3.39 / 10.176
r ≈ -0.333
Therefore, the correlation coefficient between the scores of students in courses Alb. X and Cae is approximately -0.333.
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what is m (angle) 2 if m (angle) 4 = 120°?
Answer:
if 4 is 120° then 2 is also 180° since they are opposite angles.
Freddie is reading a book that has
440 pages. He has already read
140 pages. What decimal
describes the part of the book that
Freddie has already read?
PLEASE HELP ME
Answer:
140÷440(you can write it as a fraction)
=0.318181882
=0.32
Help, help, help! Quick Answer, please!
Answer:
3. 102
4. c
Step-by-step explanation:
For number 3:
(c^2 - a) + b becomes (10^2 - 3) + 5
10^2 = 100
100 - 3 + 5 = 102
For number 4:
Usually when you see distributive property, you're multiplying into parentheses. This time you're factoring out.
20c - 8d
It's not a, because that would be 20c + 8d
It's not b, because that would be 20c -32d
It is c, because 4 x 5c = 20 c and 4 x -2d = -8d
It's not d, because that would be 20c- 8cd
6) The cost of a computer system increases with increased processor speeds. The cost C
of a system as a function of processor speed is estimated as C = 1234 - 35 + 1900,
where S is the processor speed in MHz. Find the processor speed for which cost is at a
minimum. [4 marks]
Answer:
The speed is: 0.0142MHz
Step-by-step explanation:
Given
[tex]C(s) = 1234s^2 - 35s + 1900[/tex]
Required
The processor speed when the cost is at minimum
First, differentiate C(s) with respect to s
[tex]C(s) = 1234s^2 - 35s + 1900[/tex]
[tex]C'(s) = 2468s - 35[/tex]
The cost is at minimum when [tex]C'(s) = 0[/tex]
So, we have:
[tex]C' = 2468s - 35 = 0[/tex]
[tex]2468s - 35 = 0[/tex]
Solve for s
[tex]2468s = 35[/tex]
[tex]s=\frac{35}{2468}[/tex]
[tex]s\approx0.0142[/tex]