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Find the matrix exponentia M(t) = etA The eigenvalues of A are X1 = 1 and X2 = 2. Please denote exponentiation with exp(a*t rather than e**(a*t or e^(a*t) This is a symbolic input so use exact values (e.g. ) rather than decimal approximations (0.5) Enter the matrix componentwise below M11(t)= M12t)= M21(t)= M22(t)
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The matrix exponential M(t) = exp(t*A) are: M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
To find the matrix exponential M(t) = exp(t*A), we first need to find the eigenvectors of A corresponding to the eigenvalues X1 = 1 and X2 = 2.
For X1 = 1, we solve the equation (A - I)*v = 0, where I is the identity matrix:
(A - I)*v = (1 1; 2 2 - 1)*v = 0
RREF([A - I, zeros(2,1)])
ans =
0 0 -1
0 0 0
So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = -1/2. So the eigenvector corresponding to X1 is v1 = (-1/2; 1).
For X2 = 2, we solve the equation (A - 2*I)*v = 0:
(A - 2*I)*v = (-1 1; 2 -2)*v = 0
RREF([A - 2*I, zeros(2,1)])
ans =
0 0 -1
0 0 0
So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = 1/2. So the eigenvector corresponding to X2 is v2 = (1/2; 1).
Now we can construct the matrix exponential M(t) = exp(t*A) using the formula:
M(t) = c1*exp(X1*t)*v1*v1' + c2*exp(X2*t)*v2*v2'
where c1 and c2 are constants determined by the initial conditions. Since we don't have any initial conditions given, we can choose c1 = 1 and c2 = 0 for simplicity.
So we have:
M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
So the matrix exponential M(t) is:
M(t) = ( (1/4)*exp(t) + (1/2)*exp(2*t) (1/2)*(exp(2*t) - exp(t));
(1/2)*(exp(2*t) + 1) exp(t) + exp(2*t) )
To find the matrix exponential M(t) = exp(tA) given that the eigenvalues of matrix A are λ1 = 1 and λ2 = 2, we first need to find the eigenvectors corresponding to each eigenvalue, and then form the matrix P of eigenvectors and the diagonal matrix D of eigenvalues. Finally, we can compute M(t) using the formula:
M(t) = P * exp(tD) * P^(-1)
After finding the eigenvectors and forming the matrices P and D, compute exp(tD) by taking the exponentiation of each diagonal element:
exp(tD) = | exp(tλ1) 0 |
| 0 exp(tλ2) |
Now, compute M(t) by multiplying P, exp(tD), and the inverse of P. The resulting matrix M(t) will have the following components:
M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)
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What is the value of r?
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Answer:
its 40
Step-by-step explanation:
180-100=80
80/2= 40
if ⃗a ·⃗b = √3 and ⃗a ×⃗b = ⟨1, 2, 2⟩, find the angle between ⃗a and ⃗b
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The angle between \(\(\vec{a}\) and \(\vec{b}\) is \(60^\circ\).\)
To find the angle between two vectors\(\(\vec{a}\) and \(\vec{b}\)\), we can use the dot product formula:
\(\(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)\),\)
where \(\(|\vec{a}|\) and \(|\vec{b}|\)\) are the magnitudes of the vectors and\(\(\theta\)\)is the angle between them.
Given that \(\(\vec{a} \cdot \vec{b} = \sqrt{3}\),\) we can rewrite the equation as:
\(\(\sqrt{3} = |\vec{a}| |\vec{b}| \cos(\theta)\).\)
We are also given that \(\(\vec{a} \times \vec{b} = \langle 1, 2, 2 \rangle\),\)which represents the cross product of the vectors.
The magnitude of the cross product is given by:
\(\(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta)\).\)
Substituting the given values, we have:
\(\(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta) = |\langle 1, 2, 2 \rangle| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3\).\)
We can rearrange the equation to solve for \(\(|\vec{a}| |\vec{b}| \sin(\theta)\):\(3 = |\vec{a}| |\vec{b}| \sin(\theta)\).\)
Now, we have two equations:
\(\(\sqrt{3} = |\vec{a}| |\vec{b}| \cos(\theta)\),\(3 = |\vec{a}| |\vec{b}| \sin(\theta)\).\)
To eliminate the magnitudes \(\(|\vec{a}|\) and \(|\vec{b}|\)\), we can square both equations and add them together:
\(\((\sqrt{3})^2 + 3^2 = (|\vec{a}| |\vec{b}|)^2 (\cos^2(\theta) + \sin^2(\theta))\)\).
Simplifying, we get:
\(\(3 + 9 = (|\vec{a}| |\vec{b}|)^2\).\(12 = (|\vec{a}| |\vec{b}|)^2\).\)
Taking the square root of both sides:
\(\(\sqrt{12} = |\vec{a}| |\vec{b}|\).\(\sqrt{12} = |\vec{a}| |\vec{b}| = |\vec{a}| |\vec{b}| \sqrt{\cos^2(\theta) + \sin^2(\theta)}\).\)
Since \(\(\cos^2(\theta) + \sin^2(\theta) = 1\)\), we have:
\(\(\sqrt{12} = |\vec{a}| |\vec{b}| \cdot 1\).\(\sqrt{12} = |\vec{a}| |\vec{b}|\).\)
Now, we can substitute this back into the first equation:
\(\(\sqrt{3} = \sqrt{12} \cos(\theta)\).\)
Simplifying, we get:
\(\(\cos(\theta) = \frac{\sqrt{3}}{\sqrt{12}} = \frac{\sqrt{3}}{2\sqrt{3}} = \frac{1}{2}\).\)
To find the angle \(\(\theta\)\), we take the inverse cosine (
arc cosine) of \(\(\frac{1}{2}\):\)
\(\(\theta = \cos^{-1}\left(\frac{1}{2}\right)\).\)
Using the unit circle or trigonometric identities, we find that\(\(\theta = \frac{\pi}{3}\) or \(60^\circ\).\)
Therefore, the angle between \(\(\vec{a}\) and \(\vec{b}\) is \(60^\circ\).\)
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In Ms. Rossi’s orchard, 20% of the fruit trees are apple. If the orchard has 15 apple trees, how many fruit trees are in the orchard?
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Answer:
100/20×15
75trees
Step-by-step explanation:
please help will mark brainliest
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Answer: 1 7/16
Step-by-step explanation: If you look at a ruler 1 7/8 is just a little behind 1 7/16 while 1 3/4 is far behind. Also 2 1/8 is past where the green line stops.
Answer:
option 1) 1 7/8"
Step-by-step explanation:
its clearly visible that the green line covers 1 unit completely.
but it stops before it can reach 2 units.
check the attachment
"mid point" is the mid point of 1 and 2 unitsit will be at a distance of 1/ 2 units from the "unit 1".
( since 1 and 2 are 1 unit apart their mid point will lie exactly at rhe center that is 1/ 2 = 0.5)
"second mid point" is the mid point of "mid point" and "unit 2".since the "mid point" is at a distance of 0.5 units from "unit 2", second mid point will be at a distance of (1/ 2)/ 2 = 1/ 4 units from "mid point".
now, third mid point is the mid point of "second mid point" and "unit 2"since they're 1/ 4 units apart the third mid point will be (1/ 4)/2 units away from "second mid point"
and this is where the green line stops!
we got these points now well just add them up, to get the distance of required point from unit 1.
1/ 2 + 1/ 4 + 1/ 8
= 7/ 8
and this is in centimeters so add a " after it
1 7/ 8"
that is option 1.
one and six tenths is to four as two and four tenths is to six
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Yes one and six tenths is to four as two and four tenths is to six
Proportions
In some contexts, "is to" can mean division. For example, if you say "5 is to 10 as x is to 20," you could interpret that as a proportion, where the ratio of 5 to 10 is equal to the ratio of x to 20. This can be rewritten as:
5/10 = x/20
Now
To solve this problem, we can use proportions.
One and six tenths is equivalent to 1.6, and two and four tenths is equivalent to 2.4.
Let x be the number that corresponds to "is to" in the second part of the statement. Then, we can set up the following proportion:
1.6/4 = 2.4/x
We can cross-multiply to get:
1.6x = 9.6
And then solve for x:
x = 6
So in this case, "is to" did involve division. However, "is to" doesn't always mean division and its meaning can depend on the context in which it is used.
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Victoria is studying the population of the town of Oak Hills for a class project. Based on census data, she found that the equation p = 660n + 14,400 can be used to estimate the population, p, of Oak Hills n years after the year 1900. What will be the estimated population of Oak Hills in the year 2100?
A. 3,012,000
B. 146,400
C. 15,300
D. 132,000
Answers
Answer:
Step-by-step explanation:
2100 - 1900 = 200
660(200) + 14,400
p = 146400
146,400 people
The answer is B
Maleeha buys 3 identical packs of pens at the Office superstore for $8.55. The next day, she sees an advertisement for the same pens Plenty O' Paper. The ad says the pens are $2.79 per pack. Which is the better deal?
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what is the slope of (16,9), (17,9)
Answers
Answer:
0
Step-by-step explanation:
It's a horizontal line, it has no slope.
Cylinder a has a radius 2 cm and contains water to a height of 10cm cylinder b has a radius 5cm and is empty. Some of the water is poured from cylinder a to b the height is now the same
Answers
Answer:
h = 1.38 cm
Step-by-step explanation:
The question is at what value is the height of both cylinders the same:
The area of the circular base on each cylinder is:
\(Area=\pi r^2\\A=4\pi \ cm^2\\B=25\pi \ cm^2\)
The initial volume in cylinder A is:
\(V=4\pi *10\\V=40\pi\ cm^3\)
We have that Va + Vb = 40π. The height of water in each cylinder as a function of volume is:
\(h_A=\frac{V_a}{4\pi}\\h_B=\frac{V_b}{25\pi}\)
If both heights are the same:
\(\frac{V_a}{4\pi}=\frac{V_b}{25\pi}\\V_b=\frac{25}{4}V_a \\V_a+V_b=40\pi\\V_a+\frac{25}{4}V_a=40\pi\\V_a=5.5172\pi\ cm^3\)
The height 'h' is:
\(h=\frac{5.5172\pi}{4\pi}\\ h=1.38\ cm\)
I am doing Polygons with Algebra and I have a irregular shape with letters on each point
A=x
B=130°
C=x+20°
D=120°
E=140°
F=x-20°
G=150°
Find the value of x, and use the 7 block chart to figure out answer for interior and exterior, and the equation.
Answers
Answer:
Step-by-step explanation:
Number of side = n = 7
Sum of interior angles of 'n' sided polygon = (n - 2) *180
= (7 - 2)* 180
= 5 * 180
= 900
x + 150 + x -20 + 140 + 120 + x + 20 + 130 = 900
Combine like terms
x +x +x + 150 - 20 + 140 + 120 + 20 +130 = 900
3x + 540 = 900 {Subtract 540 from both sides}
3x = 900 - 540
3x = 360 {Divide both sides by 3}
x = 360/3
x = 120
Interior angle Exterior angle
∠A = 120 180 - 120 = 60
∠B = 130 180 - 130 = 50
∠C = 120 + 20 = 140 180 - 140 = 40
∠D = 120 180 - 120 = 60
∠E = 140 180 - 140 = 40
∠F = 120 - 20 = 100 180 -100 = 80
∠G = 150 180 - 150 = 30
9320 is what percent of 15821?
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Answer:
9320 is 58.9090449% of 15,821
Step-by-step explanation:
please can you help with this
Answers
Answer:
9.34Step-by-step explanation:
(6.2-3.51)² +³√9.261 = 7.24+2.1 = 9.34Answer:
9.3361
Step-by-step explanation:
Any scientific or graphing calculator, and any spreadsheet can evaluate this expression for you.
__
The value is 9.3361.
_____
Additional comment
Most of the calculators that will evaluate this expression will not require that you determine or write down any intermediate values. If you do, you should use the full calculator precision (4 decimal places).
The finance charge on Lauren’s credit card bill last month was
$13.50. Her APR is 18%. What was her average daily balance?
Answers
Answer:
$900
Step-by-step explanation:
You want to know Lauren's average daily balance if she was charged a $13.50 finance charge on a card with an APR of 18%.
Finance chargeThe finance charge is the product of the monthly interest rate and the average daily balance. The monthly rate is 1/12 of the annual rate. This tells us ...
finance charge = ADB × 18%/12
$13.50 = ADB × 0.015
ADB = $13.50/0.015 = $900
Lauren's average daily balance was $900.
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i need help with some revision
Answers
Answer:
angle y = 60
Step-by-step explanation:
They are equal due to the rule that vertical angles are always equal
"A pair of vertically opposite angles are always equal to each other."
hope this helps
Write the event as set of outcomes. We flip three coins and obtain more tails than heads.
A. {ttt}
B. {ttt, tth, tht, htt}
C. {ttt, tth}
D. {tth, tht, htt}
Answers
Answer:
B.
Step-by-step explanation:
All the possible outcomes are listed on choice B.
The event is a set of outcomes. if we flip three coins and obtain more tails than heads is E = {ttt, tth, tht, htt} option (B) is correct.
What is set?A set is a collection of clearly - defined unique items. The term "well-defined" applies to a property that makes it simple to establish whether an entity actually belongs to a set. The term 'unique' denotes that all the objects in a set must be different.
We have three coins.
As we know, in a coin there are two sides head and a tail.
If we flip three coins then the set of all the possible outcomes:
O = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
The set of outcomes has more tails than heads.
E = {ttt, tth, tht, htt}
We can find the probability, the probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.
Probability = 4/8 = 1/2
Thus, the event is a set of outcomes. if we flip three coins and obtain more tails than heads is E = {ttt, tth, tht, htt} option (B) is correct.
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Statins are used to keep cholesterol in check and are a top-selling drug in the U.S. The equation:
S-1.9 = 5
gives the amount of sales (S) of statin in billions of dollars x years after 1998. According to this equation, how
much will/did people in the U.S. spend on statins in the year 2014?
Answers
Hence we can say that the money that the people would spend on statins is 35.4 billion dollars
How to solve the equationWe have the equation to be
S-1.9x = 5
Now the variable x can be defined to be the time period that exists from 1998 to 2014
When we check this is a period of 16 years
2014 - 1998 = 16 years
We have to put the value of x in the formula
Hence we would have
S-1.9 * 16 = 5
s = 5 + 30.4
s = 35.4
Hence we can say that the money that the people would spend on statins is 35.4 billion dollars
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A living room is 8 feet long and 16 feet wide. If the ceiling is 10 feet high, what is the total surface area of the four walls?
Answers
The living room is similarly shape to a rectangular prism. However, we will only solve for the four walls of the room.
\(\begin{gathered} \text{The surface area is now calculated as} \\ SA=2(10\text{ ft}\cdot16\text{ ft})+2(10\text{ ft}\cdot8\text{ ft}) \\ \text{Note: The top and bottom surface area are not counted.} \end{gathered}\)\(\begin{gathered} SA=2(10\text{ ft}\cdot16\text{ ft})+2(10\text{ ft}\cdot8\text{ ft}) \\ SA=2(160\text{ ft}^2)+2(80\text{ ft}^2) \\ SA=320\text{ ft}^2+160\text{ ft}^2 \\ SA=480\text{ ft}^2 \end{gathered}\)Therefore, the total surface area of the four walls is 480 square feet.
Use Distributive Property to write an equivalent expressions 7(x+6)
Answers
Answer:
7x+42
Step-by-step explanation:
Charles spent $2.85 on train tickets every day for 15 days. What is the amount of money Charles spent on these train tickets?
Answers
Answer:
charles spent $42.75
Step-by-step explanation:
2.85*15+=42.75
What is the standard form for the equation y-5= 3(x + 2)
Answers
Step-by-step explanation:
\(y - 5 = 3x + 6 \\ y = 3x + 11 \\ if \: y \: equals \: to \: 0 \\ 0 = 3x + 11 \\ 3x = - 11 \\ x =3 \times \frac{2}{3} \)
Whoever answers first gets 5 stars, and a thanks. And if possible a brainliest
Answers
Answer:14.518
Step-by-step explanation:
y = -2 x - 5 what is the answer help
Answers
Answer:
x = -5/2
Step-by-step explanation:
0 = -2x - 5
2x = -5
Divide^
x = -5/2
Hope this helped
Brainliest is appreicated./
Qual é o resultado da multiplicação a seguir:
Answers
Answer:
que????? no entiendo??????
If it took 4 hours to mow 7 lawns, then at that rate, how many lawns could be mowed in 16 hours?
Answers
Answer:
28 lawns
Step-by-step explanation
If there is 7 lawns for 4 hours there will be 14 lawns in 8 hours. and 28 lawns in 16 hours
make a conjecture (prediction) about the table. what will happen if we continue the table, going down?
Answers
FILL IN THE BLANK a/an ____________________________ diagram can be used to show how the tables in a database are defined and related..
Answers
A/an database schema diagram can be used to show how the tables in a database are defined and related.
This type of diagram provides a visual representation of the structure and organization of the database. It illustrates the tables ,and their attributes, and the relationships between them.
The database schema diagram helps in the understanding the logical design of the database, including primary keys, for foreign keys, and the connections between different tables. It allows developers, database administrators, and stakeholders to visualize the database structure and serves it as a reference for in designing, modifying, and querying the database effectively.
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Two consecutive angles of a parallelogram measure 3 x+42 and 9 x-18 . What are the measures of the angles?
A. 13,167
C. 39,141
B. 58.5,31.5
D. 81,99
Answers
Answer:
enjooooooooooooyyyyyyyyyyy
Given the function f(k)= 6k4 + 2k³ - 6, find the value of f(-4)
Answers
Answer: 1402
Step-by-step explanation:
Im assuming u mean \(6k^{4} +2k^{3} -6\)
sub in the k value when k = -4
\(6(-4)^{4} +2(-4)^{3} -6\)
1536−128−6=1402
In order to solve for f(-4) we know what the value is for K
So in the formula f(k)=6k^4 +2k^3 -6 we would replace K with (-4)
f(-4)= 6(-4)^4+2(-4)^3-6
Then start solving the exponents first
(-4)^4=256
(-4)^3=-64
f(-4)= 6(256)+2(-64)-6
= 1536-128-6
=1408-6
f(-4)= 1402
1/2x - (3x^2 - 1/2x +8) + 11+(5x^2 - 22)
Answers
Answer:
1/2x is 0.5x. 3x^2 is 9x^2. 9x^2 - 0.5x is 7.5x^2 + 8, which is 15.5x^2.
Step-by-step explanation:
Answer:
I think the answer is 45 but im not sur