I don't understand this please help :)

a. Φ(t)= P.D.[tex]P^{-1}[/tex] is the fundamental matrix of the homogeneous system.

b. [tex]P^{-1}[/tex]= 1/√5 || 1 2 || || -2 1 || is the inverse

c. x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).[tex]x_0[/tex]

d. The solution to the system is x(t)= || e-t/√5 (1-e2t)/2√5 - (2e2t -3)/2√5 || || 1/√5 e-t/√5 +2/√5 (e2t-1)/2√5 ||

In this problem, we will solve the nonhomogeneous system.

Firstly, we will write a fundamental matrix for the associated homogeneous system 1 1 2 0.

a) Fundamental matrix for the homogeneous system

The homogeneous system of equations can be written as:

X'=AX

Here, X is the vector of unknowns, and A is a matrix of constants.

For the given homogeneous system of equations, we have:

1 1 2 0X1' X2'= 1 2 X1X2

X'= AX

Thus, the matrix A can be written as:

A= 1 2 1 0

Now, let us solve the eigenvalue problem for matrix A as follows:

| A - λI |= 0

⇒| 1 - λ 2 || 1 || λ - 0 || 1 || 2 || 0 - λ || 0 |

⇒ (1-λ)(0-λ)-2⋅1=0

⇒ λ2 - λ - 2 = 0

On solving, we get:

λ1 = -1 and λ2 = 2

Thus, the eigenvectors corresponding to these eigenvalues are:

For λ1 = -1, A - (-1)I= 2 2 || x || = 0 1 || y ||

⇒2x+2y = 0y = -2x

Thus, the eigenvector corresponding to λ1 is:

x1= || 1 || and x2 = || -2 ||

The normalized eigenvector corresponding to λ1 is:

x1= || 1/√5 || and x2 = || -2/√5 ||

For λ2 = 2, A - 2I= -1 2 || x || = 0 -2 || y ||

⇒-x+2y = 0y = 1/2x

Thus, the eigenvector corresponding to λ2 is:

x1= || 2 || and x2 = || 1 ||

The normalized eigenvector corresponding to λ2 is:

x1= || 2/√5 || and x2 = || 1/√5 ||

Thus, the matrix P of eigenvectors is:

P= || 1/√5 -2/√5 || || 2/√5 1/√5 ||

Hence, the fundamental matrix of the homogeneous system is:

Φ(t)= P.D.[tex]P^{-1}[/tex]

where D= diagonal matrix of eigenvalues= || -1 0 || || 0 2 ||and [tex]P^{-1}[/tex] is the inverse of matrix P.

We will now find the inverse of matrix P.

b) Computation of inverse of matrix P

The inverse of matrix P is:

[tex]P^{-1}[/tex]= 1/√5 || 1 2 || || -2 1 ||

c) Multiplication by and integration

Multiplying Φ(t) by e

At, we have:e AtΦ(t)= PeDt.

[tex]P^{-1}[/tex].eAt= P.|| e-1t 0 || || 0 e2t ||.[tex]P^{-1}[/tex]

The solution to the system is:

x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).[tex]x_0[/tex]

where b(t) is the vector of inhomogeneous terms, and [tex]x_0[/tex] is the initial condition.

d) Solution to the system

The nonhomogeneous system of equations can be written as:

X'=AX + b

where b= 1 [tex]e^{-t}[/tex]

The solution to the system is:

x(t)= Φ(t)∫ Φ(t)-1.b(t) dt + Φ(t).x0= P.|| e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.P-1 .|| 0 || || 1 ||+ P.|| 1/√5 -2/√5 || || e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.P-1 .|| -1/√5 || || 2/√5 ||= || 1/√5 -2/√5 || || e-t/√5 (1-e2t)/2√5 || || (e2t-3)/2√5 e-t/√5 ||.|| 0 || || 1 ||+ || 1/√5 -2/√5 || || e-t/√5 (e2t -1)/2√5 || || -2(e2t-1)/2√5 e-t/√5 ||.|| -1/√5 || || 2/√5 ||
The final answer is: x(t)= || e-t/√5 (1-e2t)/2√5 - (2e2t -3)/2√5 || || 1/√5 e-t/√5 +2/√5 (e2t-1)/2√5 ||

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For the equation [tex]F(x) = x^2 - 3x + 9[/tex], when x = 2, F(x) = 5. For the equation f(x) = kx + 1, when x > 2, F(x) = kx + 1. For the two equations to be continuous at x = 2, k must equal 4. Yes, f(x) is continuous at x = 2.

For the equation [tex]F(x) = x^2 - 3x + 9[/tex], when x = 2, [tex]F(x) = 2^2 - 3(2) + 9 = 4 + (-6) + 9 = 5.[/tex]

For the equation f(x) = kx + 1, when x > 2, F(x) = kx + 1. We must find the value of k such that F(x) is continuous at x = 2. To do this, we must set F(2) = f(2).

5 = 4k + 1

4k = 4

k = 1

Therefore, for the two equations to be continuous at x = 2, k must equal 1. Yes, f(x) is continuous at x = 2.

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Answer: The point-slope form of an equation of a line is: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

We are given the y-intercept at (0, -3), which means that this point is on the line and that the line crosses the y-axis at y = -3. Therefore, the y-coordinate of the y-intercept is -3.

We are also told that the line passes through the point (4, 5), which means that this point is also on the line. Therefore, x1 = 4 and y1 = 5.

To find the slope of the line, we can use the two points (0, -3) and (4, 5):

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

slope = (5 - (-3)) / (4 - 0)

slope = 8 / 4

slope = 2

Now that we have the slope and a point on the line, we can use the point-slope form to write the equation of the line:

y - y1 = m(x - x1)

y - 5 = 2(x - 4)

This is the equation of the line in point-slope form that has a y-intercept at (0, -3) and also contains the point (4, 5).

Step-by-step explanation:

Answer:

Step-by-step explanation:

smelly a becasue

There is a potential association between hip fracture and active exercise, as a smaller proportion of women with hip fracture reported active exercise compared to the controls

Based on the provided information, we can say that there is a potential association between hip fracture and active exercise. The fact that a smaller proportion of women with hip fracture reported active exercise compared to the controls suggests that active exercise may have a protective effect against hip fracture.

However, it's important to note that this comparison alone does not establish a cause-and-effect relationship between active exercise and hip fracture. Other factors, such as age, bone density, and overall health status, may also play a role in the development of hip fractures.

To further evaluate the relationship between hip fracture and active exercise, additional research would be needed, including studies with larger sample sizes, more detailed information on exercise habits, and adjustments for potential confounding factors.

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A highway superintendent states that four bridges into a city are used in the ratio 2:3:3:4 during the morning rush hour.

A highway study of an SRS of 6000 cars indicates that main answer in 40 words and explanation in 140 words.According to the question, there are four bridges into the city, and they are used in the ratio of 2:3:3:4.

That means that for every 2 cars coming from the first bridge, there are 3 cars coming from the second and the third bridges and 4 cars coming from the fourth bridge.According to the SRS of 6000 cars, the total number of cars from the first bridge is: 2/12 * 6000 = 1000

The total number of cars from the second bridge is: 3/12 * 6000 = 1500The total number of cars from the third bridge is: 3/12 * 6000 = 1500The total number of cars from the fourth bridge is: 4/12 * 6000 = 2000

Therefore, the number of cars coming from the first, second, third, and fourth bridges are 1000, 1500, 1500, and 2000, respectively, during the morning rush hour.

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Check the picture below.

The missing value in the table for x = –1 is Y=6 is the value that x=-1 lacks.

An equation is a mathematical statement that two expressions are equal. It consists of two expressions that contain numbers, variables, and/or operators, and is written in a symbolic form. Equations are used to explain a variety of real-world phenomena and also used to describe the behavior of physical systems.An equation can be used to solve for a single unknown variable or to find relationships between multiple variables.

How  to resolve the equation?

The equation's unknown value can be obtained by inserting known values for them.

Provided is the equation y=2-4x.

discover y when x=-1

Replace x=-1 in the given equation,

We obtain y=2-4 (-1)

y=6

Therefore, the value of y at x=-1 is 6.

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Complete questions as follows-

The table represents the equation y = 2 – 4x The x-values are the The y-values are the The missing value in the table for x = –1 is y =

The equivalent expressions for the expression 9x + 5(6/5x+3)-30 are (15x-15) and 15(x-1).

Equivalent expressions are mathematical expressions that have the same value, even though they may look different.

Here are some examples:

2x + 4 and 4 + 2x are equivalent expressions.

(Proof: 2x + 4 = 4 + 2x for any value of x)

3(x + 2) and 3x + 6 are equivalent expressions.

(Proof: Distributing the 3, we get 3(x + 2) = 3x + 6)

Here we have

9x + 5(6/5x+3)-30

The above expression can be simplified as follows

=> 9x + 5(6/5x) +5(3) -30      [ Multiply 5 with (6/5x+3) ]

=> 9x + (6x) + 15 -30    

=> 15x - 15                              [ Add like terms ]

=> 15(x - 1)                              [ take 15 as common ]

Hence,

The equivalent expressions for the expression 9x + 5(6/5x+3)-30 are (15x-15) and 15(x-1).

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The center of the specified circle is at (-7, 4). Thus, the updated center will be (-7 - 3, 4 - 4) = (-10, 0).

In order to move the circle 3 units to the right and 4 units below, we must subtract 3 from x and 4 from y. Thus, the updated center will be (-7 - 3, 4 - 4) = (-10, 0).

The equation for the resulting circle can be discovered by adding the new center and simplifying. The standard version of the circle's equation is [tex](x - h)^2 + (y - k)^2 = r^2[/tex], with (h, k) denoting the circle's center and r denoting the radius. We can use the new center (-10, 0) and the original equation [tex](x+7)^2+(y-4)^2=36[/tex]to replace the original equation.

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In conclusion the value of a3 is 166.

How to solve and what does recursive mean?

We can use the given recursive formula to find the values of the sequence:

a1 = 4

a2 = (a1)²2 - 3 = 13

a3 = (a2)²2 - 3 = (13)²2 - 3 = 166

Therefore, the value of a3 is 166.

Recursive refers to a process or function that calls itself repeatedly until a specific condition is met. In mathematics, a recursive sequence or formula is a sequence that is defined in terms of its previous terms, such that the value of each term depends on the value of the previous term(s).

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The center of dilation, point C, is located at (4, 3/2). The center of dilation, point C, is the point of intersection of the lines joining each vertex of the original triangle to the corresponding vertex of the dilated triangle.

Since triangle XYZ is dilated by a scale factor of 0.5, each point of the dilated triangle is located halfway between the corresponding point of the original triangle and the center of dilation, point C.

Therefore, to find the location of point C, we need to find the midpoint of each line segment joining the corresponding vertices of the original and dilated triangles.

Plotting the vertices of the original triangle at points X(2, 4), Y(6, 2), and Z(3, 5) on the coordinate plane, we can use the midpoint formula to find the midpoint of each segment.

Midpoint of segment XX' = (2 + 6)/2, (4 + 2)/2 = (4, 3)

Midpoint of segment YY' = (6 + 18)/2, (2 + 4)/2 = (12, 3)

Midpoint of segment ZZ' = (3 + 9/2)/2, (5 + 7/2)/2 = (15/4, 11/4)

The point of intersection of the lines joining the original vertices to the dilated vertices is point C(4, 3/2).

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Step-by-step explanation:

x = 45000 - (Discount % * 45000/100)

x = 45000 - (Discount % * 450)

Given that the selling price after adding 13% VAT is Rs. 40680, we can write:

x + (13% * x/100) = 40680

Simplifying the above equation, we get:

x = 36000

Therefore, the selling price of the TV without VAT will be x, which is Rs. 36000.

So the final answer is that the selling price of the TV without VAT is Rs. 36000

please mark brainliest

Answer:

the answer for this question is A

The estimated total population size of stickleback fish in the lake is approximately 530 individuals.

The mark and recapture method can be used to estimate the total population size of a species in a particular area. Based on the information provided, we can use the Lincoln-Petersen method to estimate the population size.

The formula for the Lincoln-Petersen method is

N = (M x R) / C

Where

N = total population size

M = number of individuals in the first sample (marked individuals)

R = number of individuals in the second sample (recaptured individuals)

C = number of marked individuals recaptured in the second sample

Using the given information, we can plug in the values to the formula

N = (255 x 162) / 78

N = 529.62

N = 530

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Answer:

4 if x < 3

Step-by-step explanation:

First, find the equation of the line. From the graph we can tell the y-intercept is 4 and the slope is 0. Using y=mx+b, we find that y=(0)x+4 (same as y=4). f(x) is the same as y, so all you have to write is 4.

To find the domain, you must look at where the function is defined. It is only defined to the left of the point (3,4), where x < 3. We know its only defined there because the arrow and line only go in that direction. We know that it should be a < not a ≤ because the point at (3,4) is hollow, so (3,4) is not a point in the function and not included in the domain.

Answer:

An even function is symmetric about the y-axis. An odd function is symmetric about the origin. As the graphed function is neither symmetric about the origin, nor symmetric about the y-axis, it is neither even nor odd.

Step-by-step explanation:

Even and odd functions are special types of functions.

An even function is symmetric about the y-axis. An odd function is symmetric about the origin. As the graphed function is neither symmetric about the origin, nor symmetric about the y-axis, it is neither even nor odd.

Let's first calculate the probability of getting the question right if you eliminate one of the six choices. Since there are six options, initially you have a 1/6 chance of getting the question right. If you eliminate one of the options, you are left with five choices, so your probability of getting the question right increases to 1/5.

Now, let's calculate the expected value of the number of points you get. If you get the question right, you earn one point. If you get the question wrong, you lose 1/3 point. So, we can write the expected value as:

Expected value = Probability of getting the question right x Points for a correct answer + Probability of getting the question wrong x Points for a wrong answer

Let's plug in the values:

Expected value = (1/5) x 1 + (4/5) x (-1/3)

Expected value = 1/5 - 4/15

Expected value = -1/15

So, the expected value of the number of points you get is -1/15. This means that, on average, you can expect to lose about 0.067 points for each question you attempt.

When x = 3, y = 60 + 20(3) = 120. So, we can plot the point (3, 120).

An equation is a mathematical statement that uses symbols, numbers, and operations to show that two expressions are equal. It usually contains one or more variables, which represent unknown values that need to be solved for. Equations are used in a wide range of mathematical fields, such as algebra, geometry, calculus, and physics, to model real-world situations and solve problems. Some common examples of equations include:

Linear equation: y = mx + b

he total expenses for Myra's phone plan include the initial cost of buying the phone plus the monthly cost of the phone plan. Therefore, the total cost can be represented by the equation:

y = 60 + 20x

where y is the total cost and x is the number of months on the phone plan.

To graph this equation, we can plot a point at (0, 60) to represent the initial cost of buying the phone. Then, we can plot additional points by plugging in different values of x into the equation and finding the corresponding values of y. For example:

When x = 1, y = 60 + 20(1) = 80. So, we can plot the point (1, 80).

When x = 2, y = 60 + 20(2) = 100. So, we can plot the point (2, 100).

When x = 3, y = 60 + 20(3) = 120. So, we can plot the point (3, 120).

Connecting these points gives us a straight line, since the equation is linear.

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The value of x in the diagram given in the question above is 61°

The value of x can be obtained as illustrated below:

(x - 11)° + (2x + 8)° = 180° (angle on a straight line)

Clear the brackets

x - 11 + 2x + 8 = 180°

Rearranging the above, we have:

x + 2x - 11 + 8 = 180

3x - 3 = 180

Collect like terms

3x = 180 + 3

3x = 183

Divide both sides by 3

x = 183 / 3

x = 61°

Thus, from the above calculation, we can conclude that the value of x is 61°

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Answer:

60 and 100 degrees.

Step-by-step explanation:

Sum of the other 2 angles = 180-20 = 160 degrees.

3 + 5 = 8

160 / 8 = 20

- so the other 2 angles are:

3*20 = 60 and

5*20 = 100 degrees.

Rounding to the nearest inch, the length of side W (w) is approximately 588 inches.

Here, we have,

To find the length of side W (w) in triangle WXY, we can use the Law of Sines, which states:

a/sin(A) = b/sin(B) = c/sin(C)

where:

a, b, c are the side lengths of the triangle.

A, B, C are the opposite angles to sides a, b, c, respectively.

In this case, we have:

Side XY = y = 940 inches

Angle Y = ∠Y = 100°

Angle W = ∠W = 38°

We want to find the length of side W (w).

Using the Law of Sines:

w/sin(∠W) = y/sin(∠Y)

Now, substitute the given values:

w/sin(38°) = 940/sin(100°)

Now, solve for w:

w = (940 * sin(38°)) / sin(100°)

Using a calculator:

w ≈ (940 * 0.6157) / 0.9848

w ≈ 588.388

Rounding to the nearest inch, the length of side W (w) is approximately 588 inches.

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The value of the 7th term of the geometric progression is 223.9488

Based on the information given,

First term = 75

Second term = 90

Common ratio = 90/75 = 1.2

The nth term of a geometric progression is calculated as:

[tex]a_n=ar^{r-1}[/tex]

The 7th term will then be:

[tex]= 75 \times (1.2)^6[/tex]

[tex]=75\times2.985984[/tex]

[tex]=223.9488[/tex]

Therefore, the value of the 7th term of the geometric progression is 223.9488.

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Answer:

Step-by-step explanation:

If the mother wants to get the amount of ice cream bars that will not contain any leftovers for her family of four, the least amount of boxes the mother could purchase to receive her goal is 2 boxes

MATH-

1 box= 6 bars

2 boxes= the bars(6)x2, which is equal to 12 ice cream bars

If you split 12 evenly between the members of her family which is 4 people,

12÷4=3 or 4x3=12

each family member would receive 3 ice cream bars each.

1. The perimeter is 41.16 cm

2. The perimeter is 165.7 cm

Perimeter is a term used in geometry to describe the total length of the boundary or the distance around the edge of a two-dimensional shape.

1. Here, radius = 4cm,

Circumference of quarter circle = [tex]\frac{1}{4}[/tex] ×2πr = [tex]\frac{1}{4}[/tex] ×2×3.14×4 = 6.29 cm

Circumference of four quarters circle= 4×6.29 = 25.16 cm,

and 4 extra portion 4×4 = 16 cm

So, the Perimeter = 25.16 + 16 = 41.16 cm

2. The diameter = 40 cm then, Radius= 20 cm

here, 2 quarter circle and 1 half circle is equal to 1 complete circle.

Then circumference of circle equal to 2π r = 2×3.14×20 = 125.71,

So the perimeter = 125.71 + 20 + 20 = 165.7 cm

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Answer : The answer is B= -12

The price rose by [tex]2.5\%[/tex]

A percentage is a number or ratio that represents a fraction of 100.

The change in price is

[tex]2.90 - 2.83 = 0.07[/tex]

The increase percentage is

[tex]\text{p}=\dfrac{0.07\times100}{2.83}[/tex]

[tex]\text{p}=2.473[/tex]

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Answer:

x < -7

Step-by-step explanation:

3(x-4)-2>5x

3x - 12 - 2 > 5x

3x - 14 > 5x

-2x > 14

x < -7

Answer: x<-7

Step-by-step explanation: 3(x-4)= 3x-12

3x-14>5x

5x-3x=2x

-14>2x

-14/2=-7

-7>x

The equation would be: ΔT = 2 - (-12) = 14 degrees Celsius, ΔT = T2 - T1, where ΔT is the change in temperature, T2 is the final temperature (-12 degrees Celsius)

A. d = rt, where d = 12 meters, r = 2 meters per second (rate), and t is the time it takes for the whale to get to an elevation of 12 meters. The equation would be:

12 = 2t

B. r = d/t, where r is the rate of diving, d = 12 meters (depth), and t = 2 minutes (time). The equation would be:

r = 12/(2 x 60) = 0.1 meters per second

C. ΔT = T2 - T1, where ΔT is the change in temperature, T2 is the final temperature (2 degrees Celsius), and T1 is the initial temperature (-12 degrees Celsius). The equation would be:

ΔT = 2 - (-12) = 14 degrees Celsius

D. ΔT = T2 - T1, where ΔT is the change in temperature, T2 is the final temperature (-12 degrees Celsius), and T1 is the initial temperature (2 degrees Celsius). The equation would be:

ΔT = -12 - 2 = -14 degrees Celsius

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