The solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:
[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]
To solve the given differential equation using Laplace transform, we will apply the Laplace transform to both sides of the equation and then solve for the transformed variable.
Let's denote the Laplace transform of y(t) as Y(s).
Taking the Laplace transform of both sides of the differential equation, we get:
[tex]s^2Y(s) + 2sY(s) + Y(s) = (s^2 + 2s + 1)/(s^2 + 4)[/tex]
Now, let's solve for Y(s):
[tex]Y(s)(s^2 + 2s + 1) = (s^2 + 2s + 1)/(s^2 + 4)\\Y(s) = (s^2 + 2s + 1)/(s^2 + 4)(s^2 + 2s + 1)[/tex]
Factoring the denominator:
[tex]Y(s) = (s^2 + 2s + 1)/((s + 1)^2(s^2 + 4))[/tex]
Now, we need to decompose the fraction into partial fractions. Let's express the numerator in terms of A, B, C, and D:
[tex]s^2 + 2s + 1 = A/(s + 1) + B/(s + 1)^2 + (Cs + D)/(s^2 + 4)[/tex]
To find the values of A, B, C, and D, we can equate the numerators:
[tex]s^2 + 2s + 1 = A(s + 1)(s^2 + 4) + B(s^2 + 4) + (Cs + D)(s + 1)^2[/tex]
Expanding and equating coefficients:
[tex]s^2 + 2s + 1 = A(s^3 + 5s^2 + 4s) + B(s^2 + 4) + (C(s^2 + 2s + 1) + D(s + 1)^2)[/tex]
Simplifying:
[tex]s^2 + 2s + 1 = (A + C)s^3 + (5A + C + D)s^2 + (4A + 2C + D)s + (4A + D)[/tex]
Equating coefficients:
A + C = 0 (coefficient of [tex]s^3[/tex])
5A + C + D = 1 (coefficient of [tex]s^2)[/tex]
4A + 2C + D = 2 (coefficient of s)
4A + D = 1 (constant term)
Solving these equations simultaneously, we find A = -1/10, B = 11/10, C = 1/10, and D = 3/10.
Now, substituting these values back into Y(s):
[tex]Y(s) = (-1/10)/(s + 1) + (11/10)/(s + 1)^2 + (1/10)(s + 3)/(s^2 + 4) + (3/10)/(s^2 + 4)[/tex]
To find y(t), we need to take the inverse Laplace transform of Y(s). Fortunately, we can use a Laplace transform table to find the inverse Laplace transform of each term.
The inverse Laplace transform of (-1/10)/(s + 1) is [tex]-e^{-t}/10.[/tex]
The inverse Laplace transform of (11/10)/(s + 1)² is (11/10)t*[tex]e^{-t}.[/tex]
The inverse Laplace transform of (1/10)(s + 3)/(s² + 4) is (1/10)cos(2t).
The inverse Laplace transform of (3/10)/(s² + 4) is (3/10)sin(2t).
Combining these results, the solution y(t) is:
[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]
Therefore, the solution to the given differential equation with the initial conditions y(0) = 1 and y'(0) = 1 is:
[tex]y(t) = -e^{-t}/10 + (11/10)t*e^{-t} + (1/10)cos(2t) + (3/10)sin(2t)[/tex]
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What is the area of a regular hexagon with an apothem of 28m in length
Step-by-step explanation:
can you check again the question , seems like something is missing.
For a multistate lottery, the following probability distribution represents the cash prizes of the lottery with their corresponding probabilities. Complete parts (a) through (c) below. P(x) 0.00000000821 0.00000014 200,000 10,000 0.000001746 100 0.000153924 7 0.005426433 4 0.006847638 3 0.01791359 0 0.96965652079 (a) If the grand prize is $16,000,000, find and interpret the expected cash prize. If a ticket costs $1, what is your expected profit from one ticket? The expected cash prize is $ (Round to the nearest cent as needed.)
Your expected profit from one ticket, after accounting for the ticket cost, is $2. This means that, on average, you can expect to make a profit of $2 per ticket if you were to play the lottery multiple times.
To find the expected cash prize, we multiply each cash prize by its corresponding probability and sum up the results.
Expected cash prize = (0.00000000821 * $16,000,000) + (0.00000014 * $1,000,000) + (0.000001746 * $200,000) + (0.000153924 * $10,000) + (0.005426433 * $100) + (0.006847638 * $7) + (0.01791359 * $4) + (0.96965652079 * $3) + (0.01791359 * $0)
Calculating this, we get an expected cash prize of $3.00025908719.
Interpreting the result, we can say that, on average, the expected cash prize for one ticket is approximately $3. This means that if you were to play the lottery multiple times, the average amount you could expect to win per ticket would be around $3.
To calculate the expected profit from one ticket, we subtract the cost of the ticket ($1) from the expected cash prize:
Expected profit = $3 - $1 = $2.
Therefore, your expected profit from one ticket, after accounting for the ticket cost, is $2. This means that, on average, you can expect to make a profit of $2 per ticket if you were to play the lottery multiple times.
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El auto que se va a comprar Pablo necesita que un cambio de aceite cada 60.000km y de neumáticos cada 90.000km. ¿En cuántos kilómetros coincidirá por primera vez el cambio de aceite y de neumáticos?
Answer:
El cambio de aceite y de neumáticos coincidirá por primera vez en 180.000 km.
Step-by-step explanation:
Como tenemos que encontrar cuándo se realizarán los dos cambios al mismo tiempo por primera vez, esto significa que tenemos que encontrar el mínimo común múltiplo de los dos números dado que este es el múltiplo más pequeño que tienen en común esos números y es donde coinciden por primera vez. Para encontrar el mínimo común múltiplo, puedes escribir los múltiplos de cada número y encontrar los que son comunes y seleccionar el más pequeño.
Múltiplos de 60.000= 60.000, 120.000, 180.000, 240.000, 300.000, 360.000, 420.000
Múltiplos de 90.000= 90.000, 180.000, 270.000, 360.000, 450.000, 540.000, 630.000
De acuerdo a esto, el mínimo común múltiplo es 180.000 dado que es el múltiplo más pequeño en el que coinciden y la respuesta es que el cambio de aceite y de neumáticos coincidirá por primera vez en 180.000 km.
A random sample of n items is to be taken from a distribution with mean μ and standard deviation o. Use the central limit theorem to determine the smallest number of items n that must be taken in order to satisfy the following relation: P(|Xn-μ<0/4) ≥ 0.99.
The given relation is P (|Xn - μ| / σ < 0.25) ≥ 0.99. We need to determine the smallest value of n that satisfies the given relation using the central limit theorem.
Step 1: We know that the standard normal distribution is used in the central limit theorem to approximate the distribution of the sample means. The standard normal distribution has a mean of zero and a standard deviation of one. We need to standardize the given relation so that we can use the standard normal distribution.
Step 2: We substitute the values from the given relation and simplify as follows: P(|Xn - μ| / σ < 0.25) ≥ 0.99P(|Xn - μ| / (o/√n) < 0.25) ≥ 0.99P((Xn - μ) / (o/√n) < 0.25) ≥ 0.99P(-0.25 < (Xn - μ) / (o/√n) < 0.25) ≥ 0.99P(Z < 0.25√n) - P(Z < -0.25√n) ≥ 0.99where Z is the standard normal random variable.
Step 3: We look up the values of -0.25√n and 0.25√n from the standard normal distribution table and find their difference. We use the absolute value of the difference since we are dealing with probabilities. We get: |P (Z < 0.25√n) - P (Z < -0.25√n) | = 0.99. Since the standard normal distribution is symmetric, we have P (Z < 0.25√n) - P (Z > 0.25√n) = 0.99We can rearrange this as: P (Z < 0.25√n) = 0.995P(Z > 0.25√n) = 0.005
Step 4: We look up the value of 0.995 from the standard normal distribution table and find its corresponding z-score. We get: z = 2.58. Using the z-score formula, we can solve for the value of 0.25√n. We get:2.58 = 0.25√nn = (2.58 / 0.25) ²n ≈ 107.58We round up to the nearest integer to get n = 108. Therefore, the smallest number of items n that must be taken in order to satisfy the given relation is 108.
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How many solutions does the system of linear equations have? Use the drop-down menus to explain your answer.
y = 3/4x + 12
y = 4/3x
The system of linear equations will have
Choose...
no solution
exactly one solution
infanite many solutions
because the slopes of the equations are
Choose...
the same
different
, so the lines will
Choose...
not intersect at all
intercect at one point
both be the same line
Answer:
There is one equation, the slopes are different so they will intersect at one point
Step-by-step explanation:
Simplify the expression 7 − 1x −1(−5x) − 10 + 4x
Answer: 3x-1(-5x)-3
Step-by-step explanation:
−3-x-1(-5x)+4x
-3+3x-1(-5x)
3x-1(-5x)-3
Select the expression that is equivalent to (m2-16)
A
(m-4)2
B
(m+4)(m-4)
C
(m2+8m+16)
D
(m2-8m+16)
A portion of the Quadratic Formula proof is shown. Fill in the missing statement.
A. x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 2 times a
B. x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 4 times a
C. x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 2 times a squared
D. x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over a
The missing statement in the Quadratic Formula proof is: A. x plus b over 2 times a equals plus or minus the square root of b squared minus 4 times a times c, all over 2 times a
This statement represents the quadratic formula, where x is the variable we are solving for in the quadratic equation ax^2 + bx + c = 0. The formula gives the solutions for x in terms of the coefficients a, b, and c of the quadratic equation.
The expression (b^2 - 4ac) represents the discriminant, which determines the nature of the solutions (real, imaginary, or equal). The square root of the discriminant is taken, and then the entire expression is divided by 2a to obtain the values of x. The "plus or minus" indicates that there are two possible solutions.
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here is my question.
Answer:
the answer is 18
Step-by-step explanation:
9X6=54
54/3=18
Answer:
18
Step-by-step explanation:
If an argument has a self-contradictory statement as a premise, then the counterexample set of the argument is:
The counterexample set of an argument with a self-contradictory statement as a premise is empty, as there are no valid counterexamples that can be presented to contradict the argument.
If an argument has a self-contradictory statement as a premise, then the counterexample set of the argument is empty, meaning there are no counterexamples that can be provided to disprove the argument. A self-contradictory statement is one that inherently contradicts itself, containing both a proposition and its negation. Since a self-contradictory statement cannot be true, any argument that relies on such a premise is inherently flawed and cannot be logically valid.
In logic, a counterexample is a specific example or case that demonstrates the falsity or invalidity of a general statement or argument. However, when the premise itself is self-contradictory, it is impossible to find a counterexample that would refute the argument because the premise itself is contradictory.
Therefore, the counterexample set of an argument with a self-contradictory statement as a premise is empty, as there are no valid counterexamples that can be presented to contradict the argument.
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A basket contains 41 heads of lettuce, 9 of which are spoiled. If a sample of 3 is drawn and not replaced, what is the probability that all in the sample are spoiled?
The probability is approximately 0.0079, or 0.79%.
To find the probability that all three heads of lettuce in the sample are spoiled, we need to calculate the ratio of favorable outcomes to the total number of possible outcomes.
The total number of possible outcomes is the number of ways to choose 3 heads of lettuce from the 41 available in the basket without replacement. This can be calculated using the combination formula (nCr):
Total possible outcomes = 41 C 3 = (41!)/(3!(41-3)!) = (414039)/(321) = 412013 = 10,660.
The number of favorable outcomes is the number of ways to choose 3 spoiled heads of lettuce from the 9 spoiled ones in the basket:
Favorable outcomes = 9 C 3 = (9!)/(3!(9-3)!) = (987)/(321) = 84.
Therefore, the probability that all three heads of lettuce in the sample are spoiled is:
Probability = Favorable outcomes / Total possible outcomes = 84 / 10,660 ≈ 0.0079 (rounded to four decimal places).
So, the probability is approximately 0.0079, or 0.79%.
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15 POINTS PLSS HELP THIS IS MY 3RD TIME POSTING THIS BC NO ONE WILL HELP ME PLSSS AND NO WRONG ANSWERS PLS I REALLY NEED HELPP
Answer:
to find the equation of the line you use y=mx+b
Step-by-step explanation:
m means the slope and the b is the y intercept of the line. so in this case the b is 5. the m is 6 so the answer is y=6x+5
What is the image of (-10,-8) after a dilation by a scale factor of 1/2 centered at the
origin?
Answer:
(-5, -4)
Step-by-step explanation:
A dilation at the origin would just result with you multiplying the coordinates by the scale factor.
-10 * 1/2 and -8 * 1/2
Caleb has a guaranteed minimum salary of $1,200 per month, or 5.5% of his total monthly sales (as commission), whichever is higher. Last month, his total sales were $45,000. What was his gross pay?
Answer: $1,200 + 5.5% + $45,000= 46200.055 or 46200
Step-by-step explanation: All you have to do is just add because if you read the text it has a key word total.
Caleb has a guaranteed minimum salary of $1,200 per month, or 5.5% of his total monthly sales (as commission), whichever is higher. Last month, his total sales were $45,000. What was his gross pay?
def:This is what we call the money you get back when you pay for something.
term:
Answer:
Change
Step-by-step explanation:
I call what I get back in return, change. The money we get back if you pay extra for something you buy.
Hope this helps!
Which of the following is an example of a physical property?(55 points)
A. nail rusting
B. camp fire burning
C. Table salt melting
D. silver spoon tarnishing
you make a one time deposit of 2000 years that earn 5% simple interest. how much interest would you earn in 2 years ?
Answer:
$2205
Step-by-step explanation:
equation set up
2000 (1.05)^2
How is 0.00400068 expressed in standard scientific notation?
Answer:
4.00068 × [tex]10^{-3}[/tex]
What is 14 + 6 =
+11
Since, 14 + 6 = 20
Therefore, 14 + 6 = 9 + 11
dont forget to like and mark me
Answer:
31
Step-by-step explanation:
HURRY PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer:
option 3
Step-by-step explanation:
Answer:
wouldn't it be one, someone lmk if I'm right
Step-by-step explanation:
48 is half of 96 and the ratio says there are less trucks. 36 is the closest to 48 but still less than it.
|13 + 5| – |7 – 10|
PLEASE STEP BY STEP
Answer:
1
Step-by-step explanation:
|13 + 5| – |7 – 10| =1
Answer:
I18I-I-3I
18-3
15
Step-by-step explanation:
Let u(t) = 2t³i + (t²-1)j-8k. Compute the derivative of the following function. (+19+21) u(t) Select the correct choice below and fill in the answer box(es) to complete your choice. OA. The derivative is the scalar function OB. The derivative is the vector-valued function i+ Di+ k
The correct choice is:
B. The derivative is the vector-valued function i + Di + k
The given function is u(t) = 2t³i + (t²-1)j - 8k, which represents a vector-valued function.
To compute the derivative of (19 + 21)u(t), we need to differentiate each component of the vector function with respect to t.
The scalar function (19 + 21) is a constant multiple, and when we differentiate a constant multiple of a vector function, we can simply differentiate each component of the vector function.
Taking the derivative of each component separately, we get:
d/dt (2t³i) = 6t²i
d/dt ((t²-1)j) = 2tj
d/dt (-8k) = 0
Putting the derivatives of each component together, we have:
(6t²i + 2tj + 0k) = 6t²i + 2tj
Hence, the derivative of the function (19 + 21)u(t) is the vector-valued function 6t²i + 2tj.
Therefore, the correct choice is:
B. The derivative is the vector-valued function i + Di + k
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14. Write the ratio of 2 cups of apple juice to 5 cups of orange juice the 3 different ways.
Answer:
1. 2:5
2. 5:2
3. 2,5
Step-by-step explanation:
Hope you have a great day
A serving of walnuts is 5/6 of a cup. How many servings are there in a 2 1/2-cup bag of walnuts?
Answer:
3
Step-by-step explanation:
5/2 ÷ 5/6 = 5/2 × 6/5
5/2 × 6/5 = 30/10 or 3
The maximum weight of a shipping container is 125 pounds. What is the maximum weight in kilograms?
a. 56.7 kg
b. 62.5 kg
c. 75 kg
d. 100 kg
The maximum weight in kilograms is approximately 56.7 kg. Hence, the correct option is (a) 56.7 kg.
The maximum weight of a shipping container is 125 pounds.
We need to find out what is the maximum weight in kilograms.
Step 1: Find out 1 pound weight in kilograms We know that 1 pound = 0.45359237 kilograms (we already know that)
Step 2: Convert the maximum weight in pounds to kilograms
Maximum weight in pounds = 125 Maximum weight in kilograms
= 125 x 0.45359237
= 56.69904625≈ 56.7 kg
Therefore, the maximum weight in kilograms is approximately 56.7 kg.
Hence, the correct option is (a) 56.7 kg.
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How many teams have played 20 times or more?
Answer:
3
Step-by-step explanation:
2. Columbia Records unvelled the LP (a vinyl record) in the Waldorf Astoria on June 18, 1948,
In two fomats: 10 inches in diameter and 12 Inches in diameter. If the thickness of one
vinyl record is 0.112 in, then determine the difference in volumes between the 10 inch and
12 inch records.
Write a ratio of squares to circles
Answer:
The ratio should be 2:3
Step-by-step explanation:
Hope this helped!
Answer:
2:3 or 2/3 depending on which form they want the answer in.
Step-by-step explanation:
There are 2 squares and 3 circles.
Which equation correctly compares the tens place and ones place in 9,999?
A.
90 ÷ 9 = 10
B.
900 ÷ 9 = 100
C.
9,000 ÷ 90 = 100
D.
900 ÷ 90 = 10
Answer:
I'm pretty sure the answer is A.
Step-by-step explanation:
Since 90 plus 9 = 99 and 99 is the tens and ones place.
So, the answer should be A.
Hope this helps! :)
90 / 9 = 10 compares the tens place and one's place in 9,999. Option A is correct.
Given that,
To determine the equation correctly compare the tens place and one's place in 9,999.
A number system is described as a technique of composing to represent digits. It is the mathematical inscription for describing numbers of a given set by using numbers or other characters in a uniform method. It delivers a special presentation of every digit and describes the arithmetic structure.
Here,
In the number 9999,
let the number on the ten places be x times the number on unit place,
90 = x * 9
x = 90 / 9
x = 10
Now,
90 / 9 = 10
Thus, 90 / 9 = 10 compares the tens place and one's place in 9,999. Option A is correct.
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The index rises 4.9% over the course of the day. What is the value of the index at the end of the day? Round your answer to the nearest hundred.
Answer: $70400
Step-by-step explanation:
Attached is the question:
Based on the information given in the question:
Value of Stock X = 5000 × $4.30 = $21500
Value of Stock Y = 2000 × $3.20 = $6400
Value of Stock Z = 8000 × $4.90 = $39200
Total Stock Value = $67100.
Since there's a 4.9% increase in value of index, the value of the index at end of the day will be:
= $67100 × (100% + 4.9%)
= $67100 × 104.9%
= $67100 × 1.049 =
= $70388
= $70400 approximately
Answer:
$70,400
Explanation:
Ap3x