∫ t 0 f(τ) dτ = = ℒ−1 {F(s)/s}, to evaluate the given inverse transform. (Write your answer as a function of t.) ℒ−1 {1/s^3(s − 1)}
Answers
Hence, the answer is "The inverse Laplace Transform of ℒ⁻¹{1/s³(s - 1)} is t²/2 - t + 1".
The inverse Laplace Transform is given by, `ℒ⁻¹{F(s)/s} = ∫ from 0 to t f(τ)dτ
`To evaluate the given inverse transform `ℒ⁻¹{1/s³(s - 1)}`,
let's write the given function `1/s³(s - 1)` in partial fractions form.
`1/s³(s - 1) = A/s + B/s² + C/s³ + D/(s - 1)`
Multiplying both sides by s³(s - 1),
we get;`1 = As²(s - 1) + B(s(s - 1)) + Cs³ + Ds³`Substituting `s
= 0` in the above equation,
we get;`1 = 0 + 0 + 0 + D
`Therefore, `D = 1`Substituting `s = 1` in the above equation,
we get;`1 = A(1) + B(0) + C(1) + D(0)`Therefore, `A + C = 1`
Multiplying both sides by s and substituting `s = 0` in the above equation,
we get;`0 = A(0) + B(0) + C(0) + D(-1)`
Therefore, `D = 0`Substituting `s = infinity` in the above equation,
we get;`0 = A(infinity) + B(infinity) + C(infinity) + D(infinity)`
Therefore, `A = 0`Therefore, `C = 1`Therefore, `B = 0`Therefore, `1/s³(s - 1) = 1/s³ - 1/s² + 1/s`
Now, let's find the inverse Laplace Transform of the above partial fraction form;`ℒ⁻¹{1/s³(s - 1)} = ℒ⁻¹{1/s³} - ℒ⁻¹{1/s²} + ℒ⁻¹{1/s}`
We know that the inverse Laplace Transform of `1/sⁿ` is `tⁿ-1/Γ(n)sⁿ`.
Using this, we get;`ℒ⁻¹{1/s³(s - 1)} = t²/2 - t + 1`
Therefore, the function of `t` is `t²/2 - t + 1`.
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Related Questions
Rewrite the rational expressions x³+x²-7/x-3A) x²+4x+12+29A) x²+4x+12+29/x-3A) x²+4x+12+19/x-3A) x²+3x+12
Answers
We have the expression:
\(\frac{x^3+x^2-7}{x-3}\)We can simplify it by dividing the numerator by the denominator (division of polynomials):
\(\begin{gathered} \frac{x^3+x^2-7}{x-3} \\ \frac{x^3-3x^2+3x^2+x^2-7}{x-3}=\frac{x^2(x-3)}{x-3}+\frac{4x^2-7}{x-3}=x^2+\frac{4x^2-7}{x-3} \\ x^2+\frac{4x^2-12x}{x-3}+\frac{12x-7}{x-3}=x^2+\frac{4x(x-3)}{x-3}+\frac{12x-7}{x-3}=x^2+4x+\frac{12x-7}{x-3} \\ x^2+4x+\frac{12x-36}{x-3}+\frac{36-7}{x-3}=x^2+4x+\frac{12(x-3)}{x-3}+\frac{29}{x-3} \\ x^2+4x+12+\frac{29}{x-3} \end{gathered}\)The answer is: x²+4x+12+29/(x-3)
The puppy needs to catch a train that is moving at 8km/hr. she is 100 meters away and running at 10km/hr. How far will the train have traveled before the puppy is able to jump on?
Answers
The rabbit will jump on the train after the train moves 0.4km
The given parameters are:
\(d = 100m\) -- distance between the puppy and the train
\(s_1 = 8km/hr\) -- the speed of the train
\(s_2 = 10km/hr\) --- the puppy's speed
Required
The distance where the puppy catch up with the train
First, we calculate the relative speed (rs) of the puppy to the train
\(rs =s_2 - s_1\)
\(rs =10km/hr - 8km/hr\)
\(rs =2km/hr\)
Calculate the time to catch up with the train using:
\(Time = \frac{Distance}{Speed}\)
\(t = \frac{100m}{2km/hr}\)
Convert km to m
\(t = \frac{100m}{2*1000m/hr}\)
\(t = \frac{100m}{2000m/hr}\)
\(t = 0.05hr\)
Calculate the distance to catch up with the train using:
\(Distance = Speed * Time\)
\(d= s_2* t\) --- speed of the train multiplied by the time
So, we have:
\(d = 8km/hr * 0.05hr\)
\(d = 8km * 0.05\)
\(d = 0.4km\)
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the area of a rectangular room is 45 CM square if the length had been 3 M less and breadth 1 m more it would have been a square find the length and breadth of the room
Answers
The length of this rectangular room is equal to 9 meters.
The breadth of this rectangular room is equal to 5 meters.
How to calculate the area of a rectangle?In Mathematics and Geometry, the area of a rectangle can be calculated by using the following mathematical equation:
A = xy
Where:
A represent the area of a rectangle.y represent the breadth of a rectangle.x represent the length or height of a rectangle.If length would become 3 meters less and breadth would become 1 meter more, we have the following expressions;
y = x - 3
x = y + 1
45 = xy
When the rectangle turns to a square, all of the side lengths become equal;
y + 1 = x - 3
x = y + 4
From the area of a rectangle formula, we have:
45 = (y + 4)y
45 = y² + 4y
y² + 4y - 45 = 0
y² + 9y - 5y - 45 = 0
y(y + 9) - 5(y + 9) = 0
y = 5 or -9 meters.
For the length, we have:
x = y + 4
x = 5 + 4
x = 9 meters.
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simplify 9^3 * 3^6 in exponential form
Answers
Answer:
=[(2
2
)
3
×3
6
]×5
6
=2
6
×3
6
×5
6
=64×729×15625
=729000000
=3
6
×10
6
=30
6
3 friends are sitting around a table for Friendsgiving. The perimeter of the table is 96 inches. Find the length of EACH SIDE. 2(3x+3) is the short side of the rectangle and 3(x+5) is the long side.
Answers
X=3
sbzhshsbsnjsjsjsnsnsnns
please help will give brainly
Answers
Answer:
Angle A: 52.5
Angle B: 68.9
Angle C: 58.6
Step-by-step explanation:
Brainliest?
:)
Find the area of the triangle below to the nearest tenth.
Answers
Answer:
The answer to this question is 97.5
Step-by-step explanation:
A= 15x13/2=97.5
If r is the position vector (with components x1, x2, x3) and v is any vector, show that
Answers
The position vector r (with components x1, x2, x3) and any vector v can be expressed as r = x1*i + x2*j + x3*k and v = a*i + b*j + c*k, respectively.
How can we show that the dot product of the position vector r and the vector v is equal to the sum of the products of their corresponding components?The dot product of two vectors is defined as the sum of the products of their corresponding components. Therefore, the dot product of r and v can be calculated as:
r · v = (x1*i + x2*j + x3*k) · (a*i + b*j + c*k)
= (x1*a) + (x2*b) + (x3*c)
Here, we multiplied the corresponding components and summed them up. The dot product represents the scalar projection of one vector onto the other, which indicates the magnitude of one vector in the direction of the other.
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Question 9 (2 points)
Evaluate the function f(x), if f(x) = 4x - 2, what is the value of f(3)?
Enter your answer in the box below.
f(3) =
Answers
If the function be f(x) = 4x - 2 then the value of f(3) exists 10.
What is meant by function?An expression, rule, or law that establishes a connection between an independent variable and a dependent variable.
In mathematics, a variable is referred to as the alphabetic symbol used to represent a number or numerical value. An unknown quantity is represented by a variable in algebraic equations.
In mathematics, a function is a relationship where the values of one or more independent variables determine the values of a single dependent variable. The term "function" denotes that the independent variable influences the dependent variable.
Let the function be f(x) = 4x - 2
substitute the value of x = 3, then
f(3) = 4 × 3 - 2
simplifying the above equation, we get
f(3) = 10
Therefore, the value of f(3) exists 10.
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Someone help me on answer 6 and 7 pls
Answers
The unit rate of the distance of the yellow jacket is 0.5 meter per seconds and unit writing rate of Lydia is 0.075 pages per minute
Unit rateYellow jacket:
Distance = 4.5 meters
Time = 9 seconds
Unit rate = Distance / time
= 4.5 / 9
= 0.5 meter per seconds
Lydia:
Number of pages = 4 1/2
Time = 1 hour
Convert hour to minutes
1 hour = 60 minutes
Unit writing rate = Number of pages / Time
= 4 1/2 ÷ 60
= 9/2 × 1/60
= 9/120
= 0.075 pages per minute
Therefore, the unit rate of the distance of the yellow jacket is 0.5 meter per seconds and unit writing rate of Lydia is 0.075 pages per minute
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In the xy-plane(not shown), a right triangle has its right angle at the origin and has its hypotenuse along the line y=7x−1. If none of the sides of the triangle are vertical, what is the product of the slopes of the three sides of the triangle? A. −7 B. −1 C. -1/7 D. 1/7 E. 1
Answers
The product of the slopes of the three sides of the triangle, we need to determine the slopes of each side. Therefore, the product of the slopes of the three sides of the triangle is -1, which corresponds to option B.
Given that the hypotenuse of the right triangle is along the line y = 7x - 1, we can determine its slope by comparing it to the slope-intercept form, y = mx + b. The slope of the hypotenuse is 7.
Since the right angle of the triangle is at the origin, one side of the triangle is a vertical line along the y-axis. The slope of a vertical line is undefined.
The remaining side of the triangle is the line connecting the origin (0,0) to a point on the hypotenuse. Since this side is perpendicular to the hypotenuse, its slope will be the negative reciprocal of the hypotenuse slope. Therefore, the slope of this side is -1/7.
To find the product of the slopes, we multiply the three slopes together: 7 * undefined * (-1/7). The undefined slope doesn't affect the product, so the result is -1.
Therefore, the product of the slopes of the three sides of the triangle is -1, which corresponds to option B.
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33 divided by 4 quotient and remainder
Answers
2a-6=2(a-3) is their a solution
Answers
Find the slope and the equation of the tangent line to the graph of the function at the given value of x. y=x 4
−10x 2
+9;x=1 The slope of the tangent line is (Simplify your answer.) The equation of the tangent line is
Answers
The equation of the tangent line represents a straight line that passes through the point of tangency and has a slope of -16.
The slope of the tangent line to the graph of the function y = x^4 - 10x^2 + 9 at x = 1 can be found by taking the derivative of the function and evaluating it at x = 1. The equation of the tangent line can then be determined using the point-slope form.
Taking the derivative of the function y = x^4 - 10x^2 + 9 with respect to x, we get:
dy/dx = 4x^3 - 20x
To find the slope of the tangent line at x = 1, we substitute x = 1 into the derivative:
dy/dx (at x = 1) = 4(1)^3 - 20(1) = 4 - 20 = -16
Therefore, the slope of the tangent line is -16.
To find the equation of the tangent line, we use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Given that the point of tangency is (1, y(1)), we substitute x1 = 1 and y1 = y(1) into the equation:
y - y(1) = -16(x - 1)
Expanding the equation and simplifying, we have:
y - y(1) = -16x + 16
Rearranging the equation, we obtain the equation of the tangent line:
y = -16x + (y(1) + 16)
To find the slope of the tangent line, we first need to find the derivative of the given function. The derivative represents the rate of change of the function at any point on its graph. By evaluating the derivative at the specific value of x, we can determine the slope of the tangent line at that point.
In this case, the given function is y = x^4 - 10x^2 + 9. Taking its derivative with respect to x gives us dy/dx = 4x^3 - 20x. To find the slope of the tangent line at x = 1, we substitute x = 1 into the derivative equation, resulting in dy/dx = -16.
The slope of the tangent line is -16. This indicates that for every unit increase in x, the corresponding y-value decreases by 16 units.
To determine the equation of the tangent line, we use the point-slope form of a linear equation, which is y - y1 = m(x - x1). We know the point of tangency is (1, y(1)), where x1 = 1 and y(1) is the value of the function at x = 1.
Substituting these values into the point-slope form, we get y - y(1) = -16(x - 1). Expanding the equation and rearranging it yields the equation of the tangent line, y = -16x + (y(1) + 16).
The equation of the tangent line represents a straight line that passes through the point of tangency and has a slope of -16.
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a professor gives a special tutoring session to students who scored less than 60 percent on the first test. on the second test, their mean score rises to 70 percent. what conclusion can we draw, if any, and why?
Answers
We can conclude that the special tutoring session seems to have had a positive impact on the students' performance, as their mean score increased from below 60% to 70%.
Determine the conclusionWe can conclude that the special tutoring session has been effective since the students' mean score increased from 60 percent to 70 percent on the second test.
The students who scored less than 60 percent on the first test were given a special tutoring session, which resulted in their mean score increasing to 70 percent on the second test.
Therefore, we can conclude that the special tutoring session has been effective in improving the students' performance on the second test.
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Use the triangle below to determine which of the following statements are true a. (Sin A)^2 + ( Cos A)^2=1.
Answers
Answer 4
Step-by-step explanation:
A 4500 square foot roll of plastic wrap costs $23.95. If 120 square feet is need for a party game, what is the monetary value of that 120 square foot piece of plastic wrap? Round to the nearest cent.
Answers
$0.64 is the monetary value.
We are given that a 4500 square foot roll of plastic wrap costs $23.95 and we are to determine the monetary value of a 120 square foot piece of plastic wrap.
Let us find the cost per square foot by dividing the total cost of the roll by the number of square feet in the roll:
$23.95/4500 = $0.005322 per square foot
Now, we can multiply the cost per square foot by the number of square feet needed for the party game (120) to find the monetary value of the piece of plastic wrap.
Therefore, the monetary value of that 120-square-foot piece of plastic wrap is:
$0.005322 x 120 = $0.64 (rounded to the nearest cent)
Hence, the monetary value of that 120-square-foot piece of plastic wrap is $0.64.
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yeah help plz
What value of x will make 1/2x = 25 true?
Answers
Answer:
50
Step-by-step explanation:
\(\frac{1}{2}\)x = 25
You can multiply both sides by 2 so that you can find the value of x.
\(\frac{2}{2}\)x = 25 × 2
Find x.
x = 50
Explanation: given,
1/2x = 25
1 = 25 * 2x
1 = 50x
1/50 = x
So, if the value of x is 1/50 the equation will be true.
Checking:
(x = 1/50)
1/2 X (1/50)
= 1/2 X 50/1 [reciprocal]
=25
DUE IN 52 MINUTES PLEASE HELP IT IS MY LAST QUESTION
Answers
Answer:
(-5.5, -4.5)
Step-by-step explanation:
The table shows the length and width of a rectangle.
Dimension
-
Length
Width
Write a simplified expression for the perimeter of the rectangle.
+
units
I
X
÷
**
Measurement
(units)
3x + 6
2x - 4
O|C
dº √0 90 = = < > ≤ ≥ (0)
T
Answers
The simplified expression for the perimeter of the rectangle is 10x + 4.
What is Perimeter?Perimeter of a straight sided figures or objects is the total length of it's boundary.
It is defined for, squares, rectangles, triangles and other polygons.
Given is a rectangle.
Length of the rectangle = 3x + 6
Width of the rectangle = 2x - 4
The formula to calculate the perimeter of the rectangle is,
Perimeter of a rectangle = 2 (Length + Width)
= 2 (3x + 6 + 2x - 4)
= 2 (5x + 2)
= 10x + 4
Hence the perimeter of the rectangle is 10x + 4 units.
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Consider the right triangle below.
6x - 44
2x - 20
3x + 24
If the perimeter of the triangle is 224 cm, find the area of the triangle.
The area of the triangle is
cm
Answers
Answer:
Area = 1,344 cm²
Step-by-step explanation:
First, find the value of x and the numerical side length of all sides of the triangle before finding the area.
Thus:
Perimeter of a triangle = sum of all its side
Therefore,
(6x - 44) + (2x - 20) + (3x + 24) = 224
Solve for x
6x - 44 + 2x - 20 + 3x + 24 = 224
Add like terms
11x - 40 = 224
Add 40 to both sides
11x = 224 + 40
11x = 264
Divide both sides by 11
x = 24
Find the measurement of each side length by plugging in the value of x:
Hypotenuse = 6x - 44 = 6(24) - 44 = 100 cm
Height = 2x - 20 = 2(24) - 20 = 28 cm
Base = 3x + 24 = 3(24) + 24 = 96 cm
Find the area:
Area = ½×base×height
Plug in the values
= ½×96×28
Area = 1,344 cm²
How do you multiply fractions with a common denominator?
Answers
When multiplying fractions, multiply the numerator (top number), then the denominator (bottom number), and finally reduce to the lowest term if necessary.
Fractions are subsets of wholes. They are made up of a numerator (the number on top) and a denominator (the number on the bottom). The numerator represents the number of parts, while the denominator represents the total number of available parts.
To multiply two fractions, follow these three simple steps:
Step 1: Multiply the numerators of each fraction by themselves (the numbers on top). The answer's numerator is the result.
Step 2: Add the denominators of each fraction together (the numbers on the bottom). The answer's denominator is the result.
Step 3: Simplify or reduce your response.
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What is the leading coefficient? - 3x3 + 8x2 + 9x - 4
A.
-4
B.
-3x3
C.
-3
D.
3
If you answer this I'll give you brainliest
Answers
Answer:
c
Step-by-step explanation:
the highest exponent is 3 so the leading coefficient is -3
The scale on a map is 1 : 300,000.
The length of a river on the map is 2 cm.
What is the length of the river in real life?
Give your answer in kilometres.
Answers
The length of the river in real life is 600000.
What is the scale?A map's scale is the ratio of a distance on the map to its equivalent distance on the ground. The curvature of the Earth's surface complicates this simple concept by forcing scale to vary across a map. Because of this variation, the concept of scale takes on two distinct meanings.
From the information, the scale on a map is 1 : 300,000 and the length of a river on the map is 2 cm.
In this case, the length on real life will be calculated thus:
= 2 ×300000
= 600000
It is 600000 in real life.
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18-9.2x10 to the power of -4 subtract the following in.Write your answer in standard notation
Answers
The subtraction of the numbers, involving scientific notation, is given as follows:
18 - 9.2 x 10^(-4) = 17.99908.
What is scientific notation?A number in scientific notation is given according to the rule presented as follows:
\(a \times 10^b\)
The base is \(a \in [1, 10)\), meaning that it can assume values from 1 to 10, with an open interval at 10 meaning that for 10 the number is written as 10 = 1 x 10¹, meaning that the base is 1.
In this problem, we are given the following subtraction:
18 - 9.2 x 10^(-4).
When two numbers are subtracted, they must have the same base, hence we are going to convert 9.2 x 10^(-4) to base 0.
This means that 4 has to be added to the exponent, meaning that the base has to be divided by 10000 to compensate, then:
9.2 x 10^(-4) = 0.00092.
Then the result of the subtraction is given as follows:
18 - 9.2 x 10^(-4) = 18 - 0.00092 = 17.99908.
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upper and lower bounds question
Answers
The upper bound and lower bound of a using the equation will result to
upper bound value 136015.499
lower bound value 136014.500
How to find the upper bound and lower bound valuesEvaluating the given expression a = b + 2c
where
b = 15 to 2 significant figures
c = 68 000 to 2 significant figures
solving for a
a = b + 2c
a = 15 + 2(68000)
a = 136015
The bound values
upper bond value is the largest possible vale of a
upper bond value = 136015.499
lower bound value is the lowest possible value of a
lower bound value = 136014.500
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Use a trigonometric ratio to solve for y. Round to two decimal places as necessary!!
HELP, and can someone explain how to get the answer?
Answers
On solving the provided question, we can say that by trigonometry so sin 90 = p/h => y = 19
what is trigonometry?The area of mathematics known as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, around the third century BC. from the use of geometry in astronomical study. The area of mathematics known as exact methods deals with specific trigonometric functions and how they might be used in calculations. There are six popular trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics of triangles, particularly right triangles, is called trigonometry. The study of geometry, however, is the characteristics of all geometric figures
here,
it is a right angled triangle
by trigonometry
so sin 90 = p/h
1 = 19/y
y = 19
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nationally, the proportion of red cars on the road is 12%. a statistically minded fan of the philadelphia phillies (whose team color is red) wonders if phillies fans are more likely to drive red cars. one day during a home game, he takes a random sample of 210 cars parked at citizens bank park (the phillies home field), and counts 35 red cars. does the data provide convincing evidence that phillies fans are more likely to drive red cars? researchers found the p-value for the test in to be 0.0187
Answers
The p-value is the probability of observing a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. We are given that the p-value is 0.0187.
The null hypothesis is that the proportion of red cars among Phillies fans is the same as the national proportion, i.e., p = 0.12. The alternative hypothesis is that the proportion of red cars among Phillies fans is greater than the national proportion, i.e., p > 0.12.
We can use a one-sample proportion test to test this hypothesis. The test statistic is:
z = (P - p) /\(\sqrt(p*(1-p)/n)\)
where P is the sample proportion, p is the hypothesized proportion under the null hypothesis, n is the sample size, and sqrt is the square root function.
In this case, p = 0.12, P = 35/210 = 0.1667, and n = 210. Plugging these values into the formula, we get:
z = (0.1667 - 0.12) / \(\sqrt(0.12*(1-0.12)/210)\) = 2.072
The p-value is the probability of observing a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. We are given that the p-value is 0.0187.
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is convincing evidence that Phillies fans are more likely to drive red cars than the national proportion of 12%.
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What is the length of the diameter?
Answers
is there any instructions above the circle?
a ray of sunlight forms a 24°-angle with the ground. what is the length of the shadow cast by a person 1.82 m tall?
Answers
The length of the shadow cast by a person 1.82 m tall will be 4.09m
A ray of sunlight forms a angle of Elevation to the sub is 24°.
The angle from horizontal to the point where we stopped our head is called angle of elevation.
On this note, the shadow cast on the floor shows the adjacent of the angle of Elevation.
Hence, the length of the shadow cast can be calculated from Tangent identify as;
Tan 24° = 1.82/l
l = 1.82/Tan 24
l = 1.82/0.445
Thus, Length of shadow = 4.09m
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