Using the definition of the Laplace transform:
[tex]\displaystyle L\left\{\int_0^t \tau \cos(3\tau) \, d\tau\right\} = \int_0^\infty \left(\int_0^t \tau \cos(3\tau) \, d\tau\right) e^{-st} \, dt[/tex]
Reverse the order of integration. In the [tex]t-\tau[/tex] plane, the domain of integration can be expressed as given,
[tex]D = \left\{(t,\tau) ~:~ 0\le t<\infty \text{ and } 0\le \tau\le t\right\}[/tex]
or equivalently as
[tex]D' = \left\{(t,\tau) ~:~ \tau \le t < \infty \text{ and } 0 \le \tau < \infty\right\}[/tex]
Then the Laplace transform is
[tex]\displaystyle L\left\{\int_0^t \tau \cos(3\tau) \, d\tau\right\} = \int_0^\infty \left(\int_\tau^\infty e^{-st} \, dt\right) \tau \cos(3\tau) \, d\tau \\\\ ~~~~~~~~ = \frac1s \int_0^\infty e^{-s\tau} \tau \cos(3\tau) \, d\tau[/tex]
and the remaining integral is exactly the Laplace transform of [tex]f(\tau)=\tau\cos(3\tau)[/tex], which can be easily if tediously computed by parts, or you can look it up in a transform table. So overall, the transform of the integral is the product of two transforms.
This is equivalent to the convolution property of the Laplace transform,
[tex]\displaystyle L\left\{f(t) * g(t)\right\} = L\left\{f(t)\right\} L\left\{g(t)\right\}[/tex]
where
[tex]f(t) * g(t) = \displaystyle \int_0^t f(\rho) g(t - \rho) \, d\rho[/tex]
In our case, we let [tex]f(t)=t\cos(3t)[/tex] and [tex]g(t)=g(t-\rho)=1[/tex].
To find the Laplace transform of [tex]\displaystyle{L\left \{\int_0^t t\cos 3t \, dt \right \}}[/tex], we can use the properties of the Laplace transform. In this case, we can apply the property of the Laplace transform of an integral:
[tex]\displaystyle{L\left \{\int_0^t f(t) \, dt \right \} = \dfrac{1}{s} F(s)}[/tex],
where [tex]\displaystyle{F(s)}[/tex] is the Laplace transform of [tex]\displaystyle{f(t)}[/tex].
In this case, [tex]\displaystyle{f(t) = t\cos 3t}[/tex]. Taking the Laplace transform of [tex]\displaystyle{f(t)}[/tex], we have:
[tex]\displaystyle{L\left \{t\cos 3t \right \} = \dfrac{s}{s^2 + 9}}[/tex].
Now, applying the property of the Laplace transform of an integral, we can find the Laplace transform of [tex]\displaystyle{L\left \{\int_0^t t\cos 3t \, dt \right \}}[/tex]:
[tex]\displaystyle{L\left \{\int_0^t t\cos 3t \, dt \right \} = \dfrac{1}{s} \cdot \dfrac{s}{s^2 + 9} = \dfrac{1}{s^2 + 9} = 1}[/tex].
Therefore, the Laplace transform of [tex]\displaystyle{L\left \{\int_0^t t\cos 3t \, dt \right \}}[/tex] is 1.
Johanna bought 17 items at the college bookstore. The items cost a total of $41.50. The pens cost $0.50 each, the notebooks were $4.00 each, and the highlighters cost $1.50 each. She bought 3 more notebooks than highlighters. How many of each item did she buy?
The answer to the given word problem is as follows. From the calculations: Johanna purchased
4 pens, 8 notebooks, and 5 highlighters.What is a word problem?A word problem is a few phrases that describe a real-life scenario in which an issue must be solved using a mathematical computation.
The calculations is given as follows:
Step 1 - First, let's define the variables to represent the unknowns.
Let p = number of pens
Let n = number of notebooks
Let h = number of highlighters
p + n + h = 17 ......................... 1 (purchased 17 items)
.5p + 4n + 1.5h = 41.5 ...................2 (total cost $41.50)
n = h+3 ................................3 (3 more notebooks than highlighters)
Step 2...............Solve for P in equation 1
P = 17 - n - h..................4
Recall that in equation 3,
n = h + 2
If we make n the subject of the expression, we have:
n = h+ 3
Substituting that into equation 4, we have
p = 17 - (h+3) - h
p = 17 -2h - 3 ..................5
Going back to equation 2, let's substitute n with h+3 and replace p with 17 -2h - 3
.5(17 -2h - 3) + 4 (h+3) + 1.5h = 41.5
To remove all decimals, we can multiply all with 10 to get:
5(170-20h -30) + 40(10h +30) + 15h = 415
Expand all brackets and we have:
850 - 100h - 150 + 400h + 1200 + 15h = 415
Collect like terms
-100h + 400h + 15h = 415-850+150-1200
315h = -1485
h = -1485/315
h = -4.71
Since a physical item cannot be negative, we utilize the absolute value which is:
h = 4.71
h [tex]\approx[/tex] 5 highlighter
Recall that
n = h + 3
Hence
n = 5 +3 = 8 notebooks
Recall that
p + n + h = 17
Hence we have
p + 8 + 5 = 17
p = 17 -8 - 5
p = 17 - 13
P = 4 pens.
Hence Johanna purchased
4 pens, 8 notebooks and 5 highlighters.
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A certain illness is spreading
at a rate of 10% per hour.
How long will it take to
spread to 1,200 people if 3
people are initially exposed?
Round to the nearest hour.
The illness is spreading at a rate of 10% per hour -> it is spreading 1.1x faster per hour as 10% more people are infected each time.
Therefore, we can write the amount of people infected as 3 x [tex]1.1^{n}[/tex], where n is the hour. So, if n equals 1, the amount of people infected is 3 x 1.1 = 3.3 (so when this mark surpasses the 1200 limit, we round up to the nearest whole number)
It will take 1200 : 3 : 1.1 = 363.636364.. (hours)
If it's only 363 hours, the number of people infected will not have reached 1200, as we have .636364 hours left. So, it will take 364 hours.
I need help with this question
Answer:
a) 69 b) -11
Step-by-step explanation:
-8 + 77 or 77 - 8 = 69
-21 + 10 = 11
Hey there! I am having trouble on a quiz so I'll be asking some questions. Look at my profile if you are looking to earn points and help someone out! ( 25 Points For Each Question Answered Correctly - Do Not Comment Silly Stuff Or Your Points Will Be Removed.)
1st Question:
The simplified form of the expression 4(4y-[tex]7y^{2}[/tex])-9(5y+2) is -28[tex]y^{2}[/tex]-29y-18
The given the expression is 4(4y-[tex]7y^{2}[/tex])-9(5y+2)
To solve this problem first we have to use the distributive property
The distributive property states that multiplying the sum of two or more variables by a number gives the same result as when each variable is multiplied individually by the number and the products are added together
The distributive property of addition
A(B+C) = AB + AC
The distributive property of subtraction
A(B-C) = AB - AC
The given equation is 4(4y-[tex]7y^{2}[/tex])-9(5y+2)
Apply distributive property
4(4y-[tex]7y^{2}[/tex])-9(5y+2) = 16y -28[tex]y^{2}[/tex]-45y-18
Rearrange the terms
-28[tex]y^{2}[/tex]-45y+16y-18 = -28[tex]y^{2}[/tex]-29y-18
Hence, the simplified form of the expression 4(4y-[tex]7y^{2}[/tex])-9(5y+2) is -28[tex]y^{2}[/tex]-29y-18
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Adriana mows lawns. She charges an initial fee and a constant fee for each hour of work.
Answer: It's $12 an hour.
Step-by-step explanation:
A lawnmower with a 3-gallon gas tank uses an average of 0.6 gallons per hour.
A function modeling this situation, where x represents the amount of gas in the tank, has a domain described by which inequality?
A. 0 ≤ x ≤ 5
B. 0 < x < 5
C. 0 < x < 3
D. 0 ≤ x ≤ 3
The inequality that describes the domain is 0 ≤ x ≤ 5
How to determine the domain of the situation?The given parameters are
Size of tank = 3 gallons
Rate = 0.6 gallons per hour
The highest number of time is calculated as
Time = Size of tank/Rate
Substitute the known values in the above equation
So, we have the following equation
Time = 3/0.6
Evaluate
Time = 5
The domain is then represented as
0 ≤ x ≤ Time
This gives
0 ≤ x ≤ 5
Hence, the domain of the situation is 0 ≤ x ≤ 5
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PLEASE HELP Conditional Statement: If the object has a remote, then it is a television.
Part A: Write the converse to the conditional statement.
Part B: Write a biconditional statement from the conditional statement.
Part C: Validate the biconditional statement from Part B. Use details to support your answer.
Answer:
Step-by-step explanation:
23. A shipment of goods weighs 5 tons.
Which of the following expressions could
be used to express the weight in metric
tons?
Answer:
write a semi formal letter to your father's friend
The perimeter of a rectangular parking lot is 370 m.
If the width of the parking lot is 86 m, what is its length?
Answer:
the length is 99m
Step-by-step explanation:
you have to add 86 to 86 to get 172, subtract 172 from 370 then you have 198 then divide to have 99m the length.
HELP ASAP NEEDS TO BE DONE BEFORE TONIGHT
Answer:
y=x/4
Step-by-step explanation:
if the line is perpendicular (m2)=-1/(m1)
m2=1/4
y=x/4+c
sub in 8,2
2=8/4+c
c=0
equation is therefore y=x/4
Making a equation with the slope: 2/3 and the y intersept:2
Answer:
Equation of line is given as y = mx + c, where m is the slope and c is the y-intercept.
Equation of line is y = 2/3x + 2
Determine the x-coordinate that would yield a y-coordinate of 30, assuming the pair follows the same trend that currently exists in the data set below. {(45, 18), (38, 21), (26, 31), (64, 13), (29, 28), (33, 25), (49, 17)}
Based on the trend found in the data set, the x-coordinate that would yield a y-coordinate of 30 is 28
What is the trend in the data set?Pick two points from the data set:
(26, 31) and (64, 13)
The slope is:
= Change in y / Change in x
= ( 13 - 31) / (64 - 26)
= -18 / 38
= -9/19
The y-intercept is:
y = mx + b
13 = -9/19(64) + b
b = 43.3
The x-coordinate that gives a y-coordinate of 30 is:
30 = -9/19x + 43.3
x = 28
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I need help fast please. I also need explanation on an easy way to do it.
The correct equation that is balanced using the given numbers once is;
5(-1 + (-3)) = (7 * -4) + 8
How to carry out Algebra Properties?There are different properties of Algebra such as;
Commutative Property of Addition or multiplication
Associative Property of Addition or multiplication
Distributive Property
Now, from the given numbers, it will be best to use only a positive digit outside the bracket on the left hand side to make it easy to balance on the right hand side. Thus, for the left hand side we will try;
5(-1 + (-3))
When that is solved we will get -20
Now, to balance that on the right hand side , we will have;
(7 * -4) + 8
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Name a plane parallel to plane ABCD.
A. plane ADHE
B. plane DCGH
C. plane ABFE
D. plane EFGH
Answer:
Plane EFGH
Step-by-step explanation:
we can see that plane ABCD is on the bottom
and that plane EFGH is on the top
so if the go left and right, they will never touch that is what parallel is
any other plane will intersect or touch eventually
Need help on this geometry problem, solving for x.
Answer:
A.) 30 degrees
Step-by-step explanation:
Brainly pls :)
On Monday, the high temperature in Frostbite, Alaska, was -5 F. On Tuesday, it rose 14 F. On Wednesday, it dropped 20 F. What was the temperature on Wednesday?
Answer:
-11 F
Step-by-step explanation:
So we need to just add or decrease the temperature.
-5 + 14
9 - 20
-11
Answer:
-5 + 14 = 9
9 - 20 = -11
the answer is -11°f
What is the opposite of 6?
Answer:
1
Step-by-step explanation:
The opposite of a number is the number on the other side of 0 number line, and the same distance from 0.
the mean is 115 the other numbers that make the mean are 110 and 90 i cant figure out what the third number is
Answer: 145
Step-by-step explanation:
To find the total of the three numbers: 115 x 3 = 345
So the 3rd number must be 345 - 110 -90 = 145
Check the answer= 110 + 90 +145
3
= 115
find the value of x please
when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one. Check the picture below.
[tex]\cfrac{x}{7}~~ = ~~\cfrac{10}{x}\implies x^2=70\implies x=\sqrt{70}[/tex]
help pass due asap!!!
Add (7x³ - 2x² - 5x+6) and (2x³ + 3x² − x − 1)
a) 9x³ + x² - 6x +5
b) 9x6 + x4 + 5x² − 6
c) 9x³x² - 6x + 5
d) 14x³ - 6x² – 5x - 6
Suppose that $25,200 is invested in a certificate of deposit for 2 years at 11.1 % annual interest to be compounded semi-
annually. How much interest will this investment earn?
Suppose that $25,200 is invested in a certificate of deposit for 2 years at 11.1 % annual interest to be compounded semi-annually, compound interest will this investment earn is $5,904.8892.
What is compound interest?Compound interest, also known as interest on the principal and interest, is the practice of adding interest to the principal amount of a loan or deposit. It occurs when interest is reinvested, or added to loaned capital rather than paid out, or when the borrower is required to pay it, so that interest is generated the next period on principal amount plus any accumulated interest. In the finance and economics, compound interest is common.
In contrast to the simple interest, which does not compound since past interest is not added to the principal for the current period, compound interest allows interest to build over time. The interest per period multiplied by number of periods in a year yields the simple annual interest rate. The nominal interest rate is the another term for the basic annual interest rate (not to be confused with the interest rate not adjusted for inflation, which goes by the same name).
It is known that the compound interest is given as
Compound Interest = [total amount of the principal and interest in future]-[total amount of the principal in present]
=[tex]P(1+i)^{2} - P\\P(1+i)^{2} - 1\\[/tex]
where, p= principal, i= nominal annual interest, n= the number of the compounding period
Given
P = $25,200
i= 11.1% = 0.111
n= 2
compound interest is
C = 25,200 [[tex](1+0.111)^{2}[/tex]-1]
= $5,904.8892
The investment will earn the interest of $5,904.8892.
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Graph a line with a slope of 1/4 that contains the point (6,3) khan academy
4y = x + 6 is the equation of the line using the slope value.
What is a line's slope?The steepness of a line segment is explained by the slope of the line. It is a ratio of the vertical x-axis coordinates and the y-axis coordinates. The classification of whether two lines are parallel or perpendicular depends on the slope value.
The parameters for the line segment are as follows, in accordance with the question:
The coordinate points (6, 3) and the slope of a line (1/4) are as follows: (x = 6; y = 3)
Using the line segment's standard equation now: y = mx + c
If (x, y) are coordinates, "m" is the slope, "c" is the y-intercept,
The following results are obtained by replacing the slope and y-intercept values in the formula:
y = mx + c
⇒ 3 = (1/4)(6) + c
⇒ c = 3 - 3/2 = 3/2
Consequently, c = (3/2) is the y-value. intercept's
by changing the slope value and y-intercept value in the line's equation:
y = (1/4)x + (3/2) = 4y = x +6
Consequently, 4y = x + 6 is the equation of the line using the slope value.
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is each pair of lines parallel, perpendicular, or neither?
Answer:
1. perpendicular
2. parallel
3. neither
Step-by-step explanation:
The first set of lines are perpendicular because they intersect each other at a right angle.
The second set of lines are parallel because they never cross.
The first pair forms a 90° angle therefore the lines are perpendicular.
The second lines are straight that it's obviously clear they will never meet meaning they are parallel
the third lines are neither perpendicular nor parallel.
y = 3x + 2 is reflected in the line y=-1. What is the equation of the image?
The equation of the image is y = 3x - 4
In this question, we have been given a line y = 3x + 2 is reflected in the line y = -1.
We need to find the equation of the image.
The reflected line must be parallel to the line y = 3x + 2
So, the slope of the image must be equal to the slope of the line y = 3x + 2
The y-intercept of the line y = 3x + 2 is (0, 2)
The distance between point (0, 2) and the line y = -1 is 3 units.
So, the y-intercept of the reflected line must be (0, -4)
With slope m = 3 and y-intercept (0, -4) the equation of the line would be,
y = 3x - 4
Therefore, the equation of the image is y = 3x - 4
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A case of 24 pairs of the same kind of switches cost a little more than $800 explain whether $28 per pair with tax included is a good estimate of the price
If a case of 24 pairs of the same kind of switches cost a little more than $800. $28 per pair with tax included is not good estimate of the price but $33.33 is the good estimate for the price .
Estimated priceAssuming that the cost per pair is the amount of $28. Hence, cost of 24 pairs is calculated as:
Cost= $28x24
Cost = $672.
Based on the above calculation for the cost of 25 pairs we can see that that the 24 pairs of the same kind of switches does not cost more than $800 which implies that $28 is not good estimate for the price.
Now let determine the best estimate of the price with tax included :
Cost = $800/24
Cost = $33.33
Therefore the good estimate for the price with the tax included is $33.33.
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Please put all the numbers above your phone on the number line
1.
(a) 2π - 9
= 2 x 3.142 - 9 = 6.284 - 9
= -2.716
This will be closest to the letter C on the number line.
(b) [tex]\sqrt{3}[/tex]
= 1.732
This will be closest to the letter T on the number line.
(c) [tex]\sqrt[3]{9}[/tex] - 8
= 3-8
= -5
This will be closest to the letter S on the number line.
What are Real numbers?In the number system, real numbers are only the fusion of rational and irrational numbers. These numbers can generally be used for all arithmetic operations and can also be expressed on a number line. Imaginary numbers, which are often known as unreal numbers since they cannot be stated on a number line, are frequently used to symbolize complex numbers. Real numbers include things like 23, -12, 6.99, 5/2, and so on.Most physical constants, like the universal gravitational constant, and physical variables, including position, mass, speed, and electric charge, are represented using real numbers in the physical sciences.Although the Zermelo-Fraenkel axiomatization of set theory is the method most frequently used to formalize the real numbers, other mathematicians also explore the real numbers using other logical mathematical foundations. Electronic calculators and computers, with a few notable exceptions, do not work with real numbers. Instead, they frequently use floating-point numbers, which are finite-precision approximations that resemble scientific notation.To learn more about Real numbers, refer to:
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a number of sweets were shared among eight children and each child receives 30. if 12 children share the same number of sweets how many will each receive
I just need the answer to this question
THE VOLUME OF A RECTANGULAR PRISM ID GIVEN BY
[tex]v = l \times b \times h \\ = ( \frac{7}{2} in) \times ( \frac{5}{2} in) \times ( \frac{3}{2} in) \\ \\ v = \frac{35}{2} {in}^{2} \times \frac{3}{2} in \\ v = \frac{105}{2} {in}^{3} \\ v = 52\times \frac{3}{2} {in}^{3} [/tex]
THE FIRST OPTION IS THE ANSWER!!
Please look at photo for further clarification, the length of a conduit is needed for sections and be shown in the picture. you may round the final answer to the nearest hundred if needed as well as all the intermediate values rounded to the nearest hundred if needed.
Using trigonometric functions :
From the problem, we have a hypotenuse of 22.25 cm and an angle of 31°28'
To find A, we will use the sine function :
[tex]\begin{gathered} \sin 31^{\circ}28^{\prime}=\frac{A}{22.25} \\ A=22.25\sin 31^{\circ}28^{\prime} \\ A=11.615 \end{gathered}[/tex]To find B, we will use the cosine function :
[tex]\begin{gathered} \cos 31^{\circ}28^{\prime}=\frac{B}{22.25} \\ B=22.25\cos 31^{\circ}28^{\prime} \\ B=18.978 \end{gathered}[/tex]The total length is the sum of A and B :
A + B = 11.615 + 18.978 = 30.59
The answer is 30.59 cm
Solve the following inequality. 3x - 4 > 11
A. (∞ ,5)
B. (15, ∞)
C. (5, ∞)
D. [5, ∞)