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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Which term describes a line segment that connects a vertex of a triangle to a point on the line containing the opposite side, so that the line segment is perpendicular to that line?
Answer: C
Step-by-step explanation:
-x= x – 10
How many solutions does this equation have?
A.Infinitely many solutions
B.No solutions
C.Two solutions
D. One solution
Answer:
D
Step-by-step explanation:
Answer: One if any.
-x = x - 10 Add x to both sides
-x+x = x + x - 10 Combine
0 = 2x - 10 Add 10 to both sides
10 = 2x - 10+10 Combine
10 = 2x Divide by 2
10/2=2x/2
x = 5
Definite Answer: One Solution
The number 100 can be written as the sum of a 1-digit prime number and a 2-digit prime number. What is the product of these prime numbers?
Answer:
Answer: 3 + 97
Step-by-step explanation:
Just off the top of my head, I'm pretty sure that the answer you want is 3 and 97.
The single digit primes are 2 3 5 7
2 + 98 is what you are forced to use. 98 is not prime
3 + 97 is the answer I gave.
5 and 95 Ninety five is not prime (5 divides into it).
7 and 93 but 93 is not prime. 3 will divide into it.
Using your answer from #7, find the perimeter of the triangle.
Answer From #7: 9
Answer:
58
Step-by-step explanation:
x=9
3(9) + 1 + 4(9) -8 + (9) - 7 = 58
Step by Step solution please.
Thanks
Derivation of f(x):
[tex]f'(x) = \frac{g(x)*8x-4x^2*g'(x)}{(g(x))^2} -\frac{1}{x+2}[/tex]
Since g(3) = 6 and g'(3) = 3
[tex]f'(3) = \frac{g(3)*8*3-4(3)^2*g'(3)}{(g(3))^2} -\frac{1}{3+2} \\f'(3) = \frac{6*8*3-4*9*3}{36} -\frac{1}{5} \\f'(3) = \frac{36}{36} -\frac{1}{5} =1-\frac{1}{5} =\frac{4}{5}\\ f'(3) = 0.8[/tex]
Thus f'(3) = 0.8
Hope that helps!
A couple of identities I used in derivation:
[tex]\frac{d}{dx} (ln x) = \frac{1}{x}[/tex]
Find the surface area of a 6*6 h cube
Answer:
Surface Area = 6×62 = 216 inches2
Step-by-step explanation:
Formula for surface area of cube is 6 into side²
EXPLANATION
Cube has all sides of equal dimensions and like that it has 6 faces.
So total surface area of cube is equal to the combined area of all the six faces.
Now each face has equal dimensions so same area
Therefore,
area of a single face =side²
Therefore, area of entire cube = 6 times the area of a single face
= 6 * side²
PLS ANSWER THIS PLS PLS PLS
Answer:
20 shirts = $445.00
40 shirts = $541.00
60 shirts = $637.00
Step-by-step explanation:
multiply 20, 40 and 60 by 4.8 then add 349.
20*4.8 = 96
96+349 = 445
40*4.8 = 192
192+349 = 551
60*4.8 = 288
288+349 = 637
A ratio of boys to girls in the classroom is 4 to 5 if there are 20 girls how many boys would there be
Answer:
16 to 20
Step-by-step explanation:
Let's do our math.
4 to 5
8 to 10
12 to 15
16 to 20
In order to get to 20, you'd have to multiply 5 by 4. So in order to get an equal ratio, you multiply 4 by 4 and get 16.
I hope this helps!
-No one
A trapezoidal prism undergoes a dilation. The surface area of the pre-image is 35 m². The surface area of the image is 315 m². The height of the pre-image is 5. 8 m. What is the height of the image? Enter your answer as a decimal in the box. M.
The original object is called pre-image, and its dilated version is called image. The height of the image for the considered case is 17.4 m
How does dilation affect length, area, and volume of an object?Suppose a figure (pre image) is dilated (dilated image) by scale factor of k.
So, if a side of the figure is of length L units, and that of its similar figure is of M units, then:
L = k × M
where 'k' will be called as scale factor.
The linear things grow linearly like length, height etc.
The quantities which are squares or multiple of linear things twice grow by square of scale factor. Thus, we need to multiply or divide by k² to get each other corresponding quantity from their similar figures' quantities.
So area of first figure = k² × area of second figure
Similarly,
Volume of first figure = k³ × volume of second figure.
It is because we will need to multiply 3 linear quantities to get volume, which results in k getting multiplied 3 times, thus, cubed.
For this case, we're given that;
Surface area of pre-image = 35 m²Surface area of dilated image = 315 m²Height of pre-image = 5.8 mHeight of dilated image.Let the scale fator of dilation be k.
Then:
So area of pre-image = k² × area of dilated image
Thus, we get:
[tex]35 = k^2 \times 315\\\\k = \sqrt{\dfrac{35}{315}} = 1/3[/tex]
(took positive root as scaling is done by non-negative factors as only magnitude matters mostly)
Thus, we get:
Height of pre-image = (1/3) × Height of dilated image (say h)
[tex]5.8 = (1/3) \times h\\h = 3 \times 5.8 = 17.4 \rm \: m[/tex]
Thus, the original object is called pre-image, and its dilated version is called image. The height of the image for the considered case is 17.4 m
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How many liters are in 950mL
Answer:
0.950 L
Step-by-step explanation:
ml/1000 = L
Scientists put marks on 72 moose as part of an effort to estimate the local moose population. Later, the scientists flew over the area and counted a total of 430 moose, of which 43 were marked. To the nearest whole number, what is the best estimate for the moose population?
Answer:
72x430/43
Step-by-step explanation:
Find the average squared distance between the points of r{(x,y): 0x, 0y} and the point (,)
The average squared distance between the points is their variance
The average squared distance is 8/3
How to determine the average squared distance?The given parameters are:
R={(x,y): 0<=x<=2, 0<=y<=2}
The point = (2,2).
f(x,y) = (x-2)² + (y-2)²
The squared distance is calculated as:
D² = f(x,y) = (x-2)² + (y-2)²
Where the area (A) is:
A = xy
Substitute the maximum values of x and y in A = xy
A = 2 * 2
A = 4
The minimum values of x and y are 0.
So, the limits for the integrals are 0 to 2
The integral becomes
[tex]D\² = \int \int (x-2)\² + (y-2)\² dx dy[/tex]
Expand
[tex]D\² = \int \int x\² -4x + 4 + (y-2)\² dx dy[/tex]
Evaluate the inner integral with respect to x from 0 to 2
[tex]D\² = \int \frac 13x^3 -2x^2 + 4x + x(y-2)\² |\limits^2_0 dy[/tex]
Expand
[tex]D\² = \int [\frac 13(2)^3 -2(2)^2 + 4(2) + (2)(y-2)\²] dy[/tex]
Simplify
[tex]D\² = \int \frac 83 + (2)(y-2)\² dy[/tex]
Expand
[tex]D\² = \int \frac 83 + 2y^2-8y + 8 \ dy[/tex]
Evaluate the integral with respect to y from 0 to 2
[tex]D\² = [\frac 83y + \frac 23y^3- 4y^2 + 8y ]|\limits^2_0[/tex]
Expand
[tex]D\² = \frac 83*2 + \frac 23*2^3- 4*2^2 + 8*2[/tex]
D² = 32/3
The average squared distance (AD²) is calculated as:
AD² = D²/A
So, we have:
AD² = 32/3 [tex]\div[/tex] 4
Evaluate the quotient
AD² = 32/12
Simplify
AD² = 8/3
Hence, the average squared distance is 8/3
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The average squared distance between the points of R = {(x,y): 0<=x<= h, 0<=y<=k } is 8/3.
How do determine the average squared distance?
From the given parameters R = {(x,y): 0<=x<= h, 0<=y<=k }
The squared distance from point (h, k) is calculated as
D² = f(x,y) = (x-2)² + (y-2)²
Where the area (A) will be
A = xy
Substitute the maximum values of x and y
A = 2 × 2 = 4
The minimum values of x and y are 0.
So, The integral becomes
[tex]D^{2} = \int \int(x-2)^2 +(y-2)^2 dx dy\\\\D^{2} = \int \int\limits^2_0 (x)^2 -4x + 4+(y-2)^2 dx dy\\\\D^{2} = \int1/3(x)^3-2x^{2} +4x + 4+x(y-2)^2 dy\\\\D^{2} = \int8/3+2y^{2} -8y +8 dy[/tex]
[tex]D^2 = {8/3y +2/3y^2 -4y^2+8y[/tex]
By expanding we get,
D^2 = 32/3
The average squared distance (AD²) is calculated as:
AD² = D²/A
AD² = 32/3 4
AD² = 32/12
AD² = 8/3
Hence, the average squared distance is 8/3
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Manny has 36 coins in
his piggybank.
3 of the coins are silver dollars.
• The remaining coins are dimes (10€), quarters (25€) and nickels (5¢).
• There is an equal number of dimes, quarters and nickels.
How many nickels does Manny have?
Answer:
q=50
d=22
Step-by-step explanation:
54 x 8181 x 12 x 8271 x 1301 x 812
Answer:
4.6320447e+16
I used a calculator for this LOL.
Please answer the question in the picture :) tysm
Answer:
a= 50
b=50
c=110
d=20
Answer:
Step-by-step explanation:
This is a kite. So the two triangles formed by the longest diagonal being drawn in form 2 congruent triangles.
Therefore c = 110 degrees.
d = 20 degrees.
a and b are found by noting that every triangle has 180 degrees.
110 + 20 + a = 180 Combine the left.
130 + a = 180 Subtract 130 from both sides
a = 180 - 130
a = 50 degrees
b also equals 50 degrees. Congruent triangles
Answer
a = 50b = 50c = 110d = 20
The residents of a downtown apartment building are starting a rooftop garden. They will plant 4 beds of vegetables for every 1 bed of flowers, plus have 5 extra beds of vegetables. They plan to plant
a total of 30 garden beds. In the system of equations shown, v is the number of vegetable beds and f is the number of flower beds.
v=4f +5
v+f= 30
How many vegetable beds and how many flower beds will be in the rooftop garden?
Answer:
89
Step-by-step explanation:
Evaluate:
-5-(x+y)
2
where x=
3 and y = 6
answer
For each
x
in the expression you substitute
−
3
and for each
y
you substitute
5
. Make sure to handle the signs correctly. Remember
(
−
−
=
+
)
(
(
2
⋅
−
3
)
−
5
)
3
−
(
5
−
(
−
3
⋅
5
)
−
6
)
(
−
6
−
5
)
3
−
(
5
−
(
−
15
)
−
6
)
−
11
⋅
3
−
(
5
+
15
−
6
)
−
33
−
(
20
−
6
)
−
33
−
14
−
47
Answer linkStep-by-step explanation:
Let's see
-5-(x+y)Put x and y
-5-(3+6)-5-9-14Done!
Which effect did World War II have on the United States?
a) The United States became the richest, most powerful nation in the world.
b) The war caused the United States to enter another severe depression.
c) The war had little effect on the United States.
d) The United States became second in power only to England.
Answer:
the anwer is A cause jobs and industials had 96 percent and corperate profits after taxes doubled
Answer:
a
Step-by-step explanation:
world War 2 did in fact cause the United States to become the richest and most powerful
Find the value of c that makes the trinomial a perfect square. .
x² – 24x7c
Answer:
144
Step-by-step explanation:
The equation to find c is:
[tex](\frac{b}{2} )^{2}[/tex]
So put -24 into b:
[tex](\frac{-24}{2} )^{2}[/tex]
Which simplifies to:
144
So if you want to test this:
x^2 - 24x + 144
is
(x-12)^2
A long solenoid with radius r = 7 cm and n = 3,170 turns/m carries a current i = 6. 7 a. What is the magnitude of the magnetic field in the central area inside the solenoid, in t?
The magnitude of the magnetic field in the central area inside the solenoid, in T is 0.0267 T
Magnetic field inside solenoidThe magnetic field inside the central area of the solenoid is given by B = μ₀ni where
μ₀ = permeability of free space = 4π × 10⁻⁷ Tm/A, n = number of turns per unit length = 3,170 turns/m and i = current in solenoid = 6.7 ASince B = μ₀ni
Substituting the values of the variables into the equation, we have
B = μ₀ni
B = 4π × 10⁻⁷ Tm/A × 3,170 turns/m × 6.7 A
B = 4π × 10⁻⁷ Tm/A × 21239 A-turns/m
B = 84956π × 10⁻⁷ T
B = 266897.15 × 10⁻⁷ T
B = 0.026689715 T
B ≅ 0.0267 T
So, the magnitude of the magnetic field in the central area inside the solenoid, in T is 0.0267 T
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The magnitude of the magnetic field in the central area inside the considered solenoid (assuming in vaccum), is approx 2.67 × 10^{-2} T
What is the measurement of the magnetic field inside a long solenoid?Magnetic field = permeability constant x relative permeability x turn density x current
B = [tex]k\mu_0nI = \mu nI[/tex]
where turn density is calculated as:
n = N/L
Where N represents the number of turns of the wire and
L represents the length of the solenoid.
Assuming that the solenoid is placed in vaccum, we have:
[tex]\mu = k\mu_0 = \mu_0 = 4\pi \times 10^{-7} \: \rm T m/A[/tex]
And it is given that:
n = 3174 turns/mI = 6.7 amperesThus, we get:
[tex]B = \mu n I = 4\pi \times 10^{-7} \times 3170 \times 6.7 \approx 2.67 \times 10^5 \times 10^{-7} \\\\B = 2.67 \times 10^{5 + (-7)}B = 2.67 \times 10^{-2} \: \tm T[/tex]
Thus, the magnitude of the magnetic field in the central area inside the considered solenoid (assuming in vaccum), is approx 2.67 × 10^{-2} T
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Alexis bought 50 collectible stamps at a garage sale. Each week, she purchases 30 stamps to add to her
collection. How many weeks will she need to purchase stamps to have more than 500 stamps?
Answer:
14 (exactly 500) or 15 (over 500)
Step-by-step explanation:
We know that Alexis has already bought 50 stamps.
50 + 30 = 80
We need to subtract to see how many stamps she needs.
500 - 80 = 420
We divide by 30 to see about how many weeks it would take.
420 / 30 = 14
We can add 1 more week because she wants to have more than 500 stamps.
hope this helps :)
The base of a solid S is the region bounded by the parabola x^2=8y
Cross-sections perpendicular to the yyy-axis are equilateral triangles.
Determine the exact volume of solid SSS.
Step-by-step explanation:Updated answer is the second one
Maggie makes a scale drawing of a deck. The scale for the actual deck to the scale drawing is 7ft to 1in. The dimensions of the deck are 28ft by 17.5ft. What are the dimensions of the deck in the drawing? show your work.
Answer:
4 inches by 2.5 inches
Step-by-step explanation:
The scale factor is 7 feet/1 inch.
This can be written as 7'/1" or as 1"/7'
28 feet converts to (28')*(1"/7') = 4"
17.5 feet converts to (17.5)*(1"/7') = 2.5"
The drawing dimensions are 4 inches by 2.5 inches.
Answer:
Step-by-step explanation:
A fuel tank is in the shape of a right circular cylinder. The radius of the base and height of the tank are
10
feet and
25
feet, respectively. Due to safety measures, the tank can be filled to only
80
%
of its capacity.
If
100
cubic feet of fuel are equal to
2. 78
gallons, what is the maximum number of whole gallons of fuel the tank can contain due to safety measures?
The maximum number of whole gallons of fuel the tank can contain due to safety measures is 174.584 gallons
Volume of a cylinderv = πr²hr = radius
h = height
Therefore,
r = 10 ft
h = 25 ft
The tank can only be 80% filled due to safety measures. Therefore,
v = 3.14 × 10² × 25
v = 3.14 × 100 × 25
v = 3.14 × 2500
v = 7850 ft³
80% capacity of 7850 ft³ is as follows:
80 / 100 × 7850 = 628000 / 100 = 6280 ft³
100 ft³ = 2.78 gallons
6280 ft³ = ?
cross multiply
maximum number of whole gallons = 6280 × 2.78 / 100 = 17458.4 / 100 = 174.584 gallons
Therefore, the maximum number of whole gallons of fuel the tank can contain due to safety measures is 174.584 gallons
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The bases of a trapezoid are 11 meters and 14 meters. It’s height is 10 meters. What is the area of the trapezoid?
Answer:
120 m²
Step-by-step explanation:
A = (1/2) · (p + q) · h
A = (1/2)(14 + 10)(10)
A = (1/2)(24)(10)
A = 120
Hope this helps :)
please help me out with this I will give brainliest if the answer is right
Answer:
60 cm
Step-by-step explanation:
The perimeter of a polygon means adding up all the side lengths.
In this case, we need to add up 13, 12, 8, 8, and 19
Adding all of these gets us 60
simply 2x + x + x + 2y x 3y x y
Answer:
[tex]4x + 6y^{3} x^{2}[/tex]
Step-by-step explanation:
Simplify
In ΔNOP,the measure of ∠P=90°, NP = 16, ON = 65, and PO = 63. What ratio represents the cosecant of ∠N?
Answer: 65/63
Step-by-step explanation:
The cosecant of angle N is the reciprocal of the sine of angle N.
Using SOHCAHTOA, the sine of angle N is 63/65.
This means that the cosecant of angle N is 65/63.
Two basketballs fell, without bouncing, from a net and a chair at the same time. The ball which fell from the net rolled 30 feet. Note: Figure is not drawn to scale The horizontal and vertical distances traveled by the balls formed similar triangles. How far did the second ball roll from the chair? A. 6 feet B. 22 feet C. 4 feet D. 3 feet
Answer:
c
Step-by-step explanation:
Please help fast! (I'm not trying to rush ya)
Answer:
"LOCATION: Most temperate, deciduous (leaf-shedding) forests are located in the eastern United States, Canada, Europe, China, Japan, and parts of Russia. Deciduous forests are broken up into five zones."
Step-by-step explanation:
✖ Brazil
✓ China
✓ United States
✓ Russia
✖ Chile
Hopefully this helps you!
-Keira