The integral ſ sin(x - 2) dx is transformed into L 9(t)dt by applying an appropriate change of variable, then g(t) is: g(t) = sin )= g(t) = cos This option This option g(t) = cos (3) g(t) = sin This option This option

Answer

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

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°

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14: a

1

=

39

/2                              0.25               313%        16%

15:     54

Answer:

hi

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

Answer:

Step-by-step explanation:

1

Answer:

56

Step-by-step explanation:

i just is smart

yhhhhhhhhhhh

Answer:

c

Step-by-step explanation:

Answer:

d

Step-by-step explanation:

Note that

> means greater than

< means less than

for example : 2 < 3 means 2 is less than 3

3 >2 means 3 is greater than 2

To determine which comparison is better we have to convert litre to millimetre

1 litre = 1000mm

A 5 mm > (50 x 1000)

5 mm > 50,000

5 is not greater than 50,000

B. (2 x 1000) < 200 mm

2000 < 200

2000 is not less than 200

C. (100 x 1000) < 1,000

100,000 < 1000

100,000 is not less than 1000

D. 3200 > (3 X 1000)

3200 > 3000

3200 is greater than 3000

Answer:

12/13

Step-by-step explanation:

Given that MN = 5, NO = 12, and MO = 13, find cos O.

Since the reference angle is P, hence;

MN is the opposite = 5

MO is the hypotenuse = 13 (longest side)

NO is the adjacent = 12

Cos O = adj/hyp

Substitute the given values

Cos O = 12/13

Hence the value of Cos O is 12/13

(a) The probability of a computer failure from Company A is 0.001; from Company B is 0.002; and from Company C is 0.005.

Therefore, the probability that a computer will experience a hard drive failure within one year is:(0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005)= 0.0012. The probability of a randomly selected computer experiencing a hard drive failure within one year is 0.0012 or 0.12%.
(b) Bayes' theorem will be used to calculate this probability:Let A be the event that the computer's hard drive was manufactured by Company C. Let B be the event that the computer experienced a hard drive failure. P(A|B) is the probability that the hard drive was manufactured by Company C given that a hard drive failure was experienced.

P(A|B) = P(B|A) P(A) / P(B) Where: P(B|A) = 0.005 (the probability of failure if the hard drive was manufactured by Company C)P(A) = 0.20 (the proportion of hard drives that the computer manufacturer gets from Company C)P(B) = (0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005) = 0.0012 (as in part a)

Therefore: P(A|B) = (0.005 x 0.20) / 0.0012 = 0.0833 or 8.33%.
(c)Let A be the event that both hard drives were manufactured by Company A; B be the event that both hard drives were manufactured by Company B; and C be the event that both hard drives were manufactured by Company C. Then we need to find the probability of event A or B or C, given that a hard drive failure was experienced:P(A U B U C|F) = P(F|A U B U C) P(A U B U C) / P(F)where F is the event that the hard drive in the replacement computer fails.P(F|A U B U C) = P(F) = (0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005) = 0.0012P(A U B U C) = (0.50)^2 + (0.30)^2 + (0.20)^2 = 0.46P(F) = P(A U B U C) P(F|A U B U C) + P(A' n B n C) P(F|A' n B n C)= 0.46 x 0.0012 + 0.04 x 0.3 = 0.000552P(A U B U C|F) = P(F|A U B U C) P(A U B U C) / P(F)= (0.0012 x 0.46) / 0.000552 = 1.00or 100%. Therefore, the probability that the original and replacement computers were produced by the same company is 100%.
(d) Bayes' theorem will be used to calculate this probability:Let A be the event that the hard drive was manufactured by Company C. Let B be the event that the computer did not experience a hard drive failure. P(A|B) is the probability that the hard drive was manufactured by Company C given that no hard drive failure was experienced.P(A|B) = P(B|A) P(A) / P(B)Where:P(B|A) = 1 - 0.005 = 0.995 (the probability that the hard drive did not fail if it was manufactured by Company C)P(A) = 0.20 (as in part b)P(B) = 1 - (0.50 x 0.001) - (0.30 x 0.002) - (0.20 x 0.005) = 0.9988

Therefore:P(A|B) = (0.995 x 0.20) / 0.9988 = 0.1989 or 19.89%. Therefore, the probability that the hard drive was manufactured by Company C given that it did not fail is 19.89%.

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The probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year is approximately 0.256.

(a)Probability that a randomly chosen computer purchased from this manufacturer will experience a hard drive failure within one year = 0.5 x 0.001 + 0.3 x 0.002 + 0.2 x 0.005 = 0.0016

(b)Let's denote the event that a computer failure is experienced within one year by F and the event that the hard drive is made by company C by C.

Then we are required to calculate P(C | F), which is the probability that the hard drive was manufactured by company C given that a failure was experienced by the computer within one year. This can be found by using the Bayes' rule as follows:

[tex]$$P(C|F) = \frac{P(F|C)P(C)}{P(F|A)P(A) + P(F|B)P(B) + P(F|C)P(C)}$$[/tex]
where P(C) = 0.2, P(A) = 0.5 and P(B) = 0.3.$$P(F|A) = 0.001, P(F|B) = 0.002, P(F|C) = 0.005$$

Thus, we have:[tex]$$P(C|F) = \frac{0.005 \times 0.2}{0.001 \times 0.5 + 0.002 \times 0.3 + 0.005 \times 0.2} \approx 0.476$$[/tex]

Therefore, the probability that the hard drive was manufactured by company C given that a failure was experienced by the computer within one year is approximately 0.476.

(c)Let's denote the event that the original hard drive is manufactured by company A, B and C by A, B, and C respectively.

Similarly, let's denote the event that the replacement hard drive is manufactured by company A, B, and C by A', B', and C' respectively.

We are required to calculate P(A = A', B = B', C = C' | F), which is the probability that the hard drives in the original and replacement computers were manufactured by the same company given that a failure was experienced by both computers within one year.

This can be found by using the Bayes' rule as follows:

[tex]$$P(A = A', B = B', C = C'|F) = \frac{P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C')}{P(F)}$$[/tex]

where: [tex]$$P(F) = P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C') + P(F|A \ne A', B \ne B', C \ne C')P(A \ne A')P(B \ne B')P(C \ne C')$$[/tex]

Here, we are assuming that the probabilities of computer failure are independent of each other and the company that manufactured the hard drives of the two computers are independent of each other. Therefore, we have:

[tex]$$P(F|A = A', B = B', C = C') = P(F|A)P(F|B)P(F|C) = 0.001 \times 0.002 \times 0.005$$[/tex]

[tex]$$P(F|A \ne A', B \ne B', C \ne C') = 0$$[/tex]

Also, we have:$$P(A = A') = P(B = B') = P(C = C') = \frac{1}{3}$$

[tex]$$P(A \ne A', B \ne B', C \ne C') = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}$$[/tex]

Thus, we have:$$P(A = A', B = B', C = C'|F) = \frac{0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3}{P(F)}$$

[tex]$$P(A \ne A', B \ne B', C \ne C'|F) = \frac{P(F) - 0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3}{\frac{8}{27}}$$[/tex]

Now, we need to find P(F). This can be done as follows:

[tex]$$P(F) = P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C') + P(F|A \ne A', B \ne B', C \ne C')P(A \ne A')P(B \ne B')P(C \ne C')$$$$= 0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3 + 0 = 4.6296 \times 10^{-8}$$Thus, we have:$$P(A = A', B = B', C = C'|F) = 0.0296$$[/tex]

[tex]$$P(A \ne A', B \ne B', C \ne C'|F) = 0.9704$$[/tex]

Therefore, the probability that the hard drives in the original and replacement computers were manufactured by the same company given that a failure was experienced by both computers within one year is 0.0296.(d)Let's denote the event that the hard drive is made by company C by C and the event that a computer failure is not experienced within one year by F'. We are required to calculate P(C | F'), which is the probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year. This can be found by using the Bayes' rule as follows:

[tex]$$P(C|F') = \frac{P(F'|C)P(C)}{P(F'|A)P(A) + P(F'|B)P(B) + P(F'|C)P(C)}$$[/tex]

where P(C) = 0.2, P(A) = 0.5 and P(B) = 0.3.$$P(F'|A) = 0.999, P(F'|B) = 0.998, P(F'|C) = 0.995$$

Thus, we have: [tex]$$P(C|F') = \frac{0.995 \times 0.2}{0.999 \times 0.5 + 0.998 \times 0.3 + 0.995 \times 0.2} \approx 0.256$$[/tex]

Therefore, the probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year is approximately 0.256.

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Given:

The end points of the line segment AB are A(-2,-3) and B(4,-1).

The rule of translation is:

[tex](x,y)\to (x+4,y-3)[/tex]

To find:

The coordinates of the end points of the line segment A'B'.

Solution:

It is given that the end points of the line segment AB are A(-2,-3) and B(4,-1).

We have,

[tex](x,y)\to (x+4,y-3)[/tex]

By using the above translation rule, we get

[tex]A(-2,-3)\to A'(-2+4,-3-3)[/tex]

[tex]A(-2,-3)\to A'(2,-6)[/tex]

And

[tex]B(4,-1)\to B'(4+4,-1-3)[/tex]

[tex]B(4,-1)\to B'(8,-4)[/tex]

Hence, the endpoint of the line segment A'B' are A'(2,-6) and B'(8,-4).

The instantaneous rate of change for the function f(x) at x = 3 is 16.9068.

To determine the instantaneous rate of change for the function;  f(x) = 5x^ln(x), where x = 3, we can use the formula for the instantaneous rate of change, approximating the limit by using smaller and smaller values of h.

The instantaneous rate of change of the function f(x) at x = 3 can be found as follows:

Let h be a small increment of x that approaches zero. Then the formula for the instantaneous rate of change is given by:

f'(3) = lim[h→0] {(5(3+h)^(ln(3+h))-5(3^(ln3)))/h}

For the above formula, we have: Let f(x) = 5x^ln(x)

Then, f'(x) = 5x^ln(x) * [(d/dx) ln(x)] + 5*ln(x)*x^(ln(x) - 1)

Now, for the given problem, we can substitute 3 for x, and solve as follows: f'(3) = 5(3^ln(3)) * [(d/dx) ln(x)] + 5*ln(3)*3^(ln(3) - 1)

f'(3) = 5(3^ln(3)) * [(1/x)] + 5*ln(3)*3^(ln(3) - 1)

f'(3) = 5(3^ln(3)) * [(1/3)] + 5*ln(3)*3^(ln(3) - 1)

f'(3) = 5(3^ln(3) / 3) + 5*ln(3)*3^(ln(3) - 1)

Then, f'(3) = 5e ln(3) + 5 ln(3) / 3= (5e ln(3) + 5 ln(3) / 3)= 16.9068 (rounded to four decimal places).

Therefore, the instantaneous rate of change for the function f(x) at x = 3 is 16.9068.

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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

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

To plot an image of your own choosing, you can follow these steps. First, create a list containing a set of ordered pairs that represent the points needed to create your image.

To create the image, you need to define the set of ordered pairs that form the outline of the desired shape. The points in the list should be ordered in a way that connects them together to form the shape. Including (0,0) at the beginning and end ensures that the last point connects back to the first point.

Once you have the list of ordered pairs, you can convert it into a matrix using the "HGMatrix" command. This step is necessary for plotting the image using the "Plotimage" command. The matrix, named "mylistmat," will represent the points in a format suitable for plotting.

Finally, using the "Plotimage" command, you can plot the image based on the points in "mylistmat." The command will take the ordered pairs and connect them together to create the desired image.

By following these steps, you can plot your own image using the specified list, matrix conversion, and image plotting commands.

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Answer:

$27.50

Step-by-step explanation:

First, find the cost with the 15% off sale

29.95(0.85)

= 25.46

Find the price with the sales tax:

25.46(1.08)

= 27.5

So, the sale price of the game with tax is approximately $27.50

By filling in the elements accordingly and comparing the Venn diagram with the calculated results, the accuracy of the union and intersection can be confirmed i.e. AUB = {1, 2, 3, 4, 5, 6, 7, 8, 9} and An B = {4, 8}.

To find the union (AUB) and intersection (An B) of the sets A = {1, 4, 6, 8, 9} and B = {2, 3, 4, 5, 7, 8}, we compare the elements in the sets. The union (AUB) is the combination of all unique elements from both sets, while the intersection (An B) consists of the elements common to both sets.

(a) The union (AUB) is {1, 2, 3, 4, 5, 6, 7, 8, 9}, which includes all the distinct elements present in both sets.

(b) The intersection (An B) is {4, 8}, representing the elements that are common to both sets A and B.

To verify these results, you can draw a Venn diagram. Draw two overlapping circles representing sets A and B, and fill in the elements accordingly. The overlapping region represents the intersection, and the combination of all elements in both circles represents the union. Comparing the Venn diagram with the calculated results confirms the accuracy of the union and intersection.

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Answer:

3 AND 6 - 2 AND 5!

Step-by-step explanation:

hope i helped!

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

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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To calculate the flux of a vector field through a surface, we can use the surface integral of the dot product between the vector field and the outward-pointing normal vector of the surface.

Let's first calculate the flux of the vector field F = 6i - 7k through a sphere of radius 5 centered at the origin, oriented outward.

The equation of the sphere centered at the origin is [tex]x^2 + y^2 + z^2 = 5^2.[/tex]

To find the outward-pointing normal vector at each point on the sphere's surface, we normalize the position vector (x, y, z) by dividing it by the magnitude of the vector.

The outward-pointing normal vector is given by N = (x, y, z) / [tex]\sqrt{(x^2 + y^2 + z^2).}[/tex]

Now, we calculate the flux using the surface integral:

Flux = ∬S F · dS,

where S is the surface of the sphere.

The dot product F · dS can be expanded as F · N dS, where dS represents the differential area vector.

The magnitude of the differential area vector on the sphere's surface is given by dS = [tex]r^2[/tex]sin(θ) dθ dφ, where r is the radius of the sphere, and θ and φ are the spherical coordinates.

Since the sphere is symmetric about the origin, the flux will be the same for all points on the surface, and we can simplify the integral as:

Flux = F · N ∬S dS.

To find the flux, we need to calculate the dot product F · N and evaluate the surface integral over the sphere's surface. Let's calculate it:

F = 6i - 7k

N = (x, y, z) /[tex]\sqrt{(x^2 + y^2 + z^2)}[/tex] = (x, y, z) / 5

F · N = (6i - 7k) · (x/5, y/5, z/5) = (6x/5) - (7z/5)

Now, let's evaluate the surface integral over the sphere's surface:

Flux = ∬S F · dS = ∬S (6x/5 - 7z/5) dS

To evaluate the integral, we can use spherical coordinates. The limits of integration will be:

θ: 0 to 2π (complete rotation around the z-axis)

φ: 0 to π (from the positive z-axis to the negative z-axis)

Flux = ∫(φ=0 to π) ∫(θ=0 to 2π) (6r sin(φ) cos(θ)/5 - 7r sin(φ) sin(θ)/5) [tex]r^2[/tex]sin(φ) dθ dφ

Simplifying and evaluating the integral will give you the flux of the vector field through the sphere.

Now, let's move on to calculating the flux of the vector field F = i - 3j + 9k through a cube of side length 5 with sides parallel to the axes, oriented outward.

Since the sides of the cube are parallel to the coordinate axes, the normal vector to each side will be aligned with the corresponding unit vector.

For example, the normal vector to the side with a normal vector i will be (1, 0, 0), and the normal vector to the side with a normal vector j will be (0, 1, 0), and so on.

To calculate the flux, we need to find the dot product between the vector field F and the outward-pointing normal vectors of each side, and then sum up the flux for all six sides of the cube.

Let's calculate the flux for each side of the cube and then sum them up to get the total flux.

Side 1: Outward normal vector = (1, 0, 0)

Dot product = (i - 3j + 9k) · (1, 0, 0) = 1

Side 2: Outward normal vector = (-1, 0, 0)

Dot product = (i - 3j + 9k) · (-1, 0, 0) = -1

Side 3: Outward normal vector = (0, 1, 0)

Dot product = (i - 3j + 9k) · (0, 1, 0) = -3

Side 4: Outward normal vector = (0, -1, 0)

Dot product = (i - 3j + 9k) · (0, -1, 0) = 3

Side 5: Outward normal vector = (0, 0, 1)

Dot product = (i - 3j + 9k) · (0, 0, 1) = 9

Side 6: Outward normal vector = (0, 0, -1)

Dot product = (i - 3j + 9k) · (0, 0, -1) = -9

Now, sum up all the dot products to get the total flux:

Flux = 1 + (-1) + (-3) + 3 + 9 + (-9) = 0

The total flux of the vector field through the cube is zero.

I hope this helps! Let me know if you have any further questions.

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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.

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 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?

To know more about system of equations, refer here:

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Answer:

75.4

Step-by-step explanation:

Hello There!

We can easily solve for the volume of the cone using this formula

[tex]V=\pi r^2\frac{h}{3}[/tex]

where r = radius and h = height

we are given that the radius had a length of 3 ft and the height has a length of 8 ft

Because we know the values all we have to do is plug them into the formula

so

[tex]V=\pi 3^2\frac{8}{3}\\3^2=9\\9\pi =28.27433388\\28.27433388*\frac{8}{3} =75.39822369[/tex]

so we can conclude that the volume of the cone is 75.39822369 ft³

Finally we round to the nearest hundredth and get that the answer is 75.4

Answer:

Step-by-step explanation:

Rewrite this quadratic in standard form:  3x^2 + 7x - 1.

The coefficients of x are {3, 7, -1}, and so the discriminant is b^2 - 4ac, or

7^2 - 4(3)(-1), or 49 + 12, or 61.  Because the discriminant is positive, this quadratic has two real, unequal roots