Problem 5 (a) Find the Laplace transform of f(t) = [1 − H(t = 10)]et – e¹⁰8(t – 10) where H(t) is the Heaviside function and 8(t) is Dirac's delta function. (b) Knowing that 1 1 s+ 5 = (s 1) (s² + 4s + 5) 10(S-1) 10(s² + 4s + 5)' find the inverse Laplace transform of each of the following functions: 1 (i) (s - 1)(s² + 4s + 5) e-10s (ii) s² + 4s + 5 e-10s (iii) (s - 1)(s² + 4s + 5) (c) Solve the following initial value problem. You may use the results you obtained in the above questions. y" + 4y' + 5y = [1 − H(t — 10)]et - e¹⁰8(t - 10) y(0) = 0, y'(0) = 1,
Answers
a) The Laplace transform is L{f(t)} = 1/s - 1/s x 1/s - e¹⁰8 x 1/(s + 10)
b) The inverse Laplace transform of the given function is
i) L⁻¹{ (s - 1)(s² + 4s + 5) \(e^{(-10s)\) } = (δ'(t) - δ(t)) (\(e^{(-2t)\) sin(t)) (\(e^{(-2t)\) cos(t))
ii) L⁻¹{ s³ + 4s² + 5s - s² - 4s - 5 } = t² + 3t - 5
(a) To find the Laplace transform of f(t) = [1 − H(t = 10)]et – e¹⁰8(t – 10), we can use the linearity property of the Laplace transform.
L{f(t)} = L{[1 − H(t = 10)]et} - L{e¹⁰8(t – 10)}
Applying the Laplace transform to each term separately:
L{1} - L{H(t = 10)} x L{et} - L{e¹⁰8(t – 10)}
The Laplace transform of the constant 1 is 1/s.
L{H(t = 10)} represents the Heaviside function, which is 0 for t < 10 and 1 for t ≥ 10. Its Laplace transform is 1/s.
L{et} is the Laplace transform of et, which is 1/(s - a) where a is the constant in the exponential term.
In this case, a = 0, so L{et} = 1/s.
L{e¹⁰8(t – 10)} can be rewritten as e¹⁰8 \(e^{-10s\) using the time-shift property of the Laplace transform. Then, using the transform of e^-as, where a = 10, we get 1/(s + 10).
Putting it all together:
L{f(t)} = 1/s - 1/s x 1/s - e¹⁰8 x 1/(s + 10)
(b) To find the inverse Laplace transform of the given functions, we can use partial fraction decomposition and the inverse Laplace transform table.
(i) For (s - 1)(s² + 4s + 5) \(e^{-10s\):
We can factor the denominator as (s - 1)(s + 2 + i)(s + 2 - i) using the quadratic formula.
Applying the inverse Laplace transform to each term, we get:
L⁻¹{ s - 1 } = δ'(t) - δ(t)
L⁻¹{ (s + 2 + i) } =\(e^{(-2t)\) sin(t)
L⁻¹{ (s + 2 - i) } = \(e^{(-2t)\) cos(t)
Therefore, the inverse Laplace transform of the given function is:
L⁻¹{ (s - 1)(s² + 4s + 5) \(e^{(-10s)\) } = (δ'(t) - δ(t)) (\(e^{(-2t)\) sin(t)) (\(e^{(-2t)\) cos(t))
(ii) For s² + 4s + 5\(e{^-10}s\):
L⁻¹{ (s - 1)(s² + 4s + 5) } = L⁻¹{ s³ + 4s² + 5s - s² - 4s - 5 }
Taking the inverse Laplace transform of each term, we get:
L⁻¹{ s³ + 4s² + 5s - s² - 4s - 5 } = t² + 3t - 5
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Related Questions
y = x-2
y = 3x+4. Solutions of system of equation
Answers
Answer:
x=-3 and y=-5
Step-by-step explanation:
hope this helps!
Answer:
Y = x-2; y= 3x + 4
Step-by-step explanation:
the base of a solid is bounded by the graph of x^2 y^2=a^2 where a 0
Answers
The base of the solid is bounded by the graph of \(\(x^2 y^2 = a^2\)\), where\(\(a > 0\).\) This equation represents a hyperbola in the xy-plane, centered at the origin and symmetric about both the x-axis and y-axis.
To understand the shape of the solid, let's consider the different values of x and y. For any positive value of x, we can find two corresponding y-values that satisfy the equation: one positive and one negative. Similarly, for any positive value of y, we can find two corresponding x-values. This indicates that the base of the solid consists of two separate branches of the hyperbola, one in the first quadrant and the other in the third quadrant. When we revolve this base around the x-axis, we obtain a three-dimensional solid known as a hyperboloid of revolution. The resulting solid has a curved surface that resembles a double cone or an hourglass shape. The vertex of the solid is at the origin, and the height of the solid extends infinitely along the y-axis. In summary, the base of the solid is defined by the equation \(\(x^2 y^2 = a^2\)\) and represents a hyperbola in the xy-plane. When revolved around the x-axis, it forms a hyperboloid of revolution, a three-dimensional solid with a curved surface resembling a double cone or an hourglass.
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escribir una funcion cuadratica f cuyos ceros sean -13 y -2 por favor ayudaaa
Answers
Answer:
x²+15x+26
Step-by-step explanation:
zeros are x=-13, x=-2
x=-13 then x+13=0
x=-2 then x+2=0
(x+13) (x+2)
x²+15x+26
y=ax²+bx+c
100:131 IN THE FORM 1:N
Answers
The area of a circle is 8,340 m2. Find its diameter to the nearest hundredth of a meter.
Answers
Answer:
103.05m
Step-by-step explanation:
There are 5 positions available in the new school. Of the applicant, 12 are men and 8 are women. In how many ways can 3 men and 2 women be chosen if they are equally considered?
Answers
There are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting
What is the multiplication principle of countingThe multiplication principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.
To find the number of ways to choose 3 men from the 12 men, we can use the formula for combination, which is: ⁿCᵣ = n! / (r! (n-r)!).
where n is the total number of men and r is the number of men chosen
so, the number of ways to choose 3 men from the 12 men = ¹²C₃ = 1.
Similarly, we evaluate the number of ways to choose 2 women from the 8 women
as = ⁸C₂ = 14
Now, using the multiplication principle, we can find the total number of ways 3 men and 2 women be chosen if they are equally considered.
220 x 14 = 3080
Therefore, there are 3080 ways 3 men and 2 women can be chosen if they are equally considered, using the multiplication principle of counting
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a painter mixes (2 + (1/2)) pints of yellow point with 4 pints of red paint to make a certain shade of orange of red paint. how many pints of yellow paint should be mixed with 10 pints of red paint to make this shade of orange.
Answers
Answer: 5 pints of yellow paint
Step-by-step explanation:
We can start by using the proportion of yellow paint to red paint in the original mixture to figure out how much yellow paint should be mixed with 10 pints of red paint.
Let x be the number of pints of yellow paint that should be mixed with 10 pints of red paint to make the same shade of orange.
We know that:
(2 + (1/2))/(4 + (2 + (1/2))) = x/10
Simplifying the proportion, we get:
(5/4) / (11/4) = x/10
Cross-multiplying, we get:
(5/4) * 10 = x * (11/4)
Solving for x, we get:
x = (5/4) * 10 * (4/11)
x = (5/11) * 10
x = 5/11 * 10
x = 5
So, 5 pints of yellow paint should be mixed with 10 pints of red paint to make the same shade of orange as the original mixture.
The solid below is similar to a larger solid.
The surface area and volume of the smaller solid is given along
with the scale factor between the two solids. Find the surface area (S) and Volume (V) of the larger solid. Show
all your work.
Answers
Surface Area of Larger Solid: S2 = k^2 * S1 and Volume of Larger Solid: V2 = k^3 * V1. Substitute in the given values for S1, V1, and k before solving for S2 and V2.
To solve this problem, we need to use the fact that similar solids have corresponding sides that are proportional in length.
Let's call the scale factor between the two solids "k". This means that if the smaller solid has a side length of "s", then the corresponding side length in the larger solid is "ks".
Now let's use the given information to find the surface area and volume of the larger solid. Let's call the surface area and volume of the smaller solid "S1" and "V1", respectively.
Surface Area:
The surface area of a solid is proportional to the square of its linear dimensions. Since the scale factor between the two solids is "k", the surface area of the larger solid is k^2 times the surface area of the smaller solid. Therefore, the surface area of the larger solid (S2) can be found using the formula:
S2 = k^2 * S1
Volume:
The volume of a solid is proportional to the cube of its linear dimensions. Using the same logic as above, the volume of the larger solid (V2) can be found using the formula:
V2 = k^3 * V1
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(please do step by step so I can understand)
What are the measures of Angles a, b, and c? Show your work and explain your answers.
Answers
Angle a=35° (Vertically Opposite Angles)
Angle b= 180°-90°-35° =55° (Sum of angles in a triangle)
Angle c=180°-70°=110° (Angles on a straight line)
Need some help with the 5 by 5 puzzle. I keep getting stuck :(
Answers
Answer:
not sure
Step-by-step explanation:
Here you go...
It attached below
a sample of 153 ultimate tensile strength observations were considered. the sample mean was 135.39 and sample standard deviation was 4.59. calculate a 99% lower confidence bound for true average ultimate tensile strength.
Answers
The 99% lower confidence bound for the true average ultimate tensile strength is 134.09.
To calculate the 99% lower confidence bound for the true average ultimate tensile strength, we need to use the formula:
Lower confidence bound = sample mean - (critical value * (sample standard deviation / √(sample size)))
First, we need to determine the critical value for a 99% confidence level. Using a t-distribution table with 152 degrees of freedom (sample size - 1), we find a critical value of -2.602.
Plugging in the values, we get:
Lower confidence bound = 135.39 - (-2.602 * (4.59 / √(153)))
Lower confidence bound = 134.09
In simpler terms, we can say that we are 99% confident that the true average ultimate tensile strength is at least 134.09 or higher based on our sample data.
This means that if we were to take many different samples of the same size and calculate the lower confidence bound for each sample, we would expect 99% of them to contain the true average ultimate tensile strength value.
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When dividing 5 by 4, is a answer a rational number explain
Answers
Dividing the number 5 by 4 gives a rational number 5/4.
What is defined as the rational numbers?The word 'rational' comes from the word 'ratio.'
As a result, rational numbers are attributable to the concept of fractions, which represent ratios. In other words, a rational number can be represented as a fraction in which both the numerator and denominator are integers.Rational numbers also include integers, whole numbers, natural numbers, or fractions with integers.If a number's decimal form is terminating or recurring, as in 5.6 or 2.141414, we know it is a rational number.Irrational numbers are those in which the decimals appear to be never-ending or non-recurring.For the given expression;
When 5 is divided by the number 4, 4 will not completely divide the number 5 as 5 is not the multiple of 4.
Thus, dividing 5 by 4 gives the rational number 5/4.
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could you help withABCD is similar to EFGH And BC=26, CD=13, FG=10 And GH=x-7find the value of x?
Answers
As per similar rule:
\(\text{ABCD}\approx\text{EFGH}\)So:
\(\begin{gathered} \frac{BC}{FG}=\frac{CD}{GH} \\ \frac{26}{10}=\frac{13}{x-7}^{} \\ x-7=\frac{10\times13}{26} \\ x-7=5 \\ x=12 \end{gathered}\)Calculator
Bookwork code: M56
X not allowed
a) What is the y-intercept of this line?
b) What is the gradient of this line?
Answers
Answer:
6 and 2
Step-by-step explanation:
(a)
the y- intercept is the point on the y- axis where the line crosses.
the line crosses at (0, 6 ) , then y- intercept = 6
(b)
the gradient m ( slope) of the line is calculated using the gradient formula
m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)
with (x₁, y₁ ) = (- 3, 0) and (x₂, y₂ ) = (0, 6) ← 2 points on the line
m = \(\frac{6-0}{0-(-3)}\) = \(\frac{6}{0+3}\) = \(\frac{6}{3}\) = 2
From the adjacent image we can identify the points of the graph.
From the points we can easily find y-intercept and gradient.
How can we identify the points.
Line where cut the x-axis and y-axis. Take that location as point.
For writing point observe the distance of x-axis and y-axis
Let say point A=(0,6) and point B=(-3,0)
For find y-intercept we must know gradient first
Gradient formula for two points m=\(\frac{y_{2} -y_{1 }}{x_{2} -x_{1 }}\)
Submit Point A and Point B m=\(\frac{0 -6}{-3 -0\\}\)
m=\(\frac{-6}{-3 \\}\)
m=2
Therefore, Gradient of slope m=2
To find y-intercept, we must solve lin equation
Line equation y=mx+b
m = gradient and b=y-intercept
Submit m in line equation y=2x+b
Submit any one point in the line equation
Take pointA=(0,6)
6=2(0)+b
b=6-2
b=4
Therefore ,y-intercept equals to 4
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find two rational numbers between 3 and 5
Answers
Answer:
4 is a rational number
Step-by-step explanation:
rational numbers are numbers that can be divided by 2 and itself
Answer:4, 3.6
Step-by-step explanation: there are multiple answers to this questions.
Which statement correctly compares the spreads of the distributions? Team A's scores ㅏ ㅁ 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 Team B's scores ㅏ 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
Answers
Answer:
5
Step-by-step explanation:
Help me step by step plz
Answers
Answer:
x=-6
Step-by-step explanation:
Divide both sides by -4
You will get an answer of x=-6.
X=-6
Divide both sides by -4
-2.5f+ 0.8f-16 -3 help please
Answers
31.875 as a power of 2
Answers
Answer:
1016.015625 is the correct answer
Step-by-step explanation:
31.875 to the second power is the same as 31.875 squared.
Equation - 31.875 x 31.875
Solve and your answer should be 1016.015625
You and a friend go out to dinner and each pay for your own meal. The total cost of the two
meals is $25. Your meal cost $4 more than your friend’s meal. How much does each meal cost?
Answers
The cost of the meal from the friend is $10 and the meal for your friend is $14.50.
What is an expression?An expression simply consist of one or more numbers or variables that are along with one or more operation.
Let the friends meal be x.
Your meal = x + 4.
Total cost = 25
This will be expressed as:
x + x+4 = 25
2x + 4 = 25
Collect like terms
2x = 25 - 4.
2x = 21
Divide.
x = 21/2
x = 10.5
Friend's meal = $10.5
Your meal = x + 4 = $10.5 + $4 = $14.5
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The reporter asked students how much time they spent at the beach each week end displays the information below. Which interval represents a peak?
Answers
Explanation:
3-4hours has the most amount (peak)
The bar graph shows the differences in a city’s rainfall from the historical average. What is the difference in rainfall between the wettest and the driest months? Write your answer as a decimal.
The difference in rainfall between the wettest and the driest months is inches.
Answers
Answer:
The difference is 3.97 inches of rainfall
Step-by-step explanation:
Here, we want the difference between the wettest and driest months.
From the question, we can see that the wettest month has a rainfall height of 2.3 inches
The driest month has a rainfall of -1.67 inches
So the difference between the two will be;
2.3 - (-1.67)
= 2.3 + 1.67 = 3.97 inches
Answer:
4.03
Step-by-step explanation:
Have to add 2.36 and 1.67 and it would be a positive.
Oriana has 3 identical fish tanks in the shape of rectangular prisms. Each tank is 16 inches tall, 2 feet wide. and 12 inches long. How many cubic inches will Oriana's fish tanks hold?
Answers
The cubic inches that the fish tanks will hold is 13,824 inches³.
How many cubic inches can the fish tanks hold?In order to determine the cubic inches that the fish tanks would hold, multiply the volume of one of the fish tanks by 3.
Volume of a rectangular prism = length x width x height
The feet has to be converted to inches
1 feet = 12 inches
2 x 12 = 24 inches
Volume = 16 x 24 x 12 = 4,608 inches³
Volume of the fish tanks = 4,608 inches³ x 3 = 13,824 inches³
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Write in y= mx + b form: a line through (-3,1) having the same y-intercept as the graph of x-2y=-4
Answers
Step-by-step explanation:
Aight, so the same intercept
\( - 2y = - 4 - x = = = > \\ y = \frac{1}{2} x + 2\)
m=½
\(y = \frac{1}{2} x + b = = = > \\ now \: let \: us \: replace \: the \: point \\ 1 = \frac{1}{2} ( - 3) + b = = = > \\ \frac{5}{2} = b\)
soooo
\(y = \frac{1}{2} x + \frac{5}{2} \)
How
do I show significant difference using superscript between these
values? (anova single factor test)
Answers
Yes, you can show significant differences using superscripts in an ANOVA (Analysis of Variance) single-factor test.
In an ANOVA test, superscripts are commonly used to indicate significant differences between the means of different groups or treatments.
Typically, letters or symbols are assigned as superscripts to denote which groups have significantly different means. These superscripts are usually presented adjacent to the mean values in tables or figures.
The specific superscripts assigned to the means depend on the statistical analysis software or convention being used. Each group or treatment with a different superscript is considered significantly different from groups with different superscripts. On the other hand, groups with the same superscript are not significantly different from each other.
By including superscripts, you can visually highlight and communicate the significant differences between groups or treatments in an ANOVA single-factor test, making it easier to interpret the results and identify which groups have statistically distinct means.
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For an activity in class, students used pieces of string to measure roads on a map. They used a map scale to determine the distance between two cities. One group measured the distance between Fort Myers and Miami on a Florida state map. The string distance was 8 inches, which represents a real-life distance of about 440 miles. Another group measured the distance between Tampa and Orlando on the same map and found that distance to be 5.5 inches. What is the real-life distance to the nearest mile between Tampa and Orlando?
Answers
Answer:
303 miles.
Step-by-step explanation:
We have:
8 inches = 440 miles
To find the real-life distance to the nearest mile between Tampa and Orlando we need to see the number of miles in one inch, by using the above relation:
\( miles = \frac{440 miles}{8 inches}*1 inches = 55 miles \)
Hence, in 1 inch we have 55 miles, so 5.5 inches is equal to:
\( miles = \frac{55 miles}{1 inch}*5.5 inches = 302.5 miles \)
Therefore, the real-life distance between Tampa and Orlando is 303 miles.
I hope it helps you!
Answer:
okay go to my page and answer the question that I posted about 50 and you will get it
how can i describe the expression 3x−2
please help
Answers
Answer:
Step-by-step explanation:
3X-2=0
3X=2
X=2/3 OR X=0.66
Answer:
so -2 is the constant(integer without any variable) and 3 is the coefficient(what the variable is being multiplied by) and x is the variable(the unknown term)
we can't solve/simplify this expression since
a) it's an expression, it's impossible to solve since there's no equal sign
b)there are no like terms to simplify this
Please Help! I'll give a lot of points!
Find the simple interest Jolie owes on a 2-year loan for $9,500 to buy a used car at the rate of 9% interest.
This is a TWO part question, make sure to answer both part A. and B.
A. Find the amount of interest. Show the math computation use to find the amount of interest.
B. What was the monthly payment? Show the work used to find this answer.
Answers
Answer:
A. Interest = $1710
B. Monthly payment = $467.08
Step-by-step explanation:
Simple interest formula:
\(\sf A = P(1+rt)\)
where:
A = final amountP = principalr = annual interest rate (in decimal form)t = time (in years)Given:
P = $9,500r = 9% = 0.09t = 2 yearsSubstitute given information into the formula:
\(\begin{aligned} \sf \implies A & = \sf 9500(1+0.09 \cdot 2)\\& = \sf 11210 \end{aligned}\)
Interest = final amount - principal
= 11210 - 9500
= $1710
term = 2 years
1 year = 12 months
⇒ term = 2 × 12 = 24 months
Monthly payment = final amount ÷ months in term
= 11210 ÷ 24
= $467.08
So
\(\\ \rm\rightarrowtail I=\dfrac{PRT}{100}\)
\(\\ \rm\rightarrowtail I=\dfrac{9500(2)(9)}{100}\)
\(\\ \rm\rightarrowtail I=95(18)\)
\(\\ \rm\rightarrowtail I=\$ 1710\)
#B
Total months=24
Monthly payment:-
Principle+Interest/months9500+1710/2411210/24467.08≈467$Find the total thickness of two pieces of wood that a carpenter glued together if one is 5/16 inches and the other is 7/8 inches thick?.
Answers
Answer:
19/16 or 1 and 3/16 of an inch
Step-by-step explanation:
In order to add fractions, we must first find a common denominator, which in this case would be 16 because 8 times 2 = 16.
Whatever we do to the denominator must be done to the numerator, so we have to also multiply 7 by 2 to give us the fraction 14/16.
Now we can add the fractions: 5/16 + 14/16 = 19/16 or 1 3/16
I need help asap pls
Answers
Answer:
C = 8π
Area = 32π
Step-by-step explanation:
Radius = 4 cm
Circle Circumference Formula: 2πr
C = 2π*4 = 8π cm
Area Circle Formula: 2πr^2
Area = 2π*4^2 = 2π*4*4 = 2π*16 = 32π cm