Compute the Laplace transform of [tex]f(t)=t^n[/tex], where [tex]n[/tex] is a non-negative integer.
[tex]\displaystyle L\left\{t^n\right\} = I_n = \int_0^\infty t^n e^{-st} \, dt[/tex]
Integrate by parts with
[tex]u=t^n \implies du = nt^{n-1} \, dt[/tex]
[tex]dv = e^{-st} \, dt \implies v = -\dfrac1s e^{-st}[/tex]
[tex]\displaystyle I_n = \frac ns \int_0^\infty t^{n-1} e^{-st} \, dt = \frac ns I_{n-1}[/tex]
By substitution,
[tex]\displaystyle I_n = \frac ns I_{n-1} = \frac{n(n-1)}{s^2} I_{n-2} = \cdots = \frac{n(n-1)\cdots(n-(k-1))}{s^k} I_{n-k}[/tex]
so that for [tex]k=n[/tex], we end up with
[tex]\displaystyle I_n = \frac{n(n-1)\cdots1}{s^n} I_0 = \frac{n!}{s^n} \int_0^\infty e^{-st} \, dt = \frac{n!}{s^{n+1}}[/tex]
We then have the inverse transform relation
[tex]\displaystyle L^{-1}\left\{ \frac1{s^n} \right\} = \frac{t^{n-1}}{(n-1)!}[/tex]
and so
[tex]L^{-1}\left\{ \dfrac1{s^3} + \dfrac1s \right\} = \dfrac{t^2}{2!} + \dfrac{0!}{t^0} = \boxed{\dfrac{t^2}2 + 1}[/tex]
a cellular phone company offers several different service options. one option, for people who plan on using the phone only in emergencies, costs the user $4.90 per month plus $0.52 per minute for each minute the phone is used. (a) write a linear function for the monthly cost y of the phone in terms of the number of minutes x the phone is used.
52x - 100y + 490 = 0 , is the required linear function.
What does the term "linear function" mean?
A mathematical function where the variables only occur in the first degree, where the constants are multiplied, and where the only possible combinations are addition and subtraction.
Given, the monthaly cost = y
the number of minute used = x
Hence,
the monthly cost of the phone = (4.90 + 0.52 × the number of minute used)
Given the monthly cost = y
the number of minute used = x
Hence y = 4.90 + 0.52 * x
or y = [tex]\frac{490}{100} + \frac{52}{100 } * x[/tex]
or 100y = 455 + 52x
or 52x - 100y + 490 = 0 , is the required linear function.
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Solve w=--Z
solve for x
When you solve w = __Z,
you will get the value of X ( but there should be a variable' X' in the blank space). This is the method in Algebra.
What is Algebra?When we employ words and numbers to answer a specific mathematical issue, this is known as an algebraic expression.Example , X = 5 + Z (If the value of Z is given we can easily find out the value of X)The alphabets are known as variables.To learn more about Algebra refer:
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Please help me ill give brainly to best answer
Assuming they work until their cost become the same, obviously.
The difference between the first charge of painter A and painter B = 376 - 280 = 96.
The hourly cost difference between painter B and painter A = 15 - 12 = 3.
So, painter A in the first hour has 96 more dollars than painter B, and as every hour pass, the difference goes down by 3 (as painter B gets 3 more dollars every hour)
Therefore, it would take 96 : 3 = 32 hours for the cost to be the same
Recheck : After 32 hours, painter A has 376 + 32 x 12 = 760 (dollars)
painter B has : 280 + 15 x 32 = 760 (dollars)
list two ways you can maximize a tax-sheltered retirement program
The ways that you can maximize a tax-sheltered retirement program include:
1. when the contributions made by the employer are not taxable
2. when the employee has started making contributions early in life,
What is a tax-sheltered retirement?A tax-sheltered is a retirement savings plan that enables self-employed individuals and workers of tax-exempt organizations to invest pretax money to create retirement income. A dependable source of income in retirement, tax-sheltered annuities are made to pay out consistently throughout time.
The fact that employer contributions are tax-free is one of the best features of retirement programs. Only when an employee takes money out of their retirement accounts do taxes need to be paid. If the employee starts making contributions early in life, the plans also allow the employee the chance to maximize returns, which is another benefit that the employee can get from them.
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SECOND TIME ASKING HELP!!!
EASY POINTS WILL GIVE BRAINLIST TO BEST ANSWER Determine if the sequence is arithmetic. If it is, find the common difference, the 21st term, the explicit rule, the recursive rule, and the three terms in the sequence after the last one given. 7, 4, 1, -2, ...
Answer:
Yes, this is an arithmetic sequence.
The difference is -3, so a(n+1) = a(n) - 3
The 21st term is -53, calculated as follows:
a(21) = 7 + (21 - 1)(-3) = -53
The next three terms are -5, -8, and -11.
What numbers are greater than 4/7 as a fraction
Answer: 1/2
Step-by-step explanation:
its greater than 4/7
1/7 would be greater
HELP PLS I NEED THIS TODAY
What are the similarities and differences between number 1 and 3?
What are the similarities and differences do you see between 3 and 4
For example:
Similar: both arithmetic/linear/common difference
Different: #1 is discrete #3 is continuous
WILL GIVE BRAIN
The correct responses for the arithmetic and exponential sequence and function are;
1. First part; 32 pennies
Second part; Approximately 50 days
2. 24.6 gallons
3,738.5 minutes
3. $67.2
Approximately 24 months
4. 1106.14 gallons
66.56 minutes
The difference is 1 is an arithmetic sequence while 3 is an exponential function
What is a an arithmetic sequence?An arithmetic sequence is a group of numbers in which the difference between subsequent terms is a constant
1. The given parameters are;
The initial amount Savannah placed in the piggy bank = 5 pennies
The amount given to Savannah each day = 3 pennies
Required;
First part;
The amount Savannah will have after 10 days
Solution;
The mathematical model for the situation is an arithmetic sequence or progression
The equation for an arithmetic progression is;
[tex]a_n = a_1 + (n - 1) \times d[/tex]
Where;
[tex]a_n[/tex] = The amount after n days
[tex]a_1 [/tex] = The first amount Savannah placed in the piggy bank = 5 pennies
n = The number of days
d = The common difference between the amount present in consecutive days = 3 pennies
The amount after 10 days is therefore;
[tex]a_{10} = 5 + (10 - 1) \times 3=32[/tex]
The money after 10 days is 32 pennies
Second part;
The number of days it would take for there to be $1.50 in the piggy bank.
Solution;
100 pennies = $1.00
Therefore;
$1.50 = 150 pennies
Which gives;
150 = 5 + (n - 1) × 3
(n - 1) × 3 = 150 - 5 = 145
(n - 1) × 3 = 145
n - 1 = 145 ÷ 3
n = 145 ÷ 3 + 1 ≈ 49.3 ≈ 50
It will take approximately 50 days for the amount in the piggy bank to be $1.50
2. The volume of the pool = 1,500 gallons
The initial amount of water in the pool = 5 gallons
The rate at which water enters the pool = 2 gallons in 5 minutes
Required;
First part
The volume of water in the pool after 50 minutes
Solution;
2 gallons in 5 minutes = (2/5) gallons/minute = 0.4 gallons/minute
Therefore;
[tex]a_{50} = 5 + (50 - 1) \times 0.4 =24.6[/tex]
After 50 minutes, 24.6 gallons will be in the pool
Second part;
The time it will take to fill the pool is given by the equation;
[tex]1500 = 5 + (n - 1) \times 0.4[/tex]
number
Which gives;
n = 3738.5
It will take 3,738.5 minutes to fill the pool
3. The formula for finding the amount in the account is the compound interest formula;
[tex]\displaystyle{A = P\cdot (1 + r)^n }[/tex]
Where;
A = The amount in the account after n months
P = The initial amount in the account = $50.0
r = The interest rate = 3.0%
n = The number of periods (months)
After 10 months, we have;
[tex]\displaystyle{A = 50 \times (1 + 0.03)^{10} \approx 67.2 }[/tex]
The amount in the account after 10 months is therefore, approximately $67.2
Second part;
Double the initial amount is 2×$50 = $100
Which gives;
[tex]100 = \displaystyle{50 \times (1 + 0.03)^{n} \approx 67.2 } [/tex]
Which gives;
n ≈ 23.45
Therefore it will take approximately 23.45 months or approximately 24 months for the amount to double
4. Using the exponential function formula, we have;
Number of 5 minutes in 50 minutes = 10
[tex] \displaystyle{1500 \times (1 - 0.03)^{10} \approx 1106.14} [/tex]
The water remaining after 50 minutes is approximately 1106.14 gallons
The time it takes for there to be less than 1,000 gallons in the pool is given as follows;
[tex] \displaystyle{ 1000= 1500 \times (1 - 0.03)^{n}} [/tex]
Which gives;
n = 13.3
Therefore;
The water in the pool will be less than 1000 gallons after 13.3, 5 minute periods or 13.3 × 5 minutes≈ 66.56 minutes
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Agda, Britta and Cornelia must pay a bill together. Britta scabetala half as much as Agda. Cornelia must pay SEK 1,500 less than Britta. Agda must pay five times as much as Cornelia. Calculate how much Britta will pay.
The amount of money that Britta will pay is; SEK2500
How to solve Algebra Word Problems?Let the amount paid by Britta scabetala be x
Now, we are told that Britta scabetala half as much as Agda. Thus;
Amount paid by Agda = 2x
Now, Cornelia must pay SEK 1,500 less than Britta. Thus,
Amount paid by Cornelia = x - 1500
Agda must pay five times as much as Cornelia. Thus;
2x = 5(x - 1500)
2x = 5x - 7500
5x - 2x = 7500
3x = 7500
x = 7500/3
x = SEK2500
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Hunter paid $44.20 for 5.2 pounds of shrimp. What is the price per pound of the shrimp?
Answer:
$8.50 per pound
Step-by-step explanation:
[tex]44.20 \div 5.2 = 8.50[/tex]
Find the value of f(4) for the function.
f(x) = -7(x + 2)
=(Simplify your answer.)
f(4)=
Answer:
Substitute x = 4 into f(x).
f(4) = -7(4+2)
= -28-14
= -42
What is the slope of the line?
You can use the formula to help.
◄) slope =
11
3/2 - Y1
2₂-1
?
10043
90
80
70
60
50
40
30
20
10
0
0 1
2
3
(4, 100)
(3,75).
4
5 6 7 8
9
H
10
The slope of the given line is m =25.
The two points of the given line is (3,75) and (4,100).
The slope of the given line can be found using the formula point-slope equation.
y₂-y₁ = m(x₂-x₁)
We need to find the slope, so let us rearrange the equation,
m = y₂-y₁ / x₂-x₁
where (x,₁y₁) and (x₂,y₂) is (3,75) and (4,100) respectively
So let us substitute these values in the above equation, we get
m = 100 - 75 / 4 -3
m = 25/1
m = 25
Therefore, the slope of the line passing through the point (3,75) and (4,100) is 25
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PLEASE ANSWER QUICKLY FIRST CORRECT ANSWER WILL GET BRAINLIEST
On a sunny day, a 3-foot red kangaroo casts a shadow that is 5 feet long. The shadow of a nearby eucalyptus tree is 7 feet long. Find the height of the tree.
the kangaroo real height is 3 feet and shadow height is 5 feet
if the height of the tree is 7 feet so the real height of the tree is 5 feet
slope and y intercept of −0.75x + y = 4
pls help!!!! will give brainliest
Answer:
x(-5.3,0)
y(0,4)
slope 0.75
Step-by-step explanation:
Helllllppppp plssssssssss
Answer:
A. Opens up with a vertex at (-3, -4)
Step-by-step explanation:
y = (x + 3)² - 4
Vertex: (-3, -4)
Min: y = -4
min = up
I hope this helps!
The given equation represents the equation of a parabola and it represents the curve open up with vertex (-3, -4).
Hence, option A is correct.
A parabola is a U-shaped plane curve in which any point is an equal distance from both a fixed point (known as the focus) and a fixed straight line (known as the directrix).
The given equation is:
y = (x +3)² - 4
Since we know that,
An equation of the parabola is
y = a(x - h)² + k
Where (h, k) is the vertex.
Comparing these equations we get
a = 1
h = -3
k = -4
So, its vertex is (h, k) = (-3, -4)
Hence,
The graph opens up with vertex (3, -4).
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If p(x) = ax²-3x+10, for what value(s) of a is p(-2) = f(g(-2))? May you explain how you did this.
Answer:
If p(x) = ax²
-3x+10
p(-2) =
f(g(-2))
What is the product of 28x32
Answer:
896
Step-by-step explanation:
28x32 gives u 896 so that will b your product
Richard and teo have a combine age of 19 Richard is 4 years older than twice teo age how old are Richard and teo
The points A(0,5), B(4,2), and C(0,2) form the vertices of a right triangle in the coordinate plane.
What is the equation of the line that forms the hypotenuse?
The equation of the line that forms the hypotenuse is y = -3/4 x + 5
Equation of a lineA line is defined as the shortest distance between two points. The equation of a line in slope-intercept form is expressed as:
y = mx + b
where:
m is the slope
b is the intercept
The coordinate points that gives the largest distance will be used. the points form the largest distance;
Using the coordinate points A(0,5), B(4,2)
Slope = 2-5/4-0
Slope = -3/4
The intercept is (0,5)
Determine the equation
y = mx + b
y = -3/4 x + 5
This gives the required equation.
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please help with question fast and provide explanation so I can understand
Answer:
A
Step-by-step explanation:
C means the number of miles into the city, while H means the miles on the highway. If (c,h) becomes (138,200), this means that c=138 and h=200. So, 138 miles can be traveled into the city while 200 can be traveled on the highway. hope this helps!
The two-way table shows information about the subjects studied by
some people.
a)
b)
Male
Female
Total
Chemistry Biology Physics
12
14
16
16
28
20
34
12
28
Total
42
48
90
How many people are male OR study Biology OR both?
What is the probability that any person is male OR
studies Biology OR both?
Part (a)
We can add the total number of males and the total number of students who take biology, and then subtract the intersection.
[tex]42+34-14=62[/tex]
Part (b)
There are 90 people in all, so the probability is [tex]\frac{62}{90}=\frac{31}{45}[/tex]
Which table shows a proportional relationship? Responses b 2 3 8 9 14 c 4 6 16 18 28b 2 3 8 9 14 c 4 6 16 18 28 , x 1 2 3 4 5 y 4 5 6 7 8x 1 2 3 4 5 y 4 5 6 7 8 , p 1 3 5 7 9 q 3 7 11 15 19p 1 3 5 7 9 q 3 7 11 15 19 , r 2 3 4 5 6 s 3 5 6 8 9
The tables that shows a proportional relationship include the following:
x 1 2 3 4 5
y 4 5 6 7 8
p 1 3 5 7 9
q 3 7 11 15 19
What is a proportion?A proportion can be defined as an equation which is typically used to represent (indicate) the equality of two ratios. This ultimately implies that, proportions can be used to establish that two ratios are equivalent and solve for all unknown quantities.
In Mathematics, the graph of any proportional relationship is characterized by a straight line because as the values on the x-axis (x-coordinate) increases or decreases, the values on the y-axis (y-coordinate) increases or decreases simultaneously as shown in tables B and C.
In conclusion, tables B and C comprises data values that shows proportional relationship in a graph.
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Graph the inequality on the axes below. y≥3/2x-1
The graph will be a shaded region above and including the straight line y = (3/2)x-1.
We are asked to graph the inequality y ≥ (3/2)x-1. Describing the graph of an inequality, it will be a region on one side of a straight line.
So first we need to sketch the straight line, y = (3/2)x-1.
So find two points (x, y) satisfying this equation of line because a line need at least two points.
x y
0 -1
2/3 0
So first sketch a line joining the points (0, -1) and (2/3,0). This line will lie below the origin. That is, origin lies on the region above this line.
So the line y = (3/2)x-1 divided the plane into two regions, one of which represents the inequality y ≥ (3/2)x-1. So to find this required region we will check if origin lies in the region for inequality.
Substituting (0, 0) into y ≥ (3/2)x-1,
0 ≥ -1, which is true.
Thus in the graph, the region for inequality is the region where the origin lies. Hence shade the region above the line y = (3/2)x-1. This is the graph of the inequality y ≥ (3/2)x-1.
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Please give me some help
Sum of 8 consecutive numbers is 124. Find the numbers
Give breif detail and description this question contains 15 points (will also mark as brainliest)
The 8 consecutive numbers are 12, 13, 14, 15, 16, 17, 18, 19 respectively.
What are algebraic expressions?Algebraic expressions are simply described as arithmetic expressions that are made up of arithmetic terms, constants, variables, factors and coefficients.
These expressions are also made up of arithmetic operations which includes;
ParenthesesBracketDivisionMultiplicationAdditionSubtraction, etcBased on the information given, we have;
x + x + 1 + x + 2 + x + 3 + x + 4 + x + 5 + x + 6 + x + 7 = 124
Now, collect like terms, we get;
8x + 28 = 124
8x = 124 - 28
Subtract like terms
8x = 96
To determine the value of x, make it the subject by dividing both sides by 8, we have;
8x/8 = 96/8
x = 12
Then,
x = 12
x + 1 = 13
x + 2 = 14
x + 3 = 15
x + 4 = 16
x + 5 = 17
x + 6 = 18
x + 7 = 19
Hence, the values are 12, 13, 14, 15, 16, 17, 18 and 19 respectively.
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A scientist is determining the mass of 22000 molecules of oxygen. How many significant digits should be in the solution?
Based on the mass of the molecules of oxygen that the scientist determined, the number of significant digits in the solution is 2.
How to count significant digits?When counting the significant figures in a number, the first rule is that any digit that isn't a zero would be considered a significant digit.
Zeros that are trailing in a number however, and are not considered decimals, are not counted as significant digits. This means that both the 2 number 2s in the figure will be significant figures.
The three zeros are trailing but not decimals so they are not considered significant. This means that in the mass of oxygen according to the scientist, has only 2 significant figures.
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Complete each ordered pair so that it is a solution of the given linear equation.
x-2y=5 ( ,1) (15, )
The first ordered pair is ?
The second ordered pair is?
Answer:
it
should be something along the lines of math
The sum of three mixed numbers is 24 4/5. Two of the number are 8 2/3 and 6 5/6. What is the third number?
The numerical value of the third number in an improper and mixed fraction form is 93/10 and 9 3/10 respectively.
What is the value of the third number?Given the data in the question;
Let 'x' represent the third number
Sum of the three numbers = 24 4/5First number = 8 2/3Second number = 6 5/6Third number = xFirst, convert the numbers form mixed fractions to improper fractions
Sum of the three numbers = 24 4/5 = ( 24×5 + 4 )/5 = 124/5
First number = 8 2/3 = ( 8×3 + 2 )/3 = 26/3
Second number = 6 5/6 = ( 6×6 + 5 )/2 = 41/6
Now, to determine the third value, equate the sum of the three numbers to the total and solve for x
26/3 + 41/6 + x = 124/5
( 156 + 123 )/18 + x = 124/5
( 279 )/18 + x = 124/5
31/2 + x = 124/5
x = 124/5 - 31/2
x = ( 248 - 155 )/10
x = ( 93 )/10
x = 93/10
x = 9 3/10
Therefore, the value of the third number is 93/10 or 9 3/10.
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If 5.2 cm= 96% what does 4cm equal?
We calculate the ratio (in percentage) of 5.2cm to 4cm
5.2 : 4 = 1.3
Because 4cm is 1.3x less than 5.2, its percentage must also be 1.3x less than 96%
-> 4cm = 96% : 130% = approx. 73.85% (2 d.p)
What is an equation of the line that passes through the points (−5,−7) and (-5, 6)?
Answer: x = -5
Both points have the same x value and only have different y values, meaning that the equation of the line must be x = -5.
The size of a cell is typically found by capturing an image under a microscope then using software to measure its diameter. Two cells are measured using this method: Cell C: 7.76\times 10^{-2}7.76×10 −2 centimeters Cell D: 8.4\times 10^{-4}8.4×10 −4 centimeters How many times larger is the diameter of cell C than the diameter of cell D? Write your answer in standard notation, rounding to the nearest tenth.
The number of times larger that the diameter of cell C is more than the diameter of cell D is 9.2 × 10^-6
How to illustrate the information?It should be noted that standard notation simply means writing in standard form. This is illustrated as a × 10ⁿ.
In this case, the cells are measured using this method: Cell C is 7.76 × 10^ {-2} and Cell D is 8.4×10^ −4 centimeters
It should be noted that in order to know the number of times that the cell is larger will be gotten by dividing. This will be;
= (7.76 × 10^-2) / (8.4 × 10^-4)
= 9.2 × 10^-6
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