(1 point) Consider the initial value problem my" + cy' + ky = 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)=20 sin(6t) Newtons. 1. Solve the initial value problem. y(t)=?
Answers
The solution to the initial value problem is
y(t) = (3/13) sin(6t) + (5/26) cos(6t).
The characteristic equation associated with the homogeneous part of the differential equation is
\(mr^2 + cr + k = 0,\)
where m = 2, c = 8, and k = 80. Solving for r, we find that
r = (-c ± √(\(c^2\) - 4mk)) / 2m.
Since \(c^2 - 4mk = 8^2\) - 4(2)(80) = -256 < 0, the roots are complex conjugates:
r = (-c ± i√(4mk - \(c^2\))) / 2m.
We can write this in the form r = -α ± iβ, where α = c / 2m and β = √(4mk - \(c^2\)) / 2m. Thus, the general solution to the homogeneous equation is
\(y_{h(t)\) = \(e^{(-\alpha t)(c_1 cos \beta t + c_2 sin \beta t),\)
where \(c_1\) and \(c_2\) are constants determined by the initial conditions y(0) = 0 and y'(0) = 0. Plugging in these values, we get
\(c_1\) = 0
\(c_2\) = 0
so the homogeneous solution is \(y_h(t)\) = 0.
To find a particular solution to the non-homogeneous equation, we can use the method of undetermined coefficients. Since the forcing function is F(t) = 20 sin(6t), we can try a particular solution of the form
\(y_p(t)\) = A sin(6t) + B cos(6t).
Taking the first and second derivatives of \(y_p\), we have
\(y_p'(t)\) = 6A cos(6t) - 6B sin(6t)
\(y_p''(t)\) = -36A sin(6t) - 36B cos(6t)
Substituting these into the differential equation, we get
(-36A sin(6t) - 36B cos(6t)) + 8(6A cos(6t) - 6B sin(6t)) + 80(A sin(6t) + B cos(6t)) = 20 sin(6t).
Simplifying, we get
(80A - 36B) cos(6t) + (36A + 80B) sin(6t) = 20 sin(6t).
Since the sine and cosine functions are linearly independent, the coefficients of sin(6t) and cos(6t) must both be equal to 0. Thus, we have the system of equations
80A - 36B = 0
36A + 80B = 20
Solving for A and B, we get
A = 3/13
B = 5/26
Therefore, the particular solution is
\(y_p(t)\) = (3/13) sin(6t) + (5/26) cos(6t).
The general solution to the non-homogeneous equation is then
y(t) = \(y_h(t)\) + \(y_p(t)\) = (3/13) sin(6t) + (5/26) cos(6t).
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Related Questions
Determine f(a+h) for f(x)=2x^3
Answers
f(a + h)
= 2(a + h)³
= 2( a³ + 3a²h + 3ah² + h³ ) ← distribute
= 2a³ + 6a²h + 6ah² + 2h³
for what x value(s) does f(x)=2? HELP ANSWER FAST!!!!!!
Answers
Answer:
x = 2
Step-by-step explanation:
Given
See attachment
Required
What is the value of x?
The vertical axis represents f(x) while the horizontal represents x.
From the attachment:
f(x) = 2 when x = 2
The circled point on attachment 2 represents the required point
Assume that each box is 1 unit.
PLEASE HELP I HAVE 5 MIN TO TURN THIS IN!!!!!!!!!!!!!
Perform the operation below
(-5x^2-9x+6)+(6x^2+10)
Answers
Answer:
The answer is x²-9x+16
Step-by-step explanation:
Answer:
x² - 9x + 16Step-by-step explanation:
(-5x² - 9x + 6) + (6x² + 10)
= -5x² - 9x + 6 + 6x² + 10
= 6x² - 5x² - 9x + 6 + 10
= x² - 9x + 16
what is the angle of P?
Answers
The answer is 35 hopefully I helped:)
q1. you observe that a numerical variable in your project follows a normal distribution. what percent of observations do you expect to be contained within 1.25 standard deviations of the mean
Answers
In a normal distribution, approximately 89% of the observations are expected to be contained within 1.25 standard deviations of the mean.
This can be determined using the empirical rule, also known as the 68-95-99.7 rule, which states that:
- Approximately 68% of the observations fall within 1 standard deviation of the mean.
- Approximately 95% of the observations fall within 2 standard deviations of the mean.
- Approximately 99.7% of the observations fall within 3 standard deviations of the mean.
Since 1.25 standard deviations is between 1 and 2 standard deviations, we can estimate that about 89% of the observations will fall within this range.
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Which point lies on the line with point slope equation y-3=4(x+7)
Answers
Answer:
-7,3
Step-by-step explanation:
Which inequality is NOT true if y= -5?
Answers
Answer: A
Step-by-step explanation:
2y - 10 > 13 y = -5
2(-5) - 10 > 13
-10 - 10 > 13
-20 > 13
False
suppose that an agency collecting clothing for the poor finds itself with a container of 20 unique pairs of gloves (40 total) randomly thrown in the container. if a person reaches into the container, what is the probability they walk away with two of the same hand?
Answers
The probability that a person walks away with two gloves of the same hand is approximately 0.0256 or 2.56%.
To calculate the probability that a person walks away with two gloves of the same hand, we can consider the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
When a person reaches into the container and randomly selects two gloves, the total number of possible outcomes can be calculated using the combination formula. Since there are 40 gloves in total, the number of ways to choose 2 gloves out of 40 is given by:
Total possible outcomes = C(40, 2) = 40! / (2! * (40 - 2)!) = 780
Number of favorable outcomes:
To have two gloves of the same hand, we can choose both gloves from either the left or right hand. Since there are 20 unique pairs of gloves, the number of favorable outcomes is:
Favorable outcomes = 20
Probability:
The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Favorable outcomes / Total possible outcomes = 20 / 780 ≈ 0.0256
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Find an example of a 2×3 matrix A and a 3×2 matrix B such that, letting T(x)=Ax and U(x)=Bx, the composition T∘U is a reflection over the line y=x.
Answers
The example of 2×3 matrix A and a 3×2 matrix B such that, letting T(x)=Ax and U(x)=Bx is reflection Ab.
Matrices, the plural form of matrix, are the groupings of numbers, variables, symbols, or phrases in a rectangular table with varying rows and columns. These are rectangular arrays with specified operations such as addition, multiplication, and transposition. The elements of the matrix are the numbers or entries in it. The horizontal entries of matrices are referred to as rows, whereas the vertical elements are referred to as columns.
Let,
\(B = \left[\begin{array}{cc}0&1\\1&0&0&0\end{array}\right] , A = \left[\begin{array}{ccc}1&0&0\\0&1&0\\\end{array}\right]\)
Then,
\(AB =\left[\begin{array}{ccc}1&0&0\\0&1&0\\\end{array}\right] \left[\begin{array}{cc}0&1&1&0&0&0\\\end{array}\right] \\\\AB = \left[\begin{array}{cc}0&1&1&0\\\end{array}\right]\)
Therefore, Ab is reflection about y = x .
As U = Bx and T∘U
A matrix is a rectangular array of integers, variables, symbols, or expressions that are defined for subtraction, addition, and multiplication operations. The number of rows and columns in a matrix determines its size (also known as the order of the matrix).
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A foam cylinder, with a diameter of 3 inches and height of 6 inches, is carved into the shape of a cone. What is the maximum volume of a cone that can be carved? Round your answer to the hundredths place.
Answers
The formula for the volume (V) of the cone is
\(V=\frac{1}{3}\pi r^2h\)Given
\(\begin{gathered} r=\frac{diameter}{2}=\frac{3in}{2}=1.5in \\ h=6in \end{gathered}\)Therefore,
\(V=\frac{1}{3}\pi(1.5)^2(6)=14.13716\approx14.14\)Hence, the maximum volume is
\(V=14.14in^3\)What do the coordinates of an undefined slope have in common?
Answers
The coordinates of an undefined slope are points that are either the same or have no x-value. In both cases, the slope of a line between these points would be undefined because it would involve dividing by 0, which is not allowed in mathematics. This is because the slope of a line is calculated by dividing the difference in y-coordinates by the difference in x-coordinates, and if the x-coordinates are the same or do not exist, this division would result in an undefined value.
Please answer need asap!!
Answers
Answer:
Step-by-step explanation:
answer is c
If two sides of a triangle measure 19 cm and 34 cm, check all possible values for the third side.
Answers
Answer:
Step-by-step explanation:
The length of the third side of a triangle must always be between (but not equal to) the sum and the difference of the other two sides.
19 + 34 = 53
34 - 19 = 15
from 16 cm to 52 cm
(-2) elevado a -4 respuesta porfavor
Answers
Answer:
16
Step-by-step explanation:
-2 to the 4th power is expressed as;
= (-2)^-4
= -2 × -2 × -2 × -2
= (-2×-2)×(-2×-2)
= 4×4
= 16
Hence -2 to the 4th power is equal to 16
what is the population being studied warts
Answers
The average population being studied is people with warts. Specifically, this population includes individuals of all ages who have been diagnosed with any type of wart caused by a virus, such as common warts, plantar warts, and flat warts.
The population being studied is people with warts. Warts are caused by a virus and can occur anywhere on the body. Common warts, plantar warts, and flat warts are the most common types. Common warts are usually found on the hands and fingers, and are usually round and raised. Plantar warts are found on the soles of the feet and can be painful. Flat warts are usually found on the face, arms, and legs, and are generally flat and smooth. People of all ages can be affected by warts, and they are highly contagious and can be spread through contact with an infected person or surface. Treatment options include over-the-counter medicines and prescription medications, as well as cryotherapy and laser treatments. This population is being studied to better understand the causes, risk factors, and treatments for warts.
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The complete question is
what is the population being studied warts and the prevalence of warts in the population being studied?
Cheryl finds a bag containing 50 coins. The value of the coins is £1.81. If the bag only contains two-pence and five-pence coins, how many more five-pence coins are there than two-pence coins?
Answers
Answer:
4
Step-by-step explanation:
2 p= x, 5 p= y, total = £1.81= 181 p
-------------
x+y=50 ⇒ x=50-y2x+5y=181 ⇒ 2(50-y)+5y=181 ⇒100+3y=181 ⇒ 3y=81 ⇒ y=27x=50-y=50-27=2327-23=4 more 5 p coins
The Miller family bought 4 crates of eggs last week. Each crate had 18 eggs. Since then, they have eaten 27 of the eggs. How many eggs are left?
Answers
Answer:
45 eggs left
Step-by-step explanation:
if they had 4 crates and each crate has 18, then we have a total of 72 eggs
subtract 27 eaten eggs from 72
72-27=45
they have 45 eggs left
Answer:
The answer is 45.
Step-by-step explanation:
How I got 45 is.....
1. I multiplied 4 x 18 to get 72
2. Then I subtracted 72 - 27 to get 45
I hope this helps! :0
what is 0 divide by 0?
Answers
Answer: 0
Step-by-step explanation: Here is an example:
Cookie Monster has 0 cookies you are trying to split the 0 cookies among zero friends. How many cookies do the 0 friends get? See it just doesn’t make sense. And Cookie Monster is sad because he has no cookies and you are sad because you have no friends. (Answer by SIRI, if you ask her, this will be her response! >:( Don’t worry, you have friends! :3!)
Evaluate 5 |3x- 11| + 2 when x=
5
Answers
Answer:
22
Step-by-step explanation:
the absolute value function always give a positive value , that is
| - a | = | a | = a
given
5 | 3x - 11 | + 2 ← substitute x = 5
= 5 | 3(5) - 11 | + 2
= 5 | 15 - 11 | + 2
= 5 | 4 | + 2
= 5 × 4 + 2
= 20 + 2
= 22
Ken went to the store for his mother. He spent $2.37 on groceries. In addition, his mother said he could spend $0.35 on candy. What was his change from a five dollar bill? Enter one digit in each box below.
Answers
Answer:
the awser should be 2.28
Step-by-step explanation:
Is 8/9 the multiplicative inverse of -1⅛? why or why not?
Answers
Answer:
-8/9
Step-by-step explanation:
-1 1/8
=> - 9/8
=> Multiplicative inverse of 9/8 is:
=> - 8/9 [Sign does not change only the numerator goes to the denominator and denominator to the numerator]
=> -8/9 is the multiplicative inverse of -1 1/8
There are 9 green beads and 11 blue beads in a box.
Jim adds 10 more beads to this box.
Jim is going to take at random a bead from the 30 beads in the box.
The probability that he will take a green bead is 2/5.
Work out how many green beads Jim adds to the box.
Answers
Answer:
lil boi
Step-by-step explanation:
(25 POINTS)Please help me, and show all of your work step by step.
Now say you invest $6,500 and the highest interest rate you can find is 2.5% compounded annually, but you would have to leave this investment in the account for a minimum of 5 years. If you decide to wait 5 years to buy the car, how much more money will you have to save to buy a car at the price of $8,000? Use the compound interest formula A = P(1 + i)^n.
A = accumulated amount
P = principle
i = interest rate
n = number of years
Answers
Answer:
$ 645.85
Step-by-step explanation:
First you would plug 6,500 into P. The plug 2.5% as 0.025 into i. And plug the years (5) into n.
A = 6500 (1 + 0.025)^5 = 7354.15
That is the amount that would be in the account at the end of the 5 years.
Then you take the cost of the car (8,000) and subtract the money left in the account (7,354.15).
8000 - 7354.15 = 645.85
Your final answer would be rounded to the nearest cent, or the hundredths place.
AE=EC and BF = FC
M
EF=10
M
Answers
Answer:
92
Step-by-step explanation:
E is the midpoint of AC.
F is the midpoint of BC.
FE and AB are parallel by a theorem.
m<DBF = m<EFC = 92 by corresponding angles of parallel lines cut by a transversal.
Answer:
92°
Step-by-step explanation:
50 points!! :D
not sure how to do this :/
Answers
Answer:
1
Step-by-step explanation:
x - 5 = -2( x + 1 )
=> x - 5 = -2x - 2
=> x + 2x = 5 - 2
=> 3x = 3
∴ x = 1
picking up one part from a group of 4. the number of parts we are talking about is the numerator of the fraction.
Answers
The numerator of the fraction representing the selected part is 1.
When we pick up one part from a group of 4, we have 4 parts in total. Let's represent this situation using a fraction.
The fraction can be written as:
\(\[\frac{{\text{{Number of parts selected}}}}{{\text{{Total number of parts available}}}}\]\)
In this case, the number of parts selected is 1, since we are picking up one part. And the total number of parts available is 4.
So, the fraction representing the situation is:
\(\[\frac{{1}}{{4}}\]\)
The numerator of the fraction is 1, which represents the number of parts selected from the group of 4.
Hence, the solution is: The numerator of the fraction is 1.
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how to write the following things:
million billion trillion quadrillion zillion gazillion
Answers
Numerical values of,
Million: 1,000,000
Billion: 1,000,000,000
Trillion: 1,000,000,000,000
Quadrillion: 1,000,000,000,000,000
Zillion and Gazillion: These are informal and exaggerated terms that do not have specific numerical values
Million: A million is a number equal to 1,000,000 (one followed by six zeros). It is often used to express large quantities of things, such as the population of a city or the number of dollars in a large fortune.
Billion: A billion is a number equal to 1,000,000,000 (one followed by nine zeros). It is a larger quantity than a million and is often used in economic contexts, such as the value of a company or the national debt of a country.
Trillion: A trillion is a number equal to 1,000,000,000,000 (one followed by twelve zeros). It is an even larger quantity than a billion and is often used in scientific and astronomical contexts, such as the distance between stars or the number of cells in the human body.
Quadrillion: A quadrillion is a number equal to 1,000,000,000,000,000 (one followed by fifteen zeros). It is an extremely large quantity and is rarely used in everyday contexts.
Zillion and Gazillion: These are informal and exaggerated terms that do not have specific numerical values. They are often used to express an extremely large or unknown quantity in a humorous or hyperbolic way.
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please help with this question?!?
Answers
Answer:
416 mi
Step-by-step explanation:
In order to find the surface area, you must find the area of each square. There are a total of 6 that make up the prism.
Remember to find the area it is length x width
9 x 8 = 72 which is the area for 4 of the squares. 2 on the side and 2 on the bottomLastly for the last 2 squares to find the area you multiply 8 x 8 = 64In total, you will add 72+72+72+72+64+64 which equals 416(8b^2/3*9t^3/5)(8b^5/3*9t^3/5)
Answers
Answer:
5,184•b•t^6/5
Step-by-step explanation:
Here, we want to multiply the expressions
According to indices law, if we have same bases, we add up the exponents
Thus, we have that;
8 *8b^(2/3+5/3) * 9 * 9 * t^(3/5+3/5)
= 64 * b * 81 * t^6/5
= 5,184bt^6/5
for the function shown, what is the range of the function when the domain is {-3,2,5}
f(x)=x^2 +3x-7
A:{-7,33}
B:{3,11,21
C:{-7,3,33}
D:{-7,11,33}
Answers
The range of the function when the domain is {-3,2,5} is {1,9,33}.
To find the range of the function f(x) = x^2 + 3x - 7 when the domain is {-3, 2, 5}, you need to substitute each value from the domain into the function and evaluate the result.
Step 1: Plug in x = -3
f(-3) = (-3)^2 + 3(-3) - 7 = 9 - 9 - 7 = -7
Step 2: Plug in x = 2
f(2) = (2)^2 + 3(2) - 7 = 4 + 6 - 7 = 3
Step 3: Plug in x = 5
f(5) = (5)^2 + 3(5) - 7 = 25 + 15 - 7 = 33
So, the range of the function for the given domain is {-7, 3, 33}. Your answer is C: {-7, 3, 33}.
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