The Laplace transform of the given differential equation is (s^2 + 36)Y(s) = s/(s^2 + 36) + 3s + 6.
Solving for Y(s), we get Y(s) = (s/(s^2 + 36)) + (3s + 6)/(s^2 + 36).
Taking the inverse Laplace transform of Y(s), we obtain y(t) = sin(6t) + 3cos(6t) + 2sin(6t).
The Laplace transform of the given differential equation is s^2Y(s) + 36Y(s) = L{cos(6t)}.
Solving this algebraic equation, we find Y(s) = L{y(t)} = L{3} + 6s + L{cos(6t)} / (s^2 + 36).
Finally, taking the inverse Laplace transform of Y(s) gives us y(t).
a) Taking the Laplace transform of both sides of the given differential equation, denoting the Laplace transform of y(t) by Y(s), the equation becomes:
s^2Y(s) + 36Y(s) = L{cos(6t)}
b) Solving the algebraic equation for Y(s), we get:
Y(s) = L{y(t)} = L{3} + 6s + L{cos(6t)} / (s^2 + 36)
c) Taking the inverse Laplace transform of both sides of the equation obtained in part (b), we can solve for y(t):
y(t) = L^(-1){Y(s)}
a) We take the Laplace transform of both sides of the given differential equation, which involves transforming each term individually. The Laplace transform of the second derivative y''(t) is s^2Y(s), and the Laplace transform of 36y(t) is 36Y(s). The Laplace transform of cos(6t) can be obtained from the Laplace transform table.
b) By rearranging the equation from part (a), we isolate Y(s) to solve for it. The Laplace transform of y(0) is L{3}, which is equal to 3/s (since the Laplace transform of a constant is 1/s).
Similarly, the Laplace transform of y'(0) is L{6}, which is equal to 6. We substitute these values into the equation and simplify, resulting in Y(s) = L{y(t)} = L{3} + 6s + L{cos(6t)} / (s^2 + 36).
c) To find y(t), we need to take the inverse Laplace transform of Y(s). This involves finding the inverse Laplace transform of each term in Y(s) individually. The inverse Laplace transform of L{3} is 3 (since the inverse Laplace transform of a constant is the constant itself).
The inverse Laplace transform of 6s is 6δ(t), where δ(t) represents the Dirac delta function. The inverse Laplace transform of L{cos(6t)} / (s^2 + 36) can be obtained from the inverse Laplace transform table. Combining these terms gives us the expression for y(t).
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1. A package of sticky notes is in the shape of
a parallelogram. The dimensions of one
sticky note are shown. What is the area of
one sticky note?
A. 248.4 cm?
B. 124.2 cm2
C. 62.1 cm2
D. 22.7 cm2
Answer:
B. 124.2
Step-by-step explanation:
Area= b*h
Area= 13.5*9.2
Area = 124.2
Please Note* = Multiply
A right triangle has a 30° angle and the adjacent side to
the angle of length 6, draw the triangle and label all
sides and angles. Round to the nearest tenth.
Answer
let the triangle be supposed ABC
where angle BAC is 30 degree and angle ACB is 60 degree and where we know that angle ABC is right angled triangle which means angle ABC = 90 degree
the base side of triangle is 6cm which means ;
By taking angle A as reference angle
Hypotenuse = AC = x(let)
perpendicular = BC
base = AB = 6 cm
then by taking base and hypotenuse
b/h = AB/AC
or, cos (30 degree)= 6/ x
x =[tex]4\sqrt{x} 3[/tex] cm
that concluded that AC = [tex]4\sqrt{3}[/tex] cm
Now,
by phythagorous theorem,
[tex]h^{2} = p^{2} + b^{2}[/tex]
[tex]AC =\sqrt{AB^{2} + BC^{2} } AC = \sqrt{6^{2} + 4\sqrt{3} ^{2}} \\AC = 2\sqrt{21 } cm[/tex]
so, the length of side AB is 6 cm , BC is 4[tex]\sqrt{3}[/tex] cm and AC is 2[tex]\sqrt{21}[/tex] cm
Step-by-step explanation:
what are the answers to a, b, c, d?
Answer:
Step-by-step explanation:
A student government organization is interested in estimating the proportion of students who favor a mandatory "pass-fail" grading policy for elective courses. A list of names and addresses of the 645 students enrolled during the current quarter is available from the registrar's office. Using three-digit random numbers in row 10 of table 7. 1 and moving across the row from left to right, identify the first 10 students who would be selected using simple random sampling. The three-digit random numbers begin with 816, 283, and 610
Simple random sampling is a statistical method in which every member of the population has an equal chance of being chosen as a subject for the survey. In this case, a student government organization wants to estimate the proportion of students in favor of a mandatory "pass-fail" grading policy for elective courses, and they have a list of names and addresses of the 645 students enrolled in the current quarter from the registrar's office.
They can use simple random sampling to select a sample of students to participate in the survey. The first 10 students who would be selected using simple random sampling using three-digit random numbers in row 10 of table 7.1 and moving across the row from left to right are as follows:816283610752991768275354233410 The procedure for selecting a simple random sample of size n from a population of N subjects is as follows: Assign a unique identification number to every member of the population Obtain a list of identification numbers of the population. Use a random number generator to select n random numbers from 1 to N, without replacement, to identify the members of the sample. Identify the members of the sample using the randomly selected identification numbers. Simple random sampling is the most straightforward sampling method, and it produces samples that are unbiased and representative of the population. It is important to note that the size of the sample chosen should be large enough to make accurate inferences about the population.For such more question on organization
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8^x=1 Please solve for x.
Answer:
[tex] {8}^{x} = 1[/tex]
The exponent 0 makes the number as 1.
[tex]x = 0[/tex]
Answer:
the answer for x is = 0
Step-by-step explanation:
hope this helps
The volume of a square pyramid is 48 in^3 with a height of 4 inches. Find the length of one of the side of the base.
Answer:
6 inches
Step-by-step explanation:
area of square = 48 x 3 / 4 = 36 in^2
side = 6 in
Answer:
6 in
Step-by-step explanation:
The volume formula for a square-base pyramid of side length x and height h is V = (1/3)x²h. We want to solve this for the base side length x:
Multiplying both sides by (3/h) results in:
3V
----- = x² and so the side length, x, is √(3V/h).
h
Substituting 4 in for h and 48 in³ for V, we get: x = √(144 in³/4 in) = 6 in
Check: Does the volume formula given above result in 48 in³ when x = 6 in and h = 4 in?
V = (1/3)x²h = (1/3)(6 in)²(4 in) = 48 in³ YES: x = 6 in is correct.
Please help I’m struggling
Mario is buying a number of hamburgers from the local store that cost $2.90 each. He is also buying one packet of hamburger rolls at a cost of $4.75. He has $39.55 to spend at the store. Write and solve an inequality that shows how many hamburgers, h, Mario can afford to buy.
Write the inequality.
solve the inequality
Answer:
I think hamburgers=12
Step-by-step explanation:
Please help.
Is algebra.
Answer:
1 is C
2 is B
3 is C
Step-by-step explanation:
Answer:
give em brainliest because i have no idea
Step-by-step explanation:
could someone help pls
Answer:
40%
Step-by-step explanation:
6/15=favorable amount/total amount
6/15=2/5
2/5*20
Answer: 40%
6/15 can be simplified to 2/5. multiply by 20 on both sides to get 40/100 (40%)
7 ⅓ ÷ 225 =
answer this question plz
Answer:
0.03259..
Step-by-step explanation:
Prove the following:
(2cos x) / (cos 2x + 1) = sec x
The equation (2cos x) / (cos 2x + 1) = sec x is proven to be true.
We start with the expression on the left-hand side and simplify it step by step to show that it is equivalent to sec x.
(2cos x) / (cos 2x + 1) = (2cos x) / ((2cos^2 x - 1) + 1) [Using the double-angle formula cos 2x = 2cos^2 x - 1]
= (2cos x) / 2cos^2 x [Simplifying the numerator and denominator]
= cos x / cos^2 x [Cancelling out the factor of 2]
Now, we can simplify the expression further using the identity sec x = 1 / cos x:
cos x / cos^2 x = 1 / cos x = sec x.
Therefore, we have proved that (2cos x) / (cos 2x + 1) is equal to sec x.
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Which statement explains the type of function that is represented by the equation
y=x2 +9?
(q11) Find the center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively.
The center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively is located at the point (1.6, 2.2).
To find the center of mass of the system of objects, we need to consider the masses and positions of each object. Let's go through the steps to calculate the center of mass:
Assign variables to the masses and coordinates of the objects:
Object 1: Mass = 2, Coordinates = (-1, 2)
Object 2: Mass = 3, Coordinates = (1, 1)
Object 3: Mass = 5, Coordinates = (3, 3)
Calculate the total mass of the system by summing the masses of the objects:
Total mass = 2 + 3 + 5 = 10
Calculate the weighted average of the x-coordinates and y-coordinates to find the center of mass:
Center of mass (x-coordinate) = (m1 * x1 + m2 * x2 + m3 * x3) / Total mass
Center of mass (y-coordinate) = (m1 * y1 + m2 * y2 + m3 * y3) / Total mass
Substituting the values:
Center of mass (x-coordinate) = (2 * -1 + 3 * 1 + 5 * 3) / 10 = ( -2 + 3 + 15) / 10 = 16 / 10 = 1.6
Center of mass (y-coordinate) = (2 * 2 + 3 * 1 + 5 * 3) / 10 = (4 + 3 + 15) / 10 = 22 / 10 = 2.2
Therefore, the center of mass of the system of objects is located at the point (1.6, 2.2).
The center of mass represents the average position of the system, taking into account the masses and positions of the objects. It is a point where the system can be considered to be balanced. Calculating the center of mass is important in various areas of physics, such as studying the motion, stability, and collisions of objects.
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an aquarium measures 16 in. × 8 in. × 10 in. how many liters of water does it hold? how many gallons?
The aquarium with dimensions 16 in. × 8 in. × 10 in. can hold approximately 30.6 liters of water and approximately 8.09 gallons.
To calculate the volume of the aquarium, we multiply its length, width, and height. Since the dimensions are given in inches, we need to convert the volume to liters and gallons.
First, let's calculate the volume in cubic inches:
Volume = Length × Width × Height = 16 in. × 8 in. × 10 in. = 1280 cubic inches.
To convert cubic inches to liters, we divide the volume by 61.024:
Volume in liters = 1280 in³ / 61.024 = 20.96 liters (rounded to two decimal places).
To convert liters to gallons, we divide the volume by 3.78541:
Volume in gallons = 20.96 liters / 3.78541 = 5.53 gallons (rounded to two decimal places).
Therefore, the aquarium can hold approximately 30.6 liters of water and approximately 8.09 gallons.
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The graph solves this system of equations. Enter the solution in the boxes. –2y = –4x + 8 x + y = 5
Answer:
x=2-y/2
Step-by-step explanation:
The solution of system of equations –2y = –4x + 8 and x + y = 5 is (3, 2).
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The given system of equations are –2y = –4x + 8
x + y = 5
x=5-y
Nu=ow plug in this in the first equation
-2y=-4(5-y)+8
-2y=-20+4y+8
Take the variable terms on one side and constant on other side.
-6y=-12
Divide both sides by 6
y=2
Now put in the equation of x
x=5-2=3
Hence, the solution of system of equations –2y = –4x + 8 and x + y = 5 is (3, 2).
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HELP!!! answer quickly pls
Find the area of the figure
Answer:
A=153.94
Step-by-step explanation:
My Samsung charger broke so I went to Five Below to get a new one. They were on sale for $15,35 (not so five below). There
was a 6% tax too. How much did the new charger cost?
( I don’t need the explanation, only the right answer)
Answer:
The answer to your question is 16.27
Explanation:
You would start with the equation,
15.35 + 6% = C (Cost)
15.35 · 6% = 0.921
15.35 + 0.921 = 16.271
Round the one down, since it is less than four, and you get 16.27
Hope this helps.
waffletowne
Which expression represents the perimeter of the rectangle below?
A. 13x - 31
B. 13x - 55
C. 26x - 62
D. 26x-124
The perimeter of the rectangle given is P = 26x - 62.
What is the perimeter of a rectangle?
The perimeter of a rectangle is the sum of its sides. Mathematically -
Perimeter = [P] = 2(L + B)
Given is a rectangle with its length and breadth.
We can write the perimeter of the rectangle as follows -
[P] = 2(L + B)
L = 5x + 12
B = 8x - 43
Therefore -
P = 2(L + B)
P = 2(5x + 12 + 8x - 43)
P = 2(13x - 31)
P = 26x - 62
Therefore, the perimeter of the rectangle given is P = 26x - 62.
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To finish a board game Yanis needed to land on the last square rolling a sum of 6 with two dice. She was dismayed that it took her eight tries. Should she have been surprised?
Yanis should not have been surprised that it took her eight tries to roll a sum of six with two dice to finish the board game.
When rolling two dice, the total number of possible outcomes is 36 (6 sides on the first die multiplied by 6 sides on the second die). Out of these 36 possible outcomes, there are five ways to obtain a sum of six: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). This means that the probability of rolling a sum of six is 5/36.
Since each roll is independent of the previous rolls, the probability of not rolling a sum of six in a single roll is 31/36 (36 possible outcomes minus the 5 favorable outcomes). To calculate the probability of not rolling a sum of six in eight consecutive rolls, we raise this probability to the power of eight: (31/36)^8 ≈ 0.282.
Therefore, there was approximately a 28.2% chance that Yanis would not roll a sum of six in eight tries. This is a significant probability, indicating that it was not unlikely for her to take eight attempts to land on the last square. Thus, she should not have been surprised by the outcome.
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PLEASE HELP ME! 20 POINTS! NO BOTS -.-
Simplify
(2½)² x (0,5)²
Answer:
1.5625
Step-by-step explanation:
2½ = 5/2
(5/2)² = 25/4
25/4 = 6.25
0.5² = 0.25
6.25 x 0.25 = 1.5625
What Statistic information we can get from a histogram?
A-Standard Deviation
B-Interquartile Range
C-Mean
SOMEONE PLS HELP SHEHSJHSHS
Answer:
1. Quadrant II
2. Quadrant IV
3. Y-axis
Step-by-step explanation:
Describe the attributes of An acute triangle: An obtuse triangle: A right triangle: How many equal sides does an equilateral triangle have: isosceles: scalene pls help I will give you 15 pointss
Answer:
An acute triangle - interior angles are always less than 90 degrees with different side measures.
An obtuse triangle - one of the interior angles is more than 90 degrees. if one angle measures more than 90 degrees, then the sum of the remaining two angles is less than 90 degrees.
An right triangle - a triangle with a right angle (90 degrees)
Equilateral triangle has three equal sides. Isosceles triangles has two equal side lengths. Scalene triangles has no equal side lengths
Show step-by-step solution. Compute manually.
1. Carlo borrows 100,000 pesos at an annual interest rate of 12% compounded quarterly. The loan is to be repaid by equal quarterly payments for 2 years. Determine each payment. Then make an amortization schedule for this loan.
Carlo's loan of 100,000 pesos at a 12% annual interest rate compounded quarterly for 2 years requires equal quarterly payments of approximately 7,974.51 pesos.
The amortization schedule shows the breakdown of each payment, including the interest and principal portions, over the 8-payment period.
To compute the equal quarterly payments for Carlo's loan, we can use the formula for the equal payment amount in an amortizing loan:
Payment = (Principal * Interest Rate) / (1 - (1 + Interest Rate)^(-n))
Where:
Principal = 100,000 pesos (loan amount)
Interest Rate = 12% per year (convert to quarterly rate by dividing by 4: 0.12/4 = 0.03)
n = number of payments (2 years * 4 quarters per year = 8 payments)
Let's calculate the payment amount:
Payment = (100,000 * 0.03) / (1 - (1 + 0.03)^(-8))
Payment = 7,974.51 pesos
Therefore, each quarterly payment for Carlo's loan is 7,974.51 pesos.
To create an amortization schedule, we can calculate the interest and principal portion of each payment for each quarter:
Quarter | Beginning Balance | Payment | Interest | Principal | Ending Balance
1 | 100,000 | 7,974.51| 3,000 | 4,974.51 | 95,025.49
2 | 95,025.49 | 7,974.51| 2,851.27 | 5,123.24 | 89,902.25
3 | 89,902.25 | 7,974.51| 2,697.07 | 5,277.44 | 84,624.81
4 | 84,624.81 | 7,974.51| 2,537.87 | 5,436.64 | 79,188.17
5 | 79,188.17 | 7,974.51| 2,373.66 | 5,600.85 | 73,587.32
6 | 73,587.32 | 7,974.51| 2,204.37 | 5,769.14 | 67,818.18
7 | 67,818.18 | 7,974.51| 2,029.89 | 5,944.62 | 61,873.56
8 | 61,873.56 | 7,974.51| 1,850.13 | 6,124.38 | 55,749.18
This amortization schedule shows the payment number, beginning balance, payment amount, interest portion, principal portion, and ending balance for each quarter.
Note: The values in the amortization schedule have been rounded for simplicity, but it's advisable to use the exact values for accurate calculations.
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On a certain hot summer's day, 344 people used the public swimming pool. The daily prices are $1.50 for children and $2.00 for adults. The receipts for admission totaled $537.00. How many children and how many adults swam at the public pool that day?
Answer: 302 children and 42 adults
Step-by-step explanation:
x= Children y= Adults
x+ y = 344
1.50x + 2y= 537
Then I would use elimination and multiply the first equation by -2.
-2x-2y= -688
+1.5x+2y= 537
Add the equations and get
-0.5x= -151
Divide by -0.5
x= 302
Then plug x back into the equation.
302+y=344
Subtract 302 from both sides.
y=42
Simplify the following expressions to have fewer terms (MIDDLE SCHOOL)
5x-3+(4x-6)+2
4y+3-(8x-2)
Answer:
Install math calculator
Thank you
Answer:
x=1\9
Step-by-step explanation:
first part:
5x-3+4x+2
5x+4x-3+2
9x-1
9x=1
x=1\9 answer.
5. What might be the dimensions of a
cylindrical container that holds 750 mL
of juice?
Answer:
radius of 10.9254843 and height of 10.9254843
Step-by-step explanation:
The equation for the volume of a cylinder is V = 2 pi r h
If V (volume) = 750, find for r and h
(I'm just going to make the radius and height the same thing)
750 = 2 pi r r
375 = pi r^2
119.366207 = r^2
10.9254843 = r