Consider the following initial-value problem. Y'' − 5y' = 8e4t − 4e−t, y(0) = 1, y'(0) = −1 find ℒ{f(t)}, for f(t) = 8e4t − 4e−t. (write your answer as a function of s. )
The Laplace transform of the non-homogeneous second order differential equation is [tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex].
How to determine the Laplace transform of a non-homogeneous second order differential equationA Laplace transform is a frequency-based algebraic substitution method used to determine the solutions of differential equations in a quick and efficient manner.
In this question we shall use the following Laplace transforms:
[tex]\mathcal {L} \{f(t) + g(t)\} = \mathcal {L} \{f(t)\} + \mathcal {L}\{g(t)\}[/tex] (1)
[tex]\mathcal {L} \{\alpha\cdot f(t)\} = \alpha\cdot \mathcal {L} \{f(t)\}[/tex] (2)
[tex]\mathcal{L} \left\{y^{(n)} \right\} = s^{n}\cdot \matcal {L}\{f(t)\}-s^{n-1}\cdot y(0) -...-y^{(n)}(0)[/tex] (3)
[tex]\mathcal {L} \{e^{-a\cdot t}\} = \frac{1}{s+a}[/tex] (4)
Now we proceed to derive an expression fo the Laplace transform of the solution of the differential equation:
[tex]y'' -5\cdot y' = 8\cdot e^{4\cdot t}-4\cdot e^{-t}[/tex]
[tex]s^{2}\cdot \mathcal {L}\{f(t)\}-5\cdot y(0) - y'(0) - 5\cdot s \cdot \mathcal {L} \{f(t)\} +5\cdot y(0) = \frac{8}{s-4}-\frac{4}{s+1}[/tex]
[tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex]
The Laplace transform of the non-homogeneous second order differential equation is [tex]\mathcal {L} \{f(t)\} = \frac{4\cdot (s+6)}{s\cdot (s-4)\cdot (s+1)\cdot (s-5)} +\frac{1}{s} -\frac{1}{s\cdot (s-5)}[/tex]. [tex]\blacksquare[/tex]
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The Laplace transform Y'' − 5y' = 8e4t − 4e−t when y(0) = 1, y'(0) = −1 for f(t) = 8e4t − 4e−t would be [tex]\frac{4(s+6)}{s.(s-4)(s+1).(s-5)} + \frac{1}{s} - \frac{1}{s.(s-5)}[/tex].
What is Laplace transform?A Laplace transform is a frequency-based algebraic substitution method used to determine the solutions of differential equations quickly and efficiently.
The Laplace transform
L[ f(t) + g(t) ] = Lf(t) + Lg(t)
Also,
L [[tex]y^{n}[/tex]] = [tex]s^{n} . L[ f(t) ] - s^{n-1} .y(0) .....y^{n}(0)[/tex]
We have
Y'' − 5y' = 8 . e^4t − 4 . e−t
[tex]s^{2} . L[ f(t) ] - 5. y(0)- y'(0) .....y^{n}(0)[/tex]
y(0) = 1, y'(0) = −1
for f(t) = 8e4t − 4e−t.
= [tex]\frac{8}{s-4} - \frac{4}{s+1}[/tex]
L [f(t)] = [tex]\frac{4(s+6)}{s.(s-4)(s+1).(s-5)} + \frac{1}{s} - \frac{1}{s.(s-5)}[/tex]
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Suppose $500 is divided into two bank accounts. One account pays 15% simple interest per year and the other pays 10%. After three years there is a total of $200 in interest between the two accounts. How much was invested into the bank account that pays 15% simple interest (rounded to the nearest cent)?
Answer:
x = amount invested in the 15% account
x*.15*3 + (500-x)*.1*3 = 200
x = 333.33
so, 500-333.33 = 166.67 at 10%
Step-by-step explanation:
Answer:
x = amount invested in the 15% account
x*.15*3 + (500-x)*.1*3 = 200
x = 333.33
so, 500-333.33 = 166.67 at 10%
Step-by-step explanation:
sorry if i did it wrong
Find the value of the unknown angle x.
The diagram is not drawn to scale.
Please answer. Thank you!
Answer:
x = 40°Step-by-step explanation:
This is a quadrilateral. A quadrilateral is a plane figure that has four sides or edges, and also has four corners or vertices.
The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc.
Answer:
Angle x = 40°
Step-by-step explanation:
All angles of any quadrilateral sum up to 360°.
So here, already measure of 3 angles are given.
So if we add up all angles + Angle x then it sums up to 360°.
So, the following steps will lead you to the answer:
73° + 157° + 90° + Angle x = 360°
(Now add up the measure of the three given angles)
320° + Angle x = 360°
(Now, through transposition moves 320° to the RHS) (Remember when transposing the signs change)
Angle x = 360° - 320°
Finally,
Angle x = 40°
Hope it helps!!!
How many 1/6 pound servings are there in 4/5 of a pound of chicken?
Answer: [tex]4\frac{4}{5}[/tex]
Step-by-step explanation:
To see how many [tex]\frac{1}{6}[/tex] pound servings are in [tex]\frac{4}{5}[/tex] of a pound of chicken we must divide [tex]\frac{4}{5}[/tex] by [tex]\frac{1}{6}[/tex]
[tex]\frac{4}{5}[/tex]÷[tex]\frac{1}{6}[/tex]
When dividing by a fraction you can take the reciprocal (flip the numerator and denominator) and you can multiply instead of dividing
[tex]\frac{4}{5}[/tex]×[tex]\frac{6}{1}[/tex]=[tex]\frac{24}{5}[/tex]=[tex]4 \frac{4}{5}[/tex]
Answer:
24/5, or 4.8, servings
Step-by-step explanation:
Let s represent the number of servings. We have:
[tex] \frac{1}{6} s = \frac{4}{5} [/tex]
[tex]s = \frac{24}{5} = 4.8[/tex]
Please help I was able to do the hard part but I don’t know how to crack the code
Step-by-step explanation:
what do you mean by "the hard part" ?
anyway,
we need to sum the numbers in every column creating 5 sums.
then we need to multiply these 5 numbers (product) and divide the result by 16 (we need 1/16 of that product).
so, we have
5 + 1 = 6
1 + 6 + 4 + 8 = 19
2 + 1 + 3 = 6
2 + 5 = 7
1 + 0 + 7 = 8
6×19×6×7×8 = 38,304
38,304 / 16 = 2394
that is your code : 2394
Marcelle and Amia each buy two “Treat Dispenser” shirts from Ava’s website for $22 each. They live in South Dakota, where the sales tax is 6.4%. Marcelle calculates his total bill by finding the sales tax for each individual shirt, multiplying by 2, and then adding it to the retail price of both shirts. Amia calculates her bill by adding up the total cost of both shirts, finding the sales tax of the sum, and then adding it to the retail price of both shirts. Whose method is correct, and what was the total bill for each of their orders?
Answer: Amia is correct
Step-by-step explanation:
the total bill would be $46.82
The height of a triangle is six times the base. Write an expression for the area of the triangle and let x be the base.
Answer:
If
Hight = 6×base
And
Base = X
Then
Hight = 6 × X
Which is equivalent to
Hight = 6X
Why every values in maths has an approximation ?
( Example : The value of π,... )
Answer:
Approximating has always been an important process in the experimental sciences and engineering, in part because it is impossible to make perfectly accurate measurements. Approximation also arises because some numbers can never be expressed completely in decimal notation. In these cases approximations are used.
Answer:
Approximating has always been an important process in the experimental sciences and engineering, in part because it is impossible to make perfectly accurate measurements. Approximation also arises because some numbers can never be expressed completely in decimal notation. In these cases approximations are used.
Step-by-step explanation:
Identify the area of the rhombus.
Answer:
A=130cm2
Step-by-step explanation:
13x10=130
Which counterexample shows that the conjecture "Every parallelogram is also a rectangle" is false?
A. A square is both a rhombus and a parallelogram.
B. A pentagon has five sides but is not a parallelogram.
C. A rectangle is a parallelogram.
D. A rhombus is a parallelogram but is not a rectangle.
Answer:
D
Step-by-step explanation:
A doesn't work because a square is not a rhombus
B is out of context
C is true
D makes the most sense to the original question
20,10,5, ...
Find the 9th term.
Answer:
0.078125.
Step-by-step explanation:
This is a geometric sequence with first term a1 = 20 and common ratio r = 10/20 = 5/10 = 0.5.
nth term = a1 r^(n -1) so here:
9th term = 20(0.5)^(9-1)
= 20(0.5)^8
= 0.078125.
Karissa rides her bicycle for 4 hours and is 22 miles from her house. After riding for 8 hours, she is
42 miles away
What is Karissa's average rate during her trip?
__ miles per hour
Answer:
5.5
Step-by-step explanation:
We nned to divide to solve this!!
22/4 = 5.5
She is going 5.5 mph!!
Have an amazing day!!
Please rate and mark brainliest!!!!
HELP: What is another name for Angle 1?
Answer:
it could be many answers but my guess is acute
Answer:
(3) We can also name an angle by placing any number or symbol at the vertex in the INTERIOR of the angle. So, angles can also be called angle 1 or angle 2 or angle 4, etc. Sides AB and AC of the given angle are actually rays.
Step-by-step explanation:
Ex 65 : Étagères , merci d’ avance
Answer:
Voir ci-dessous.
Step-by-step explanation:
1.
OA/OD = 36/64 = 9/16
OB/OC = 27/48 = 9/16
m<AOB = m<COD
triangle AOB ~ trianlge DOC
m<BAO = m<CDO
AB || CD
2.
AB/CD = OA/OD
AB/80 = 36/64
AB/80 = 9/16
16AB = 9 × 80
AB = 9 × 5
AB = 45
AB = 45 cm
3.
AB = 45
BC = OC + OB = 48 + 27 = 75
AB² + AC² = BC²
45² + AC² = 75²
AC² = 5625 - 2025
AC = √3600
AC = 60 cm
hauteur = 5 × 2 cm + 4 × 60 cm
hauteur = 10 cm + 240 cm
hauteur = 250 cm
1. Three friends are at a restaurant for dinner. They decide to split the $90. 00 check equally among the three of them. A. If they decide to leave a 20% tip, what will the total bill be?
b. What will each person pay if the total bill including tip is divided by three?
Answer:
1A.)The tip would be $18, thus meaning the total will shoot up to 108$.
1B.) Divide 3 by 18, and you get $6 per person.
Step-by-step explanation:
Hope this helps!~
The total bill with the tip will be $108.00. Considering this value each person will pay $36.00
PercentageThe percentage is the given definition for a fraction whose denominator is equal to 100. It is represented by %. See the example: [tex]\frac{20}{100}=\frac{1}{5}=0.2*100=20\%[/tex]
The question gives:
3 friendstotal bill without the tip - $90.0020% tipLetter AFor solving this item you should sum the value of the total bill without the tip with the value equivalent to a 20% tip.
Tip=[tex]\frac{20*90}{100} =\frac{1800}{100}=18[/tex]
Total bill= 18+90=108
Letter BFor solving this item you should divide the value of the total bill with the tip by 3 (friends).
Individual Payment= 108/3 = 36
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A middle school took all of its 6th grade students on a field trip to see a ballet. The
students filled 702 seats, leaving 35% vacant. How many total seats were in the
theater?
By working with percentages, we will see that there are 1,080 seats in the theater.
How many total seats were in the theater?
We know that the students filled 702 seats, leaving 35% vacant.
Then 702 seats represents the (100% - 35% = 65%) of the total number of seats.
So, if the total number of seats is S, we can write:
(65%/100%)*S as the 65% of S.
Then we solve:
(65%/100%)*S = 702
0.65*S = 702
S = 702/0.65 = 1,080
There are 1,080 seats in the theater.
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Help me please. Don't mind that I answered it, I put whatever so that I can move on to the next page. Show how you did the work!
Answer:
true false false
Step-by-step explanation:
3,4,5 is a common triangle (proof sqrt(3^2+4^2) = 5)
5,6,7 cant exist sqrt(5^2+6^2) doesnt equal 7
5,5,10 cant exist sqrt(5^2+5^2) doesnt equal 10
Please help I will give brainliest.
Answer:
See below
Step-by-step explanation:
1. Two numbers that multiply to 17 and add to 18.
17 x 1 = 1717 + 1 = 181 and 172. Two numbers that multiply to 36 and add to -20.
-18 x -2 = 36-18 + -2 = -20-18 and -2Find the area of each figure.
[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]
Area of a Parallelogram is ~
[tex]\qquad \sf \dashrightarrow \:base \times height[/tex]
[tex]\qquad \sf \dashrightarrow \:7 \times 5[/tex]
[tex]\qquad \sf \dashrightarrow \:35 \: {cm}^{2} [/tex]
What is the value of x?
Answer:
20
Step-by-step explanation:
3x=x+40
Because Vertical Angles
2x=40
x=20
simplify the expressions by applying the properties of potentiation...
Answer:
[tex] \dfrac{x^5}{y^{10}} [/tex]
Step-by-step explanation:
[tex] \dfrac{x^2y^{-6}}{x^{-3}y^4} = [/tex]
[tex] = x^{2 - (-3)}y^{-6 - 4} [/tex]
[tex] = x^5y^{-10} [/tex]
[tex] = \dfrac{x^5}{y^{10}} [/tex]
simplify 1/2+3/4-2/7×(1/3+2)
Answer:
the answer is 7/12..
thank u
In the 2004 Summer Olympics, the U.S. won 11 more medals than Russia. Together they won 195 medals. How many medals did the U.S. win?
Answer:
The U.S. won 101 medals.
Step-by-step explanation:
101
Hope it helped and have a great day!
help me please i so need help
Answer:
9
It is 9 because 9 is the only number that multiplied by itself is 81.
The following structure is composed of three right rectangular prisms: two that each measure 12 inches by 10 inches by 5 inches and one right rectangular prism that measures 10 inches by 8 inches by 36 inches. What is the total volume of the structure? Show your work and/or explain your reasoning..
The volume of the stucture made from three rectangular prism is 4080 ft³
What is volume?Volume is the amount of space occupied by a three dimensional shape or object.
The volume of the rectangular prism = length * width * height = 12 * 10 * 5 = 600 ft³
The volume of the other prism = length * width * height = 10 * 8 * 36 = 2880 ft³
Volume of the structure = 600 ft³ + 600 ft³ + 2880 ft³ = 4080 ft³
The volume of the stucture made from three rectangular prism is 4080 ft³
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Bonjour !
Please solve My maths problem.
If X = 2y +6, then find the value of x³ - 8y³ - 36xy -216.
Hint - answer is 0
Kindly Show steps
I desperately need help with this question.
Correct person will be appreciated and brainlist will be awarded
Irellevant answer won't be tolerated and will be reported immediately.
Thank you in. advance !
y = 1
Step-by-step explanation:
X = 2y + 6
X³ - 8y³ - 36xy - 216 = 0
(subsitute)
(2y + 6)³ - 8y³ - 36(2y + 6)y -216 = 0
(concentrate on first part)
(2y + 6)(2y + 6)(2y + 6)
(4y² + 12y + 12y + 36)(2y + 6)
(4y² + 24y + 36)(2y + 6)
8y³ + 24y² + 48y² + 144y + 72y + 216
(back to full equation)
8y³ + 72y² + 216y + 216 - 8y³ - 36(2y + 6)y - 216 = 0
(factor the last part)
-36(2y + 6)y
y(-72y - 216)
-72y² - 216y
(back to main equation)
8y³ + 72y² + 216y + 216 - 8y³ - 72y² - 216y - 216 = 0
(start to cancel out)
8y³ and -8y³ cancel
72y² + 216y + 216 - 72y² - 216y - 216 = 0
72y² and -72y² cancel
216y + 216 - 216y - 216 = 0
216y and -216y cancel
216 - 216 = 0
those cancel
0y = 0
add 1 to both sides
y = 1.
(I hope this helped)
Guys can you help me with this please
Answer:
AC=18.1cm
Step-by-step explanation:
When we subtract 11-4=7cm we get one cathetus of the right triangle a=7cm also we know hypotenuse which is c=16cm.
We will use Pythagorean theorem to find the second cathetus b
b=√16∧2-7∧2= √256-49= √207 ≈ 14.39cm =DC =>
AC =√(DC)∧2+(AD)∧2= √14.39∧2+11∧2= √207+121= √328= 18.1cm
AC=18.1cm
find the sum of first 40 terms of 11+14+17+...
Answer:
S₄₀ = 2780
Step-by-step explanation:
there is a common difference between consecutive terms in the sequence, that is
14 - 11 = 17 - 14 = 3
this indicates the sequence is arithmetic with sum to n terms
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2a₁ + (n - 1)d ]
where a₁ is the first term and d the common difference
here a₁ = 11 and d = 3 , then
S₄₀ = [tex]\frac{40}{2}[/tex] [ (2 × 11) + (39 × 3) ]
= 20( 22 + 117)
= 20 × 139
= 2780
Find the area of the figure below. Use 3.14 to approximate .
Round your answer to the nearest tenth.
Answer:
145.1 m^2
Step-by-step explanation:
(8x15)+(3.14x8)
120+25.12
145.12
round
145.1
if a new baseball is labeled $36 what would the new price be if the sign above said 41% off
Answer:
$24.24
Step-by-step explanation:
0.41 x 36 = 14.76
36 - 14.76 =21.24
Hope this helps :)