Consider the system of linear differential equations
x_1’ (t) = -4 x_1 (t) – 8 x_2(t)
x_2’ (t) = 1 x_1 (t) + 5 x_2 (t)
We want to determine the stability of the origin.
a) This system can be written in the form X'=AX, where X(t) = (x_1 (t)) and
A = ______

b) Find the eigenvalues of A. List them between square brackets and separated by commas if there are more than one.
Eigenvalues: ______
c) From (b), we can conclude that the origin is
O unstable
O stable

because

• all the eigenvalues are non-positive with at least one of them null
• the absolute value of each eigenvalue is less than one
• all eigenvalues are negative
• both of the eigenvalues have the same sign
• at least one of the eigenvalues is positive.

Rounding to the nearest whole value, we get the estimated number of observations above 118 as 2940.

To determine the number of observations expected to be below a certain value using the 68-95-99.7 Rule, we need to find the area under the normal distribution curve up to that value.

For the first scenario:

Mean (μ) = 162

Standard deviation (σ) = 11

Value to evaluate (x) = 195

We want to find the area under the curve to the left of x = 195. This corresponds to the cumulative probability of a value being less than 195 in a normal distribution.

Using a standard normal distribution table or a calculator, we can find that the cumulative probability for x = 195 is approximately 0.961. This means that about 96.1% of the observations are expected to be below 195.

To find the number of observations, we multiply the cumulative probability by the total number of observations:

Number of observations = 0.961 * 2400 = 2306.4

Rounding to the nearest single observation, we get the estimated number of observations below 195 as 2306.

For the second scenario:

Mean (μ) = 137

Standard deviation (σ) = 19

Value to evaluate (x) = 118

We want to find the area under the curve to the right of x = 118. This corresponds to the cumulative probability of a value being greater than 118 in a normal distribution.

Using the same approach, we can find that the cumulative probability for x = 118 is approximately 0.735. This means that about 73.5% of the observations are expected to be above 118.

To find the number of observations, we multiply the cumulative probability by the total number of observations:

Number of observations = 0.735 * 4000 = 2940

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The solution to the 1-dimensional heat equation problem with the given initial and boundary conditions is \(u(x, t) = \sum_{n=1}^\infty \frac{(-1)^n}{n\pi} \sin\left(\frac{n\pi x}{5}\right) e^{-(n\pi)^2 \frac{2t}{25}}\).

We are given the one-dimensional heat equation:

\(\frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}\)

We need to find the solution u(x, t) subject to the initial condition u(x, 0) = f(x) and boundary conditions u(0, t) = u(5, t) = 0 for t > 0.

To solve this problem, we use separation of variables:

u(x, t) = X(x)T(t)

Substituting this into the heat equation and dividing by u(x, t), we obtain:

\(\frac{1}{T(t)}\frac{dT}{dt} = \frac{2}{X(x)}\frac{d^2X}{dx^2}\)

Both sides are equal to a constant, which we denote by -λ². We then obtain two ordinary differential equations:

\(\frac{d^2X}{dx^2} + \lambda^2X = 0\)

\(\frac{1}{T} \frac{dT}{dt} = -\frac{\lambda^2}{2}\)

The general solution to the first equation is:

\(X(x) = A \sin\left(\frac{\pi x}{L}\right) + B \cos\left(\frac{\pi x}{L}\right)\)

where L = 5 is the length of the interval and A, B are constants determined by the boundary conditions.

The second equation has the solution:

\(T(t) = Ce^{-\lambda^2t/2}\)

where C is a constant determined by the initial condition.

The general solution to the heat equation is then:

\(u(x,t)=\sum\limits_{n=1}^{\infty}\left(A_n\sin\left(\frac{n\pi x}{5}\right) + B_n\cos\left(\frac{n\pi x}{5}\right)\right)e^{-\frac{n^2\pi^2t}{5^2}}\)

where the sum is taken over all positive integers n.

Using the initial condition, we can determine the coefficients A_n and B_n as:

A_n = \((-1)^n \frac{(4n\pi\sin(n\pi) - 3\sin(2n\pi))}{n^3\pi^3}\)

B_n = 0

Therefore, the solution to the given problem is:

\(u(x, t) = \sum_{n=1}^\infty \frac{(-1)^n}{n\pi} \sin\left(\frac{n\pi x}{5}\right) e^{-(n\pi)^2 \frac{2t}{25}}\)

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Answer:

Step-by-step explanation

Mikhail is 37

Gabrielle is 30

37+30= 67

Since girls and boys have equal probability, iin a family with four children, they have probability equal to

so, the probability of having 3 boys and 1 girls is

that is, the answer is 3/16 which is equal to 0.19.

The max margin classifier, also known as the Support Vector Machine (SVM), can be formulated as an optimization problem:

Maximize: γ

Subject to:

yi(w·xi + b) ≥ γ, for all training examples (xi, yi)

||w|| = 1

Here, xi represents the feature vector of the training example, yi is the corresponding label (+1 or -1), w is the weight vector, b is the bias term, and γ represents the margin.

The SVM classifier aims to find the optimal hyperplane that maximizes the margin (γ) between the positive and negative training examples while satisfying the constraint that all examples are correctly classified. The margin is defined as the perpendicular distance from the hyperplane to the nearest training example.

To solve for the optimal SVM classifier, we need to find the values of w and b that maximize the margin while satisfying the constraints. This can be achieved through various optimization techniques such as quadratic programming or gradient descent.

The optimal solutions will be the hyperplane and margins that separate the positive and negative examples with the maximum possible margin. The value of the optimal w will depend on the specific dataset and the result of the optimization process. The optimal w represents the normal vector to the hyperplane, indicating the direction of separation between the two classes. The exact value of w will vary based on the dataset and its distribution, and cannot be determined without solving the optimization problem specific to the dataset at hand.

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The area of this rectangle is x*y. But, this problem requires that you represent the variable "y" in terms of "x" only.

A hint here is that the variable "y" represent a point on the line that connects the end points (0,b) and (a,0).

The first step to solve this problem is to get the expression of this line.

The slope of this line is:   m = (0 - b)/(a - 0) = -b/a

Because the point (x,y) is on this line we have:   y - b = m(x - 0)

                                                                      y = (-b/a)x + b

Solution: Area of the rectangle = x*y = x*((-b/a)x + b) = (-b/a)x2 +bx

Answer:

angle pts is a 45 degree angle

Step-by-step explanation:

To eliminate the x variable from the given system of equations, we can multiply the first equation by −3, then add the two equations. This will give us a new equation in terms of y, which we can solve for y. Once we have the value of y, we can substitute it into either of the two original equations to get the value of x.

Given the system of equations:2x − 3y = −117x − 2y = −5We want to eliminate the x variable from these equations. One way to do this is to use the elimination method. We can eliminate one of the variables by adding or subtracting the two equations so that the coefficient of one variable is the same but the opposite signs are used. In this case, we can eliminate the x variable by multiplying the first equation by −3 and then adding the two equations:−

6x + 9y = 3511x − 2y = −5Adding these two equations gives us:5x + 7y = 30Now we have an equation in terms of y only. We can solve for y by isolating it on one side of the equation:

5x + 7y = 30⇔ 7y = −5x + 30⇔ y = −(5/7)x + (30/7)Now that we have the value of y, we can substitute it into one of the two original equations to get the value of x. Let's use the first equation:

2x − 3y = −11⇔ 2x − 3(−(5/7)x + (30/7)) = −11⇔ 2x + (15/7)x − 30/7 = −11⇔ (29/7)x = (93/7)⇔ x = 3Thus, the solution to the system of equations is (x, y) = (3, 2).

To eliminate the x variable from a system of equations, we can use the elimination method. We can do this by adding or subtracting the two equations so that the coefficient of one variable is the same but the opposite signs are used. Once we have eliminated one variable, we can solve for the other variable.

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Answer:

14

Step-by-step explanation:

hope this helps

Answer:

Step-by-step explanation:

tan34=h/25

h=25tan34

h=16.9

tanx=(16.9+10)/25

tanx=26.9/25

x=arctan(26.9/25)

x=47.1 degrees

Answer:

answer 1: 16.9 ft. Answer 2: 47.1°

Step-by-step explanation:

Answer:

360 feets

Step-by-step explanation:

Given :

Scale : 1 inch = 12 feets

This means 1 inch on paper (model) represents 12 feets in real life

If the model height = 30 inches ;

The actual height of the Saturn V rocket will be :

1 inch = 12 feets

30 inch = x

x = (30 * 12)

x = 360 feets

Actual height = 360 feets

The most direct way to measure how dispersed a set of scores is, is by using the range. The range is calculated by subtracting the lowest score from the highest score in the dataset, which gives a quick and easy measure of the spread of the scores.

However, the range can be influenced by extreme scores and may not provide a complete picture of the variability. Other measures of variability, such as the standard deviation and variance, take into account the variability of all scores in the dataset and provide a more robust measure of variability.

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Answer:

a) 640 ml

b) 40 ml

Step-by-step explanation:

The ratio of lime to lemonade for the fizzy drink is 5 : 3 or as a fraction that would be

\(\dfrac{\text{Lime juice}}{\text{Lemonade}}= \dfrac{5}{3}\)

\(\text{Therefore the ratio of lemonade to lime }\\\\ = \text{reciprocal of $ \dfrac{5}{3} $} }\\\\= \dfrac{3}{5}\)

Part a)

For  all 400 ml of  lime juice we would need
\(\dfrac{3}{5} \times 400 \;ml = 3 \times 80 = 240 \;ml\)  

Total amount of fizzy drink that can be maade
= amount of lime juice + amount of lemonade

= 400 + 240

= 640 ml

This is the answer to Part a)

Part b)

If Gianni has only 280 ml and is using all 400 ml of lime juice then the amount of lemonade used as calculated in part 1) is 240ml

That means the amount of lemonade left over = 280 - 240 = 40 ml

We plot the means on a number line to see which means are different. Plotting the sample means on a number line serves to illustrate this when the null hypothesis that all population means are equal is rejected.

This criteria, known as (alpha) in null hypothesis testing, is typically set to.05. The null hypothesis is rejected if there is less than a 5% likelihood that a result as dramatic as the sample result would occur if the null hypothesis were true. The outcome is referred to as statistically significant when this occurs.

Plotting scientific data is used to highlight correlations between variables or to visualize variance, but not all data sets need one. If there are only one or two points, it is simple to look at the numbers directly, and adding a graph adds little to nothing.

hence we plot the means on a number line to see which means are different. Plotting the sample means on a number line serves to illustrate this when the null hypothesis that all population means are equal is rejected.

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Subscription: $5.95 per month

One print: $0.19

January charge: $15.83

Number of prints: ?

We need to know the amount of money of the January charge that correspond to the prints. Since it has two components (subscription and prints), we find that value substracting

$15.83 - $5.95 = $9.88

We find the number of prints dividing that value into the cost of each print

$9.88/$0.19 = 52

Answer: D (52 prints)

Answer:

  295 months

Step-by-step explanation:

You want to know the number of months required to pay off a 30-year loan of $125000 at 6% if an extra payment is made each year.

The number of payments required will depend on the actual payment schedule, and the way the lender computes interest due.

The attached calculation shows it takes 319 payments. The assumption is that they are evenly spaced at 13 payments per year (one every 4 weeks), and the interest is computed on the interval between payments.

If 13 payments are made per year, the number of years it takes to pay off the loan will be ...

  319/13 = 24 7/13

24 years and 7 months is 295 months.

__

Additional comment

If the first payment is made on Jan 28, and payments are made every 4 weeks, there will be 2 payments in Dec. That is, except for the last month of the year, there will be 1 payment per month.

If regular monthly payments are made, but one of the payments is doubled, the result may be different.

Answer:

4 cans of paint will cover 8/9 of the surface

Step-by-step explanation:

One can of paint covers 2/9 of the house.

Garvin has used 4 of those cans.

To get the fraction of the whole surface that has been covered by those 4 cans of paint, we can multiply, following the logic below:

1 can of paint covers = 2/9 of the surface

4 cans of paint will cover = x of the surface

x = 4 X 2/9 = 8/9

Therefore, 4 cans of paint will cover 8/9 of the surface

Answer:

5

Step-by-step explanation:

let width be x

and because length is 11 times more than the width, therefore length equals to 11x

hence the equation equals to

area = length x width

60 = 11x + x

60 = 12x

x = 60/12

therefore, the width of the rectangle is 5

Answer:

20x^3/4x7

20/4x^4

5/x^4

Step-by-step explanation:

Answer:

  y = 18

Step-by-step explanation:

You want the value of y, given a rectangle with diagonals of lengths (y+36) and (3y).

The diagonals of a rectangle are congruent, so ...

  3y = y +36

  2y = 36 . . . . . . . subtract y

  y = 18 . . . . . . . . . divide by 2

According to Pythagorean theorem:

To find the missing length of the leg we'll convert the equation:

Simplify:

\(\large\bf{\underline{\underline{\mathfrak{Question}:}}}\)

The hypotenuse of a right triangle has a length of 14 units and side that is 9 units long. which equation can be used to find the length of the remaining side?

\(\large\bf{\underline{\underline{\mathfrak{Solution}:}}}\)

To solve the problems of a right angle triangle we commonly used Pythagoras theorem. It is the most widely used theorem which helps in finding the missing length of a right angle triangle.

According to Pythagoras theorem, the sum of square of base and the square of perpendicular give us square of hypotenuse.

\(:{\Longrightarrow{\large{\rm{H^2=P^2+B^2}}}}\)

In a right angle triangle there are two legs and a hypotenuse.

Given that,

\({:\Longrightarrow{\small{\rm{Hypotenuse\:of\:a\:right\:triangle=14\:units.}}}}\)

\(:{\Longrightarrow{\small{\rm{A\:side\:of\:a\:right\:triangle=9\:units.}}}}\)

By putting these values in the Pythagoras theorem we get;

\(:{\Longrightarrow{\small{\rm{14^2=9^2+B^2}}}}\)

Let B = x

\(:{\Longrightarrow{\small{\rm{196=81+x^2}}}}\)

\(:{\Longrightarrow{\small{\rm{196-81=x^2}}}}\)

\(:{\Longrightarrow{\small{\rm{115=x^2}}}}\)

\(:{\Longrightarrow{\small{\rm{\sqrt{115}=x}}}}\)

Answer:

g(x) = 5^3x is a horizontal stretch of a factor of 3. g(x)= 1/3(5)^x is a vertical stretch of a factor of 3. g(x) = 5^1/3x is a vertical compression of a factor of 1/3. g(x) = 3(5)^3 is a horizontal compression of a factor of 1/3.

the equation describes the pattern shown in the table is y = x² + 2x - 3 (option D)

Explanation:

To determine which of the equation describes the pattern shown in the table, we need to insert the values of x and confirm if we will get corresponding value of y in any of the equation.

Let's check the equations in the options: when x = -4, -2

a) y = x² – 11

when x = -4

y = (-4)² - 11

y = 16 - 11 = 5

when x = -2

y = (-2)² - 11

y = 4 - 11 = -7

equation is wrong

b) x² +3x -1

when x = -4

y = (-4)² -3(-4) - 1

y = 16 + 12 -1 = 27

when x = -2

y = (-2)² -3(-2) - 1

y = 4 + 6 -1 = 9

equation is wrong

c) y = x² – 1

when x = -4

y = (-4)² - 1

y = 16 - 1 = 15

equation is wrong

d) y = x² + 2x - 3

when x = -4

y = (-4)² +2(-4) - 3

y = 16 - 8 -3 = 5

when x = -2

y = (-2)² +2(-2) - 3

y = 4 - 4 - 3

y = -3

equation is right

e) y=x²-x-1

when x = -4

y = (-4)² -(-4) - 1

y = 16 +4 -1

y = 19

equation is wrong

Therefore, the equation describes the pattern shown in the table is y = x² + 2x - 3 (option D)

Answer: a= 1 b=-8 c=15

Step-by-step explanation:

A is the number before \(x^{2}\)

B is the number (including the sign) before X

c is the number without a variable

Given:

The highest point on the graph of y =f(x) is (-3, 6)

The highest point of the graph of :

The new function is translated to the right by 1 unit and translated upwards by 5 units.

Applying this to the highest point of the original function y = f(x), we have:

Answer:

(-2, 11)

The ratio of blue paint to yellow paint that Pedro has after adding 2 more containers of each color is (1 + (1/2)x):(1 + (3/4)x).

To create his favorite shade of green paint, Pedro used a ratio of 2 to 3. For every 2 containers of blue paint, there are 3 containers of yellow paint.

Pedro added 2 more containers of each color.

The initial ratio of blue paint to yellow paint is 2:3. This implies that, for every 2 containers of blue paint, there are 3 containers of yellow paint.

Let's suppose that Pedro had x containers of blue paint initially, then he had (3/2)x containers of yellow paint. Therefore, the total quantity of paint he had initially is:

x + (3/2)x = (5/2)x.

Since Pedro added 2 more containers of each color, he now has (2 + 2) = 4 containers of blue paint and (3 + 2) = 5 containers of yellow paint.

Thus, the total quantity of paint he has now is:

4 + 5 = 9.

The question is asking us to find the ratio of blue paint to yellow paint that Pedro has after adding 2 more containers of each color.

Pedro added 2 containers of blue paint, so he now has a total of (2 + x) containers of blue paint.

Similarly, he added 2 containers of yellow paint, so he now has a total of (2 + (3/2)x) containers of yellow paint.

Thus, the ratio of blue paint to yellow paint that Pedro has after adding 2 more containers of each color is:

(2 + x):(2 + (3/2)x).

To get this ratio in its simplest form, we need to divide both sides by 2:

(2 + x):(2 + (3/2)x) = (1 + (1/2)x):(1 + (3/4)x).

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the work that you're required to do has to contain all three requirements in order to earn the maximum number of points, and it also has to be creative (meaning no copying other people's work)

The solutions for the equation 4 sin(2x) = sin(x) are x ≈ 0.4596π, π and 1.539π

From the question, we have the following parameters that can be used in our computation:

4 sin(2x) = sin(x)

Expand sin(2x)

So, we have

4 * 2sin(x)cos(x) = sin(x)

Evaluate the products

8sin(x)cos(x) = sin(x)

Divide both sides by sin(x)

This gives

8cos(x) = 1 and sin(x) = 0

Divide both sides by 8

cos(x) = 1/8 and sin(x) = 0

Take the arc cos & arc sin of both sides

x = cos⁻¹(1/8) and x = sin⁻¹(0)

Using the interval 0 < x < 2π, we have

x ≈ 0.4596 π, π and 1.539 π

Hence, the solutions for the equation are x ≈ 0.4596π, π and 1.539π

The graph is attached

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