The solution of the differential equation is 1 = c₁v₁ + c₂v₂ and 5 = c₁v₁ + c₂v₂
We are given a system of two differential equations:
x₁' = – 5x₁ + 0x₂
x₂' = – 16x₁ + 3x₂
To solve this system, we can use several methods, such as substitution or matrix methods. In this explanation, we will use the substitution method.
We can write the given system of differential equations in matrix form as follows:
X' = AX
where X is the column vector [x₁, x₂], X' is the derivative of X, and A is the coefficient matrix:
A = [–5 0]
[–16 3]
To find the eigenvalues λ and eigenvectors v, we solve the characteristic equation:
|A - λI| = 0
where I is the identity matrix. Solving this equation will give us the eigenvalues and eigenvectors.
A - λI = [–5-λ 0]
[–16 3-λ]
Setting the determinant of A - λI to zero, we get:
(–5-λ)(3-λ) - (0)(–16) = 0
Simplifying, we have:
(λ + 5)(λ - 3) = 0
Solving this equation, we find two eigenvalues:
λ₁ = -5
λ₂ = 3
For each eigenvalue, we need to find its corresponding eigenvector. For λ₁ = -5, we solve the system of equations:
(A - (-5)I)v₁ = 0
Substituting the values of A and λ₁, we have:
[0 0] v₁ = 0
[–16 8]
Simplifying the equation, we get:
0v₁ + 0v₂ = 0
-16v₁ + 8v₂ = 0
From the first equation, we can see that v₁ can take any value. Let's choose v₁ = 1 for simplicity. Substituting this value into the second equation, we get:
-16(1) + 8v₂ = 0
-16 + 8v₂ = 0
8v₂ = 16
v₂ = 2
So, for λ₁ = -5, the corresponding eigenvector is v₁ = [1, 2].
Similarly, for λ₂ = 3, we solve the system of equations:
(A - 3I)v₂ = 0
Substituting the values of A and λ₂, we have:
[-8 0] v₂ = 0
[–16 0]
Simplifying the equation, we get:
-8v₁ + 0v₂ = 0
-16v₁ + 0v₂ = 0
From the first equation, we can see that v₁ can take any value. Let's choose v₁ = 1 for simplicity. Substituting this value into the second equation, we get:
-16(1) + 0v₂ = 0
-16 = 0
This equation has no solution. However, this means that v₂ can take any value. Let's choose v₂ = 1 for simplicity.
So, for λ₂ = 3, the corresponding eigenvector is v₂ = [1, 1].
The general solution of the system of differential equations can be expressed as:
X(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂
where c₁ and c₂ are constants that need to be determined.
We are given the initial conditions x₁(0) = 1 and x₂(0) = 5. Substituting these values into the general solution, we get two equations:
x₁(0) = c₁e(λ₁(0))v₁ + c₂e(λ₂(0))v₂
x₂(0) = c₁e(λ₁(0))v₁ + c₂e(λ₂(0))v₂
Simplifying, we have:
1 = c₁v₁ + c₂v₂
5 = c₁v₁ + c₂v₂
Solving this system of equations, we can find the values of c₁ and c₂.
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What is the equation of the line that passes through the point (7,-6) and has a slope of -2?
Answer:
y=-2+8
Step-by-step explanation:
The answer is y=-2+8 because of course the slope has to be -2 as you stated in your Question. So all you have to do is change the Y-Intercept until you reach the point. Since the Slope is negative, the Y-Intercept will rise until you reach your point. You may double check my answer to see if it is right, it is up to you. If you find any fault in my answer please let me know. Have a good day!
What is the vertex of f(x) = -2|x + 1| + 2?
Answer:
(-1,2) i think
Step-by-step explanation:
It's been found that there is a 15% chance (3 out of 20) that you can
win a particular game. How many "wins” would you have if you
played 80 times?
The augmented matrix of a system of linear equations AX = B was reduced to upper-triangular form so that 2 1 02 [AB] 3 0 where m and n are real numbers. State all values of m and/or n such that the following statements are true. (a) Matrix A is invertible. (b) The system AX- B has no solutions. (c) The system AX = B has an infinite number of solutions. (a) Columns of the augmented matrix (AB) are linearly independent. (e) The system AX = 0 has a unique solution. (f) At least one eigenvalue of the matrix A is zero. (g) Columns of the matrix A form a basis in R3.
a. Matrix A is invertible when |A| = -m ≠ 0 then statement true.
b. The system AX = B has no solution when m = 0 and n ≠ 0 has a real number then statement true.
c. The system AX = B has an infinite number of solutions when m = n = 0 then statement true.
d. Columns of the augmented matrix (AB) are linearly independent when m ≠ 0 and n= 0 then statement true.
e. The system AX = 0 has a unique solution when m ≠ 0 then statement true.
f. At least one eigenvalue of the matrix A is zero when m = 0 then statement true.
g. Columns of the matrix A form a basis in R³ when m ≠ 0 then statement true.
Given that,
The augmented matrix of a system of linear equations AX = B was reduced to upper-triangular form so that
[A|B] = [tex]\left[\begin{array}{ccc}2&1&0 \ | \ 2\\0&-1&3 \ | \ 1 \\0&0&m \ | \ n\end{array}\right][/tex]
Where m and n are real numbers.
We know that,
a. We have to prove matrix A is invertible.
For A to be invertible.
|A| ≠ 0
|A| is the determinant of the matrix A.
|A| = 2(-m) -1(0) + 0(0) = -m
Here, m is the real number.
So, |A| = -m ≠ 0
Therefore, Matrix A is invertible when |A| = -m ≠ 0 then statement true.
b. We have to prove the system AX = B has no solution.
When Rank[A|B] > Rank[A]
m = 0 and n ≠ 0 has a real number
Therefore, The system AX = B has no solution when m = 0 and n ≠ 0 has a real number then statement true.
c. We have to prove the system AX = B has an infinite number of solutions.
When m = n = 0, and Rank[A] < 3
Therefore, The system AX = B has an infinite number of solutions when m = n = 0 then statement true.
d. We have to prove columns of the augmented matrix (AB) are linearly independent.
When m ≠ 0 and m∈R and n= 0
Therefore, Columns of the augmented matrix (AB) are linearly independent when m ≠ 0 and n= 0 then statement true.
e. We have to prove the system AX = 0 has a unique solution.
When [tex]\left[\begin{array}{ccc}2&1&0 \\0&-1&3 \\0&0&m \end{array}\right]\left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}0\\0\\0\end{array}\right][/tex]
The equation are 2x + y = 0, -y + 3z = 0 and mz = 0
m ≠ 0 should be any real number except zero.
Therefore, The system AX = 0 has a unique solution when m ≠ 0 then statement true.
f. We have to prove at least one eigenvalue of the matrix A is zero.
When λ = 2, 1, m
m = 0 then eigen value is zero
Therefore, At least one eigenvalue of the matrix A is zero when m = 0 then statement true.
g. We have to prove columns of the matrix A form a basis in R³.
When m ≠ 0
Therefore, Columns of the matrix A form a basis in R³ when m ≠ 0 then statement true.
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Solve for x.
PLEASE ANSWER I WILL GIVE BRAINLIEST!!
Answer:
16 + 5 =21 21 is your answer
Step-by-step explanation:
Which of the following sets of ordered pairs does not define a function?
{(1,2),(5,6),(6,7),(10,11),(13,14)}
{(−1,4),(0,4),(1,4),(2,4),(3,4)}
{(1,1),(2,2),(3,3),(4,4),(5,5)}
{(1,3),(5,2),(6,9),(1,12),(10,2)}
Answer: D
Step-by-step explanation: As we can see, D is the only one that has matching inputs, but those inputs have separate outputs. If they don't have the same output, it is not a function
Hope this helps :)
Solve for x.
20
8
4x+3
38
Answer:
x = 18
Step-by-step explanation:
The "Angle Bisector" Theorem says that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle.
So, 20/8 = (4x + 3)/30
8(4x + 3) = 20(30)
32x + 24 = 600
32x = 576
x = 18
Convert the point from Cartesian to polar coordinates. Write your answer in radians. Round to the nearest hundredth.
(−10,1)
The point (-10, 1) in Cartesian-Coordinates can be represented in polar coordinates as approximately (10.05, 3.0416 radians).
To convert the point (-10, 1) from Cartesian-Coordinates to polar coordinates, we can use the formulas:
r = √(x² + y²)
θ = arctan(y / x)
We know that, the point is (-10, 1), we substitute the values into the formulas:
We get,
r = √((-10)² + 1²) = √(100 + 1) = √101 ≈ 10.05, and
The point lies in second-quadrant, so, the angle is measured counterclockwise from the positive x-axis, which means it is between π/2 and π radians.
Therefore, The adjusted θ is : θ = π + arctan(1/-10) ≈ 3.0416 radians.
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which would result in an integer
Answer:
c I think but I am not sure but I hope you have a good day
Simplify. Use only one symbol between terms. Use standard form. 6x + 3 - 8 + x
Answer:
7 is the answer
Step-by-step explanation:
Because 6x + 3 -8 + x = x is 6
Help please show work how to get the answer.
Answer:
A or D
Step-by-step explanation:
plllleeeasssw help scams are reporteddd
PLEASE HELP
WILL GIVE BRAINLIEST
Identify the situation that each graph could represent.
A ray is graphed in the first quadrant. The horizontal axis is labeled Time. The ray starts at the bottom left and continues to the upper right.
A. the length of a necklace that you make at a rate of 10 cm per hour without taking a break
B. the height of a balloon as it rises, gets caught in a tree for a few minutes, and then continues to rise
C. the total distance you are from home if you ride your bicycle three miles per hour for one hour, and then stop and take a rest
D. The volume of water in a bath tub as it is draining.
Answer:
The answer your looking for is, C.
A structural steel rod 1-1/2 in. in diameter and 20 ft long supports a balcony and is subjected to an axial tensile load of 30,000 lb. Compute: (a) the total elongation (b) the diameter of the rod required if the total elongation must not exceed 0.10 in. A. a. Elongation = 0.2358in. b. Use a1-1/2" dia. Rod B. a. Elongation = 1.1358in. b. Use a 1-1/4" dia. Rod C. a. Elongation = 0.1358in. b. Use a 1-3/4" dia. Rod D. a. Elongation = 0.1458in. b. Use a 3/4" dia. Rod
The diameter of the rod required to limit the total elongation to 0.10 inches is approximately 1-1/2 inches (or 1.441 inches to be more precise). Hence, the correct answer is option (A) with an elongation of 0.2358 inches and using a 1-1/2" diameter rod.
(a) To compute the total elongation, we can use the formula:
Elongation = (P * L) / (A * E)
where P is the axial tensile load, L is the length of the rod, A is the cross-sectional area of the rod, and E is the modulus of elasticity for the material.
Given:
P = 30,000 lb
L = 20 ft = 240 in
Diameter of the rod = 1-1/2 in
First, we need to calculate the cross-sectional area:
Area = π * (diameter/2)^2
Area = π * (1.5/2)^2
Area ≈ 1.767 in^2
Next, we need to determine the modulus of elasticity for the material. Assuming it's a standard structural steel, we can use a typical value of 29,000,000 psi.
Now we can plug the values into the formula:
Elongation = (30,000 * 240) / (1.767 * 29,000,000)
Elongation ≈ 0.2358 in
Therefore, the total elongation is approximately 0.2358 inches.
(b) If the total elongation must not exceed 0.10 inches, we need to determine the diameter of the rod that satisfies this requirement.
We can rearrange the formula for elongation to solve for the cross-sectional area:
A = (P * L) / (E * Elongation)
Using the given values:
A = (30,000 * 240) / (29,000,000 * 0.10)
A ≈ 2.069 in^2
To find the corresponding diameter, we use the formula:
Diameter = √(4 * A / π)
Diameter = √(4 * 2.069 / π)
Diameter ≈ 1.441 in
Therefore, the diameter of the rod required to limit the total elongation to 0.10 inches is approximately 1-1/2 inches (or 1.441 inches to be more precise). Hence, the correct answer is option (A) with an elongation of 0.2358 inches and using a 1-1/2" diameter rod.
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If an apple pie recipe calls for 3 pounds of candy apples then how many cups of canned apples required
Answer:
Seven cups of canned apples are required to make apple pie recipe
Step-by-step explanation:
The weight of one canned apple is 0.45 pounds
Weight of total canned apple required to make the apple pie recipe is 3 pounds.
Total number of cups of canned apples required
[tex]= \frac{3}{0.45} \\= \frac{300}{45} \\= \frac{20}{3} \\[/tex]
So approximately seven cups of canned apples are required to make apple pie recipe
A child toy is made by removing a triangular prism from the center of a wooden rectangular prism The triangular base of the triangular prism has a base length of 1 inch and a height of 1 inch. Write and solve an equation to find the volume of the toy.
*see attachment for the diagram given
Answer:
Volume of the toy = 68 in.³
Step-by-step explanation:
The equation to find the volume of the toy = volume of the wooden rectangular prism - volume of the triangular prism removed form the center
Volume of the toy = (L*W*H) - (½*bhl)
Where,
L = 8 inches
W = 3 inches
H = 3 inches
b = 1 inch
h = 1 inch
l = 8 inches
Plug in the values into the equation
Volume of the toy = (8*3*3) - (½*1*1*8)
Volume = 72 - 4
Volume of the toy = 68 in.³
Mark sorted a set of shapes into two different categories. Explain, what two attributes were used to sort the shapes. help please!!
Group A parallelogram, Group B Quadrilateral.
Answer: Parallelogram and Quadrilateral.
The two ways of classifying shapes are: Parallelogram and Quadrilateral.
There are different ways to classify an item.
How do one identify the type of quadrilateral?Quadrilaterals can be known by;
It is a polygon with four sides.
Since rectangle is known to be a parallelogram that has four right angles.
A trapezoid is regarded as a quadrilateral with only one pair of parallel sides.
And Parallelograms are known to be shapes that has four sides with only two pairs of sides that are known to be parallel.
So we conclude that Group A parallelogram, and Group B Quadrilateral.
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If m(x) = x+5/x-1 and n(x)=x-3, which function has the same domain as (m o n)(x)?
it's simple it's really easy so the answer is 2.0 1682
PLEASE HELP:
The distance d, in kilometers, that a car travels at a speed of 80 km per hour, for t hours, is given by the equation d= 80t. What is the inverse to represent time, t as a function of distance, d?
Choices:
1. t= d/80
2.t= 80/d
3.t= 80d
Answer:
The car was traveling for 1.5 hours.
Step-by-step explanation:
Given that distance d, in kilometers, that a car travels at a speed of 80 km per hour , for t hours, is given by the equation d=80t.
Here wee need to find the time if the car has gone 120 kilometers.
That is
d = 120 km
we need to find t.
d=80t
120 = 80 x t
The car was traveling for 1.5 hours.
What is 5x÷6=20
Pls help I can't figure it out
Answer:
x= 24
Step-by-step explanation:
5x=20*6
x=120/5
x=24
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Least:
Greatest:.
Median
Lower Quartile Range:
Upper Quartile Range:
thx :)
Answer:
Below :)
Step-by-step explanation:
Least/Minimum: 0
Greatest/Maximum: 6
Median: 2
Lower Quartile Range: 1
Upper Quartile Range: 3
Find median:
0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6
Find Lower Quartile:
0, 1, 1, 1, 1, 1, 1, 2, 2, 2
Find Upper Quartile:
2, 2, 3, 3, 3, 3, 4, 4, 4, 6
Combine the like terms to create an equivalent expression for −n+(−4)−(−4n)+6
Answer:
3n + 2
Step-by-step explanation:
−n+(−4)−(−4n)+6
-n - 4 + 4n + 6
3n + 2
In the above figure, m∠AOC = 30° and m∠BOD = (2x + 39)°. If ∠AOC and ∠BOD are vertical angles, what is the value of x? A. x = 69 B. x = -9 C. x = 34.5 D. x = -4.5
i need help asap
Answer:
bestie thats hard
Step-by-step explanation:
Answer:
D. x=-4.5
Step-by-step explanation:
Since they are both vertical angles, m∠BOD must also be equal to 30 degrees, and if you input -4.5 as x, (2 x -4.5x) + 39, that is rewritten as -9 and 39. 39 - 9 is 30 degrees.
[tex] \frac{ - 48 + 6}{ - 7} + ( - 3)( - 4)( - 2)[/tex]
Step-by-step explanation:
[tex] \frac{ - 48 + 6}{ - 7} + ( - 3)( - 4)( - 2) \\ = \frac{ - 42}{ - 7} + 12( - 2) \\ = 6 + ( - 24) \\ = 6 - 24 \\ = - 18[/tex]
Find the length of the diagonal of
rectangle whose length
is 12ft and whose
width 5 ft
Answer:
13 ft
Step-by-step explanation:
The formula to find the length of the diagonal of a rectangle =
Diagonal² = Length² + Width²
Diagonal = √Length² + Width²
Length = 12 ft
Width = 5ft
Diagonal = √12² + 5²
Diagonal = √144 + 25
Diagonal = √169
Diagonal = 13 ft
The length of the diagonal of the rectangle = 13 ft
Correct Given the sample data, find the mean (round to 2 decimals): 23, 27, 35, 44 1.00 points out of 1.00 Flag question Answer: 32.25 Check Correct Marks for this submission: 1.00/1.00 Question 6 Incorrect 0.00 points out of 1.00 Given the data from problem 5 (sample data: 23, 27, 35, 44), find the sum of the squared deviations (the numerator of the fraction under the square root in the formula). In finding the number, round all calculations to 2 decimals (if you carry more or fewer your answer may be off enough to be marked incorrect on this system).
The sum of the squared deviation for the given sample data (23, 27, 35, 44) is 212.00.
In statistics, the squared deviation is calculated by subtracting each data point from the mean and then squaring the result. The sum of these squared differences gives us a measure of how much the individual data points vary from the mean.
Find the sum of squared deviations, we first calculate the mean of the data set. In this case, the mean is found by adding up all the values (23 + 27 + 35 + 44) and dividing the sum by the number of data points (4).
The mean turns out to be 32.25.Next, we subtract the mean from each data point:
(23 - 32.25) = -9.25
(27 - 32.25) = -5.25
(35 - 32.25) = 2.75
(44 - 32.25) = 11.75
Then, we square each of these differences:
(-9.25)² = 85.56
(-5.25)² = 27.56
(2.75)² = 7.56
(11.75)² = 138.06
Finally, we sum up these squared deviations:
85.56 + 27.56 + 7.56 + 138.06
= 212.00
Therefore, the sum of the squared deviations is 212.00 (rounded to two decimal places).
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find the lateral surface area help needed asap will give brainliest
Step-by-step explanation:
3 Area of lateral = 3 ( bh ) = 12.2 XIO +7.04X12.2 +7.04x 12.2 = 122+85.888+85.888 = 293.776
293.776 approximate to 288
In a normal distribution, approximately what percentage of scores fall between the z scores of -1.00 and + 1.00?
In a normal distribution, approximately 68% of scores fall between the z-scores of -1.00 and +1.00.
In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, the Empirical Rule (also known as the 68-95-99.7 Rule) applies. According to this rule, approximately 68% of the data falls within one standard deviation from the mean.
Since the z-scores represent the number of standard deviations a particular value is away from the mean, a z-score of -1.00 represents one standard deviation below the mean, and a z-score of +1.00 represents one standard deviation above the mean. Therefore, using the Empirical Rule, we can conclude that approximately 68% of scores fall between these two z-scores (-1.00 and +1.00).
This percentage represents the central portion of the distribution that is within one standard deviation from the mean, providing a useful measure of the spread and concentration of data in a normal distribution.
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Consider the inequality x<1. Determine whether each value of x makes the inequality trueSelect Yes or No va Yes No 3/2 13/6
12 × (3 + 2²) ÷ 2 - 10
Answer:
32
Step-by-step explanation:
Answer:
32
Step-by-step explanation:
This should help