The solution to the given initial value problem using the method of Laplace transforms, is: y(t) = -4 [tex]e^{-t}[/tex] + 5 [tex]e^{-3t/5}[/tex]
To solve the given initial value problem using the method of Laplace transforms, we will follow these steps:
Taking the Laplace transform of both sides of the differential equation.
Applying the Laplace transform to the given differential equation, we get:
5L{y''} + 2L{y'} + 3L{y} = L{u(t-[tex]\pi[/tex])}
Using the properties of Laplace transforms and the table of Laplace transforms to simplify the equation.
The Laplace transform of y'' is [tex]s^2[/tex]Y(s) - sy(0) - y'(0), where Y(s) is the Laplace transform of y(t).
The Laplace transform of y' is sY(s) - y(0), and the Laplace transform of y is Y(s).
Using these transformations and considering the initial conditions y(0) = 1 and y'(0) = 1, we can rewrite the equation as:
5([tex]s^2[/tex]Y(s) - s - 1) + 2(sY(s) - 1) + 3Y(s) = e^(-pi*s) / s
Simplifying further, we have:
(5[tex]s^2[/tex] + 2s + 3)Y(s) - (5s + 7) = [tex]e^{-\pi s}[/tex] / s
Solving for Y(s):
Rearranging the equation, we get:
Y(s) = ([tex]e^{-\pi s}[/tex] / s + (5s + 7)) / (5[tex]s^2[/tex] + 2s + 3)
Using partial fraction decomposition to express Y(s) in simpler terms.
Performing partial fraction decomposition on the right side, we can express Y(s) as:
Y(s) = A / (s + 1) + B / (5s + 3)
where A and B are constants to be determined.
Using the inverse Laplace transform, we can find the solution y(t) as:
y(t) = [tex]L^{-1}[/tex]{Y(s)} = [tex]L^{-1}[/tex]{A / (s + 1)} + [tex]L^{-1}[/tex]{B / (5s + 3)}
Taking the inverse Laplace transforms using the table of Laplace transforms, we find:
y(t) = A [tex]e^{-t}[/tex] + B [tex]e^{-3t/5}[/tex]
Substituting the initial conditions y(0) = 1 and y'(0) = 1 into the solution y(t) = A [tex]e^{-t}[/tex] + B [tex]e^{-3t/5}[/tex], we can solve for the constants A and B.
First, substitute t = 0 into the equation:
y(0) = A * [tex]e^{-0}[/tex] + B * [tex]e^{-0}[/tex] = A + B = 1
Next, differentiate the solution y(t) with respect to t:
y'(t) = -A * [tex]e^{-t}[/tex] - (3B/5) * [tex]e^{-3t/5}[/tex]
Then, substitute t = 0 and y'(0) = 1 into the equation:
y'(0) = -A * [tex]e^{-0}[/tex] - (3B/5) * [tex]e^{-0}[/tex] = -A - (3B/5) = 1
We now have a system of equations:
A + B = 1
-A - (3B/5) = 1
Solving this system of equations, we can find the values of A and B.
From the first equation, we can rewrite it as:
A = 1 - B
Substituting this expression for A into the second equation:
-(1 - B) - (3B/5) = 1
Simplifying the equation:
-1 + B - (3B/5) = 1
Multiplying through by 5 to eliminate the fraction:
-5 + 5B - 3B = 5
Combining like terms:
2B = 10
Dividing by 2:
B = 5
Substituting the value of B back into the first equation:
A = 1 - 5 = -4
Therefore, the constants A and B are -4 and 5, respectively.
The solution to the initial value problem is:
y(t) = -4 [tex]e^{-t}[/tex] + 5 [tex]e^{-3t/5}[/tex]
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if f(x) = x^2 which equation represents function g?
OA g(x) = f(2x)
OB. g(x) = f(4x)
Oc. g(x) = 2f(x)
OD. g(x) = f(1/2x)
Answer:
if f(x) = x^2 which equation represents function g?
Step-by-step explanation:
maby ( C) am not sure
The equation represents the function is g(x) = 3f(x).
What is Function?A function is an expression, rule, or law in mathematics that describes a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are common in mathematics and are required for the formulation of physical relationships in the sciences.
We have,
Function, f(x) = x²
The function g(x) is compressed horizontally by a factor of 3, to get the function g(x)
g(x) = kf(x)
Where k = 3.
So, we have:
g(x) = 3f(x)
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The temperature went down 3 F. Each hour. What was the total temperature change after 5 hours ?
Find the value of x. Round to the
nearest tenth.
X
37
6
X=
Enter
Answer:
x ≈ 4.5Steps:
tan(37°) = 0.75..
tan(37°) = x/6
x/6 = 0.75..
x = 0.75.. × 6
x = 4.52.. ≈ 4.5
Answer:
The answer is 4.5
Step-by-step explanation:
x =4.5
Fill in 4.5
Convert 312 quarts to gallons.
Answer:
78 gallons
Step-by-step explanation:
there are four quarts in one gallon. so, to convert 312 quarts to gallons you must divide by 4. The answer is 78 gallons.
*Will mark brainliest* Can someone explain this to me?
I need help on this.
Dada la recta L1:x+(m+2)y=n−5 pasa por el punto (1,2) y que es paralelo a la recta L2:3y=2x−4. Determine m+n .
Answer:
-1/2
Step-by-step explanation:
The point slope form of equation of a line is expressed as;
[tex]y - y_{0} = m(x-x_0)[/tex]
m is the slope
(x₀, y₀) is the point on the line
Given
(x₀, y₀) = (1, 2)
Given the equation 3y = 2x - 4
Rewrite the equation
y = 2/3x - 4/3
mx = 2/3x
m = 2/3
The slope is 2/3
Substitute the slope and the point into the formula above
[tex]y -y_0 =m(x-x_0)\\y - 2 = 2/3(x - 1)\\3(y-2) = 2(x-1)\\3y - 6 = 2x - 2\\Swap\\2x - 2 = 3y - 6\\2x - 3y = -6 + 2\\2x - 3y = -4\\x - 3/2 y = -4/2\\x - 3/2 y= -2[/tex]
Compare the expression to x+(m+2)y=n−5
m + 2 = -3/2
m = -3/2 - 2
m = -7/2
Similarly
n - 5 = -2
n = -2 + 5
n = 3
m + n = -7/2 + 3
m + n = -1/2
hence the sum of m and n is -1/2
Joseph has a bag filled with 2 red, 4 green, 10 yellow, and 9 purple marbles. Determine P(not purple) when choosing one marble from the bag.
a.64%
b.36%
c.24%
d.8%
To determine the probability of not choosing a purple marble when selecting one marble from a bag containing 2 red, 4 green, 10 yellow, and 9 purple marbles. Correct option is A).
The total number of marbles in the bag is 2 (red) + 4 (green) + 10 (yellow) + 9 (purple) = 25 marbles.
The number of non-purple marbles is 2 (red) + 4 (green) + 10 (yellow) = 16 marbles.
Therefore, the probability of not choosing a purple marble is P(not purple) = 16/25.
To convert this fraction to a percentage, we divide the numerator (16) by the denominator (25) and multiply by 100: P(not purple) = (16/25) * 100 = 64%.
Hence, the probability of not choosing a purple marble when selecting one marble from the bag is 64%, which corresponds to option (a) in the given choices.
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Gilbert's weekly pay in dollars, D. can be represented by the function D = 13.5h, where h represents the number of hours
he works during a week.
Answer:
she will have 27,216 dollars after 12 week
27,216 per week
Step-by-step explanation:
12 week has (12*7*24)=2,016 hours
D = 13.5h D = 13.5*(2,016 )=27,216
-2+ 3(1-4)-2 what to do ???
Answer:
-13
Step-by-step explanation:
−2+3(1−4)−2
=−2+(3)(−3)−2
=−2+−9−2
=−11−2
=−13
Step-by-step explanation:
Is this ok
If its correct, please don't forget to like and mark me
I made a fort by connecting two boxes. The first box is 5 meters long, 9 meters wide, and 9 meters high. The second box is 3 meters long, 8 meters wide, and 2 meters high.
Complete question :
I made a fort by connecting two boxes. The first box is 5 meters long, 9 meters wide, and 9 meters high. The second box is 3 meters long, 8 meters wide, and 2 meters high. How many cubic meters of space does my fort have?
Answer:
453 m³
Step-by-step explanation:
Volume of first box :
Volume = Length * width * height
Volume = 5 * 9 * 9
Volume = 405 m³
Volume of second box :
Volume = Length * width * height
Volume = 3 * 8 * 2
Volume = 48 m³
Total volume :
405m³ + 48m³ = 453m³
determine which rate should be used to complete the conversion: 12 gallons per hour to gallons per minute.
A. 1 hour per 60 minutes
B. 60 minutes per 1 hour
This is confusing me please help
Answer:
It’s the last one
Step-by-step explanation:
Did this before on a test
100 POINTSSS PLZZZZZZZZZZZ
Answer:
The correct anser is B. (i think. sorry if wrong) have a wonderfull day. dont let them bring you down. your amazing just the way you are.
Step-by-step explanation:
Step-by-step explanation:
\blue{\large \rightarrow }\: \boxed{ \sf{ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} }}
A fair coin is tossed five times. The event "A = getting all heads" has probability = 1/32. (a) Describe in words what the event AC is. At least one tail occurs in the five flips. At least four tails occur in the five flips. At least four heads occur in the five flips. Getting all tails. At least one head occurs in the five flips. (b) What is the probability of AC? (Enter your answer as a fraction.)
The event AC describes the scenario in which at least one tail occurs in the five coin flips.The probability of AC is 31/32.
(a) In words, the event AC describes the scenario in which at least one tail occurs in the five coin flips.
Event AC can also be written as {HTTTT, THTTT, TTHTT, TTTHT, TTTTH, HTTTH, HHTTT, THHTT, TTHHT, TTTHH, HTTHH, HHTTH, THHHT, TTHHH, HTHTT, HHTHT, THHTH, HTTHT, THTHT, HTTH, THTTH, TTHTH, TTHHT, THTHH, HTTHH, THHTH, HTHHT, HHTHH, THHH, TTHH, THTH, HTTH, HHTH, THHT}.
(b) The probability of AC can be calculated as follows:
There are two possible outcomes of each coin flip: heads or tails.
Since the coin is fair, both outcomes are equally likely.
Therefore, the probability of getting tails in one flip is 1/2, and the probability of getting heads in one flip is also 1/2.In order to find the probability of AC, we will use the complement rule:Prob(AC) = 1 - Prob(A)Prob(A) = 1/32Prob(AC) = 1 - 1/32Prob(AC) = 31/32
Therefore, the probability of AC is 31/32.
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What did the girl mushroom say about the boy mushroom after their first date
Answer:
what’d she say? xD
Step-by-step explanation:
The solution is, She said that he was a fun gi, the girl mushroom say about the boy mushroom after their first date.
What is mushrooms?A mushroom is the reproductive structure produced by some fungi. It is somewhat like the fruit of a plant, except that the "seeds" it produces are in fact millions of microscopic spores that form in the gills or pores underneath the mushroom's cap.
here, we have,
She said that he was a fun gi.
This is a pun because fungi is a mushroom and here it is a play on words because of "fun guy" sounding the same as "fun gi".
Since they are both mushrooms, the joke stands and these types of puns are very common among people because of their simplicity.
The solution is, She said that he was a fun gi, the girl mushroom say about the boy mushroom after their first date.
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The drawing below shows circle B and circle E.Which equation best describes the relationship between BC and EF?
Answer:
BC = 3(EF)
Step-by-step explanation:
The question stated that AC is 3 times the length of DF.
Answer:
BC = 3(EF)
Step-by-step explanation:
Remember that AC is 3 times the length of DF
Two parallel lines l and m are cut by a transversal t. If the interior angles of the same side of t are (2x−8)∘ and (3x−7)∘, find the measure of each of these angles.
Step-by-step explanation:
When two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. This means that their measures add up to 180 degrees.
In this case, the two interior angles on the same side of the transversal 't' are (2x-8) degrees and (3x-7) degrees. Since these angles are supplementary, we can write the equation (2x-8) + (3x-7) = 180.
Solving this equation for x, we get:
(2x-8) + (3x-7) = 180
5x - 15 = 180
5x = 195
x = 39
Substituting this value of x into the expressions for the two interior angles, we find that their measures are:
(2x-8) = (2*39 - 8) = 70 degrees
(3x-7) = (3*39 - 7) = 110 degrees
So, the measure of each of these angles is 70 degrees and 110 degrees.
A loan of R 180 000, granted at 15,6% p.a. compounded monthly, is amortised by means of regular equal monthly payments of R 6 300 and a final payment F (F < R 6 300) made one month after the last equal payment of R 6 300. If the first payment is made one month after the loan is granted, then the final payment F , (to the nearest cent)
The final payment F, made one month after the last equal payment, for a loan of R 180,000 at an interest rate of 15.6% p.a. compounded monthly with regular equal monthly payments of R 6,300, is R 6,345.94 (to the nearest cent).
To calculate the final payment F, we need to determine the remaining balance after the last equal payment of R 6,300. We can consider it as the present value of the remaining debt, including one month of interest.
First, we calculate the monthly interest rate by dividing the annual interest rate by 12:
r = 15.6% / 12 = 0.013
Next, we calculate the remaining balance as the present value of the remaining debt after the last equal payment:
Remaining balance = R 180,000 * (1 + r)^(n_remaining) - R 6,300
Where: n_remaining = Number of remaining months after the last equal payment (1 month)
Remaining balance = 180,000 * (1 + 0.013)^1 - 6,300 = R 180,000 * 1.013 - 6,300 = R 180,000 * 1.013 - 6,300 = R 182,340
Finally, we subtract the remaining balance from the final payment to determine the value of F:
F = Remaining balance - R 6,300 = R 182,340 - 6,300 = R 6,345.94 (to the nearest cent)
Therefore, the final payment F, made one month after the last equal payment, is R 6,345.94 (to the nearest cent).
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y = 3x - 5 and y = 4x.
Answer: x = -5
New Equation (after substituting) 4x = 3x - 5
Subtract 3x from both sides
x = -5
The Bungling Brothers Circus is in town and you are part of the crew that
is setting up its enormous tent. The center pole that holds up the tent
is 70 feet tall. To keep it upright, a support cable needs to be attached to
the top of the pole so that the cable forms a 60° angle with the ground.
a) How long is the cable?
b) How far from the pole should the cable be attached to the ground?
Answer:
a. 80.83 ft b. 40.42 ft
Step-by-step explanation:
Let h = height of pole = 70 ft, L = length of cable and x = distance of cable on ground to pole and Ф = angle between cable and ground.
a) How long is the cable?
Since L, h and x form a right-angled triangle with angle Ф and h being the opposite side to Ф and L being the hypotenuse side, by trigonometric ratios,
sinФ = h/L
L = h/sinФ
L = 70 ft/sin60°
L = 70 ft/0.8660
L = 80.83 ft
b) How far from the pole should the cable be attached to the ground?
Since L, h and x form a right-angled triangle with angle Ф and h being the opposite side to Ф and x being the adjacent side, by trigonometric ratios,
tanФ = h/x
x = h/tanФ
x = 70 ft/tan60°
x = 70 ft/1.7321
x = 40.42 ft
The length of the cable is 80.83 ft and the distance from the pole should the cable be attached to the ground is 40.42 ft and this can be determined by using the trigonometric function.
Given :
The Bungling Brothers Circus is in town and you are part of the crew that is setting up its enormous tent. The center pole that holds up the tent is 70 feet tall. To keep it upright, a support cable needs to be attached to the top of the pole so that the cable forms a 60° angle with the ground.a) The trigonometric function can be used in order to determine the length of the cable.
[tex]\rm sin\theta=\dfrac{h }{L}[/tex]
where h is the height of the pole, L is the length of the pole, and [tex]\theta[/tex] is the angle from the ground.
Substitute the known terms in the above expression.
[tex]\rm L=\dfrac{h }{sin\theta}[/tex]
[tex]\rm L = \dfrac{70}{sin60}[/tex]
L = 80.83 ft.
b) The trigonometric function can be used in order to determine the distance from the pole should the cable be attached to the ground.
[tex]\rm tan \alpha =\dfrac{h}{x}[/tex]
where x is the distance between pole and cable.
Substitute the known terms in the above expression.
[tex]\rm x = \dfrac{70}{tan 60 }[/tex]
x = 40.42 ft.
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Urgent!!!!!
12. Arrange the stages of municipal council election by writing 1-7 in the
correct place. (2 marks each)
Stages in municipal elections
Girst (1) to last (17)
People going to the polling station to
Councilors campaign for election.
The number of votes for each division
is counted.
Voters vote by secret ballot.
Candidates nominated by two main
political parties.
(PLZ I NEED HELP QUICLY) Angela lives in a remote area serviced by only two cell phone towers. Tower 1 is located 5 miles west and 2 miles south of her home and transmits a signal for 12 miles. Tower 2 is located 8 miles east and 7 miles north of her home and transmits a signal for 10 miles. Consider the location of Angela’s house to be (0, 0) on a coordinate grid. The system of inequalities represents this scenario.
Angela travels to a friend’s house, which is 4 miles east and 8 miles north of her house. Will Angela have cell phone service at her friend’s house?
A. Angela will not have any cell phone service.
B. Angela will have cell phone service from tower 1 only.
C. Angela will have cell phone service from tower 2 only.
D. will have cell phone service from both towers.
Angela will have cell phone service from tower 2 only , Option C is the answer.
What is an Inequality ?An inequality is an algebraic expression connected to another algebraic expression with an inequality operator.
The two inequalities given are
(x+5)² + (y+2)² ≤ 144
(x-8)² +(y-7)² ≤100
The inequalities are plotted on the graph
and the location of Angels's Friends ( 4 , 8) is also plotted
If this lies in the area of the Inequalities she will have signal.
At Angela's friends house she will have the signal of Tower 2 only.
Therefore Option C is the answer.
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if two samples are selected from the same population, under what circumstances are the two samples guaranteed to have exactly the same t statistic?
If two samples are selected from same population, then two samples are guaranteed to have exactly same t-statistic is (d) If samples are same-size and have same-mean and have same-variance.
If two samples are selected from the same population and have the same size, same mean, and same variance, the t statistic calculated for both samples will be exactly the same.
The t statistic is computed using the sample means, sample variances, and sample sizes. When these values are identical for both samples, the calculations for the t-statistic will yield the same result.
This scenario ensures that the t-statistic, which is used for hypothesis testing or constructing confidence intervals, will be consistent across the two samples.
Therefore, the correct option is (d).
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The given question is incomplete, the complete question is
If two samples are selected from the same population, under what circumstances are the two samples guaranteed to have exactly the same t statistic?
(a) If the samples are the same size and have the same variance.
(b) If the samples are the same size and have the same mean.
(c) If the samples have the same mean and the same variance.
(d) If the samples are the same size and have the same mean and have the same variance.
the solution of the equation (x+3)2=7 is
Answer:
x=1/2
Step-by-step explanation:
f ( x ) = − x 2 − 12 Find f ( − 3 )
Answer:
f(-3)=-21
Step-by-step explanation:
f(x)=-x2-12
f(-3)=-(-3)2-12
f(-3)=-9-12
f(-3)=-21
Answer:
-3
Step-by-step explanation:
x=-3 substitute in the equation
+3 squared-12=-3
la suma de cuatro y el producto de tres y un número x
Answer:
4+3x esa es la suma de cuatro y el producto de tres y un número x
A cylindrical tank holds 352 cu. ft. of water. The height of the tank is 7 ft. Find the
radius of the tank.
pls help me thx and show your work
Answer:
0.3334 ft
Step-by-step explanation:
Measure the height and radius of the tank. The radius is the distance from the center of the tank to its outer edge. Another way to find the radius is to divide the diameter, or width, by two. Square the radius by multiplying the radius times itself and then multiply it by 3.1416, which is the constant pi.
Given height and volume: r = √(V / (π * h)),Given height and lateral area: r = A_l / (2 * π * h),Given height and total area: r = (√(h² + 2 * A / π) - h) / 2,Given height and diagonal: r = √(h² + d²) / 2,Given height and surface-area-to-volume ratio: r = 2 * h / (h * SA:V - 2),Given volume and lateral area: r = 2 * V / A_l,Given base area: r = √(A_b / (2 * π)),Given lateral area and total area: r = √((A - A_l) / (2 * π)).The two cones are similar. The smaller cone has a
surface area of 11. 74 inches
Complete the last step to determine the surface area of
the larger cone
3. 5 in.
1. The scale factor of the larger to the smaller is
Sor
2. The surface area will change by the square of the
scale factor, which is
3. Let the surface area of the larger cone be x.
Then, the proportion is = 11
4. Solve for x and round to the nearest hundredth.
The surface area of the larger cone is about
inches
The surface area of the larger cone is about 41.09 square inches.
To determine the surface area of the larger cone, we can set up a proportion based on the scale factor between the two cones. Let's call the scale factor "k".
From the given information, we know that the surface area of the smaller cone is 11.74 square inches and the surface area of the larger cone is unknown (let's call it "x" square inches).
Using the scale factor, we can write the proportion:
(11.74 / x) = (3.5 / 3.5)
Simplifying the proportion, we have:
11.74 = (x / 3.5)
To find the value of "x", we can cross-multiply:
x = 11.74 * 3.5
x ≈ 41.09
The surface area of the larger cone is approximately 41.09 square inches, rounded to the closest hundredth.
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What is the median for the following data set? 5,7,9,11,13,13
A. 10
B. 8
C. 9
D. 11