Answer:
-36
Step-by-step explanation:
-18 - 18 = -36
SOLVE. y''+3y'+2y=4e^x cos3x
Solve the homogeneous equation
[tex]y'' + 3y' + 2y = 0[/tex]
Its characteristic equation is
[tex]r^2 + 3r + 2 = (r + 1) (r + 2) = 0[/tex]
with roots at [tex]r=-1[/tex] and [tex]r=-2[/tex], hence the characteristic solution is
[tex]y_c = C_1 e^{-x} + C_2 e^{-2x}[/tex]
For the nonhomogeneous equation, I'll use variation of parameters. We're looking for a solution of the form
[tex]y = u_1 y_1 + u_2 y_2[/tex]
to the equation
[tex]y'' + a(x) y'' + b(x) y = f(x)[/tex]
such that
[tex]\displaystyle u_1 = - \int \frac{y_2f(x)}{W(y_1,y_2)} \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{y_1 f(x)}{W(y_1,y_2)} \, dx[/tex]
The Wronskian [tex]W(y_1,y_2)[/tex] of the two fundamental solutions [tex]y_1=e^{-x}[/tex] and [tex]y_2=e^{-2x}[/tex] is
[tex]W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = -e^{-3x}[/tex]
Then we have
[tex]\displaystyle u_1 = - \int \frac{e^{-2x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = 4 \int e^{2x} \cos(3x) \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{e^{-x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = -4 \int e^{3x} \cos(3x) \, dx[/tex]
Recall Euler's identity,
[tex]e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))[/tex]
Then we have the general antiderivative
[tex]\displaystyle \int e^{(a+bi)t} \, dt = \frac1{a+bi} e^{(a+bi)t} + C = \frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C[/tex]
Taking the real parts of both sides, we have
[tex]\displaystyle \mathrm{Re}\left\{\int e^{(a+bi)t} \, dt \right\} = \mathrm{Re}\left\{\frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C\right\} \\\\ \int\,\mathrm{Re}\left\{e^{(a+bi)t}\right\} \, dt = \frac{e^{at}}{a^2+b^2} \mathrm{Re}\left\{(a-bi)(\cos(bt) + i \sin(bt))\right\} + C \\\\ \int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2+b^2} (a\cos(bt)+b\sin(bt)) + C[/tex]
so that
[tex]\displaystyle u_1 = 4 \int e^{2x} \cos(3x) \, dx = \frac{4e^{2x}}{13} (2\cos(3x) + 3 \sin(3x))[/tex]
and
[tex]\displaystyle u_2 = -4 \int e^{3x} \cos(3x) \, dx = -\frac{2e^{3x}}3 (\cos(3x) + \sin(3x))[/tex]
We've found
[tex]y = u_1 y_1 + u_2 y_2[/tex]
[tex]\displaystyle y = \frac{4e^x}{13} (2\cos(3x) + 3 \sin(3x)) - \frac{2e^x}3 (\cos(3x) + \sin(3x))[/tex]
[tex]\displaystyle y = \frac2{39} e^x (5\sin(3x) - \cos(3x))[/tex]
Then the general solution to the differential equation is
[tex]\boxed{y(x) = C_1 e^{-x} + C_2 e^{-2x} + \frac2{39} e^x (5\sin(3x) - \cos(3x))}[/tex]
Easy math, Just to lazy to do it.
Answer:
9 26/30 or 9 13/15
Step-by-step explanation:
20/30 6/30
Answer:
9 13/15
Step-by-step explanation:
7. Alicia and Dexter are each walking on a straight path. For a particular -second window of time, each has their velocity (in feet per second) measured and recorded as a function of time. Their respective velocity functions are plotted in .
Figure 1.4.14. The velocity functions and for Alicia and Damon, respectively.
Determine formulas for both and .
What is the value and meaning of the slope of ? Write a complete sentence to explain and be sure to include units in your response.
What is the value and meaning of the average rate of change of on the interval ? Write a complete sentence to explain and be sure to include units in your response.
Is there ever a time when Alicia and Damon are walking at the same velocity? If yes, determine both the time and velocity; if not, explain why.
Is is possible to determine if there is ever a time when Alicia and Damon are located at the same place on the path? If yes, determine the time and location; if not, explain why not enough information is provided.
(a) The formula for velocity functions is:
f(t)=0.4t+4
g(t)=0.9t
(b) The slope of the v-t graph gives us the acceleration which as found above is 0.4 ft/sec². The slope of the v-t graph gives us the acceleration which as found above is 0.9 ft/sec².
(c) Damon is accelerating with 0.9 ft/sec² in intervals [4,8].
(d) Both are walking at a velocity of 7.2 ft/sec at t=8 sec.
What is velocity?
The primary indicator of an object's position and speed is its velocity. It is the distance that an object travels in one unit of time. The displacement of the item in one unit of time is the definition of velocity.(a) For Alicia,
m=(Δv/Δt)=(8-4)/(10-0)=4/10=0.4
f(t)=0.4t+4
for Damon,
m=(Δv/Δt)=(9-0)/(10-0)=9/10=0.9
g(t)=0.9t
The formula for velocity functions is:
f(t)=0.4t+4
g(t)=0.9t
(b) Value and meaning of the slope is
m for Alicia =0.4
Slope is determined (Δv/Δt)=(8-4)/(10-0)=4/10=0.4 ft/sec².
So, the slope of the v-t graph gives us the acceleration which as found above is 0.4 ft/sec².
m for Damon=0.9
Slope is determined (Δv/Δt)=(9-0)/(10-0)=9/10=0.9 ft/sec².
So, the slope of the v-t graph gives us the acceleration which as found above is 0.9 ft/sec².
(c) Value and meaning of the average rate of change on the interval:
the average rate of change of Δ on the interval [4,8] =
[tex]\frac{Velocity\ at\ 8\ sec - Velocity\ at\ 4\ sec}{Time\ interval\ (8-4)\ sec}[/tex]
=(7.2-3.6)/(8-4)=(3.6/4)=0.9 ft/sec²
So, Damon is accelerating with 0.9 ft/sec² in intervals [4,8].
(d) Let both are walking with the same velocity at t sec.
So, Let's equate their velocities
0.4t+4=0.9t
t=4/0.5 = 8 sec
The velocity of Alicia at 8 sec = 0.4(8)+4 = 7.2 ft/sec
The velocity of Damon at 8 sec = 0.9(8) = 7.2 ft/sec
So, both are walking at a velocity of 7.2 ft/sec at t=8 sec.
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What are the coordinates of the point on the directed line segment from (−5,8)(−5,8) to (−1,−8) (−1,−8) that partitions the segment into a ratio of 3 to 1?
The coordinates of the point on the directed line segment is (-2, 4)
How to determine the coordinates of the point on the directed line segment?The points are given as
(-5, 8) and (-1, -8)
The ratio is given as
m : n = 3 : 1
The coordinates of the point are calculated as
Point = 1/(m + n) * (mx2 + nx1, my2 + ny1)
So, we have
Point = 1/(3+ 1) * (3 * -1 + 1 * -5, 3 * -8 + 1 * 8)
Evaluate
Point = 1/4 * (-8, 16)
So, we have
Point = (-2, 4)
Hence, the coordinates of the point on the directed line segment is (-2, 4)
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radius of a circle is 5 units what is the diameter
Write a rule to describe each transformation please I need help ASP
Answer:
The figure is reflected over y = -2
Step-by-step explanation:
If you draw a horizontal line through y = -2, so will see that every point is the same distance from y = -2 but in the opposite direction. For example, Point w is 3 units above y= -2 and w' is three units below y = -2. That is true for every point on original to the image.
Determine whether the equation defines y as a function of x. y equals = 8 Over x Does the equation define y as a function of x?
Yes , the equation y =8ˣ defines y as a function of x.
An mathematical function could be a function within the kind f (x) = aˣ, wherever “x” could be a variable and “a” could be a constant that is termed the bottom of the perform and it ought to be bigger than zero.An mathematical function is outlined by the formula f(x) = aˣ, wherever the input variable x happens as a fan. The graph depends on the mathematical function and it depends on the worth of the x.Yes, because to fulfill the condition of an equation to be a function, a particular value of x must produce only one value of y and y =8ˣ fulfills the condition. This can be also tested using horizontal line test, which only cuts the graph at one point only,
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Arun says if he adds together two mixed numbers, his answer will always be less than the sum of the whole-number parts plus 1. Say 2 counter-examples to show Arun's statement is not true. A counter statement is any example that shows a statement is false
The two examples can be 1 1/2 & 1 3/4 and 2 2/3 & 1 2/3.
According to Arun,
Sum of two mixed numbers < Sum of the whole number part + 1
Now, lets take the example as 1 1/2 and 1 3/4
Their sum will be:
1 1/2 + 1 3/4 = 3 / 2 + 7 / 4
= 13 / 4 = 3 1/4
The sum of whole parts = 1 + 1 = 2
Adding 1 to it = 2 + 1 = 3
We can clearly see that 3 1/4 > 3.
It contradicts Arun's statement.
Another example can be:
2 2/3 + 1 2/3 = 8 / 3 + 5 / 3
= 13 / 3 = 4 1/3
Sum of whole part + 1 = 2 + 1 + 1 = 4
4 1/3 > 4
It contradicts Arun's statement.
Therefore, the two examples can be 1 1/2 & 1 3/4 and 2 2/3 & 1 2/3.
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Q2. The diagram shows the position of town A.
Scale: 1 cm represents 10 km
Town B is 64 km from town A on a bearing of 070°.
Mark the position of town B, with a cross (*).
Use a scale of 1 cm represents 10 km.
The position of town B in cartesian plane is 6.4 cm in 1st quadrant at an angle of 70°.
As per the question statement, we are given that a diagram which shows the position of town A and the scale of 1 cm representing 10 km.
Town B is 64 km from town A on a bearing of 070°. We are supposed to tell the position of town B.
Let's assume that town A lies at origin in cartesian plane and as 1 cm represents 10 km therefore,
[tex]1cm = 10 km\\1 km = 0.1 cm\\64 km = 64*0.1=6.4 cm[/tex]
The town B is 6.4 cm from the origin and as the angle is given as 70° so the position of town B is in 1st quadrant.
Therefore the position of town B is 6.4 cm in 1st quadrant at an angle of 70° when town A lies at origin in the cartesian plane.
Cartesian Plane: The cartesian coordinate system includes a two-dimensional plane known as the Cartesian Plane. Numerical coordinates can be used to describe any point on a cartesian plane.To know more about cartesian plane, click on the link given below:
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motorcycle maker, says that it expects to build 312,000 motorcycles this year, up from 290,700 last year. Find the percent of increase in production.
Explanation:
A = last year's value = 290,700
B = this year's value = 312,000
C = change in values
C = B - A
C = 312,000 - 290,700 = 21,300
The positive result shows the increase of 21,300 more motorcycles made this year, compared to last year. A negative C value would represent a percent decrease.
Divide this change over the original.
C/A = (21,300)/(290,700) = 0.07327141382869
which is approximate. Let's say we rounded to four decimal places to get 0.0733 which then converts to the percentage 7.33%
Use the distributive property to remove the parentheses. -3(4v-2y-6)
-12v+6y+18
you multiply everything by -3, so -3×4v=-12v
-3×(-2y)=6y, since negative × negative=positive
-3×(-6)=18
i hope this helps :)
A TV show had 3.6 x 10 ^4 viewers in the first week and 4.1 x 10 ^4 viewers in the second week. Determine the average number of viewers over the two weeks and write the final answer in scientific notation.
3.85 x 10 ^4
7.7 x 10 ^4
3.85 x 10 ^8
7.7 x 10 ^8
The average number of viewers over the two weeks is 3.85 x 10 ^4
How to determine the average number of viewers over the two weeks?The given parameters are
Week 1 = 3.6 x 10 ^4 viewers
Week 2 = 4.1 x 10 ^4 viewers
The average is calculated as
Average = (Week 1 + Week 2)/2
This gives
Average = (3.6 x 10 ^4 + 4.1 x 10 ^4)/2
Evaluate the sum
Average = (7.7 x 10 ^4)/2
Evaluate the quotient
Average = 3.85 x 10 ^4
Hence, the average number of viewers over the two weeks is 3.85 x 10 ^4
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An open cylindrical tank is 150 feet in diameter and 180 feet high. The lateral surface area of the tank (no top and no bottom) is to be painted with paint that covers 225 square feet per gallon. When filled each cubic foot of the tank holds approximately 7.5 gallons of gasoline valued at $2.75 per gallon.
a) Find the number of gallons of paint required for a single coat of paint.
b) Find the total value of the gasoline in the filled tank.
In linear equation, the total value of the gasoline in the filled tank = $65605290
What is a formula for linear equations?
A linear equation has the slope-intercept form y = mx + b. Variables in the equation are x and y. When x is 0, the integers m and b provide the line's slope (m) and the value of y. (b). Because (0,y) is the location where the line crosses the y-axis, the value of y when x is 0 is referred to as the y-intercept.
Radius = 150/2 = 75 feet
height = 180 feet
A)
lateral surface area = 2 *pi*r*h = 2*pi* 75 *180 =84823
number of gallons of paint=84823/225 = 377 gallons
B)
Volume of cylinder= pi*r²h= pi*75^2*180 =3180862.5618
total value of the gasoline in the filled tank =3180862.5618 *7.5 *2.75 =$65605290
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c. A parallelogram
has an area of 7 square units. If the height that corresponds to
a base is unit, what is the base?
If a parallelogram has an area of 7 square units, and the height corresponding to a base is [tex](\frac{1}{4})[/tex] units, then the base is 28 units.
As per the question statement, a parallelogram has an area of 7 square units, and the height corresponding to a base is [tex](\frac{1}{4})[/tex] units.
We are required to find out the base of the parallelogram.
To solve this question, we need to know the formula to calculate the area of a parallelogram, which goes as
Area of Parallelogram = (Base * Height)
Putting our data obtain from the question statement, in the above-mentioned formula, we can calculate the value of the base of our concerned parallelogram, i.e.,
[tex]7=(base*\frac{1}{4})\\or, base=(7*4)\\or, base = 28[/tex]
Therefore, base of our concerned parallelogram is 28 units.
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Approximate 13 plus cube root of 9 to the nearest tenth.
14.5
15.1
16.0
17.3
Answer:
15.1
Step-by-step explanation:
If m∠NOL=54° then m∠FOG=__
What is the quotient of 5/6 + 2/7?
Answer:
Step-by-step explanation:
this app is helpful
If u get it correct I’ll give brainlest
Answer:
pretty sure it’s addition property of equality since you’re adding 5 to 10
ILL give BRAINIEST
What is the width of a rectangle with a length of 15 cm and area of 125 cm
No fake answers
Answer:
w≈8.33
Step-by-step explanation:
Greetings !
Firstly recall rectangle area formula
Thus,
[tex]area \: of \: rectangle \: = length \: \times \: width[/tex]
Given values:-
length = 15cmarea = 125cmrequire value:-
width =?solution/ work-out:-
[tex]width = \: \frac{area \: of \: rectangle}{length} [/tex]
[tex]w = \frac{125}{15} [/tex]
[tex]w = 8.333...[/tex]
Hope it helps !!!
What is p2 + 13 if p = −4?
29
17
−3
−21
The value of the expression p² + 13 will be A. 29.
How to compute the expression?It should be noted that the expression given is that p² + 13 and we are told that p = -4.
Therefore, we have to put the value of p into the expression. This will be:
p² + 13
= (-4)² + 13
= 16 + 13.
= 29
The value is 29.
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What is the domain of the function expressed by this formula?
Y= 1/x - 7
Answer:
x cannot equal seven, there's a mark for it in RSM menu
Step-by-step explanation:
so the answer is y = -7
A silversmith combined pure silver that cost $34.48 per ounce with 51 oz of a silver alloy that cost $25.35 per ounce. How many ounces of pure silver were used to make an alloy of silver costing
$28.87.per.ounce?
32 ounces of pure silver were used to make the silver alloy.
Here, we are given that a silversmith combines pure silver that costs $34.48 per ounce with 51 oz of a silver alloy that cost $25.35 per ounce.
Let the amount of pure silver = x oz
Cost of pure silver = $34.48 per ounce
Thus, total cost of pure silver = $34.48x
Similarly, amount of silver alloy = 51 oz
Cost of silver alloy = $25.35 per ounce
Thus, total cost of silver alloy = 25.35 x 51 = $1292.85
Now, the total weight of the new silver allow formed = (x + 51) Oz
and the cost of this new allow = $28.87 per ounce
Thus, the total weight of the new alloy = $28.87 (x + 51)
Hence, we can form the following equation-
$34.48x + $1292.85 = $28.87 (x + 51)
34.48x + 1292.85 = 28.87x + 1472.37
34.48x - 28.87x = 1472.37- 1292.85
5.61x = 179.52
x = 179.52/ 5.61
x= 32
Hence, 32 ounces of pure silver were used to make the silver alloy.
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a shelf contains 8 novels, 5 biographies, and 2 dictionaries. jane will select one book of each type. how many different ways can this be done?
Rs=3x st=6x-5 rt=58 find st
The value of st is 30.4
As S is the point in RT, it implies that RS+st=Rt. Therefore according to the question,
3x+6x-5=58
⟹9x=58-5
or
9x=53
⟹x=5.9
RS = 3×5.9=17.7
st=(6×5.9)-5=35.4-5=30.4
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is X a positive, negative or zero? x + 5 = -8
Answer:
Negative
Step-by-step explanation:
x + 5 = -8
Subtract 5 from both sides to get x by itself
x = -13
Answer:
X is negative
Step-by-step explanation:
This is the question
Answer:
x<0: 0<x<1; x>1
Step-by-step explanation:
The denominator cannot be 0 (since division by 0 is undefined), meaning x cannot be 0 or 1.
So, the domain is x<0: 0<x<1; x>1.
find the area of 8in 5in 4in 4in 10in 6in
By decomposing the figure into simpler ones, we conclude that the area of the irregular figure is 108 square inches.
How to find the area of the irregular figure?
Here we have an irregular figure that can be decomposed into 3 simpler figures, these are:
A rectangle of 6 inches by 8 inches (the top one).A rectangle of 10 inches by 4 inches (the one in the middle).A rectangle of 5 inches by 4 inches (the one at the right).Remember that the area of a rectangle is the product between the two dimensions, so the areas of these 3 rectangles are:
a = 6in*8in = 48in^2
a' = 10in*4in = 40in^2
a'' = 5in*4in = 20in^2
Adding these areas we get:
48in^2 + 40in^2 + 20in^2 = 108 in^2
The area of the irregular figure is 108 square inches.
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Find the union of the sets (22, 33, 44, 55) and (66, 44, 22}
Application of Finite Mathematics
The union of the sets is { 22, 33, 44, 55, 66 }
What is the union of sets?The union of sets can be defined as the sum of the elements of two or more sets without repetition of those elements.
It is also known as the combination of the all the elements of two or more sets.
It is denoted using the symbol , ' ∪ '
Given the sets;
(22, 33, 44, 55)(66, 44, 22}The union of the sets is:
{ 22, 33, 44, 55, 66 }
Thus, the union of the sets is { 22, 33, 44, 55, 66 }
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What is the decimal rule?
1.2 x (-5) = ?
I got -60 but it’s saying it’s the incorrect answer and I’m not sure what I’m doing wrong.
Answer:
-6
Step-by-step explanation:
1 x -5 is still -5, and the do .2 x -5 = -1 -1+-5= -6
What is the slope of the line
Answer:
its 1/3
Step-by-step explanation:
see picture for referance