Problem 0: Compute the inverse Laplace Transforms of:
[tex]F(s) = \frac{s + 1}{s(s - 1)(s - 3)} [/tex]
[tex]F(s) = \frac{1}{(s - 1)(s - 2)(s - 3)} [/tex]
![Problem 0: Compute The Inverse Laplace Transforms Of:[tex]F(s) = \frac{s + 1}{s(s - 1)(s - 3)} [/tex][tex]F(s)](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/nzCK6aJJ3sOeIOd4yctwuFX1dh67GobN.jpeg)
Answers
Decompose each given F(s) into partial fractions.
[tex]F(s) = \dfrac{s+1}{s(s-1)(s-3)}[/tex]
has partial fraction decomposition
[tex]\dfrac{s+1}{s(s-1)(s-3)} = \dfrac as + \dfrac b{s-1} + \dfrac c{s-3}[/tex]
Combine the rational terms on the right and solve for the coefficients:
[tex]\dfrac{s+1}{s(s-1)(s-3)} = \dfrac{a(s-1)(s-3) + b s(s-3) + c s(s-1)}{s (s-1) (s-3)}[/tex]
[tex]1 = a(s-1)(s-3) + bs(s-3) + c s(s-1)[/tex]
[tex]1 = 3 a + (-4 a - 3 b - c) s + (a + b + c) s^2[/tex]
[tex]\begin{cases}3a=1 \\ -4a-3b-c = 0 \\ a+b+c=0 \end{cases} \implies a=\dfrac13, b=-\dfrac12, c=\dfrac16[/tex]
Then
[tex]F(s) = \dfrac13 \times \dfrac1s - \dfrac12 \times \dfrac1{s-1} + \dfrac16 \times \dfrac1{s-3}[/tex]
Using the frequency-shifting property, the inverse transform is
[tex]\boxed{f(t) = \dfrac13 - \dfrac{e^t}2 + \dfrac{e^{3t}}6}[/tex]
The other transform can be dealt with in the same manner.
[tex]F(s) = \dfrac1{(s-1)(s-2)(s-3)} = \dfrac a{s-1} + \dfrac b{s-2} + \dfrac c{s-3}[/tex]
[tex]\implies 1 = a(s-2)(s-3) + b(s-1)(s-3) + c(s-1)(s-2)[/tex]
[tex]\implies 1 = 6 a + 3 b + 2 c + (-5 a - 4 b - 3 c) s + (a + b + c) s^2[/tex]
[tex]\implies \begin{cases}6 a + 3 b + 2 c=1 \\ -5a-4b-3c = 0 \\ a+b+c=0\end{cases} \implies a=\dfrac12, b=-1, c=\dfrac12[/tex]
[tex]\implies F(s) = \dfrac12 \times \dfrac1{s-1} - \dfrac1{s-2} + \dfrac12 \times \dfrac1{s-3}[/tex]
[tex]\implies \boxed{f(t) = \dfrac{e^t}2 - e^{2t} + \dfrac{e^{3t}}2}[/tex]
Related Questions
PLEASE HELP FAST ILL GIVE BRAINLIEST How much wrapping paper is needed to wrap a cube shaped gift with an edge measuring 15 cm?
0
O A. 1350 cm squared
B. 225 cm squared
O C. 1125 cm squared
Answers
Answer:
Its letter B.
#Carryonlearning
Can somebody help me pls 3/10 +16/100 = 19/100 ??? Is it correct
Answers
Answer:
yes it is correct 3/10+16/100=19/100
can someone please help!!! Urgent!!
Answers
explanation: this is what i’m doing in math right now and i’m pretty sure it’s correct
I will give the BRAINEST
The length of a hypotenuse of a 30° -60° - 90° triangle is 17 yards. What is the length of the
other two sides?
A. 8.5 yds: 8.5 yds
B. 8.5 yds: 8.5/3yds
C. 8.5y3yds: 8.5VZ yds
D. 1773 yds: 8.5 yds
Answers
Step-by-step explanation:
answer is B which is 8•5 yds and 8•5√3 yds
can someone answer this for me please
Answers
Answer:
36 Seconds
Step-by-step explanation:
3 x 12 equals 36
4 x 9 equals 36
what we do is find a least common multiple for both the numbers
hope this helps, if it does, can i have brainliest?
maybe a little more point will make you help me
i put 100 but you may not get that much
am i correct
also, i would give you but that is all it will let me give
Answers
Answer:
SA = 76 yd²
Step-by-step explanation:
Use the formula given:
SA = 2(4)(2) + 2(4)(5) + 2(2)(5)SA = 16 + 40 + 20SA = 76The units are yards²
SA = 76yd²
-Chetan K
The surface area of the rectangular prism is 76 square yards
The dimensions of the rectangular prism are given as:
Length (l) = 4 yards
Width (w) = 2 yards
Height (h) = 5 yards
The surface area (SA) is calculated as:
SA =2lw + 2lh + 2wh
So, we have:
SA = 2 * 4 * 2 + 2 * 4 * 5 + 2 * 2 * 5
Evaluate the expression
SA = 76
Hence, the surface area of the rectangular prism is 76 square yards
Read more about surface area at:
https://brainly.com/question/6613758
The bakery’s expenses and receipts will both total $_____ after ___ hours
PLEASE HELP ♡
Answers
Expenses and receipts will total $2800 after 200 hours.
==============================================================
Explanation:
x = number of hours
Ron spends $11 an hour for operating costs. He spends 11x dollars on top of the $600 he spent on the equipment. The cost expression is 11x+600
On the flip side, Ron earns (on average) $14 an hour from the receipts or revenue. Overall, he earns 14x dollars.
Often in business applications, the owner is curious when s/he will break even. This is the point where the profit is 0 dollars, ie when cost = revenue.
So we'll set the two expressions equal to one another and solve for x.
cost = revenue
11x+600 = 14x
600 = 14x-11x
600 = 3x
600/3 = x
200 = x
x = 200
At the 200 hour mark is when Ron is likely to break even.
This x value leads to:
cost = 11x+600 = 11*200+600 = 2200+600 = 2800 dollarsrevenue = 14x = 14*200 = 2800 dollarsWe get the same cost for each expression, helping to confirm that x = 200 is correct.
The bakery's expenses and receipts will total $2800 after 200 hours.
what is 224 split into a 5 over 2 ratio
Answers
Answer:
160:64
Step-by-step explanation:
224/7 we divide by 7 because 5 and 2 equals 7
224/7=32
32*5 and 32*2
160 and 64
160:64
Convert 356.50 base 10 to hexadecimal
Answers
Answer:
164.8
Step-by-step explanation:
I used a calculator :)
In a card game, Joseph has 2 more than 4 times as many cards as Zachary has. Zachary has z cards. Which expression can be used to find the number of cards Joseph has?
Answers
Answer:
Joseph's cards = 4z + 2
Step-by-step explanation:
4 times as many = 4 times z or 4z
2 more= 4z + 2
Answer:
2x4=6 6x2= 12
Step-by-step explanation:
how do i convert 0,00023kl to ml
Answers
Explanation:
Volume:
0.00023 Kl to mL
= 230 mL
[tex]0.00023[/tex]×[tex]1[/tex]×[tex]10^{6}[/tex]
= 0.00023 x 1×10×10×10×10×10×10
= 0.00023 x 1000000
= 230 mL
Formula: kL x 1000000 = mL
Taylor needs to put 8 hats in each box.
She has 64 hats. Which equation can
help you find how many boxes Taylor
can fill?
Answers
Hi,
64 ÷ 8 = x
X= the number of boxes
Hope I could help~ have a good day/night :]
What is 152 divided by 5
Answers
Answer:
30.4
Step-by-step explanation:
125/5 = 30.4
hope i could help
=30.4
Step-by-step explanation:
152/5
=30.4
Hope that helped
HELP PLEASEEEEE!!!!!
Answers
1/3*9/5 = 9/15 = 3/5
Hope this helps!
What is the volume of a cylinder if ,
height = 21 cm
radius = 24 cm
and the formula is
[tex]\pi \: {r}^{2} h[/tex]
Answers
We are given that :
Height of cylinder = 21 cmRadius of cylinder = 24 cmTherefore, Volume :
➙ V = πr²h
➙ V = ( 22 / 7 ) × ( 24 )² × ( 21 )
➙ V = ( 22 / 7 ) × 576 × 21
➙ V = ( 22 × 576 × 21 ) / 7
➙ V = 266112 / 7
➙ V = 38016
ㅤㅤ ㅤㅤ~Hence, the volume of given cylinder is 38,016 cm³.
⳨ⲟⲅⲙⳙⳑⲇ ⳙ⳽ⲉ∂ :The volume of cylinder is it's density which tells us the amount by which it can be filled by any material or the the amount of material which can be immersed in it.
The Formula for volume of cylinder is given by :
ㅤㅤㅤㅤㅤ➙ πr²h
Answer:
The answer is 38016 cm³
Step-by-step explanation:
Given;Height (h) = 21 cmRadius (r) = 24 cmTo Find;The Volume of cylinderFormula;V = πr²hNow,
V = πr²h
V = 22/7 × (24)² × 21 cm
V = 22 × 24 × 24 × 3 cm
V = 38016 cm³
Thus, The volume of cylinder is 38016 cm³
-TheUnknownScientist 72
Brainliest if correct
Answers
2-6 = -4 so 2x-6x would equal -4x
Answer: -4x
Question 4 of 13
if v = (-2,5) and V2 = (4,-3), then the angle between the two vectors is
Round your answer to two decimal places.
please i need help asap
Answers
Answer:
148.67
Step-by-step explanation:
Angle between two vectors is equal to
Inverse cosine of their vectors dot product/ their magnitudes multiplied.
[tex]x = \cos {}^{ - 1} ( \frac{v \times v_{2} }{ |v| |v _{2} | } )[/tex]
Let first, find the dot product
[tex] < - 2,5 > \times < 4, - 3 > = - 8 - 15 = - 23[/tex]
Next we find magnitudes
[tex] - 2 {}^{2} + {5}^{2} = x {}^{2} [/tex]
[tex]29 = {x}^{2} [/tex]
[tex]x = \sqrt{29} [/tex]
[tex] {4}^{2} + {3}^{2} = {x}^{2} [/tex]
[tex]25 = {x}^{2} [/tex]
[tex]x = 5[/tex]
So the we are now,
[tex] \cos {}^{ - 1} ( \frac{ - 23}{ 5\sqrt{29}} ) [/tex]
We then get
[tex]148.67[/tex]
3 cm 7 cm 4 cm 8 cm area
Answers
Answer:
672 square inches
with area, multiply the side(s) by themselves for a single number, or their following numbers for more than 2 numbers.
Example
18in x 18in = 324 square inches4in x 2in x 5in x 9in = 360 square incheshope this helps!
Find the mean, median, mode, and range for this set of numbers:
14, 12, 7, 13, 6, 8
Answers
Answer:
In explanation
Step-by-step explanation:
Your mean=10
Median=10
Your mode is all of these numbers because none of them was in here more than once.
Your range would be 8 because range is highest number subtract lowest number so 14 is highest and 6 is lowest whats 14-6=8
Is it 1/10, 10, 7 or 5/72 please help thank you?
Answers
A rectangular sheet of metal has a length of 42 cm and a breadth of 100 cm. It is folded to form a cylinder with the breadth becoming the height. Calculate a) the radius of the cylinder formed b) the volume of the cylinder [Take 1 = 22) ]
Answers
Si el ancho de 100 cm de la hoja se convierte en la altura del cilindro, la longitud de 42 cm debe ser su circunferencia.[matemáticas]\ C = 2 \pi r \implica r = \frac{C}{2 \pi} \approx \boxed{6.68 \ \mathrm{cm}}.[/matemáticas]
[matemáticas]V = \pi r^2h = \dfrac{C^2 \cdot h}{4 \pi} = \dfrac{\left(42 \ \mathrm{cm}\right)^2 \cdot 100 \ \ mathrm{cm}}{4 \pi} \approx \boxed{14\,037.5 \ \mathrm{cm^3}}.[/math]
Step-by-step explanatio:
What numbers do i put in what box
Answers
Answer:
And I also glue a board at the bottom. So I can lift if I want my number one. And I could have these as a stand-up. In the piece I made some little cupcakes that I'm going to put inside.
Step-by-step explanation:
And put 1, 3 5 7 9 0 in box
he number of eggs in the refrigerator e decreased by 7 equals 19.
Answers
Jack's pancake recipe calls for 2 teaspoons baking powder and 3 cups of flour.
How much baking powder is needed if Jack uses 9 cups of flour?
Answers
Answer:
6 teaspoons of baking powder
PSG has given vertices:
P - (2, 3)
S - (1, 1)
G - (4, -3)
Reflect the image across the x-axis then reflect the image across the y-axis.
Final location of the image:
Answers
Answer:
is p because if you miltiply it
Find area of the figure
Answers
The area of the figure is the length of one side times the length of one side.
Area is 2 yards * 5/4 yards = 5/2 square yards
Hope that helps!
Find the center radius form for each circle having the given endpoints of a diameter.
22. (-4,5) and (6,-9)
26. ( 0,9) and (0,-9)
Answers
Answer:
22. [tex](x-1)^2+(y+2)^2=74[/tex]
26. [tex]x^2+y^2=81[/tex]
Step-by-step explanation:
The center of a circle is the midpoint of the diameter.
[tex]\textsf{midpoint}=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]
(where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are the endpoints of the diameter)
The radius of the circle is the distance between the center and an endpoint of the diameter.
[tex]\textsf{radius}=\sqrt{(a-x_1)^2+(b-y_1)^2}[/tex]
(where [tex](a,b)[/tex] is the center of the circle, and [tex](x_1,y_1)[/tex] is an endpoint of the diameter)
The center-radius form of a circle: [tex](x - a)^2+(y-b)^2=r^2[/tex]
(where (a, b) is the center and r is the radius)
Question 22
[tex]\textsf{Let }(x_1,y_1)=(-4.5)[/tex]
[tex]\textsf{Let }(x_2,y_2)=(6,-9)[/tex]
[tex]\textsf{center}=\left(\dfrac{-4+6}{2},\dfrac{5-9}{2}\right)=(1,-2)[/tex]
[tex]\textsf{radius}=\sqrt{(1+4)^2+(-2-5)^2}=\sqrt{74}[/tex]
⇒ equation of circle: [tex](x-1)^2+(y+2)^2=74[/tex]
Question 26
[tex]\textsf{Let }(x_1,y_1)=(0,9)[/tex]
[tex]\textsf{Let }(x_2,y_2)=(0,-9)[/tex]
[tex]\textsf{center}=\left(\dfrac{0+0}{2},\dfrac{9-9}{2}\right)=(0,0)[/tex]
[tex]\textsf{radius}=\sqrt{(0-0)^2+(0-9)^2}=9[/tex]
⇒ equation of circle: [tex]x^2+y^2=81[/tex]
What is the slope of line A?
A. -4
B. 4
C. -1/4
D. 1/4
Answers
Answer:
the answer is d have a great day
Answer:
-1/4
Step-by-step explanation:
Slope formula using two points: y2-y1/x2-x1.
0-1/4-0
-1/4
Sociology graduates, upon entering the workforce, earn a mean salary of
$25,000 with a standard deviation of $5,000. Assume that the distribution of salaries is Normal.
a) What is the probability that a randomly chosen salary exceeds $35,000?
b) What is the probability that a randomly chosen salary is less than $22,500?
c) What is the probability that a randomly chosen salary lies between
$25,000 and $37,500?
d) Find the 15th percentile of the sociology salaries.
e) Find the two salaries that are the cutoff values for the middle 60% of
salaries.
Answers
Using the normal distribution, it is found that:
a) There is a 0.0228 = 2.28% probability that a randomly chosen salary exceeds $35,000.
b) There is a 0.3085 = 30.85% probability that a randomly chosen salary is less than $22,500.
c) There is a 0.4938 = 49.38% probability that a randomly chosen salary lies between $25,000 and $37,500.
d) The 15th percentile of the sociology salaries is of $19,825.
e) The salaries are $20,800 and $29,200.
Normal Probability DistributionIn a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It 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, which is the percentile of X.In this problem:
The mean is of [tex]\mu = 25000[/tex].The standard deviation is of [tex]\sigma = 5000[/tex].Item a:
The probability is 1 subtracted by the p-value of Z when X = 35000, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{35000 - 25000}{5000}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a p-value of 0.9772.
1 - 0.9772 = 0.0228.
There is a 0.0228 = 2.28% probability that a randomly chosen salary exceeds $35,000.
Item b:
The probability is the p-value of Z when X = 22500, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{22500 - 25000}{5000}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a p-value of 0.3085.
There is a 0.3085 = 30.85% probability that a randomly chosen salary is less than $22,500.
Item c:
This probability is the p-value of Z when X = 37500 subtracted by the p-value of Z when X = 25000, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{37500 - 25000}{5000}[/tex]
[tex]Z = 2.5[/tex]
[tex]Z = 2.5[/tex] has a p-value of 0.9938.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{25000 - 25000}{5000}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a p-value of 0.5.
0.9938 - 0.5 = 0.4938
There is a 0.4938 = 49.38% probability that a randomly chosen salary lies between $25,000 and $37,500.
Item d:
This is X when Z has a p-value of 0.15, so X when Z = -1.035.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.035 = \frac{X - 25000}{5000}[/tex]
[tex]X - 25000 = -1.035(5000)[/tex]
[tex]X = 19825[/tex]
The 15th percentile of the sociology salaries is of $19,825.
Item e:
Due to the symmetry of the normal distribution, it is the 20th percentile and the 80th percentile, that is, X when Z = -0.84 and X when Z = 0.84.
20th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{X - 25000}{5000}[/tex]
[tex]X - 25000 = -0.84(5000)[/tex]
[tex]X = 20800[/tex]
80th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 25000}{5000}[/tex]
[tex]X - 25000 = 0.84(5000)[/tex]
[tex]X = 29200[/tex]
The salaries are $20,800 and $29,200.
More can be learned about the normal distribution at https://brainly.com/question/24663213
PLS HELP ME, PLS THIS IS DUE FIRST CORRECT ANSWER GETS BRAINLIEST
Answers
Answer:
B
Step-by-step explanation:
5÷7 = 0.7142857142857
