It looks like the system is
[tex]x' = \begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} x + \begin{bmatrix} \sin(t) \\ -2 \cos(t) \end{bmatrix}[/tex]
Compute the eigenvalues of the coefficient matrix.
[tex]\begin{vmatrix} -1 - \lambda & 5 \\ -1 & 1 - \lambda \end{vmatrix} = \lambda^2 + 4 = 0 \implies \lambda = \pm2i[/tex]
For [tex]\lambda = 2i[/tex], the corresponding eigenvector is [tex]\eta=\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}^\top[/tex] such that
[tex]\begin{bmatrix} -1 - 2i & 5 \\ -1 & 1 - 2i \end{bmatrix} \begin{bmatrix} \eta_1 \\ \eta_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}[/tex]
Notice that the first row is 1 + 2i times the second row, so
[tex](1+2i) \eta_1 - 5\eta_2 = 0[/tex]
Let [tex]\eta_1 = 1-2i[/tex]; then [tex]\eta_2=1[/tex], so that
[tex]\begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} = 2i \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix}[/tex]
The eigenvector corresponding to [tex]\lambda=-2i[/tex] is the complex conjugate of [tex]\eta[/tex].
So, the characteristic solution to the homogeneous system is
[tex]x = C_1 e^{2it} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} + C_2 e^{-2it} \begin{bmatrix} 1 + 2i \\ 1 \end{bmatrix}[/tex]
The characteristic solution contains [tex]\cos(2t)[/tex] and [tex]\sin(2t)[/tex], both of which are linearly independent to [tex]\cos(t)[/tex] and [tex]\sin(t)[/tex]. So for the nonhomogeneous part, we consider the ansatz particular solution
[tex]x = \cos(t) \begin{bmatrix} a \\ b \end{bmatrix} + \sin(t) \begin{bmatrix} c \\ d \end{bmatrix}[/tex]
Differentiating this and substituting into the ODE system gives
[tex]-\sin(t) \begin{bmatrix} a \\ b \end{bmatrix} + \cos(t) \begin{bmatrix} c \\ d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ -1 & 1 \end{bmatrix} \left(\cos(t) \begin{bmatrix} a \\ b \end{bmatrix} + \sin(t) \begin{bmatrix} c \\ d \end{bmatrix}\right) + \begin{bmatrix} \sin(t) \\ -2 \cos(t) \end{bmatrix}[/tex]
[tex]\implies \begin{cases}a - 5c + d = 1 \\ b - c + d = 0 \\ 5a - b + c = 0 \\ a - b + d = -2 \end{cases} \implies a=\dfrac{11}{41}, b=\dfrac{38}{41}, c=-\dfrac{17}{41}, d=-\dfrac{55}{41}[/tex]
Then the general solution to the system is
[tex]x = C_1 e^{2it} \begin{bmatrix} 1 - 2i \\ 1 \end{bmatrix} + C_2 e^{-2it} \begin{bmatrix} 1 + 2i \\ 1 \end{bmatrix} + \dfrac1{41} \cos(t) \begin{bmatrix} 11 \\ 38 \end{bmatrix} - \dfrac1{41} \sin(t) \begin{bmatrix} 17 \\ 55 \end{bmatrix}[/tex]
how to do questions 28-29?
thanks!
28. 1.45l ÷ 0.32l = 4.53125
so approximately 5 glasses to leave the bottle empty
29. 5.6kg + 2.5kg = 8.1kg
8.1kg ÷ 0.48 = 16.875
he can make 16 complete doughs
there will be 0.875kg of flour left
What are the x-intercept and vertex of this quadratic function? Write each feature as an ordered pair: (a,b). The x-intercept of function g is The vertex of function g is
The x-intercept of the function g(x) = -5(x - 3)^2 is (3, 0) and the vertex of the function g(x) = -5(x - 3)^2 is (3, 0)
How to determine the x-intercept?The function is given as:
g(x) = -5(x - 3)^2
Set g(x) to 0, to determine the x-intercept.
-5(x - 3)^2 = 0
Divide through by -5
(x - 3)^2 = 0
Take the square root of both sides
x - 3 = 0
Add 3 to both sides
x = 3
Hence, the x-intercept of the function g(x) = -5(x - 3)^2 is (3, 0)
How to determine the vertex?The function is given as:
g(x) = -5(x - 3)^2
A quadratic function is represented as:
g(x) = a(x - h)^2 + k
Where, the vertex is
Vertex = (h, k)
By comparing g(x) = a(x - h)^2 + k and g(x) = -5(x - 3)^2, we have
(h, k) = (3, 0)
Hence, the vertex of the function g(x) = -5(x - 3)^2 is (3, 0)
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The following are the distances (in miles) to the nearest airport for 11 families. 9, 11, 11, 16, 16, 24, 28, 30, 34, 42, 45 Notice that the numbers are ordered from least to greatest. Give the five-number summary and the interquartile range for the data set. Five-number summary
Minimum:
Lower quartile:
Median:
Upper quartile:
Maximum:
Interquartile range:
Answer:
11, 11, 24, 34, 45 and 23===========================
Given data9, 11, 11, 16, 16, 24, 28, 30, 34, 42, 45Five-number summaryMinimum:
The smallest number of data ⇒ 11Maximum:
The largest number of data ⇒ 45Median:
The middle number of the data if data consists of odd number of elements ⇒ 24Lower quartile:
The median of the lower half of data, 9, 11, 11, 16, 16 ⇒ 11Upper quartile:
The median of the upper half of data, 28, 30, 34, 42, 45 ⇒ 34Interquartile rangeThe difference of upper and lower quartiles, 34 - 11 = 23Write an equation that represents the line.
Use exact numbers.
First point (1,2)
Second point is (4,4)
Answer: equation y= 3/2x+ 4/3
PLEASE HELP ASAP……..
Answer:
4x+2y = 14 The original equation
2y = 14-4x Subtract 4x from both sides
2y = -4x+14 Rearrange so x is first
y = -2x+7 Divide everything by 2
Explanation:
The steps above practically explain everything needed, so there's not much to add. The standard form is Ax+By = C while slope intercept form is y = mx+b
In the case of y = -2x+7, we have a slope of m = -2 and y intercept of b = 7. This means (0,7) is one point on the line. Use the slope to move down two and to the right one unit to arrive at (1, 5) as another point on this line.
Amanda increased the amount of protein she eats everyday from 48g to 54 g By what percentage did Amanda increase the amount of protein she eats
Step-by-step explanation:
she increased the amount from 48g to 54g. that is 54-48 = 6g.
so, the question really is, how many % of 48 is 6 ?
100% = 48g
1% = 100%/100 = 48/100 = 0.48g
how many % are 6g ?
as many as how often 1% (0.48g) fits into 6g :
6 / 0.48 = 12.5%
What happens when you reflect a shape over the x-axis and then the y-axis. What is the one transformation that could have been performed to achieve the same result?
Answer:
Rotate 180 degrees with the center the origin (0,0)
Step-by-step explanation:
Try drawing a small square in the first quadrant, then reflect it like the direction. If you use tracing paper and and copy the figure, you will see that if you rotate the figure at the origin, it will land on the same spot as the translations.
Determine the factors of 5x2 29x − 6. (5x − 1)(x 6) (5x − 6)(x 1) (5x − 3)(x 2) (5x − 2)(x 3)
The correct option is option(A) (5x-1)(x+6).
The factors of the given equation are (5x-1)(x+6).
What are the factors of a quadratic equation?An algebraic expression or number that divides another expression or number equally, leaving no remainder is known as factor.
The quadratic equation [tex]ax^{2} +bx+c=0[/tex] can be expressed as the product of its linear factors as (x - k)(x - h), where h, k are the roots of the equation. This is known as factoring quadratics.
What is splitting middle term of an equation?Dividing the middle term into two factors is used in quadratic factorization in this method. In splitting of the middle term, where x is the product of two factors and last term is the sum of the two factors.
Use the splitting middle term formula
[tex]5x^{2} +29x-6[/tex]
=[tex]5x^{2} +30x-x-6[/tex]
=5x(x+6)-1(x+6)
=(5x-1)(x+6)
The factors of the given equation are (5x-1)(x+6).
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The price of a new car is £27000 In a sale, the price is reduced by 30% Brodie buys the car in the sale. He pays a £8000 deposit and pays the rest over 50 monthly payments. Find the cost of each monthly payment.
Answer:
£218
Step-by-step explanation:
price of the car = £27000
reduce 30% = 30/100 x 27000 = £8100
£27000 - £8100 = £18900
£18900 - £8000 = £10900
installment per month = £10900/50 = £218
In a recent study, your town's chamber of commerce found that 73% of town residents believed that local businesses overcharged for their products. the source of this information was a telephone survey of 1,000 town residents. find a 95% confidence interval for the average percentage of everyone in your town who believes that local businesses overcharge.
The Confidence interval for 95% who believes that the local businesses overcharge = (0.7026, 0.7574)
What is meant by confidence interval?The range of values we see in our sample and hope to identify the value that most closely represents the population are referred to as a confidence interval.
According to the given information:Sample size n = 1000
73% of town residents believed that local businesses overcharged for their products over 1000 resident.
= (1000/100) x 73
= 730
Sample proportion p = 730/1000
= .73
q = 1-p = 0.23
Std error of proportion = √(pq/n)
= √((.73*0.27)/1000)
= 0.0140
95% Z critical value = 1.96
Margin of error = 1.96*0.0140
= 0.0274
Confidence interval = sample proportion ±margin of error
(0.7026, 0.7574)
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Over a period of 3 hours, the outside temperature changed an average of -2.250 Fahrenheit per hour. Which statement correctly describes the change in temperature from the beginning the end of the 3-hour period?
A.
The temperature decreased by 6.75 degrees Fahrenheit.
B.
The temperature increased by 0.75 degrees Fahrenheit.
C.
The temperature increased by 6.75 degrees Fahrenheit.
D.
The temperature decreased by 0.75 degrees Fahrenheit.
Answer: A
Step-by-step explanation: The question states that the change wasn't a positive one, it was a negative one as it states that the temperature changed an average of -2.250 Fahrenheit per hour. So, we multiply that by 3 to get -6.75. This is why A is correct.
6. In a standard deck of cards:
Event A: Card is red
Event B: Card is a heart
Events A and B are ____
A)None of these
B)Dependent
C)Independet
D)Both
Events A and B are dependent (option B).
What are dependent events?Dependent events are events that can occur together. If event A occurs then event B would occur. In a standard deck of cards, cards that are hearts are red in colour. If a heart card is picked, then it would be red in colour.
Independents events are events whose occurrence do not depend on each other. They are random events.
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A 105
5) How will you educated, uneducated people in your locality
about the environment problems of your locallly.
I will educated uneducated people on my locality
Get your associates, neighbors, friends, family, or even your local government concerned. It's much more comfortable and more useful to apply environmental attention and create a local project if you cooperate with others in our community.
How to improve the uneducated people in our locality about the environmental problems of our local?Get your associates, neighbors, friends, family, or even your local government concerned. It's much more comfortable and more useful to apply environmental attention and create a local project if you cooperate with others in our community. Encouraging environmental awareness exists a vital aspect of being an environmental guardian.
Facilitate knowledge transfer from workers approaching retirement to workers of all ages. Work with families to support children in school and control illiteracy. Provide young people from deprived backgrounds the norms to study. Improve access to education for Aboriginal people.
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how do you solve this question?
The answers to the questions are as follows:
The value of the empty box in the diagram is 10The probability that the number is in AnB = 1/5How to solve the Venn diagramWe have E = {1, 3,5, 7, 9, 11, 13, 15, 17, 19}
A= { 3, 7, 9, 11, 15}
B = {5, 7, 11, 13}
We have A n B = numbers that are contained in both of the sets
= 7, 11
a.) We have to count the total number in the dataset. That is the total number of odd numbers that are less than 20.
Hence total numbers in the set = 10
b. The probability that the number is in set AnB
= 7, 11
= 2/10
= 1/5
We can conclude that the probability that the number is in A intersect B = 1/5
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hey can someone give me the answer to the problem
a. The rocket splashes down after 63.94 seconds.
b. The rocket peaks at 7386.68 m
a. How to find when the rocket splashes down?
Since the height of the rocket is h(t) = -4.9t² + 370t + 402, the rocket splashes down when h(t) = 0.
So, h(t) = 0
-4.9t² + 370t + 402 = 0
Using the quadratic formula, we find t.
[tex]t = \frac{-b +/- \sqrt{b^{2} - 4ac} }{2a}[/tex]
where a = -4.9, b = 370 and c = 402
So, [tex]t = \frac{-b +/- \sqrt{b^{2} - 4ac} }{2a}\\= \frac{-307 +/- \sqrt{(307)^{2} - 4(-4.9)(402)} }{2(-4.9)}\\= \frac{-307 +/- \sqrt{94249 + 7879.2} }{-9.8)}\\= \frac{-307 +/- \sqrt{102128.2} }{-9.8)}\\= \frac{-307 +/- 319.58}{-9.8)}\\= \frac{-307 + 319.58}{-9.8)} or \frac{-307 - 319.58}{-9.8)}\\= \frac{12.58}{-9.8)} or \frac{-626.58}{-9.8}\\= -1.28 or 63.94[/tex]
Since t cannot be negative t = 63.94 s
So, the rocket splashes down after 63.94 seconds.
b. How to find the peak of the rocket?Since h(t) = -4.9t² + 370t + 402, to find the time the rocket reaches it peak, we differentiate h(t) with respect to t and equate to zero.
Soi, dh(t)/dt = d(-4.9t² + 370t + 402)/dt
= -9.8t + 370
dh(t)/dt = 0
-9.8t + 370 = 0
-9.8t = -370
t = -370/-9.8
t = 37.76 s
Substitung t = 37.76 into h(t), we have
h(t) = -4.9t² + 370t + 402
h(37.76) = -4.9(37.76)² + 370(37.76) + 402
h(37.76) = -4.9(1425.82) + 370(37.76) + 402
h(37.76) = -6986.518 + 13971.2 + 402
h(37.76) = 7386.682 m
h(37.76) ≅ 7386.68 m
So, the rocket peaks at 7386.68 m
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The question is down below.
The correct expressions are sinC = 12/25.5 = 24/51, <A = 62 degrees and tan A = 1.875
SOH CAH TOA identityThe given figure is a right triangle with legs 12.0 and 22.5cm and the hypotenuse 25.5cm
For sinA, <A is the reference angle, hence the opposite side will be 22.5cm
sin A = opp/hyp
sinA = 22.5/25.5
For cos C, the adjacent side will be 22.5cm, hence the measure of cos C is 22.5/25.5
cosC = 22.5/25.5
Similarly, sinC = 12/25.5 = 24/51
<A = 180-(90 +38)
<A =180 - 128
<A = 62 degrees
tan A = opp/adj
tan A = 22.5/12 = 1.875
tan C = 12/22.5
tan C = 0.533
Hence the correct expressions are sinC = 12/25.5 = 24/51, <A = 62 degrees and tan A = 1.875
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The picture below shows a pole and its shadow:
A pole is shown with a right triangle side. The right triangle has hypotenuse 113 cm and base 15 cm.
What is the height of the pole?
112 centimeters
100 centimeters
98 centimeters
64 centimeters
The answer is 112 centimeters.
As this problem represents a right triangle, here we need to apply the Pythagorean Theorem.
h = √113² - 15²h = √12769 - 225h = √12544h = 112 centimetersA rectangle is 16 cm. in length. Its width is 50% of its length. what is its perimeter?
Answer: 48 cm
Step-by-step explanation:
A perimeter(P) of a rectangle is equal to 2(L + W), or 2L + 2W, so we write the equation as:
1st way:
W = 50%(16) = 0.5 x 16 = 8
P = 2(L + W)
P = 2(16 + 8)
P = 2(24)
P = 48 cm
2nd way:
P = 2L + 2W
P = 2(16) + 2[50%(16)]
P = 2(16) + 2[0.5 X 16]
P = 32 + 2(8)
P = 32 + 16
P = 48 cm
If the ice shop has only enough cocoa to make 4 gallons of chocolate ice cream, how many gallons of strawberry ice cream can the shop efficiently produce? 0 2 8 12
The complete questions is:
The ice cream shop needs 5 pounds of strawberries for every gallon of strawberry ice cream. the shop chooses to produce 4 gallons of chocolate ice cream. how many pounds of strawberries should the shop purchase?
Pounds of strawberry ice cream needed to be bought by the shop exists 20 pounds.
How many gallons of strawberry ice cream can the shop efficiently produce?The ice cream shop requires 5 pounds of strawberries for every gallon of strawberry ice cream and the shop chooses to make 4 gallons of ice cream,
Quantity of chocolate ice cream created in the shop = 4 gallons
Quantity of strawberries needed for each gallon of ice cream = 5 pounds
Pounds of strawberries needed to be bought by the shop = Quantity of chocolate ice cream created by the shop [tex]*[/tex] Quantity of strawberries needed for each gallon of ice cream
[tex]$= 4 * 5[/tex]
= 20 pounds.
Pounds of strawberries needed to be bought by the shop exists 20 pounds.
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Answer:
12
Step-by-step explanation:
what phrases can be used to describe the line representing the relationship between the number of balloons remaining and the number of hats created? Select three options.
positive slope
negative slope
constant slope
increasing function
decreasing function
Answer: negative slope
constant slope
decreasing function
B-C-E
Step-by-step explanation:negative slope
constant slope
decreasing function
PLEASE HELP ME PLEASE
WHICH EQUATION DOES THE GRAPH REPRESENT
Answer:
Step-by-step explanation:
B is the best answer
The starting lineup for a softball team consists of 10 players, How many different batting orders are possible using the starting lineup
The number of different batting orders are possible using the starting lineup is 3628800
How to determine the number of different batting orders are possible using the starting lineup?The given parameter is
Players, n = 10
The different batting orders are possible using the starting lineup is calculated using the following permutation expression
Ways = nCn
Substitute the known values in the above equation
Ways = 10P10
Apply the permutation formula
Ways = 10!/(10 - 10)!
Evaluate the difference
Ways = 10!/0!
0! = 1
So, we have:
Ways = 10!/1
Evaluate
Ways = 10!
Expand
Ways = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Evaluate the product
Ways = 3628800
Hence, the number of different batting orders are possible using the starting lineup is 3628800
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What is the probability that the battery life for an ipad mini will be at least 11 hours?
Answer:
0.2857
Step-by-step explanation:
We want to determine,
The probability that the battery life is at least 11 hours is 0.2857.
Complete the equivalent ratio. How many ounces are in 2 cups?
StartFraction 48 ounces Over 6 cups EndFraction = StartFraction question mark ounces Over 2 cups EndFraction
Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
[tex]2 \sqrt{5} \: + \: 7 \sqrt{5} \\ \sqrt{5} \: ( \: 2 + \: 7 \: ) \\ \sqrt{5} \: ( \: 9 \: ) \\ 9 \sqrt{5} [/tex]
Option 1
The simplification of the expression 2√5 + 7√5 is: 9√5.
So, Option A. is correct.
To simplify the expression 2√5 + 7√5, we can combine the like terms.
Both terms have the same radical expression, which is √5, so we can add their coefficients together.
2√5 + 7√5 = (2 + 7)√5
Now, simplify the coefficients:
2 + 7 = 9
Therefore, the simplified expression is:
2√5 + 7√5 = 9√5
So, the final answer is 9√5.
So, Option A. is correct.
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Mr. Granger, a cyclist, rode from his home to his office at an average speed of 18 miles per. On his return home from his office, using the same route, he averaged 12 miles per hour. If the total trip took 5 hours, what was the distance from his home to his office?
Answer:
The distance is 36 miles.
Step-by-step explanation:
distance in either direction = d
total time = 5
time going = t
time returning = 5 - t
speed = distance/time
going:
speed = distance/time
18 = d/t
d = 18t Eq. 1
returning:
speed = distance/time
12 = d/(5 - t)
60 - 12t = d
d = 60 - 12t Eq. 2
Eq. 1 and Eq. 2 form a system of equations.
d = 18t
d = 60 - 12t
Since d = d, then 18t must equal 60 - 12t
18t = 60 - 12t
30t = 60
t = 2
d = 18t = 18(2) = 36
The distance is 36 miles.
BRAINLIEST!
Find the area of the figure below
Answer:
140 in^2
Step-by-step explanation:
Take the outside dimensions of the LARGE rectangle
11 x 17 = 187 in^2
subtract the two corner cutout rectangles
187 - 8x4 - 3 x 5 = 140 in^2
Diagram
The bisectors BO and CO of the angles at B and C of ABC meet at O. Prove that BOC = 90° + A/2.
The prove that BOC = 90° + A/2. is given below:
What is the bisector about?Note that, bisectors of angles B and C and that of a triangle ABC is one that pass across each other at the point O.
Hence: one need to to prove that ∠BOC = 90° + A/2
Since:
BO is said to be the bisector of angle B e.g. ∠OBC = (∠1)
CO is said to be the bisector of angle C e.g. ∠OCB = (∠2)
Looking at the triangle BOC, to interpret by the use of angle sum property, then:
∠OBC + ∠BOC + ∠OCB = 180°
∠1 + ∠BOC + ∠2 = 180° ---- ( equation 1)
Looking at triangle ABC and by the use of angle sum property, So;
∠A + ∠B + ∠C = 180°
∠A + 2(∠1) + 2(∠2) = 180°
Then we divide by 2 on the two sides, and it will be:
∠A/2 + ∠1 + ∠2 = 180°/2
∠A/2 + ∠1 + ∠2 = 90°
∠1 + ∠2 = 90° - ∠A/2 ---(equation 2)
Then one need to Substitute (2) in (1), So:
∠BOC + 90° - ∠A/2 = 180°
∠BOC - ∠A/2 = 180° - 90°
∠BOC - ∠A/2 = 90°
Hence , ∠BOC = 90° + ∠A/2
Therefore, from the above, we care able to prove that BOC = 90° + A/2.
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For a population with µ = 80 and σ = 20, the distribution of sample means based on n = 16 will have an expected value of __________
Answer:
calculate the number of atom in 3 mol helium
If the store has only 12 red balloons and only 8 blue balloons but at least 29 of each other color of balloon, how many combinations of balloons can be chosen?
The number of combinations of balloons that can be chosen is 1716.
Selections and combinations are synonyms. Combinations represent the choosing of items from a predetermined group of items. We're not trying to arrange anything here. We're going to pick them.
Utilizing the combinations formula, it is simple to determine how many distinct groups of r objects each may be created from the provided n unique objects.
Number of red balloons = 12
Number of blue balloons = 8
First, we figure out how to choose from 13 red and 9 blue balloons.
There are 7 balloons left out of the 29 total (i.e. 29 - 13 - 9)
So, n = 7+7-1
n = 13
r = 7
Number of combinations of balloons = [tex]_{13} C_{7}[/tex]
= 13!/(13-7)!7!
= 13!/6!7!
= 1716
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