As men and women​ age, their risk of death increases. There are also health issues that can greatly impair older​ citizens' abilities to participate in activities and travel. Assume you are planning a 60th high school reunion and are wondering how many of the older adults would be able to attend. If you know that 22 women and 18 men are still​ alive, and are told that exactly 8 of them will​ attend, what is the probability that you would have exactly four men and exactly four women​ attend?

Answer:

  m∠XQZ = 72°

Step-by-step explanation:

We presume the "if ..." condition is intended to mean that ray QY lies between rays QX and QZ. Then by angle addition, you get

  ∠XQY + ∠YQZ = ∠XQZ

  (4a +8) + (28) = (8a)

  36 = 4a . . . . . . . . . . . . . subtract 4a

  72 = 8a = ∠XQZ . . . . multiply by 2

To find out how much is invested in each bond, set up a system of equations and solve for the values of x and y.

To find out how much is invested in each bond, we can set up a system of equations.

Let x be the amount invested in the bond paying 14% interest, and y be the amount invested in the bond paying 12% interest.

We know that the retiree receives $5120 in interest each year. So, we have the equation:

x(0.14) + y(0.12) = 5120

Since the total amount invested is $40,000, we can also write the equation:

x + y = 40000

We can now solve this system of equations using any method, such as substitution or elimination, to find the values of x and y.

Answer:

11 * 17 = 187.

Step-by-step explanation:

11 * 14 = 154

11 * 15 = 165

11 * 16 =  176

So we have the series 154, 165, 176  which has a common difference of 11.

so the next line is 11*17 = 176 + 11 = 187.

Answer:

time period (t)  is 2.25 years

Step-by-step explanation:

Given data in question

amount (a) = $94000

principal (p) = $78870

rate (r) = 7.8 % = 0.078

to find out

time period (t)  

solution

we know that continuous compound interest formula i.e.

amount = principal [tex]e^{rt}[/tex]   ...............1

we will put all value a, p and r  in equation 1

amount = principal [tex]e^{rt}[/tex]

94000 = 78870 [tex]e^{0.078t}[/tex]

[tex]e^{0.078t}[/tex] = 94000 / 78870

now we take ln both side

ln  [tex]e^{0.078t}[/tex]  ln (94000 / 78870)

0.078 t = ln 1.19183466

0.078 t = 0.175494

t = 0.175494 /0.078  

t = 2.249923  

so time period (t)  is 2.25 years

In this complex interest problem, by organizing and substitifying the values into the continuous compound interest formula, we obtain t = ln(1.1911) / 0.078, which approximately equals 2.81 years when rounded to two decimal places.

The continuous compound interest formula is A = Pe^(rt), where P is the principal amount, r is the interest rate, t is the time in years, and A is the amount of money accumulated after n years, including interest.

In this case, we have A = $94,000, P = $78,870, r = 7.8% = 0.078, and we need to solve for t. Therefore, let's substitute the given values into the formula to solve for t:

$94,000 = $78,870 * e^(0.078t),

Solving this equation for 't' involves isolating 't' on one side. First, divide both sides by $78,870:

e^(0.078t) = $94,000 / $78,870 = 1.1911.

Take the natural logarithm (ln) of both sides:

0.078t = ln(1.1911),

Finally, divide both sides by 0.078 to solve for 't':

t = ln(1.1911) / 0.078.

With a calculator, this results in approximately t = 2.81 years when rounded to two decimal places.

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Answer:  0.6731

Step-by-step explanation:

Given : Mean : [tex]\mu = 40,000\text{ miles}[/tex]

Standard deviation : [tex]\sigma = 5,000\text{ miles}[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=36,000

[tex]z=\dfrac{36000-40000}{5000}=-0.8[/tex]

For x=46,000

[tex]z=\dfrac{46000-40000}{5000}=1.2[/tex]

The P-value : [tex]P(-0.8<z<1.2)=P(z<1.2)-P(z<-0.8)[/tex]

[tex]=0.8849303-0.2118554=0.6730749\approx0.6731[/tex]

Hence, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles =0.6731

the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).

The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 miles and a standard deviation of 5,000 miles. To find the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles, we can calculate the z-scores for 36,000 and 46,000 and then use the standard normal distribution to find the probabilities for these z-scores.

To compute the z-scores:

We then look up the corresponding probabilities for these z-scores in a standard normal distribution table or use a calculator with normal distribution functions. The probability corresponding to z=-0.8 is approximately 0.2119, and the probability corresponding to z=1.2 is approximately 0.8849. The probability of the tire's life being between 36,000 and 46,000 is the difference: 0.8849 - 0.2119 = 0.6730.

Therefore, the probability that a randomly selected set of tires will have a life of 36,000 to 46,000 miles is 0.6730 (rounded to four decimal places).

The car's velocity at time [tex]t[/tex] is

[tex]v(t)=16\dfrac{\rm m}{\rm s}+\displaystyle\int_0^t\left(\left(-0.50\frac{\rm m}{\mathrm s^2}\right)u\right)\,\mathrm du=16\dfrac{\rm m}{\rm s}+\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2[/tex]

It comes to rest at

[tex]v(t)=0\implies16\dfrac{\rm m}{\rm s}=\left(0.25\dfrac{\rm m}{\mathrm s^2}\right)t^2\implies t=8.0\,\mathrm s[/tex]

Its velocity over this period is positive, so that the total distance the car travels is

[tex]\displaystyle\int_0^{8.0}v(t)\,\mathrm dt=\left(16\dfrac{\rm m}{\rm s}\right)(8.0\,\mathrm s)+\frac13\left(-0.25\dfrac{\rm m}{\mathrm s^2}\right)(8.0\,\mathrm s)^3=\boxed{85\,\mathrm m}[/tex]

so the answer is D.

The car takes 32 seconds to stop from its initial velocity of 16m/s. During this period, the car has traveled a distance of 256 meters, which is not an option in the given choices.

The initial velocity of the car is given as 16 m/s and the acceleration is given as -0.5t m/s^2. We know that the car slows down until it stops, which means its final velocity is 0 m/s.

Firstly, we need to find out the time it takes for the car to stop. That could be calculated with the equation 'v = u + at', where v is the final velocity, u is the initial velocity, a is the acceleration and t is time. Since we know v = 0 m/s, u = 16 m/s and a = -0.5t m/s^2, we could set the equation to find the time to be '0 = 16 - 0.5t'. Solving this equation gives t=32 seconds.

Second, to find the distance traveled by the car during this time, we use the equation 's = ut + 0.5at^2', where s is the distance, u is the initial velocity, a is the acceleration and t is time. Substituting the known values into this equation, we get 's = 16(32) + 0.5*(-0.5*32)*(32)', which simplifies to s = 512 - 256 = 256 meters. Hence, the answer is not in the options given.

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Rating plays on Broadway, Poor, good, or excellent would be a type of Ordinal measurement.

You can think or ordinal like order, which could be listing something from best to worst.

The answer is Ordinal.

Ordinal measurement can shown by name, group, or rank. Poor, good, and excellent shows the ratings of the play by "rank", in other words, by order. Thus proves our answer.

Best of Luck!

6 divided by 4 is 1.5.

1.5 times 20 is 30

it would cost michelle $30 to buy 20 pieces of cookies

Answer:

$30

Step-by-step explanation:

4pcs = $6

20pcs = x

Cross multiplying, we have

x = (20pcs * $6)/4pcs

= $120/4 = $30

Answer:

B. 180 mg

Step-by-step explanation:

In order to answer the given problem we need to be aware that:

1000 milligrams = 1 milliliter

The above means that in 1 milliliter a 100% solution means 1000 milligrams. Because we have 18% solution, then:

(1000 milligrams / 1 milliliter) * 18% =

(1000 milligrams / 1 milliliter) * (18/100) =

(1000*18/100) milligrams/milliliter =

180 milligrams/milliliter

In conclusion, the answer is B. 180 mg.

Answer:

Step-by-step explanation:this is confusing for me oof

Final answer:

The maximum area that can be enclosed with 144 feet of fencing is when the enclosure is a square. Calculating the side length as 36 feet results in a maximum area of 1296 square feet.

Explanation:

Maximizing the Area of a Rectangular Region with a Given Perimeter:

To find the maximum area that can be enclosed with 144 feet of fencing in a rectangular shape, we can use the knowledge that for a given perimeter, a rectangle with equal length and width (a square) will have the maximum possible area. Let's denote the length of the rectangle as L and the width as W. Since the perimeter is twice the sum of the length and width, we have 2L + 2W = 144 feet. To form a square, which gives the maximum area, L equals W, making 4L = 144 feet or L = 36 feet. The maximum area is L squared, which is 36 feet by 36 feet, equaling 1296 square feet.

The maximum area that can be enclosed with 144 feet of fencing is 1296 square feet, which corresponds to option c) 1296 square feet.

To maximize the area enclosed by a fixed perimeter, we look to geometry, which tells us that of all the rectangles with a given perimeter, the square has the highest area.
Let's denote the perimeter of the square as P and the length of each side of the square as s. Since the square has four equal sides, we have:
P = 4s
We are given that P is 144 feet, so we can find the length of each side s by dividing the total perimeter by 4:
s = P/4 = 144/4 = 36 feet
The area A of a square is given by the formula A = s^2, where s is the length of a side of the square. We calculated above that s = 36 feet, so:
A = s^2 = (36 feet)^2 = 1296 square feet
This is the maximum area that can be enclosed by 144 feet of fencing when arranged in a square. Matching our result with the provided options, the correct answer is:
c) 1296 square feet

Answer:

The company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.

Step-by-step explanation:

Let x be the number of ounces of chicken

let y be the number of ounces of grains

We are given that Chicken has 10 grams of protein and 5 grams of fat per ounce.

Chicken has protein in 1 ounce = 10

Chicken has protein in x ounces = 10x

Chicken has fat in 1 ounce = 5

Chicken has fat in x ounces = 5x

We are also given that grain has 2 grams of protein and 2 grams of fat per ounce

Grain has protein in 1 ounce = 2

Grain has protein in y ounces = 2y

Grain has fat in 1 ounce = 2

Grain has fat in y ounces =2y

Now we are given that . A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat.

So, equation becomes:

[tex]10x+2y\geq 200[/tex] ---A

[tex]5x+2y\geq 150[/tex]  ---B

Since x and y represent the ounces .

So, [tex]x\geq 0[/tex] ---C and [tex]y\geq 0[/tex] ----D

Plot A ,B ,C and D on the graph

Refer the attached graph so , the corner points are (0,100),(10,50) and (30,0)

We are also given that chicken costs 10 cents per ounce and grain costs 1 cent per ounce

So, Cost function becomes : [tex]c=10x+y[/tex]

At(0,100)

[tex]c=10x+y[/tex]

[tex]c=10(0)+100[/tex]

[tex]c=100[/tex]

At(10,50)

[tex]c=10x+y[/tex]

[tex]c=10(10)+50[/tex]

[tex]c=150[/tex]

At(30,0)

[tex]c=10x+y[/tex]

[tex]c=10(30)+0[/tex]

[tex]c=300[/tex]

Minimum cost is 100 cents at (0,100)

So,  the company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.

Try this suggested solution.

Final answer:

The parametric equations that model the problem situation in this case are x(t) = v0x * t,  [tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex], and vy(t) = v0y - g * t.

Explanation:

To find parametric equations that model the problem situation, we need to consider the horizontal and vertical components of the motion separately.

Horizontal Component:

The horizontal velocity remains constant throughout the motion. Therefore, the horizontal position can be given by the equation:

x(t) = v0x * t

where x(t) is the horizontal position at time t and v0x is the initial horizontal velocity.

Vertical Component:

The vertical position depends on the initial velocity, acceleration due to gravity, and time. We can use the following equations:

[tex]y(t) = v0y * t - (1/2) * g * t^2,[/tex]

vy(t) = v0y - g * t

where y(t) is the vertical position at time t, v0y is the initial vertical velocity, g is the acceleration due to gravity (approximately 32 ft/s2), and vy(t) is the vertical velocity at time t.

Answer: There are 22.5 films were profitable.

Step-by-step explanation:

Since we have given that

Number of total films = 40

Percentage of comedies = 60%

Number of comedies is given by

[tex]0.6\times 40\\\\=24[/tex]

Percentage of horror films = 40%

Number of horror films is given by

[tex]0.4\times 40\\\\=16[/tex]

Percentage of comedies were profitable = 75%

Number of profitable comedies is given by

[tex]0.75\times 24=18[/tex]

Percentage of horror were unprofitable = 75%

Percentage of horror were profitable = 25%

Number of profitable horror films is given by

[tex]0.25\times 18\\\\=4.5[/tex]

So, Total number of profitable films were

[tex]18+4.5=22.5[/tex]

Hence, there are 22.5 films were profitable.

Answer: The probability that a person is not qualified if a person was approved by the manager is 0.0766.

Step-by-step explanation:

Since we have given that

Probability that a person approves qualified = 0.85× 0.85 = 0.7225

Probability that a person does not approve qualified = 0.85 × 0.15 = 0.1275

Probability that a person approves unqualified = 0.40 × 0.15 = 0.06

Probability that a person does not approve unqualified = 0.60 × 0.15 = 0.009

so, using the conditional probability, we get that [tex]p(unqualified\mid approved)=\dfrac{0.06}{0.7225+0.06}=\dfrac{0.06}{0.7825}=0.0766[/tex]

Hence, the probability that a person is not qualified if a person was approved by the manager is 0.0766.

Final answer:

To convert Emanuel's weight from pounds to kilograms on the 35th day, multiply 171 lbs by 0.4536 to get 77.52 kg. The general formula for converting function f(t) to g(t) is g(t) = f(t) × 0.4536.

Explanation:

If Emanuel's weight in pounds on the 35th day since the beginning of 2017 is 171 lbs ( f(35) = 171 ), we can find his weight in kilograms ( g(35) ) using the conversion factor from pounds to kilograms. Since 1 pound is equivalent to approximately 0.4536 kilograms on Earth, we can calculate g(35) by multiplying Emanuel's weight in pounds by this conversion factor:

g(35) = 171 lbs × 0.4536 kg/lb

This results in g(35) = 77.5156 kg. When we round this to significant figures based on the given conversion fact of pounds to kilograms (which is inexact and has 4 significant figures), Emanuel's weight would be g(35) = 77.52 kg (to 4 SFs).

The formula for g using the function f is:

g(t) = f(t) × 0.4536

Answer:

answer 1

Step-by-step explanation:

Since there's only 1 value in the 90-99 groups it is either 1 or 3.

Furthermore, there are only 2 values starting with 10.

This only fits with answer 1

The answer is A.

A stem plot shows every number with the tens place and above being on the left side while the ones place is on the right side.

For example,

9 is in the tens place while 4 is in the ones place. So, this would look like:

9 | 4

There is only one number with a "9" in the tens place and there is only two numbers with "10" in the tens place. This only sastisfys the first option.

Best of Luck!

Answer:

A. 6.0

Step-by-step explanation:

We have been given that the the inside diameter of the part on the paper is 1.50 inches.

We have been given that scale is 4:1 for actual size to drawing side.

[tex]\text{Scale}=\frac{\text{Actual size}}{\text{Map size}}=\frac{4}{1}[/tex]

Upon substituting 1.50 in our given proportion, we will get:

[tex]\frac{\text{Actual size}}{1.50\text{ inches}}=\frac{4}{1}[/tex]

[tex]\frac{\text{Actual size}}{1.50\text{inches}}\times 1.50\text{ inches}=\frac{4}{1}\times 1.50\text{ inches}[/tex]

[tex]\text{Actual size}=4\times 1.50\text{ inches}[/tex]

[tex]\text{Actual size}=6.0\text{ inches}[/tex]

Therefore, the actual size is 6.0 inches and option A is the correct choice.

To find the equation in function form for the number of cars fixed each week by the mechanic, we can use the slope-intercept form of a linear equation. The equation is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.

To write an equation in function form to show the number of cars seen each week by the mechanic, we can let the variable x represent the week number and y represent the number of cars fixed. We know that the reduction in the number of cars each week is linear, so we can use the slope-intercept form of a linear equation, y = mx + b. To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. Let's use the points (3, 391) and (13, 361) to find the slope. Plugging these values into the formula gives us m = (361 - 391) / (13 - 3) = -3. Therefore, the equation in function form is y = -3x + b. To find the y-intercept b, we can use one of the points on the line. Let's use the point (3, 391): 391 = -3(3) + b. Solving for b gives us b = 400. Therefore, the equation in function form is y = -3x + 400, where x represents the week number and y represents the number of cars fixed.

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The equation in function form to show the number of cars seen each week by the mechanic is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.

To write an equation in function form to show the number of cars seen each week by the mechanic, we can use the given information that the reduction in the number of cars each week is linear. Let's assume the number of cars fixed in week 3 as y = 391 and in week 13 as y = 361. We can use the formula for the equation of a line, y = mx + b, where m is the slope and b is the y-intercept.

Using the slope formula, m = (y2 - y1) / (x2 - x1), where (x1, y1) = (3, 391) and (x2, y2) = (13, 361), we find m = (361 - 391) / (13 - 3) = -3.

Therefore, the equation in function form to show the number of cars seen each week by the mechanic is y = -3x + b. To find the y-intercept, we can substitute the coordinates of one of the points (x, y) = (3, 391) into the equation, 391 = -3(3) + b. Solving for b gives b = 400.

Thus, the equation in function form is y = -3x + 400, where x represents the week and y represents the number of cars fixed by the mechanic.

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Answer: Option (d) is correct.

Step-by-step explanation:

Regression analysis can be used to determine if there is any linear relationship exists between the two quantitative variables.

There are one dependent variable and one or more than one independent variable in a single regression equation.

After running a simple regression, we get to know the relationship between the dependent variable and explanatory variable.

Various statistical software are used for running regression like STATA, E- Views, SPSS, etc.  

Regression analysis is used primarily to determine if a linear relationship exists between two quantitative variables. While it can identify relationships between variables, it does not imply causation. It does not establish a relationship between two categorical variables or if the difference in the population means may be zero.

Regression analysis is a statistical methodology often used in mathematics and related fields. Primarily, it is used to determine if a linear relationship exists between two quantitative variables.

For example, you could use regression analysis to see if there is a linear relationship between the age of a car (quantitative variable) and the distance it can travel on a tank of gas (another quantitative variable).

It's important to note however that while regression analysis can identify a correlation or relationship between variables, it does not necessarily imply causation, or a cause-and-effect relationship. There could be other related variables causing the effect.

For your options, regression analysis can't be used to determine if a linear relationship exists between two categorical variables or if the difference in the population means may be zero.

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Answer:

  1,556,675,366,400

Step-by-step explanation:

There are 37 possible characters, of which 8 can be chosen. Order matters, so the number is ...

  37P8 = 1,556,675,366,400

_____

This number includes 84,144,614,400 strings in which the space character is either first or last. Such strings may be ruled invalid because they are indistinguishable from 7-character strings.

_____

nPk = n!/(n-k)! . . . . the number of permutations of n things taken k at a time

The 37 allowed characters are the 26 letters of the alphabet, 10 digits, and 1 space character.

[tex]y'+2y=2e^x\Longrightarrow y'=2e^x-2y[/tex]

If [tex]f'(x)=g(x)[/tex] then [tex]y=\int{g(x)dx}[/tex]

So we extract,

[tex]y=\int{2e^x-2x}dx[/tex]

Which becomes,

[tex]y=2e^x-x^2+C[/tex]

Hope this helps.

r3t40

Answer:

(x-9)^2 + (y+3)^2 = 8^2

or

(x-9)^2 + (y+3)^2 = 64

Step-by-step explanation:

An equation for a circle can be written as

(x-h)^2 + (y-k)^2 = r^2  where (h,k) is the center and r is the radius

(x-9)^2 + (y- -3)^2 = 8^2

(x-9)^2 + (y+3)^2 = 8^2

or

(x-9)^2 + (y+3)^2 = 64

Answer:

Yes , S is a basis for [tex]P_3[/tex].

Step-by-step explanation:

Given

S=[tex]\left\{4t-12,5+t^3,5+3t,-3t^2+\frac{2}{3}\righ\}[/tex].

We can make a matrix

Let A=[tex]\begin{bmatrix}-12&4&0&0\\5&0&0&1\\5&3&0&0\\\frac{2}{3}&0&-3&0\end{bmatrix}[/tex]

All rows and columns are linearly indepedent and S span [tex]P_3[/tex].Hence, S is a basis of [tex]P_3[/tex]

Linearly independent means any row or any column should not combination of any rows or columns.

Because  a subset of V with n elements is a basis if and only if it is linearly independent.

Basis:- If B is a subset  of a vector space V over a field F .B is basis of V if satisfied the following conditions:

1.The elements of B are linearly independent.

2.Every element of vector V spanned by the elements of B.

Answer:

The required interval is (26.23 , 27.37)

Step-by-step explanation:

The mean is = 26.8

The standard deviation is = 7.42

n = 654

At 95% confidence interval, the z score is 1.96

To find the desired interval we will calculate as:

[tex]26.8+1.96(\frac{7.42}{\sqrt{654} } )[/tex]

And [tex]26.8-1.96(\frac{7.42}{\sqrt{654} } )[/tex]

[tex]26.8+0.57[/tex] and [tex]26.8-0.57[/tex]

= 27.37 and 26.23

So, the required interval is (26.23 , 27.37)

The 95% confidence interval for the mean BMI of all young women, given a sample of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42 is approximately (26.23, 27.37). This is calculated by finding the standard error and then using that to find the interval around the sample mean which covers 95% of the probable values for the population mean.

The question is asking you to calculate the 95% confidence interval for the mean BMI of all young women, given a sample size of 654 women with a mean BMI of 26.8 and a standard deviation of 7.42.

In order to do this, we first need to calculate the standard error, which is the standard deviation divided by the square root of the number of observations. This gives us 7.42 divided by the square root of 654, or about 0.29032.

The 95% confidence interval is calculated by taking the mean and adding and subtracting the standard error multiplied by the relevant z-score for a 95% confidence interval, which is 1.96 (from the standard normal distribution table). So we take 26.8 plus and minus 1.96 times 0.29032, giving us a 95% confidence interval of approximately (26.23, 27.37).

So, we can say with 95% confidence that the mean BMI for all young women is between 26.23 and 27.37.

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[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]

Divergence is easier to compute:

[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]

[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]

Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then

[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]

[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]

[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]

To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.

In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).  

The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.  

The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.

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Answer:

\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right]

Step-by-step explanation:

Given is a matrix 3x3 as

[tex]\left[\begin{array}{ccc}1&0&2\\-3&1&4\\2&-3&4\end{array}\right][/tex]

|A| =2 hence inverse exists.

Cofactors are 16   20   7

                        6     8    3

                         2     2    1

Hence inverse =

[tex]\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right][/tex]

Answer:

a.[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]

b[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]

Step-by-step explanation:

1.[tex]y''-4y'+4y=6x e^{2x}[/tex]

Auxillary equation

[tex]D^2-4D+ 4=0[/tex]

[tex](D-2)(D-2)=0[/tex]

D=2,2

Then complementary solution =[tex] C_1e^{2x}+C_2xe^{2x}[/tex]

Particular solution [tex]=\frac{6 xe^{2x}}{(D-2)^2}[/tex]

D is replace by D+2 then we get

P.I=[tex]\frac{6xe^{2x}}{0}[/tex]

P.I=[tex]\frac{e^{ax}}{D+a} \cdot .V[/tex]

where V is a function of x

P.I=[tex]\frac}x^3e^{2x}[/tex]

By integrating two times

Hence, the general solution

[tex]y(x)=c_1e^{2x}+c-2xe^{2x}+x^3e^{2x}[/tex]

b.y''+9y=5 cos x-7 sin x

Auxillary equation

[tex]D^2+9=0[/tex]

D=[tex]\pm 3i[/tex]

[tex]C.F=c_1 cos 3x+ c_2sin 3x[/tex]

P.I=[tex]\frac{5 cos x-7 sin x}{D^2+9}[/tex]

P.I=[tex]\frac{sin ax}{D^2+bD +C}[/tex]

Then  D square is replace by -a square

[tex] D^2 [/tex] is replace by - then we get

P.I=-5 cos x+7 sin x

The general solution

[tex]y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x[/tex]