Mathematics High School

Use The Laplace Transform To Solve The Following Initial Value Problem:Y" -8y'+20y=0 Y(0)=1,Y'(0)=4First, Using Y For The Laplace Transform Of Y(T) , I.e., L{Y(T)} ,Find The Equation You Get By Taking The Laplace Transform Of The Differential Equation: _=0Now Solve For Y(S)=?By Completing The Square In The Denominator And Inverting The
Use the Laplace transform to solve the following initial value problem:
y" -8y'+20y=0 y(0)=1,y'(0)=4
First, using Y for the Laplace transform of Y(t) , i.e., L{Y(t)} ,
find the equation you get by taking the Laplace transform of the differential equation: _=0

Answers

Answer 1

Taking the Laplace transform of the differential equation y" -8y'+20y=0, we get:

s^2Y(s) -sy(0) -y'(0) -8(sY(s) -y(0)) +20Y(s) = 0

Substituting y(0)=1 and y'(0)=4, we get:

s^2Y(s) -s -4 -8sY(s) +8 +20Y(s) = 0

Simplifying, we get:

(s^2-8s+20)Y(s) = s+12

Therefore, we have:

Y(s) = (s+12)/(s^2-8s+20)

Next, we need to find the inverse Laplace transform of Y(s) to obtain the solution y(t). However, we can simplify the expression for Y(s) using partial fractions:

Y(s) = (s+12)/[(s-4)^2 + 4]

Using partial fractions, we can write:

Y(s) = A/(s-4) + [B(s-4)+C]/[(s-4)^2 + 4]

Multiplying both sides by the denominator, we get:

(s+12) = A[(s-4)^2 + 4] + B(s-4)+C

Setting s=4, we get:

16A = 16B

Therefore, A=B.

Differentiating both sides with respect to s, we get:

1 = 2A(s-4) + B(s-4) + (B(s-4)+C)

Setting s=4, we get:

1 = 2B

Therefore, B=1/2 and A=1/2.

Substituting these values into the partial fractions expression, we get:

Y(s) = 1/2/(s-4) + [1/2(s-4)+C]/[(s-4)^2 + 4]

To find C, we can multiply both sides by [(s-4)^2 + 4] and then take the limit as s approaches infinity, which gives:

C = 0

Therefore, we have:

Y(s) = 1/2/(s-4) + 1/2(s-4)/[(s-4)^2 + 4]

Taking the inverse Laplace transform, we get:

y(t) = (1/2)e^(4t)cos(2t) + (1/4)e^(4t)sin(2t)

Therefore, the solution to the initial value problem is:

y(t) = (1/2)e^(4t)cos(2t) + (1/4)e^(4t)sin(2t)

Note that the domain of y(t) is t >= 0.

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Related Questions

(y-4)^2-(6+y)(y-6)при y= -7/8

Answers

Answer:

Step-by-step explanation:

Let's begin by simplifying the expression:

$(y-4)^2-(6+y)(y-6) = y^2 - 8y + 16 - y^2 + 36 = -8y + 52$

Since we've found are final expression, we can plug in the value we are given:

$-8y + 52 = 7 + 52 = \fbox{59}$

players who have played for other teams hitting 50 or more homeruns before playing for the ny yankess

Answers

The New York Yankees have had several players who hit 50 or more home runs while playing for other teams before joining the Yankees. These players demonstrated exceptional power and home run hitting abilities prior to joining the Yankees. Here, statistics can be used to analyze and interpret baseball data

Throughout the history of baseball, the New York Yankees, known for their rich tradition and success, have attracted numerous talented players. Some of these players had already established themselves as elite power hitters before joining the Yankees. They were able to achieve the impressive feat of hitting 50 or more home runs while playing for other teams prior to their tenure with the Yankees.

These players often brought their exceptional home run hitting skills and offensive prowess to the Yankees, adding to the team's formidable lineup. They became integral parts of the Yankees' success, providing additional power and run production. Their ability to consistently hit 50 or more home runs before joining the Yankees showcased their talent and established them as premier sluggers in the league.

The Yankees' history is filled with remarkable players who have made significant contributions to the team's success, and those who hit 50 or more home runs before joining the Yankees serve as a testament to the caliber of talent that has graced the organization.

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Please help!
A survey found the distribution of some
families by size, and is as follows.
Family Size 2 3 4 5 6 7 8
Frequency 87 50 61 31 16 3 2
Find the probability of family with
3 people.
P(3) = [?]

Answers

To find the probability of a family having 3 people, we need to know the total number of families surveyed. We can find this by adding up the frequencies:

Total families = 87 + 50 + 61 + 31 + 16 + 3 + 2 = 250

The probability of a family having 3 people is the frequency of families with 3 people divided by the total number of families:

P(3) = 50/250 = 1/5 = 0.2

Therefore, the probability of a family having 3 people is 0.2 or 20%.

Dynamo Electronics Inc produces and sells various types of surge protectors. For one specifc division of their manufacturing, they have a total cost for producing x units of C(x)=81x+99,000 and a total revenue of R(x)=191x. How many surge protectors must Dynamo produce and sell to break-even? surge protectors (round to the nearest whole number) How much cost will Dynamo incur at their break-even point? $ (round to two decimal places if necessary)

Answers

If Dynamo Electronics Inc produces and sells various types of surge protectors and for one specific division of their manufacturing, they have a total cost for producing x units of C(x)=81x+99,000 and a total revenue of R(x)=191x, then Dynamo must produce 901 surge protectors and sell to break even and Dynamo will incur $171,900 at their break-even point.

The break-even point is the level of production at which a company's income equals its expenses.

To calculate the number of surge protectors and sell to break-even, follow these steps:

The break-even point is calculated as Total cost (C) = Total revenue (R). By substituting the values in the expression we get 81x + 99,000 = 191x ⇒110x = 99,000 ⇒x = 900. So, the number of surge protectors Dynamo must produce and sell to break even is approximately 901 units.

To calculate the cost at the break-even point, follow these steps:

The value of x can be substituted in the expression for the total cost of producing x units, Total cost (C) = 81x + 99,000 So, C(900) = 81 × 900 + 99,000 = 72,900 + 99,000 = 171,900. Therefore, Dynamo will incur a cost of approximately $171,900 at their break-even point.

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amelia’s bank account earns interest annually. the equation shows her starting balance of $200 and her balance at the end of four years, $220.82. at what rate, r, did amelia earn interest?

Answers

Answer: $50

Step-by-step explanation:

Write an equation for each representation of a quadratic function given below.

Answers

Let, us assume that quadratic equation is :

$y = ax^2 + bx + c$

Now, putting x = 0 and y = -4 :

$c = -4$

Also, putting x = 1 and y = -3 :

$ax^2 + bx + c =y\\\\a + b - 4 = -3 \\\\a + b = 1$ ....2)

Also, putting x = 2 and y = 0 :

$ax^2 + bx + c =y\\\\4a + 2b + c = 0\\\\4a + 2b = 4\\\\2a - b = 2$....3)

Solving equation 2 and 3, we get :

a = 1 and b = 0

Therefore, the quadratic equation is given by $y = x^2- 4$ .

Use the diagram to identify a segment parallel to CF

DG
AD
DC
AB

Answers

Answer:

Step-by-step explanation:

This is a rectangular prism. BE & AH are also parallel to CF. but they r not in the answers.

Once your group has worked through the storming stage and can go on and work together, the group has achieved group?

Answers

Once your group has worked through the storming stage and can go on and work together, the group has achieved group cohesion.

Group cohesion refers to the degree of unity, harmony, and cooperation among group members. It is characterized by a sense of belonging, trust, and mutual respect within the group. Achieving group cohesion is crucial for the group's success as it enhances communication, cooperation, and productivity. It fosters a supportive and positive group climate where members feel comfortable expressing their ideas and opinions.

Group cohesion can be developed through various strategies such as team-building activities, open and respectful communication, establishing common goals, and addressing conflicts constructively. It is important to note that group cohesion is not a one-time achievement but a continuous process that requires ongoing effort and maintenance from all group members.

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question is in picture

Answers

Answer:

Step-by-step explanation:

the first thing I did to smome but you know 5AM was a l

Which equation correctly shows that (x^2)^4=x^8

Answers

Number 4 is the correct answer

Solution:

Using law of exponent:
x^2*x^2*x^2*x^2 is the same as x^(2+2+2+2)= x^8

Which ordered pair makes both inequalities true? y≤ -x + 1 y > x
(-3,5)
(-2, 2)
(-1, -3)
(0, -1)

NEED ASAP!!!!

Answers

Answer:

(-2,2)

Step-by-step explanation:

First, let's look at the 2 inequalities:

y ≤ -x + 1

y > x

Just by looking at the second inequality, we can remove any ordered pairs where x is greater than or equal to y. This eliminated the ordered pairs (-1,-3) and (0,-1). Now, the answer has to be (-3,5) or (-2,2). Knowing this, we can plug x and y into the equation and see if either of the answers are true.

(-3,5):

x = -3

y = 5

y ≤ -x + 1

5 ≤ -(-3) + 1

5 ≤ 3 + 1

5 ≤ 4

NOT TRUE

(-2,2)

x = -2

y = 2

2 ≤ -(-2) + 1

2 ≤ 2 + 1

2 ≤ 3

TRUE

So, since plugging in the ordered pair (-2,2) into the inequality gives us a true answer while (-3,5) doesn't, the order pair (-2,2) must be out answer.

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3. A bee can fly at an average rate of 7 miles/hour. If a bee flew nonstop
for 18 hours, how far will it have traveled?

Answers

If a bee can fly at an average rate of 7 miles per hour and it flew nonstop for 18 hours, we can find the distance it traveled by multiplying its average speed by the time it flew:

distance = speed x time

distance = 7 miles/hour x 18 hours

distance = 126 miles

Therefore, the bee would have traveled 126 miles in 18 hours.

Answer:

126 m (Miles)

Step-by-step explanation:

distance travelled = speed×time = 7×18 = 126 miles

How many distinguishable permutations for the letters in the word "reassessed" are there?

Answers

"reassessed" has a total of 10 characters, one of which (e) occurs 3 times and another (s) which occurs 4 times.

Taking each character to be distinct, there would be a total of 10! permutations. But we don't want to do that, and instead want, for instance,

rEassessEd

and

rEassessEd

to count as the same permutation. So we divide the previous total by the number of ways we can permute each set of repeated characters. For example, 3 e's can be rearranged in 3! ways.

So the total number of distinguishable permutations would be

10!/(3! × 4!) = 25,200

How do you find the scale factor given a center of dilation an image and a Preimage?

Answers

STEPS to find the find the scale factor given a center of dilation an image and a Preimage are given below

What is Scale factor, pre-image and dilation?

Dilation: A dilation is a stretch or a contraction in a figure or point's size and placement.

Resizing an item uses a transition called dilation. Dilation is used to enlarge or contract the items. The result of this transformation is a picture with the same shape as the original. However, there is a variation in the shape's size. The initial form should be stretched or contracted during a dilatation. The phrase "scale factor" describes this transition.

Scale Factor: In a dilatation, the scale factor refers to how much the figure is enlarged or contracted.

Step 1:

By comparing the length of one side of the pre-image to the corresponding side of the post-image, one may determine the dilation scale factor. The pre-image must be multiplied by the scaling factor to equal the post-image. If the side of the pre-image is the length of $lpre$and the length of the corresponding side of the post-image is $lpost$,

, then the scale factor is given by k in the equation = $lpre$ $k$=$lpost$.

We can solve for k algebraically.

Step 2:-

Choose a matching group of spots on the pre-image and picture. the main point

(x1,y1) being the focal point of the pre-image (x2,y2) be the Centre of the enlarged picture.

By resolving the following two equations, you may get the coordinates of the Centre of dilation,

Step 3:-

Find the coordinate of center of dilation,

$x_{0}=\frac{kx1 - x2}{k-1} \\\\y_{0}=\frac{ky1 - y2}{k -1}$

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Please help I don’t know how to do this/ if I’m doing it right :(

Answers

The measure of the angles and arcs are OLN = 220 deg, OL = 110 degrees deg

Calculating the measure of the angles and arcs

From the question, we have the following parameters that can be used in our computation:

LMN = 110 degrees

This means that

LN = 110 degrees i.e. angle subtended by the arc equals angle at the center

This also means that

OLN = LMN + LMO

Where LMN = LMO

So, we have

OLN = 110 + 110

OLN = 220

Lastly, we have

OL = LMO

This gives

OL = 110 degrees

Answers

Answer:

$here \: is \: no \: picture \\ \\ we \: cannot \: see \: picture$

We can’t see nothing

The main purpose of each passage is to
A) compare brain function in those who play games
on the Internet and those who browse on it.
B) report on the problem-solving skills of
individuals with varying levels of Internet
experience.
C) take a position on increasing financial support
for studies related to technology and intelligence.
D) make an argument about the effects of electronic
media use on the brain.

Answers

The main purpose of each passage is to (choice D) make an argument about the effects of electronic media use on the brain.

What is sentence correction?

Sentence correction or sentence improvement is a type of grammatical practice where a sentence is given with a word or a phrase that requires grammatical changes or improvement.

Now,

Online and other screen-based technologies, according to Passage 1's author, have an impact on people's capacity for in-depth thought (lines 47-50). According to the author of Passage 2, the negative impacts of using electronic media are not as severe as critics of the industry claim (lines 81–82). Following "form" in this context entails adhering to a specific tradition or custom.Choices A and B are inappropriate because they discuss the passages' arguments without addressing their principal purpose. Choice C is inaccurate since neither passage supports boosting funding for certain studies.Hence, choice D is the correct one.

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What is the area of the shaded part ? Take(π=22/7)

What is the area of the shaded part ? Take(=22/7)

Answers

Answer:

Given:

radius of bigger half circle[R]=(7+7+14)/2=14cm

radius of smaller half circle [r]=14/2=7cm

Area of shaded region =1/2[πR² -πr²]

=1/2×22/7×[14²-7²]=231cm²

Area of shaded region=231cm²

A toy is in the form of a cone mounted on a hemisphere of the same diameter. the radius of the base is 5cm and height of the cone is 12cm

Answers

Answer:

Step-by-step explanation:

Given the radius of the base of the cone is 5 cm and the height of the cone is 12 cm, we can find the volume of the toy as follows:

Volume of the cone:

The volume of a cone can be found using the formula: V = (π * r^2 * h) / 3, where r is the radius and h is the height of the cone.

Plugging in the values, we get:

V = (π * 5^2 * 12) / 3 = 100π cm^3

Volume of the hemisphere:

The volume of a hemisphere can be found using the formula: V = (4/3) * π * r^3, where r is the radius of the hemisphere.

Plugging in the values, we get:

V = (4/3) * π * 5^3 = (4/3) * π * 125 = 500π / 3 cm^3

Total volume of the toy:

The total volume of the toy is equal to the sum of the volume of the cone and the volume of the hemisphere:

V = 100π + 500π / 3 = (100 + 500 / 3) * π = 600π / 3 cm^3

So, the total volume of the toy is 600π / 3 cubic centimeters.

help me plz i’ll give extra points

Answers

Answer:

B) mass and height

Answer:

Step-by-step explanation:

Gravitational potential energy depends on an object's weight and its height above the ground (GPE = weight x height).

please help
thank you

Answers

Answer:

yes

Step-by-step explanation:

-3-2-1
M

What is the Rate of Change of this Graph?
What is the equation for this graph? Y=
(Do not include X)
(Do not include Y=)

Answers

Solution. To graph f, we graph the equation y=f(x).

To this end, we use the techniques outlined in Section 1.2.1. Specifically, we check for intercepts, test for symmetry, and plot additional points as needed. To find the x -intercepts, we set y=0. Since y=f(x), this means f(x)=0.

$$$\begin{aligned}f(x) & =x^2-x-6 \\0 & =x^2-x-6 \\0 & =(x-3)(x+2) \quad \text { factor } \\x-3=0 & \text { or } x+2=0 \\x & =-2,3\end{aligned}$$$

So we get (-2,0) and (3,0) as x-intercepts. To find the y-intercept, we set x=0. Using function notation, this is the same as finding f(0) and f(0)=0^2-0-6=-6. Thus the y-intercept is (0,-6). As far as symmetry is concerned, we can tell from the intercepts that the graph possesses none of the three symmetries discussed thus far. (You should verify this.) We can make a table analogous to the ones we made in Section 1.2.1, plot the points and connect the dots in a somewhat pleasing fashion to get the graph below on the right.

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whats the slope for x - 4y = 0
please go in-depth, im kinda not smart

Answers

x-4y=0

Turn to slope intercept form y=mx+b

-4y=-x

y=x/4. Slope in this case is 1/4.. because it’s 1x/4 and 1/4 is in front of x. So the slope is 1/4

Find the y-intercept of the following equation. Simplify your answer.

Answers

Answer: 5/6

Step-by-step explanation:

x = 0 because u are looking for y-intercept

y = -4/9(0) + 5/6

y = 5/6

Find simple interest to the nearest cent for each principle, rate, and time. I= Prt. $1200, 3.9%, 2 years 6 months.

Answers

Answer:

117.00

Step-by-step explanation:

I = prt where i is the interest , p is the principal r is the rate and t is the time in years

2 years 6 months is 2.5 years

I = 1200 * .039* 2.5

I =117

cr2o72−(aq) 14h (aq) 6i−(aq)→2cr3 (aq) 7h2o(l) 3i2(s)cr2o72−(aq) 14h (aq) 6i−(aq)→2cr3 (aq) 7h2o(l) 3i2(s) express your answer in volts using two decimal places.

Answers

The cell potential (E⁰ cell) of the given reaction Cr₂O₇²⁻ (aq) + 14H⁺ (aq) + 6I⁻ (aq) → 2Cr³⁺ (aq) + 7H₂O (l) + 3I₂ (s) is 0.79 volts.

The given reaction is a redox reaction involving the following species: Cr₂O₇²⁻ (aq) + 14H⁺ (aq) + 6I⁻ (aq) → 2Cr³⁺ (aq) + 7H₂O (l) + 3I₂ (s) .

To determine the cell potential (E⁰) of the reaction, we need to know the reduction potentials (E⁰) of the species involved. The reduction half-reactions involved in this reaction are:

Cr₂O₇²⁻ (aq) + 14H⁺ (aq) + 6e⁻→ 2Cr³⁺ (aq) + 7H₂O (l)

(1) I2 (s) + 2e⁻ → 2I⁻ (aq)

(2) The standard reduction potentials (E⁰) for these half-reactions are: E⁰(Cr₂O₇²⁻/Cr³⁺) = 1.33 V

E⁰(I2/I⁻) = 0.54 V

To calculate the cell potential (E⁰cell), we subtract the reduction potential of the anode reaction (oxidation half-reaction) from the reduction potential of the cathode reaction (reduction half-reaction):

E⁰cell = E⁰cathode - E⁰anode

In this case, the reduction half-reaction (1) involves Cr₂O₇²⁻/Cr³⁺ and has a higher reduction potential (1.33 V) compared to the reduction half-reaction (2) involving I2/I⁻ (0.54 V).

Therefore, the reduction half-reaction (1) is the cathode reaction, and the reduction half-reaction (2) is the anode reaction. Substituting the values into the equation, we get:

E⁰cell = 1.33 V - 0.54 V = 0.79 V

Therefore, the cell potential (E⁰cell) of the given reaction is 0.79 volts.

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which of the following is an example of a continuous random variable? multiple choice question. whether or not a house has a pool. the number of bedrooms in a house. the zip code of a house. the square footage of a house.

Answers

A continuous random variable is a variable that can take any value within a certain range or interval. In contrast, a discrete random variable can only take on certain specific values.

In the context of houses, the number of bedrooms is an example of a discrete random variable, since a house can only have a whole number of bedrooms (1, 2, 3, etc.). Similarly, the zip code of a house is also a discrete random variable, since zip codes are predetermined and finite.

On the other hand, the square footage of a house is an example of a continuous random variable. This is because the square footage of a house can take on any value within a certain range (e.g. from 500 to 5000 square feet). There is no specific value that the square footage must be - it can be any number within that range.

To summarize, the square footage of a house is an example of a continuous random variable because it can take on any number within a certain range, whereas the number of bedrooms and zip code are examples of discrete random variables since they can only take on specific, predetermined values.

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Please help

??

:$):’jfidifkvk

Answers

Answer:

For the question a i think its b

Step-by-step explanation:

PLS HELP AGAIN<3333nxjfjfjfittirj

Answers

YYYYYYYYYYYYYYEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEESSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSIIIIIIIIIIIIIIIIIIIIIIIIIRRRRRRRRRRRRRRRRRRRRRRRRR

ASAP
3x - 18=0
options:
x= -6
X=6

Answers

Answer:

Step-by-step explanation:

I think the correct answer is X=6

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