I suppose you're asking about the inverse Laplace transform of
[tex]F(s)=\dfrac4{(s-3)(s-2)}[/tex]
by way of the convolution theorem.
The theorem itself says that
[tex]\mathscr L^{-1}\{A(s)B(s)\}=(a*b)(t)[/tex]
where [tex]A(s)[/tex] and [tex]B(s)[/tex] are the Laplace transforms of [tex]a(t)[/tex] and [tex]b(t)[/tex], respectively.
Let [tex]A(s)=\frac1{s-3}[/tex] and [tex]B(s)=\frac1{s-2}[/tex]. Then
[tex]a(t)=\mathscr L^{-1}\left\{\dfrac1{s-3}\right\}=e^{3t}[/tex]
[tex]b(t)=\mathscr L^{-1}\left\{\dfrac1{s-2}\right\}=e^{2t}[/tex]
So we have
[tex]\mathscr L^{-1}\left\{\dfrac4{(s-3)(s-2)}\right\}=e^{3t}*e^{2t}[/tex]
Convolution is defined as
[tex](a*b)(t)=\displaystyle\int_0^t a(t-\tau)b(\tau)\,\mathrm d\tau[/tex]
The convolution of [tex]e^{3t}[/tex] and [tex]e^{2t}[/tex] is
[tex]e^{3t}*e^{2t}=\displaystyle\int_0^t e^{3(t-\tau)}e^{2\tau}\,\mathrm d\tau[/tex]
[tex]=\displaystyle e^{3t} \int_0^t e^{-\tau}\,\mathrm d\tau[/tex]
[tex]=e^{3t}(1-e^{-t})=e^{3t}-e^{2t}[/tex]
We end up with
[tex]\mathscr L^{-1}\left\{\dfrac4{(s-3)(s-2)}\right\}=\boxed{4(e^{3t}-e^{2t})}[/tex]
What is the value of y?
Answer:
y = 40
Step-by-step explanation:
The sum of the angles of a triangle are 180 degrees
y+10 + 2y +50 = 180
3y+60 = 180
Subtract 60 from each side
3y+60-60 = 180-60
3y = 120
Divide each side by 3
3y/3 = 120/3
y = 40
(c) Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect
Answer:
The probability is 0.357
Step-by-step explanation:
Given that,
The system has at least one type of defect,
Suppose, A certain system can experience three different types of defects.
Let [tex]A_{i}[/tex] (i = 1,2,3) denote the event that the system has a defect of type i.
Suppose that the following probabilities are,
[tex]P(A_{1})=0.11[/tex]
[tex]P(A_{2})=0.08[/tex]
[tex]P(A_{3})=0.05[/tex]
[tex]P(A_{1}\cup A_{2})=0.13[/tex]
[tex]P(A_{1}\cup A_{3})=0.13[/tex]
[tex]P(A_{2}\cup A_{3})=0.11[/tex]
[tex]P(A_{1}\cap A_{2}\cap A_{3})=0.01[/tex]
[tex]P(A_{1}\cap A_{2})=0.06[/tex]
[tex]P(A_{1}\cap A_{3})=0.03[/tex]
[tex]P(A_{2}\cap A_{3})=0.02[/tex]
[tex]P(A_{1}\cup A_{2}\cup A_{3})=0.14[/tex]
We need to calculate the probability that it has exactly one type of defect
Using given data
[tex]P=\dfrac{P(A_{1}\cap A_{2}'\cap A_{3}')}{P(A_{1}\cup A_{2}\cup A_{3})}+\dfrac{P(A_{1}'\cap A_{2}\cap A_{3}')}{P(A_{1}\cup A_{2}\cup A_{3})}+\dfrac{P(A_{1}'\cap A_{2}'\cap A_{3})}{P(A_{1}\cup A_{2}\cup A_{3})}[/tex]
[tex]P=\dfrac{P(A_{1})-P(A_{1}\cap A_{2})-P(A_{1}\cap A_{3})+P(A_{1}\cap A_{2}\cap A_{3})}{P(A_{1}\cup A_{2}\cup A_{3})}[/tex] + [tex]\dfrac{P(A_{2})-P(A_{1}\cap A_{2})-P(A_{2}\cap A_{3})+P(A_{1}\cap A_{2}\cap A_{3})}{P(A_{1}\cup A_{2}\cup A_{3})}[/tex]+ [tex]\dfrac{P(A_{3})-P(A_{1}\cap A_{3})-P(A_{2}\cap A_{3})+P(A_{1}\cap A_{2}\cap A_{3})}{P(A_{1}\cup A_{2}\cup A_{3})}[/tex]
P = [tex]\dfrac{P(A_{1})+P(A_{2})+P(A_{3})-2P(A_{1}\cap A_{2})-2P(A_{1}\cap A_{3})-2P(A_{2}\cap A_{3})+3P(A_{1}\cap A_{2}\cap A_{3})}{P(A_{1}\cup A_{2}\cup A_{3})}[/tex]
Put the value into the formula
[tex]P=\dfrac{0.11+0.08+0.05-2(0.06)-2(0.03)-2(0.02)+3(0.01)}{0.14}[/tex]
[tex]P=0.357[/tex]
Hence, The probability is 0.357
How do you do this problem?
Answer:
Your answer is absolutely correct
Step-by-step explanation:
The work would be as follows:
[tex]\int _0^{\sqrt{\pi }}4x^3\cos \left(x^2\right)dx,\\\\\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\=> 4\cdot \int _0^{\sqrt{\pi }}x^3\cos \left(x^2\right)dx\\\\\mathrm{Apply\:u-substitution:}\:u=x^2\\=> 4\cdot \int _0^{\pi }\frac{u\cos \left(u\right)}{2}du\\\\\mathrm{Apply\:Integration\:By\:Parts:}\:u=u,\:v'=\cos \left(u\right)\\=> 4\cdot \frac{1}{2}\left[u\sin \left(u\right)-\int \sin \left(u\right)du\right]^{\pi }_0\\\\[/tex]
[tex]\int \sin \left(u\right)du=-\cos \left(u\right)\\=> 4\cdot \frac{1}{2}\left[u\sin \left(u\right)-\left(-\cos \left(u\right)\right)\right]^{\pi }_0\\\\\mathrm{Simplify\:}4\cdot \frac{1}{2}\left[u\sin \left(u\right)-\left(-\cos \left(u\right)\right)\right]^{\pi }_0:\quad 2\left[u\sin \left(u\right)+\cos \left(u\right)\right]^{\pi }_0\\\\\mathrm{Compute\:the\:boundaries}:\quad \left[u\sin \left(u\right)+\cos \left(u\right)\right]^{\pi }_0=-2\\=> 2(-2) = - 4[/tex]
Hence proved that your solution is accurate.
Answer:
[tex]\int\limits^{\sqrt{\pi}}_0 {4x^3\cos(x^2)} \, dx=-4[/tex]
Step-by-step explanation:
So we have the integral:
[tex]\int\limits^{\sqrt{\pi}}_0 {4x^3\cos(x^2)} \, dx[/tex]
As told, let's use u-substitution first and then use integration by parts.
For the u-substitution, we can let u to be equal to x². So:
[tex]u=x^2[/tex]
Differentiate:
[tex]du=2x\, dx[/tex]
We can rewrite our integral as:
[tex]\int\limits^{\sqrt{\pi}}_0 {2x(2x^2)\cos(x^2)} \, dx[/tex]
Therefore, by making our u-substitution, our integral is now:
[tex]\int\limits {2u\cos(u)} \, du[/tex]
We also need to change our bounds. Substitute them into u. So:
[tex]u=\sqrt{\pi}^2=\pi\\u=(0)^2=0[/tex]
Therefore, our integral with our new bounds is:
[tex]\int\limits^{\pi}_{0} {2u\cos(u)} \, du[/tex]
Now, let's use integration by parts. Integration by parts is given by:
[tex]\int\limits {v}\, dy=vy-\int y\, dv[/tex]
(I changed the standard u to y because we are already using u).
Let's let v be 2u and let's let dy be cos(u). Thus:
[tex]v=2u\\dv=2\,du[/tex]
And:
[tex]dy=\cos(u)\\y=\sin(u)[/tex]
So, do integration by parts:
[tex]=2u\sin(u)-\int \sin(u)2\,du[/tex]
Simplify:
[tex]=2u\sin(u)-2\int \sin(u)\,du[/tex]
Evaluate the integral:
[tex]=2u\sin(u)+2\cos(u)[/tex]
Now, use the bounds. So:
[tex](2(\pi)\sin(\pi)+2\cos(\pi))-(2(0)\sin(0)+2\cos(0))[/tex]
Evaluate:
[tex]=(2\pi(0)+2(-1))-(0(0)+2(1))[/tex]
Simplify:
[tex]=(-2)-(2)[/tex]
Subtract:
[tex]=-4[/tex]
And we're done!
a right rectangular prism has these dimensions:
Length: 1 1/3
width: 5/6
height: 2/3
which answer is right?
Answer: D
Step-by-step explanation:
Since we know that the area of the square is 900 in², we can use that to find the length of each side.
900=x² [square root both sides]
x=30
Now that we know each side is 30 in, we know that the diameter of the circle is also 30 in. To find the radius, divide the diameter by 2.
2r=30 [divide both sides by 2]
r=15
With the radius, we can find the area of the circle.
A=π(15)² [exponent]
A=225π
Since the problem said to leave in terms of π, the area is A=225π in².
(6n-7)+(8n+2)=23 HELP FAST PLEASE
Answer:
n=2
Step-by-step explanation:
(6n-7)+(8n+2)=23
14n-5=23
14n=28
n=2
Probenecid is a drug used by some athletes to prevent the excretion of other substances into urine, thus lowering their detectable concentrations. A scientist makes three measurements of a urine sample known to contain probenecid and obtains a confidence interval of 11.8±0.5 mg/L at the 95% confidence level. What would happen to the confidence interval at the same confidence level if the number of measurements is increased to 10 assuming the standard deviation remains the same?
Answer:
The confidence interval will decrease
Step-by-step explanation:
Generally the confidence interval is mathematically represented as
[tex]\= x \pm z_c * \frac{s}{\sqrt{n} }[/tex]
Here [tex]z_c[/tex] is called the critical value of [tex]\frac{level \ of \ significance }{2}[/tex] obtained from the normal distribution table , s is the standard deviation , n is the sample size (number of measurements )
Now looking at the formula we that if the increase the sample size that the confidence level would decrease
So if the n is increased to 10
The confidence interval would decrease
......................................................
Answer:
x = 6
Step-by-step explanation:
Area of rectangle = length * width
= 10 * (x +2)
= 10*x + 10*2
= 10x + 20 square units
Area of triangle = [tex]\frac{1}{2}*bh[/tex]
[tex]=\frac{1}{2}*2*(x + 4)\\\\\\=1 * (x +4)\\\\\\[/tex]
= (x + 4) square units
Area of rectangle = 8* area of triangle
10x + 20 = 8 *( x+ 4)
10x + 20 = 8x + 32
Subtract 20 form both sides.
10x = 8x + 32 - 20
10x = 8x + 12
Subtract 8x form both sides
10x - 8x = 12
2x = 12
Divide both sides by 2
2x/2 = 12/2
x = 6
x = - 4y + 37x + 3y = 29 pls help i need this grade!!!!!!!!
A.
Write an expression for the perimeter?
B.
Write an expression for area
C.
Write Both as equations
Answer:
A. x+2+x+2
B. 2(x)
C. x+2+x+2=p
2(x)= a
The endpoints of (MP)are M(2,1) and P(12,6). If point K partitions (MP) in a ratio of MK:KP = 3:2, what are the coordinates of K?
Answer:
K(8, 4)
Step-by-step explanation:
Given:
M(2, 1), P(12, 6)
MK:KP = 3:2
Required:
Coordinates of K
SOLUTION:
Coordinates of K can be determined using the formula below:
[tex] x = \frac{mx_2 + nx_1}{m + n} [/tex]
[tex] y = \frac{my_2 + ny_1}{m + n} [/tex]
Where,
[tex] M(2, 1) = (x_1, y_1) [/tex]
[tex] P(12, 6) = (x_2, y_2) [/tex]
[tex] m = 3, n = 2 [/tex]
Plug in the necessary values to find the coordinates of K:
[tex] x = \frac{mx_2 + nx_1}{m + n} [/tex]
[tex] x = \frac{3(12) + 2(2)}{3 + 2} [/tex]
[tex] x = \frac{36 + 4}{5} [/tex]
[tex] x = \frac{40}{5} [/tex]
[tex] x = 8 [/tex]
[tex] y = \frac{my_2 + ny_1}{m + n} [/tex]
[tex] y = \frac{3(6) + 2(1)}{3 + 2} [/tex]
[tex] y = \frac{18 + 2}{5} [/tex]
[tex] y = \frac{20}{5} [/tex]
[tex] y = 4 [/tex]
The coordinates of K = (8, 4)
Wich is equivalent to 12(3 2/3)!?
Answer:
1/5 = 2/10 = 3/15 = 4/20
= 5/25 = 6/30 = 7/35 = 8/40
= 9/45 = 10/50 = 11/55 = 12/60
= 13/65 = 14/70 = 15/75 = 16/80
= 17/85 = 18/90 = 19/95 = 20/100
= 21/105 = 22/110 = 23/115 = 24/120
= 25/125 = 26/130 = 27/135 = 28/140
= 29/145 = 30/150 = 31/155 = 32/160
= 33/165 = 34/170 = 35/175 = 36/180
= 37/185 = 38/190 = 39/195 = 40/200
= 41/205 = 42/210 = 43/215 = 44/220
= 45/225 = 46/230 = 47/235 = 48/240
= 49/245 = 50/250 = 51/255 = 52/260
= 53/265 = 54/270 = 55/275 = 56/280
= 57/285 = 58/290 = 59/295 = 60/300
= 61/305 = 62/310 = 63/315 = 64/320
= 65/325 = 66/330 = 67/335 = 68/340
= 69/345 = 70/350 = 71/355 = 72/360
= 73/365 = 74/370 = 75/375 = 76/380
= 77/385 = 78/390 = 79/395 = 80/400
= 81/405 = 82/410 = 83/415 = 84/420
= 85/425 = 86/430 = 87/435 = 88/440
= 89/445 = 90/450 = 91/455 = 92/460
= 93/465 = 94/470 = 95/475 = 96/480
= 97/485 = 98/490 = 99/495 = 100/500
Step-by-step explanation:
all of these are equivalent
Answer: 36/11 Or 3 3/11
Step-by-step explanation:
when do historians believe that Euclid complete his work
Answer:
2.999 millions hours ago
Step-by-step explanation: :)
Help...plzz giving 25 points
Answer: the integers are closed under addition, multiplication, and subtraction, but NOT division.
Step-by-step explanation:
It takes Dariya 35 seconds to download 5 songs from the Internet. How can the number of seconds it would take Dariya to download 7 songs at this rate be determined? ( need help )
By dividing 35 by 7.
By multiplying 35 by 7.
By finding the unit rate and dividing it by 7.
By finding the unit rate and multiplying it by 7.
To determine the number of seconds it would take Dariya to download 7 songs at the same rate, we first find the unit rate and multiply it by 7.
Hence, option 4. By finding the unit rate and multiplying it by 7 is the right option.
Unit Rate = 7second/song.
Time to download 7 songs = 49 seconds.
What do we mean by unit rate?A unit rate is a quantity taken for a unit of another quantity.
How do we solve the given question?We are given that it takes Dariya 35 seconds to download 5 songs from the Internet. We are asked about the time it will take Dariya to download y songs at the same rate.
We will first determine the unit rate of the time taken for downloading a song.
5 songs take 35 seconds.
∴ 1 song takes 35/5 = 7 seconds.
∴ Unit rate is 7seconds/song.
Now to determine the time for downloading 7 songs, we multiply the unit rate by 7.
∴ Time taken to download 7 songs = 7 songs * unit rate
or, Time taken to download 7 songs = 7 songs * 7 second/song
or, Time taken to download 7 songs = 49 seconds.
∴ To determine the number of seconds it would take Dariya to download 7 songs at the same rate, we first find the unit rate and multiply it by 7.
Hence, option 4. By finding the unit rate and multiplying it by 7 is the right option.
Learn more about the unit rate at
https://brainly.com/question/620388
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Solve -6(4-x)≤-4(x+1) for x
No silly answers please.
Answer:
x≤2
Step-by-step explanation:
Answer:
x ≤ 2
Step-by-step explanation:
well, -6 x 4 = -24
-6 x X = -6x
so far, we have -24 -6x ≤-4(x+1)
-4 x X = -4x
and -4 x 1 = -4 , so we now have -24 -6x ≤ -4x + -4
now we subtract x on one side, and to the other side as well, and you do the same for the numbers!
A village fete has a children’s running race each year, run in heats of up to ten children. For each heat the first three contestants past the finishing line qualify for the final. There are three prizes in the final for 1st, 2nd and 3rd places. One year 29 children enter the race so there are three heats, of ten, ten and nine children. One year 29 children enter the race so there are three heats, of ten, ten and nine children. 1) What is the probability that three randomly chosen competitors win prizes? 2) What is the probability that two randomly chosen competitors win prizes? 3) How many ways are there to select ten competitors for the first heat? 4) Once the competitors have been selected for the first heat, how many different groups of three qualifiers are possible from this heat
Answer: 1) 1/3,654 2) 3/406 3) 72,684,900,288,000 4) 120
Step-by-step explanation:
1) First and Second and Third
[tex]\dfrac{3\ total\ prizes}{29\ total\ people}\times \dfrac{2\ remaining\ prizes}{28\ remaining\ people}\times \dfrac{1\ remaining\ prize}{27\ remaining\ people}=\dfrac{6}{21,924}\\\\\\.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad =\large\boxed{\dfrac{1}{3,654}}[/tex]
2) First and Second
[tex]\dfrac{3\ total\ prizes}{29\ total\ people}\times \dfrac{2\ remaining\ prizes}{28\ remaining\ people}=\dfrac{6}{812}=\large\boxed{\dfrac{13}{406}}[/tex]
[tex]3)\quad \dfrac{29!}{(29-10)!}=\large\boxed{72,684,900,288,000}[/tex]
[tex]4)\quad _{10}C_3=\dfrac{10!}{3!(10-3)!}=\large\boxed{120}[/tex]
helppp please I have limited time
Answer:
p = T - a - b
Step-by-step explanation:
Hope that helps, not sure if it is correct though.
Evaluate the expression and enter your answer in the box below.
-7 • (5+ 3) +216 ➗ 4
Answer:
= -2
-7(5+3)+216/4
-7(8)+216/4
-56+54
= -2
Introduction to Interval Notation
What is the domain and range?
2.
The domain of this function is 3≤x≤5 in interval notation that is [3,5]
The range is -3≤y≤3. In interval notation that is [-3,3]
4.
The domain of this function is -5≤x≤-1 in interval notation that is [-5,-1]
The range is 1≤y≤5. In interval notation that is [1,5]
:)
i need alot of help plzzzzzz 50 points and ill give brainliest
Answer:
y=4
y=8
y=12
y=16
2)4=19
5=22
addition of 3 to each value
3)17
4)3a^2
5)a^4
6)1
7)An arithmetic sequence is such that the difference between any term and the one immediately preceeding it is constant.
8)A geometric sequence is such in which each term is a constant multiple of its preceeding.
9)1.25
10)B.
Answer:
y=4
y=8
y=12
y=16
2)4=19
5=22
addition of 3 to each value
3)17
4)3a^2
5)a^4
6)1
7)An arithmetic sequence is such that the difference between any term and the one immediately proceeding it is constant.
8)A geometric sequence is such in which each term is a constant multiple of its proceeding.
9)1.25
10)B.
Step-by-step explanation:
A car is advertised as $288/month for 72 months. What is the cost of the car?
To
$14,400
$20,736
O
$28,800
O $17,280
Answer:
$20,736
Step-by-step explanation:
When these kind of advertisements are displayed,, it means the EMI cost per month shall be $288.
Since it provides the time period, that is 72 months,
Total cost of car in this case shall be $288 [tex]\times[/tex] 72 months = $20,736
Also, these include an interest factor, but overall it is = $20,736 only,
If one down payment for full price of the car is made the cost shall reduce by the interest amount, but since no interest rate is provided it shall be ignored.
Solve for x. 6x−24=x−2
Answer:
22/5
Step-by-step explanation:
6x-24=x-2
6x-x=-2+24
5x=22
x=22/5
If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability that all 4 cards are picture cards .
Answer:
If we defelt 4 cards from 52 cards
Step-by-step explanation:
P(b) favorable outcome /total number of outcomes
48/52
The probability that all 4 cards are picture cards is 0.00183
How many kinds are in a deck of 52?A standard 52-card deck comprises 13 ranks in each of the four French suits: clubs (♣), diamonds (♦), hearts (♥) and spades (♠). Each suit includes three court cards (face cards), King, Queen and Jack, with reversible (double-headed) images.
as, we know
A face card is any of the twelve cards in a deck which has a picture of a face. The face cards are kings, queens, and jacks.
So, the possibilities to get a picture card is are 3*4=12.
Now, he probability that all 4 cards are picture cards
=(12/52) * (11/51) * (10/50) * (9/49)
=( 11880 / 6497400 )
= ( 0.0018284237 )
= 0.00183
Learn more about deck of cards here:
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Graph the line with slope -1/2
passing through the point (-2, 4).
This image should help with that.
Evaluate the expression and enter your answer in the box below.
[(8 + 10) /3] +3 • (7-3)
Answer:
38.6
Step-by-step explanation:
(7)/(9)-2n+(1)/(9) please helpppppp
Answer:
-2n + 8/9
Step-by-step explanation:
Try that tell me I it don't work
help me please this is due today :(:
Answer:
F(-6, 3) --> F'(-6,3)
G(-4,3) --> G'(-8,3)
H(-2,4) --> H'(-10,4)
Answer:
F(-6,3)
G(-8,3)
H(-10,4)
Step-by-step explanation:
If your reflecting over the red line then all I do is first try to recreate the shape on the other side of the red line. (Like a reflection) And then look at the points it landed on. And that's your answer
State the domain and range in set notation
Answer:
Image result for State the domain and range in set notation
The domain is the set of all first elements of ordered pairs (x-coordinates). The range is the set of all second elements of ordered pairs (y-coordinates). Only the elements "used" by the relation or function constitute the range. Domain: all x-values that are to be used (independent values).
Step-by-step explanation:
Answer:
4
Step-by-step explanation:
The last option is not complete, but since the first three options are incorrect, the last option is correct
the ratio of two numbers is 3:7. the product of these 2 numbers is 189. what is the smaller of these numbers?
Answer:
9.
Step-by-step explanation:
I am assuming that the two numbers are positive.
Let the numbers be 3x and 7x.
Then 3x * 7x = 19
21x^2 = 189
x^2 = 9
x = 3.
So the smaller of these numbers is 3*3 = 9.
Answer:
9 if that does not work try - 21
Step-by-step explanation:
Let the two numbers be x and y
x/y = 3/7 which means that x is the smaller number.
x*y = 189
Cross multiply the top equation
7x = 3y
Divide by 7
x = 3/7 ^ y
x*y = 189
(3/7 y) * y = 189
Multiply by 7
3y^2 = 189*7
3y^2 = 1323
Divide by 3
y^2 = 441
Take the square root of both sides
y = 21
The question is a bit ambiguous. You could use - 9 and - 21 in which case the smallest number is -21. I think they intend 9 however.
x = 3/7 * 21
x = 3*3
x = 9
Check
x*y = 9 * 21 = 189