Simplify this expression

The solution to the system of equations is[tex]X(t) = c_1 * e^{6t} * [37 \ \ -3] + c_2 * e^{-3t} * [-37\ \ 12][/tex]. The solution curves exhibit a combination of exponential growth and decay, and as t approaches infinity, they converge towards the eigenvector associated with the negative eigenvalue.

To find the general solution to the linear system of differential equations:

[tex]X' = \left[\begin{array}{ccc}9&37\\-1&-3\end{array}\right] X[/tex]

We need to find the eigenvalues and eigenvectors of the coefficient matrix [[9   37] [-1   -3]].

Let A be the coefficient matrix.

The characteristic equation is given by:

det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.

The coefficient matrix A - λI is:

[tex]X' = \left[\begin{array}{ccc}9-\lambda&37\\-1&-3-\lambda\end{array}\right] X[/tex]

Setting the determinant equal to zero:

[tex]det (\left[\begin{array}{ccc}9-\lambda&37\\-1&-3-\lambda\end{array}\right] )[/tex]

Expanding the determinant, we get:

[tex](9-\lambda)(-3-\lambda) - (-1)(37) = 0[/tex]

Simplifying the equation, we have:

[tex](\lambda-6)(\lambda+3) = 0[/tex]

Solving for λ, we find two eigenvalues:

[tex]\lambda_1 = 6\\\lambda_2 = -3[/tex]

Next, we find the eigenvectors corresponding to each eigenvalue.

For [tex]\lambda_1 = 6[/tex]:

[tex](A - \lambda_1I)v_1 = 0[/tex]

Substituting the values, we have:

[tex]\left[\begin{array}{ccc}3&37\\-1&-9\end{array}\right] v_1 = 0[/tex]

Solving the system of equations, we find v1 = [37 -3].

For [tex]\lambda_2 = -3[/tex]:

[tex](A - \lambda_2I)v_2 = 0[/tex]

Substituting the values, we have:

[tex]\left[\begin{array}{ccc}12&37\\-1&0\end{array}\right] v_2 = 0[/tex]

Solving the system of equations, we find [tex]v_2[/tex] = [-37 12].

Therefore, the general solution to the linear system of differential equations is:

[tex]X(t) = c_1 * e^{6t} * [37 \ \ -3] + c_2 * e^{-3t} * [-37\ \ 12][/tex]

where [tex]c_1\ and\ c_2[/tex] are constants.

b) The solution curves to this linear system represent trajectories in the state space. The behavior of the solution curves depends on the eigenvalues.

Since we have [tex]\lambda_1 = 6[/tex] and [tex]\lambda_2 = -3[/tex], the system has one positive eigenvalue and one negative eigenvalue. This indicates that the solution curves will exhibit a combination of exponential growth and decay.

As t approaches infinity, the exponential term with [tex]e^{-3t}[/tex] will dominate, and the solution curves will converge towards the eigenvector associated with the negative eigenvalue, [-37   12].

On the other hand, as t approaches negative infinity, the exponential term with [tex]e^{6t}[/tex] will dominate, and the solution curves will diverge away from the origin in the direction of the eigenvector associated with the positive eigenvalue, [37   -3].

In summary, the solution curves will either converge or diverge depending on the initial conditions, and as t approaches infinity, they will converge towards the eigenvector associated with the negative eigenvalue.

Complete Question:

a) Find the metal solution to the linear system of differential equations

[tex]X' = \left[\begin{array}{ccc}9&37\\-1&-3\end{array}\right] X[/tex]

b) Give a physical description of what the solution curves to this linear system look like. What happens to the solution curves as to 12?

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Answer:  17; 13; 9; 5; 1

Step-by-step explanation:

y = -2x + 9

When x = -4, y = 17

When x = -2 , y = -2x + 9 = -2(-2) + 9 = 13

When x = 0 , y = -2x + 9 = -2(0) + 9 = 9

When x = 2 , y = -2x + 9 = -2(2) + 9 = 5

When x = 4 , y = -2x + 9 = -2(4) + 9 = 1

So I'll walk through the first 2. Then please try the other ones and let me know if you run into more problems.

5m-6

If m is equal to 7, then replace m with 7.

5(7)-6

5x7=35 so the expression is now 35-6. Solve.

35-6=29

So that was the first one.

4m+t

We just do what we do with the first problem.

4(7)+(2)

4x7=28 so the expression is 28+2.

28+2=28

---

hope it helps

p.s. When there is subtraction, add. When there is division, multiply to cancel that out. When there is multiplication, divide to cancel. etc.

Answer:

1,5(7)-6=29

2,4(7)+2=30

3,8/2=4

4,7*2=14

5,5(2)+2(7)=24

6,8*7=56

7,3(7)-5(2)=11

8,7*8/2=28

Given:

Three line segments have measures of 4 units, 6 units, and 8 units.

To find:

Will the segments form a triangle?

Solution:

We know that three line segments can form a triangle if the sum of two smaller sides is greater than the largest side.

Three line segments have measures of 4 units, 6 units, and 8 units. Here, the measure of the largest sides is 8 units.

The sum of two smaller sides is

[tex]4+6=10[/tex]

[tex]4+6>8[/tex]

Since the sum of two smaller sides is greater than the largest side, therefore the segments will form a triangle.

Answer:

hi

Step-by-step explanation:

Answer:

Step-by-step explanation:0.828282 = 0.828282/1 = 8.28282/10 = 82.8282/100 = 828.282/1000 = 8282.82/10000 = 82828.2/100000 = 828282/1000000

And finally we have:

0.828282 as a fraction equals 828282/1000000

Answer:

∠ B = 48°

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

Subtract the sum of the 2 given angles from 180° for ∠ B

∠ B = 180° - (90 + 42)° = 180° - 132° = 48°

Answer:

$7.35

Step-by-step explanation:

We have 200 shirts

Cost $2.80 each

plus a 5% charge for shipping.

Selling price

150% markup

Recall when we use percentages, we need to express them as decimals.

Michelle's cost for each shirt was

2.80 (1.05) = 2.94

The markup is

2.94 (1.50) = 4.41

The markup is added to the cost to get the final selling price.

2.94 + 4.41 = $7.35

Michelle will sell each shirt for $7.35

Answer:

its B

Step-by-step explanation: Cause I have big brain. >:)

Answer:

c

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

Answer:

64 = 0.72 (x)

Step-by-step explanation:

Answer: 64 = 0.72 (x)

Step-by-step explanation:

Answer:

1) y = -7/10x + 21

Step-by-step explanation:

1)

Points (5, 7) and (15, 0)

Slope:

m=(y2-y1)/(x2-x1)

m=(0-7)/(15-5)

m=(-7)/10

m= -7/10

Slope-intercept:

y - y1 = m(x - x1)

y - 7 = -7/10(x - 5)

y - 7 = -7/10x + 14

y = -7/10x + 21

[tex] \purple{ \tt{ \huge{ \: ✨Answer ✨ \: }}}[/tex]

[tex] \: \: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \boxed{ \tt{ \huge{{ \: 32 \: }}}}}}[/tex]

Hypotheses are always statements about the d. population parameters

Hypotheses are assertions or claims concerning population characteristics stated in statistics. An attribute or value of a population, such the population mean or percentage, is referred to as a population parameter. Based on sample data, hypothese are developed to draw conclusions or inferences about these population attributes.

The hypothesis can be expressed as comparisons of population metrics or as statements of equality or inequality. They serve as a basis for statistical studies and are used to examine certain assertions or research hypotheses. Finding pertinent solutions to the scientific inquiry is the main goal of the hypothesis. It is supported by a few evidences, and experimental methods are used to test the whole statement of the hypothesis.

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Complete Question:

Hypotheses are always statements about which of the following?  choose the correct answer below.

a. sample statistics

b. sample size

c. estimators

d. population parameters

The optimum solution for the maximization problem is Z = 0 at the point (0, 0). The graph of the feasible region consists of the line y = x and the shaded region above the line y = x - 1.

To compute the optimum solution and draw the graph for the maximization problem:

Maximize Z = x + y

Subject to the constraints:

x - y ≤ 1

-x + y = 0

x, y ≥ 0

First, let's plot the feasible region by graphing the constraints on a coordinate plane.

For the constraint x - y ≤ 1, we can rewrite it as y ≥ x - 1. This represents a boundary line with a slope of 1 and a y-intercept of -1. Shade the region above this line to satisfy the constraint.

For the constraint -x + y = 0, rewrite it as y = x. This represents a line passing through the origin with a slope of 1. Plot this line.

Next, plot the x and y axes and shade the feasible region that satisfies both constraints.

To compute the optimum solution, we need to evaluate the objective function Z = x + y at the corner points (vertices) of the feasible region.

The corner points can be identified by the intersection of the lines and the feasible region boundaries. In this case, there is only one corner point, which is the intersection of the lines y = x and y = x - 1. Solving these equations simultaneously gives x = 0 and y = 0.

Thus, the optimum solution occurs at the corner point (0, 0), where Z = 0 + 0 = 0.

Therefore, the optimum solution is Z = 0 at the point (0, 0).

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Answer:

area = 8167.14 units ²

Step-by-step explanation:

area = πr²

r = 102/2 = 51

area = (3.14)(51²) = 8167.14 units ²

the answer is 12.57

Step-by-step explanation:

The formula to find circumference is 2×pi×r. Two is the radius. So when you plug it into the formula you get 12.57.

Step-by-step explanation:

[tex]i \: \: think \: \: it \: \: is \: \\ \\ {a}^{2} - 2ab + {b}^{2} \\ \\ that \: is \: for \: \: {(a - b)}^{2} [/tex]

I hope that is useful for you :)

Answer:

17

Step-by-step explanation:

Hello There!

Once again we are going to use the distance formula to find the answer

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

This time we need to find the distance between the points (4,10) and (4,-7)

*we plug in the values into the formula)

[tex]d=\sqrt{(4-4)^2+(-7-10)^2} \\4-4=0\\-7-10=17\\d=\sqrt{0^2+(-17)^2} \\0^2=0\\-17^2=289\\\sqrt{289} =17[/tex]

so we can conclude that the distance between the points (4,10) and (4,-7) is 17 units

Answer:  210 mm²

Step-by-step explanation:

A = 1/2(long base + short base) x height

A = 1/2(18 + 10)(15)

A = 1/2(28)(15)

A = 210 mm²

Answer:

79.7°

Step-by-step explanation:

We resolve the speed of the plane into horizontal and vertical components respectively as 300cos75° and 300sin75° respectively. Since the wind blows due west at a speed of 25 miles per hour, its direction is horizontal and is given by 25cos180° = -25 mph.  We now add both horizontal components to get the resultant horizontal component of the airplane's speed.

So 300cos75° mph + (-25 mph) = 77.646 - 25 = 52.646 mph.

The vertical component of its speed is 300sin75° since that's the only horizontal motion of the airplane. So the resultant vertical component of the airplane's speed is 300sin75° = 289.778 mph

The direction of the plane, Ф = tan⁻¹(vertical component of speed/horizontal component of speed)

Ф = tan⁻¹(289.778 mph/52.646 mph)

Ф = tan⁻¹(5.5043)

Ф = 79.7°

14: a

1

=

39

/2                              0.25               313%        16%

15:     54

Solution:

Outfit         Shirt       Slacks        Tie

Outfit 1      Blue        Black         Red

Outfit 2     Blue        Grey          Red

Outfit 3    White       Black         Red

Outfit 4    White       Grey          Red

Outfit 5    Black       Black         Red

Outfit 6    Black       Grey         Red

We take the outfits 1, 2 , 5  and 6 as a subset of the sample space.

So these 1, 2, 5 and 6 consists either a blue shirt or a black shirt.

The subset consists of all the outfits that do not have a white shirt.

So the correct options are :

1. (Choice A)

The subset consists of all the outfits that do not have a white shirt.

2. (Choice C)

The subset consists of all the outfits that have a black shirt.

The code creates a scatter plot of the "gamble" variable on the "income" variable, with different plotting symbols based on sex, using the "faraway" package in R.

Here's the code to make a plot of the "gamble" variable on the "income" variable using different plotting symbols based on the sex in R:

# Load the required package and data

library(faraway)

data(teengamb)

# Create a plot of a gamble on income with different symbols for each sex

plot(income ~ gamble, data = teengamb, pch = ifelse(sex == "M", 16, 17),

    xlab = "Gamble", ylab = "Income", main = "Gamble on Income by Sex")

legend("topleft", legend = c("Female", "Male"), pch = c(17, 16), bty = "n")

This code will create a scatter plot where the "income" variable is plotted against the "gamble" variable. The plotting symbols used will be different depending on the "sex" variable.

Females will be represented by an open circle (pch = 17), and males will be represented by a closed circle (pch = 16). The legend will indicate the corresponding symbols for each sex.

Make sure to have the "faraway" package installed in R and load it using 'library'(faraway) before running this code.

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Answer: 1.3

Step-by-step explanation:

The radius of the circle that python forms is 1.3 meters.

Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.

Given that,

Length of python = 2.6π meters

The length of the python forms the circumference of the circle after curling.

Circumference of a circle = 2π r, where r is the radius.

2π r = 2.6π

2r = 2.6

r = 2.6 / 2

r = 1.3

Hence 1.3 meters is the radius of the circle.

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Answer:

(a) Similar polygons; scale factor is 2

(b) Similar polygons; scale factor is 1.5

Step-by-step explanation:

Given

See attachment for polygons

Required

Determine if they are similar or not

Solving (a): The triangle

The angles in both triangles show that the triangles are similar

To calculate the scale factor (k), we simply take corresponding sides.

i.e.

[tex]k = \frac{DF}{BA} = \frac{FE}{AC} = \frac{DE}{BC}[/tex]

[tex]k = \frac{6}{3} = \frac{8}{4} = \frac{10}{5}[/tex]

[tex]k = 2=2=2[/tex]

[tex]k = 2[/tex]

The scale factor is 2

Solving (b): The trapezium

The angles in both trapeziums show that the trapeziums are similar

To calculate the scale factor (k), we simply take corresponding sides.

i.e.

[tex]k = \frac{KN}{GJ}[/tex]

[tex]k = \frac{6}{4}[/tex]

[tex]k = 1.5[/tex]

The scale factor is 1.5

Damien's regular pay per payment period is $1,062.50, his hourly rate of pay is $26.87, his overtime rate of pay is $53.74, and his gross pay for a pay period in which he worked 8 hours overtime is $1,492.42.

a) Calculation of Damien's regular pay per payment period: Given, Damien receives an annual salary of $55,300.Damien is paid weekly and his regular workweek is 39.5 hours. Therefore, the regular pay per payment period = 55,300/52 = $1,062.50So, Damien's regular pay per payment period is $1,062.50.

b) Calculation of Damien's hourly rate of pay: Let's calculate the hourly rate of pay for Damien, we will divide the regular pay per payment period by the regular workweek hours. Hourly rate of pay = 1,062.50/39.5 = $26.87Thus, Damien's hourly rate of pay is $26.87.

c) Calculation of Damien's overtime rate of pay: The overtime rate of pay will be double the hourly rate of pay. Hence, Damien's overtime rate of pay will be: Double the hourly rate of pay = 2 × 26.87 = $53.74. Therefore, Damien's overtime rate of pay is $53.74.

d) Calculation of Damien's gross pay for a pay period in which he worked 8 hours overtime at double regular pay: Damien worked 8 hours overtime, so his gross pay for the pay period will be: Regular pay = 39.5 hours × $26.87 per hour = $1,062.50. Overtime pay = 8 hours × $53.74 per hour = $429.92Gross pay = Regular pay + Overtime pay= 1,062.50 + 429.92= $1,492.42Therefore, Damien's gross pay for a pay period in which he worked 8 hours overtime at double the regular pay is $1,492.42.

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