a) The equilibrium solution is xe = -20/9.
b) The equilibrium solution xe = -20/9 is not asymptotically stable.
c) The equilibrium solution xe = -20/9 is not Lyapunov stable.
d) The system is BIBO stable.
(a) The equilibrium solution, we set the derivative of x to zero:
-20 + x = 10x + 0u
Simplifying the equation, we get:
-20 = 9x + 0u
Since there is no input (u = 0), we can ignore the second term. Solving for x, we have:
9x = -20
x = -20/9
Therefore, the equilibrium solution is xe = -20/9.
(b) To determine if the equilibrium is asymptotically stable, we need to analyze the stability of the system. The stability can be determined by examining the eigenvalues of the system matrix.
The system can be represented as follows:
A = 10
The eigenvalues of A are simply the elements on the diagonal, so we have one eigenvalue: λ = 10.
Since the eigenvalue λ = 10 is positive, the system is unstable. Therefore, the equilibrium xe = -20/9 is not asymptotically stable.
(c) To determine if the equilibrium solution is Lyapunov stable, we need to check if the system satisfies the Lyapunov stability criterion. The criterion states that for every ε > 0, there exists a δ > 0 such that if ||x(0) - xe|| < δ, then ||x(t) - xe|| < ε for all t > 0.
Since the system is unstable (as determined above), the equilibrium solution is not Lyapunov stable.
(d) BIBO (Bounded Input Bounded Output) stability refers to the stability of the system's output when the input is bounded. In this case, the system is described by x' = Ax + Bu, where u is the input. Since the input u is specified as 0, the system becomes x' = Ax + 0u = Ax.
The system matrix A = 10 does not depend on the input u. Therefore, the system is BIBO stable since it does not rely on the input and the output remains bounded.
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help me understand please
Answer:
V=pi x (r)^2 x h
V-volume r-radius h-height
V= pi x (3)^2 x 9
V= pi x 9 x 9
V= pi x 81
V= 254.47 (rounded)
For n e N let an be the number of strings of 0's and l's such that every 0 is followed by a 1. (a) Write down the values for a¡ through as (you may need to use some scratch paper) i. ai = ii. 02 = iii. a3 = iv. 04 = V. a5 = (b) Do these numbers look familiar? Make a conjecture. (C) How might you build the set of these strings of length n from the sets of strings of length n - 1 and strings of length n - 2? (d) Prove your conjecture.
For n e N let an be the number of strings of 0's and l's,
(a) i. a₁ = 2, ii. a₂ = 1, iii. a₃ = 2, iv. a₄ = 3, v. a₅ = 5.
(b) The values of aₙ represent the nth term in the Fibonacci sequence.
(c) The set of strings of length n can be built by appending "1" or "01" to strings of length n-1 and n-2.
(d) The conjecture that the values of aₙ represent the nth term in the Fibonacci sequence is proven using mathematical induction.
(a) To find the values of aₙ, we'll calculate them one by one:
i. a₁: We have two possible strings of length 1 that satisfy the condition: "1" and "0". So a₁ = 2.
ii. a₂: For a string of length 2, the only valid option is "10". So a₂ = 1.
iii. a₃: Now, let's consider strings of length 3. We can build them by appending either "1" or "01" to valid strings of length 2. From the previous step, we know that a₂ = 1. Thus, we have two options: "101" and "100". So a₃ = 2.
iv. a₄: Similarly, for strings of length 4, we can append either "1" or "01" to valid strings of length 3. From the previous step, we know that a₃ = 2. Thus, we have three options: "1010", "1001", and "1000". So a₄ = 3.
v. a₅: Continuing the same pattern, we can append either "1" or "01" to valid strings of length 4. From the previous step, we know that a₄ = 3. Thus, we have five options: "10101", "10100", "10010", "10001", and "10000". So a₅ = 5.
(b) If we look closely at the values of aₙ that we calculated, we notice that they form the Fibonacci sequence: 2, 1, 2, 3, 5, ...
Conjecture: The values of aₙ represent the nth term in the Fibonacci sequence.
(c) To build the set of strings of length n from the sets of strings of length n-1 and n-2, we can append either "1" or "01" to the strings of length n-1 and n-2.
For example, to obtain a string of length 4, we can append "1" or "01" to the strings of length 3. Similarly, to obtain a string of length 5, we can append "1" or "01" to the strings of length 4.
(d) To prove our conjecture that the values of aₙ represent the nth term in the Fibonacci sequence, we can use mathematical induction. We would need to show that a₁ = F₁, a₂ = F₂, and assume that aₖ = Fₖ and aₖ₋₁ = Fₖ₋₁ hold for some k ≥ 2 and prove that aₖ₊₁ = Fₖ₊₁.
Since a₁ = 2 = F₁ and a₂ = 1 = F₂, the base cases hold.
Next, assume that aₖ = Fₖ and aₖ₋₁ = Fₖ₋₁ hold for some k ≥ 2.
From the construction in part (c), we know that to obtain a string of length k+1, we append "1" to a string of length k and append "01" to a string of length k-1.
So, aₖ₊₁ = aₖ + aₖ₋₁ = Fₖ + Fₖ₋₁.
Using the property of the Fibonacci sequence that Fₖ + Fₖ₋₁ = Fₖ₊₁, we can conclude that aₖ₊₁ = Fₖ₊₁.
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Find the area of each rhombus. Write your answer as an integer or a simplified radical
Step-by-step explanation:
area of rhombus =1/2×d1×d2
area of rhombus =1/2×9cm×5cm
area of rhombus =22.5cm²
area of rhombus = 1/2×d1×d2
area of rhombus =1/2×8in×17in
area of rhombus= 68in²
On a coordinate plane, a curved line with a minimum value of (1.5, negative 1) and a maximum value of (negative 1.5, 13), crosses the x-axis at (negative 3, 0), (1, 0), and (2, 0), and crosses the y-axis at (0, 6).
Which lists all of the x-intercepts of the graphed function?
(0,6)
(1,0)(2,0)
(1,0)(2,0) and (-3,0)
(1,0)(2,0)(-3,0) and (0,6)
Answer:
c
Step-by-step explanation:
find the value of two numbers if their sum is 39 and their difference is 1
Given the set { 1 , 1 , 1 , 1 , 2 , 3 } . Picking a number at random .
The probability of choosing a "1" is 2 / 3 .
True
False
Given the set { 1 , 1 , 1 , 1 , 2 , 3 } . Picking a number at random.The probability of choosing a "1" is 2 / 3 . False
In the given set {1, 1, 1, 1, 2, 3}, there are four occurrences of the number "1" and a total of six elements in the set.
To calculate the probability of choosing a "1", we need to divide the number of favorable outcomes (the occurrences of "1") by the total number of possible outcomes (the total number of elements in the set).
The number of favorable outcomes is 4 (the occurrences of "1"), and the total number of possible outcomes is 6 (the total number of elements in the set).
So, the probability of choosing a "1" is 4/6, which simplifies to 2/3.
Therefore, the statement is true: the probability of choosing a "1" from the given set is 2/3.
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A librarian noticed that 14% of the seventh graders checked out science fiction books and 12% checked out sports book. Out of 350 seventh graders, how many students should she expect NOT to check out science fiction and sports books?
Answer:
259 students
Step-by-step explanation:
First, find how many will check out science fiction books
350(0.14)
= 49
Find how many will check out sports books
350(0.12)
= 42
Subtract these amounts from 350
350 - 49 - 42
= 259
So, she should expect 259 students not to check out science fiction and sports books
11 inches in miles?
24.5 km in miles?
Answer:
0.000173611 miles is 11 inches
15.22359 miles is 24.5km
Step-by-step explanation:
Consider the function y = 8x + 3 between the limits of x = 2 and x = 8.
a) Find the arclength L of this curve:
L: ___________ Round your answer to 3 significant figures.
b) Find the area of the surface of revolution, A, that is obtained when the curve isrotated by 2π radians about the x-axis.
Do not include the surface areas of the disks that are formed at x = 2 and x = 8.
A = ___________ Round your answer to 3 significant figures.
a) We have the function given by; y = 8x + 3We need to find the arclength of the curve between the limits of x = 2 and x = 8.The arclength L of the curve is given by; L = ∫(2,8) sqrt(1 + f'(x)²)dx Here, f(x) = 8x + 3 Differentiate f(x) with respect to x;f'(x) = 8Now, substitute f'(x) in the above equation; L = ∫(2,8) sqrt(1 + 8²)dx L = ∫(2,8) sqrt(65)dxL = sqrt(65)∫(2,8)dxL = sqrt(65) [x]₂⁸L = sqrt(65) [8 - 2]L = 6sqrt(65)Therefore, the arclength L of this curve is 6sqrt(65).
b) We are given the function y = 8x + 3We need to rotate this curve by 2π radians about the x-axis to get the required surface of revolution. The formula for the surface area of the surface of revolution generated by revolving the curve y = f(x) between x = a and x = b about the x-axis is given by;A = ∫(a,b) 2πf(x) sqrt(1 + f'(x)²)dx Here, f(x) = 8x + 3f'(x) = 8We know that the limits of integration are from x = 2 to x = 8.
Substitute the values in the above equation; A = ∫(2,8) 2π(8x + 3) sqrt(1 + 8²)dxA = 16π ∫(2,8) (8x + 3) sqrt(65)dxA = 16π [∫(2,8) (8x sqrt(65))dx + ∫(2,8) (3 sqrt(65))dx]A = 16π [2/3(8sqrt(65))² - 2/3(2sqrt(65))² + 3sqrt(65)(8 - 2)]A = 16π [2/3(8sqrt(65))² - 2/3(2sqrt(65))² + 3sqrt(65)(6)]A = 192πsqrt(65)
Therefore, the area of the surface of revolution, A that is obtained when the curve is rotated by 2π radians about the x-axis is 192πsqrt(65) square units.
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1/4x + 6 = 1/2 (x+4)
Answer:
Solution x=16
Step-by-step explanation:
The degree of precision of a quadrature formula whose error term is 24 f'"'() is: 5 4 3 2
The degree of precision of the quadrature formula with an error term of 24 f‴() is 2.
To know more about the degree of precision of a quadrature formula, refer here:
The degree of precision of a quadrature formula represents the highest power of x that the formula can integrate exactly. In this case, the error term of the formula is given as 24 f‴(), where f‴() denotes the third derivative of the function f(x). The degree of precision is determined by the highest power of x that appears in the error term.
In a quadrature formula, the error term typically has the form K * h^p, where K is a constant, h is the step size, and p is the degree of precision. In this case, the error term is 24 f‴(). We can see that there is no dependence on the step size h, which implies that h^p = h^0 = 1. Therefore, the highest power of x in the error term is determined by the highest power of x that appears in f‴().
Since the error term is 24 f‴(), it indicates that the highest power of x in f‴() is 1. Thus, the degree of precision of the quadrature formula is 2, as the highest power of x in the error term is two degrees less than the highest power of x that the formula can integrate exactly.
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. Amy Company sold merchandise of $8,000 to Tory Turnbull with terms 2/10, n/30. Amy Company recorded this transaction using the gross method. If Tory Turnbull paid for all the merchandize within the discount period, the journal entry that Amy Company will make to record the collection of cash would include a: a. Credit to Sales Discount of $160 b. Credit to Account receivable $7,840 c. Debit to Sales Discount of $160 d. Credit to Cash of $160 Select-
The journal entry that Amy Company will make to record the collection of cash from Tory Turnbull, who paid within the discount period, would include a credit to Cash for $160. Therefore, option d is the correct answer.
This is because Tory Turnbull will pay $8,000 - $160 (2% of $8,000) to avail the discount. The Sales Discount account is not involved in the journal entry as the discount was taken by the customer, not given by Amy Company.
Therefore, the correct answer is d. Credit to Cash of $160. This entry reflects the cash received by Amy Company and the reduction in the Accounts Receivable balance for the amount paid by the customer.
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If X has an exponential distribution, which of the following statements is correct?
A. The exponential distribution has the property that its mean equals its variance. B. All of the given statements are correct. C. The exponential distribution is governed by two parameters that determine its shape and location. D. The cumulative density function for an exponential random variable x has a bell-shaped graph. E. The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.
option A is the correct statement regarding the exponential distribution.
What is Exponential Distribution?
The exponential distribution is a continuous probability distribution that models the time between events occurring at a constant average rate. It is often used to describe situations where events happen randomly and independently over time, such as the time between phone calls received at a call center or the time between arrivals of customers at a service counter.
The correct answer is:
[tex]\textbf{A. The exponential distribution has the property that its mean equals its variance.}[/tex]
The exponential distribution is characterized by the property that its mean is equal to its variance. This means that the average value of the distribution and the spread of the distribution are the same. In other words, the measure of central tendency and the measure of dispersion are equal in the exponential distribution.
This property holds true for all exponential distributions, regardless of the specific rate parameter or scale parameter. It is a fundamental characteristic of this distribution.
Therefore, option A is the correct statement regarding the exponential distribution.
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In the diagram, mBDA = 150°. Find mBDC.
Answer:
a. 70°
Step-by-step explanation:
m∠BDC + m∠CDA = m∠BDA
-3x + 34 - 2x + 56 = 150
-5x + 90 = 150
-5x = 60
x = -12
m∠BDC = -3x + 34 = -3(-12) + 34
= 36 + 34 = 70°
please help i’ll give brainliest
Answer:
A insects and plants
Step-by-step explanation:
the decaying things are used as fertilizer for plants to grow and insects to eat
Theoretically, if a month is chosen 300 times,
how many times would you expect a month
that starts with the letter J?
Answer:
75 times.
Step-by-step explanation:
Well there are 12 months and 3 of them start with the letter J, so theoretically, 25% of the months chosen will start with the letter J because [tex]\frac{3}{12}=\frac{1}{4}[/tex], which is 25%. 25% of 300 (or [tex]\frac{300}{4}[/tex], for those who like fractions) is 75, so theoretically, we can expect a month that starts with the letter J 75 times.
Hope this helps!
P.S: Please mark me as brainliest!
what is true about a rhombus and square?
A. all sides are congruent to each other
B. all of these are true
C. both have diagonals that are perpendicular to each other
D. both have diagonals that bisect each other
Answer:
B. all of these are true
Step-by-step explanation:
The Square is a rhombus because all the sides of a square are equal in length. Even, the diagonals of both square and rhombus are perpendicular to each other and bisect the opposite angles. Therefore, we can say the square is a rhombus.
Hope this helped!!
Lisa can mow the lawn in 3 hours. If Rhianna helps her with another mower, the lawn can be mowed in 2 hours. How long would it take Rhianna if she worked alone? PLEAsE HELPPPP
Answer:
5 hours or 300 mins
Step-by-step explanation:
Un paseo tiene una longitud de 1250 km. Se quiere plantar un árbol cada 0,025 km. ¿cuantos arboles se plantaran? ¿ y si se planta un arbol cada 50 m?
Answer:
-Se plantarán 50.000 arboles.
-Si se planta un árbol cada 50 m, se plantarán 25.000 arboles.
Step-by-step explanation:
Para encontrar cuántos arboles se plantarán, tienes que dividir la longitud el paseo entre la distancia que habrá entre los árboles:
1250/0,025=50.000
Ahora, para encontrar cuántos arboles se plantarán si se planta uno cada 50m, primero tienes que encontrar cuántos kilómetros hay en 50 m considerando que 1 km es igual a 1000 metros. Para esto, puedes utilizar una regla de tres:
1 km → 1000 m
x ← 50 m
x=(50*1)/1000
x= 0,05 km
Ahora puedes dividir la longitud del paseo entre 0,05 para encontrar el número de arboles que se plantarán:
1250/0.05=25.000
De acuerdo a esto, la respuesta es que se plantarán 50.000 arboles.
Si se planta un árbol cada 50 m, se plantarán 25.000 arboles.
How many yards are equivalent to 38 feet? Sh
Answer:
16 2/3 yards
Step-by-step explanation:
Hope this helps and have a wonderful day!!!!
In an experiment a bag contains 2 blue marbles and 5 red marbles. Two marbles are drawn from the bag.
Answer:
5
Step-by-step explanation:
2-5=5. Subscribe Single-3 FF on yt
The sample space of all possible pairs of marbles that can be drawn is:
{BB, BR, RB, RR, RR, RR}
What is sample space?It is the total number of possible outcomes from a given set.
We have,
The sample space is the set of all possible outcomes of the experiment.
In this case, we are drawing two marbles from a bag containing 2 blue and 5 red marbles, without replacement (meaning that we do not put the first marble back into the bag before drawing the second one).
The sample space consists of all possible pairs of marbles that can be drawn:
{BB, BR, RB, RR, RR, RR}
where BB means both marbles are blue, BR means the first marble is blue and the second is red, RB means the first marble is red and the second is blue, and RR means both marbles are red.
Note that there are two outcomes corresponding to drawing two red marbles because there are 5 red marbles in the bag.
Thus,
The sample space of all possible pairs of marbles that can be drawn is:
{BB, BR, RB, RR, RR, RR}
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The complete question:
In an experiment, a bag contains 2 blue marbles and 5 red marbles. Two marbles are drawn from the bag.
List the sample space.
Find mXY
X
23°
N
I
plz help
Answer:
arc XY = 46°
Step-by-step explanation:
The inscribed angle XZY is half the measure of its intercepted arc , then
arc XY = 2 × 23° = 46°
calculate the velocity of a stream if a drop of food coloring is timed traveling 32 feet in 15 seconds.
The velocity of the stream is approximately 2.13 feet per second.
To calculate the velocity of the stream, we can use the formula:
Velocity = Distance / Time
Given that the drop of food coloring travels 32 feet in 15 seconds, we can plug in these values into the formula:
Velocity = 32 feet / 15 seconds
To find the velocity, we divide 32 by 15:
Velocity ≈ 2.13 feet per second
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A 1-g antibiotic vial states "Reconstitute with 3.4 mL of sterile water for a final volume of 4 ml. * What is the powder volume in the vial?
A. 3.4 mL
B. 0.6 mL
C. 4 mL
D. 4.6 mL
The correct answer is option B. 0.6 mL which is the powder volume in the vial.
To determine that 0.6 mL of powder volume in the vial, we need to subtract the volume of the sterile water used for reconstitution from the final volume.
The vial states that it needs to be reconstituted with 3.4 mL of sterile water for a final volume of 4 mL. This means that 3.4 mL of sterile water will be added to the vial to make a total volume of 4 mL.
To find the powder volume, we subtract the volume of the sterile water (3.4 mL) from the final volume (4 mL):
Powder volume = Final volume - Volume of sterile water
Powder volume = 4 mL - 3.4 mL
Powder volume = 0.6 mL
Therefore, the powder volume in the vial is 0.6 mL.
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in class of 40 pupils,20% are absent one day. (a) how many pupils are absent?
Answer: 8 pupils are absent
Step-by-step explanation:
20 percent * 40 =
(20* 40)/100 =
(800)100 =
=8
Please help:///. Due today
Answer:
C: 108 cans
Step-by-step explanation:
just trust me bro
Please help with this question?!?
Answer:
50.3
Step-by-step explanation:
area for circle formula=πr^2
r=4
π4^2 which is 50.3 rounded
Answer:
50.3
Step-by-step explanation:
A=πr^2
π·4^2≈50.3
Need this answer as soon as possible
Answer:
x=22
Step-by-step explanation:
Rule states diagonals = same so 6x-36 = 96 in which x=22 cause 6(22)=132
132-36 = 96
In cell B23, find the required sample size to estimate the true proportion of satisfied customers within 3% margin of error, with 99% confidence. in excel??
To calculate the required sample size in Excel for estimating the true proportion of satisfied customers within a 3% margin of error with 99% confidence,
In Excel, you can use the NORM.S.INV function to find the critical value corresponding to the desired confidence level. In this case, we use 1 - (1-0.99)/2 to find the z-score for 99% confidence level.
We square this z-score and multiply it by (0.50.5)/(0.030.03), which represents the maximum variance under worst-case scenario and the desired margin of error. The CEILING function is used to round up the result to the nearest whole number since the sample size should be an integer.
you can use the formula =CEILING(MROUND((NORM.S.INV(1-(1-0.99)/2,0)^2)(0.50.5)/(0.03*0.03),1),1). The result will give you the minimum sample size needed for the desired confidence level and margin of error.
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Solve this please!!!!
Answer:
The volume of the rectangular prism is 336
The volume of the triangular prism is 24.875
Add them together, 360.875. You can round if you want