Which statement must be true?Choose all that apply.

To find a particular solution of the non-homogeneous equation and the general solution of the equation, we can use the method of variation of parameters.

First, let's find the Wronskian of the homogeneous solutions y1(x) and y2(x):

W(y1, y2) = | y1 y2 |

| y1' y2' |

We have y1(x) = sin(x) / √x and y2(x) = cos(x) / √x. Differentiating these functions, we get:

y1'(x) = (cos(x) / √x - sin(x) / (2√x^3))

y2'(x) = (-sin(x) / √x - cos(x) / (2√x^3))

Substituting these values into the Wronskian:

W(y1, y2) = | sin(x) / √x cos(x) / √x |

| (cos(x) / √x - sin(x) / (2√x^3)) (-sin(x) / √x - cos(x) / (2√x^3)) |

Expanding the determinant:

W(y1, y2) = (sin(x) / √x) * (-sin(x) / √x - cos(x) / (2√x^3)) - (cos(x) / √x) * (cos(x) / √x - sin(x) / (2√x^3))

Simplifying:

W(y1, y2) = -1 / (2√x)

Now, we can find the particular solution using the variation of parameters formula:

y_p(x) = -y1(x) * ∫(y2(x) * g(x)) / W(y1, y2) dx + y2(x) * ∫(y1(x) * g(x)) / W(y1, y2) dx

Here, g(x) = 3x√xsin(x). Substituting the values:

y_p(x) = -((sin(x) / √x) * ∫((3x√xsin(x)) * (-1 / (2√x))) dx + (cos(x) / √x) * ∫((3x√xsin(x)) / (2√x)) dx

Simplifying the integrals:

y_p(x) = -(∫(-3sin^2(x)) dx) + (∫(3xcos(x)sin(x)) dx)

Integrating:

y_p(x) = 3/2 (xsin^2(x) - cos^2(x)) - 3/2 (xcos^2(x) + sin^2(x)) + C

Simplifying:

y_p(x) = 3x(sin^2(x) - cos^2(x)) + C

The general solution of the equation is given by the sum of the homogeneous solutions and the particular solution:

y(x) = C1 * (sin(x) / √x) + C2 * (cos(x) / √x) + 3x(sin^2(x) - cos^2(x)) + C

where C1, C2, and C are arbitrary constants.

Check the picture below.

Answer:2nd one

Step-by-step explanation:

Answer:

Step-by-step explanation:

area of circle = pi*r^2

pi * 36 = 36pi = 113.04 which is about 113

Answer:

Apologies, but it seems that the table you mentioned is missing from your question. Could you please provide the table or the weights of the chicken in each package so that I can assist you further?

Step-by-step explanation:

Answer:

area= 39 square meters

Step-by-step explanation:

10×6=60m²

60m²÷2=30m² is the area of the bigger triangle

6×3=18m²

18m²÷2=9m²

9m²+30m²=39m²

Answer:

42/11

Step-by-step explanation:

11/7 = 6/v

cross multiply:

11v = 6x7 = 42.

v = 42/11 ≈ 3.81

Hello !

28   23   18   13    8  ...

Para ir de izquierda a derecha, restamos 5 cada vez.

Clearer image, please!

For the given rectangular prism dimensions, the area of each base is obtained by multiplying the length and width.

1. Area of each base:

The area of a rectangular base is calculated by multiplying the length by the width. Therefore:

- The area of the base with dimensions 8 ft by 6 ft is 48 square feet.

- The area of the base with dimensions 10 ft by 15 ft is 150 square feet.

2. Lateral area:

The lateral area of a rectangular prism is the sum of the areas of all its sides, excluding the base areas.

- For the given dimensions, the lateral area would be (8 ft + 6 ft) × 2 + (10 ft + 15 ft) × 2 = 36 ft + 50 ft = 86 square feet.

The lateral area is calculated by adding the perimeter of the bases and multiplying it by the height of the prism, excluding the base areas.

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All six trigonometric ratios of the triangle are :

sin (θ) = 3/5, cos (θ) = 4/5, tan (θ) = 3/4

cosec (θ) = 5/3, sec (θ) = 5/4, cot (θ) = 4/3

Given a triangle, which is right angled.

Length of hypotenuse = √(20² + 15²) = 25

Given an angle θ.

We have to find the six trigonometric ratios which are sin, cos, tan, cosec, sec and cot.

sin (θ) = opposite side / hypotenuse = 15/25 = 3/5

cos (θ) = adjacent side / hypotenuse = 20/25 = 4/5

tan (θ) = opposite side / adjacent side = 15/20 = 3/4

cosec (θ) = 1 / sin (θ) = 5/3

sec (θ) = 1 / cos (θ) = 5/4

cot (θ) = 1 / tan (θ) = 4/3

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The critical value of the given data set is 3.42.

The two-way chi-square analysis is used to determine if there is a relationship between two categorical variables. In this case, the two variables are shiftwork availability and the union-management relationship.

To conduct the analysis, we need to calculate the expected frequencies for each cell in the table. We can do this by multiplying the row and column totals and dividing by the total number of observations.

For the cell in the top left corner (No Shiftwork/Good Relationship):

Expected Frequency = (36 × 33) / 100 = 11.88

We can then calculate the expected frequencies for the other cells in the same way.

Once we have calculated the expected frequencies, we can then calculate the chi-square statistic using the following formula:

Chi-square = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]

Using this formula, we can calculate the chi-square statistic for the above example as follows:

Chi-square = (11 - 11.88)²/11.88 + (22 - 21.12)²/21.12 + (25 - 23.12)²/23.12 + (42 - 40.88)²/40.88 = 3.42

Finally, we can compare the chi-square statistic to the critical value of 3.84 at α = 0.05 (df = 1). Since our chi-square statistic is less than the critical value, we can conclude that there is no relationship between shiftwork availability and the union-management relationship.

Therefore, the critical value of the given data set is 3.42.

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The length of one side of the square is 36 meters.In general, the perimeter of a rectangle can be found by adding up the lengths of all four sides.

If we know the length and width of the rectangle, we can use the formula:

Perimeter = 2(length + width)

To find the length of the rectangle using the Pythagorean Theorem, we would need to be given some additional information, such as the width and the length of one of the sides or one of the diagonals. Once we have the length and width of the rectangle, we can use the formula above to find its perimeter.

If the diagonals of a square measure 36 meters, what is the length of a side of the square?(Use Pythagorean Theorem to find the side of the square)

Let s be the length of one side of the square. Since the diagonals of a square are equal in length, we have:

s² + s² = 36²

Simplifying the left-hand side:

2s² = 1296

Dividing both sides by 2:

s² = 648

Taking the square root of both sides:

s = √648 = 18√4 = 36.

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Answer:

Lucy has ridden 25% of the course so far. This means she has completed a quarter of the total distance.

Step-by-step explanation:

To determine the percentage of the course Lucy has ridden so far, we can use the following formula:

Percentage = (Distance ridden / Total distance) * 100

Given that the total distance of the bike course is 88 miles and Lucy has

ridden 22 miles, we can substitute these values into the formula:

Percentage = (22 miles / 88 miles) * 100

Simplifying the equation:

Percentage = (1/4) * 100

Percentage = 25

Therefore, Lucy has ridden 25% of the course so far. This means she has completed a quarter of the total distance.

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To find the average velocity of the ball between t = 1 and t = 2, we need to calculate the displacement of the ball during that time interval and divide it by the duration.

The displacement of an object can be determined by finding the difference in its position at the beginning and end of the interval. In this case, we can find the height of the ball at t = 1 and t = 2 using the given equation:

h(1) = -5(1)² + 30(1) + 5

= -5 + 30 + 5

= 30 meters

h(2) = -5(2)² + 30(2) + 5

= -20 + 60 + 5

= 45 meters

The displacement of the ball between t = 1 and t = 2 is:

Displacement = h(2) - h(1)

= 45 - 30

= 15 meters

The duration of the interval is 2 - 1 = 1 second.

Finally, we can calculate the average velocity using the formula:

Average velocity = Displacement / Duration

= 15 meters / 1 second

= 15 m/s

Therefore, the average velocity of the ball between t = 1 and t = 2 is 15 m/s.

The probability of this occurring is around 0.0714.

In order to determine the likelihood that an adolescent would play the lead role in each of Britney's four films, we must take into account both the total number of possible outcomes and the number of favourable possibilities.

There are 8 films scheduled in total.

Five films have had a teenager as the primary character.

Calculating the ratio of the number of favourable outcomes—all four films with teenagers—to the entire number of possible outcomes—any four films—will help us determine the probability.

The combination formula, sometimes referred to as "n choose r," shows how many different methods there are to choose 4 films from the available 8 options:

[tex]C(n, r) = n! / (r! * (n - r)!)[/tex]

In this case, we want to select 4 movies out of the 5 movies featuring a teenager:

[tex]C(5, 4) = 5! / (4! * (5 - 4)!) = 5[/tex]

The total number of ways to select any 4 movies out of the 8 total movies is:

[tex]C(8, 4) = 8! / (4! * (8 - 4)!) = 70[/tex]

Therefore, the likelihood that a teenager will play the lead role in each of the four films that Britney cleans up after is:

Probability is calculated as follows: 5 / 70 (rounded to four decimal places) = 0.0714 (Number of favourable outcomes / Total number of probable outcomes).

Therefore, the chance is around 0.0714.

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The volume of a cone is given by the formula V = (1/3)πr²h, where "V" is the volume of the cone, "r" is the radius of the base, and "h" is the height of the cone.

If the cone is exactly half full of water by volume, then the volume of water is equal to half the volume of the cone. Let's call this volume "Vw".

So, Vw = (1/2) V

Substituting the formula for the volume of a cone, we get:

(1/3)πr²h(water) = (1/2) (1/3)πr²h(cone)

Simplifying, we get:

h(water) = (1/2) h(cone)

Therefore, the depth of the water in the cone is half the height of the cone.

The value of x in the intersecting chords is determined as 5.

The value of x is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

Also this theory states that arc angles of intersecting secants at the center of the circle is equal to the angle formed at the center of the circle by the two intersecting chords.

46 = ¹/₂ ( arc 21x - 5  -  arc x + box)

you will need to substitute the value of the box (what is inside the box?), that is the only you can solve it.

Let's say the box = 3

46 = ¹/₂ ( arc 21x - 5  -  (arc x + 3)

46 = ¹/₂ (21x - 5 - x - 3)

2(46) = 20x - 8

92 = 20x - 8

20x = 100

x = 100/20

x = 5

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In the given diagram, we have a triangle with one angle labeled as 2x + 10° and another angle labeled as 3x - 5°. To solve for x, we can set up an equation using the sum of angles in a triangle, which is 180°.

The sum of the angles in a triangle is given by the equation:

Angle1 + Angle2 + Angle3 = 180°

Substituting the given angle measurements, we have:

(2x + 10°) + (3x - 5°) + 95° = 180°

Combining like terms, we get:

5x + 100° = 180°

Next, we can isolate the variable x by subtracting 100° from both sides:

5x = 180° - 100°

5x = 80°

Finally, we can solve for x by dividing both sides by 5:

x = 80° / 5

x = 16°

Therefore, the value of x is 16°.

We cannot determine the exact number of complex zeros for the given polynomial.

We have,

To determine the number of complex zeros of the polynomial

f(x) = 6x^6 - 5x^4 + 2x^3 - x^2 + 12

We need to analyze the degree and sign changes of the polynomial.

The degree of the polynomial is 6, which means it is a polynomial of degree 6.

According to the Fundamental Theorem of Algebra, a polynomial of degree n can have at most n complex zeros (counting multiplicity).

To analyze the sign changes, we can write down the coefficients of the polynomial in a row:

6, -5, 2, -1, 0, 0, 12

By observing the sign changes in the row of coefficients, we see that there are no sign changes.

This indicates that the polynomial has either zero or an even number of positive real zeros.

Since we are interested in complex zeros, this information alone does not provide conclusive evidence.

To determine the number of complex zeros, we would need more information or further analysis.

It is possible that some of the zeros are complex or imaginary, but without more details or a complete factorization of the polynomial, we cannot determine the exact number of complex zeros.

Thus,

We cannot determine the exact number of complex zeros.

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The given data set: 4, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10 is skewed right or positively skewed due to a cluster of values towards the center and a gradual increase on the right side. The correct answer is A) Skewed Right.

The given data set is: 4, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10.

To determine the shape of the data set, we can analyze its distribution.

Looking at the data set, we can observe that the values are gradually increasing from 4 to 10. However, there are multiple occurrences of the numbers 6, 7, and 8. This indicates that the data set has a cluster or mode around these values.

Since the data set has a cluster of values towards the center and a gradual increase on the right side, it suggests that the data set is skewed right or positively skewed.

Therefore, the correct answer is A) Skewed Right.

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The value of midpoint M of AB is,

⇒ M = (2, - 4)

Since, A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

We have to given that;

Coordinates of A and B are,

A = (- 3, - 1)

B = (- 7 - 7)

Hence, By definition of midpoint we get;

M = (- 3 + (- 7))/2, (- 1 + (- 7))/2

M = (4/2, - 8/2)

M = (2, - 4)

Thus, The value of midpoint M of AB is,

⇒ M = (2, - 4)

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Answer: I WOULD SAY 45%

Step-by-step explanation:

The percentage increase in miles per gallon from Toyota city driving to highway driving is 33.33%.

The percentage increase in miles per gallon from Toyota city driving to highway driving can be calculated using the formula:

Percentage increase = ((new value - old value) / old value) x 100

Let's say the old miles per gallon for city driving is 30 and the new miles per gallon for highway driving is 40:

Therefore, there is a 33.33% increase in miles per gallon from Toyota city driving to highway driving.

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1. The of jigsaw puzzles is 216 and crossword puzzle is 261.

2. 22 table lamps were left in the warehouse.

1. Crossword Puzzle Books and Jigsaw Puzzles:

Let's the number of jigsaw puzzles as "x".

then, the number of crossword puzzle books will be "x + 45"

Now, Number of sold crossword puzzle books = (5/9) (x + 45)

= (25/100) x

= x/4

It is also given that there were 91 more crossword puzzle books sold than jigsaw puzzles. Therefore, we can set up the equation:

(5/9) (x + 45) - (x/4) = 91

(5/9)x + 25 - (x/4) = 91

(20/36)x + 25 - (9/36)x = 91

(11/36)x = 91 - 25

(11/36)x = 66

x = (36/11)  66

x ≈ 216

So, the number of jigsaw puzzles is 216 and the number of crossword puzzle books is 216 + 45 = 261.

2. Table Lamps and Standing Lamps:

Let's the number of table lamps as "x".

then, the number of standing lamps will be "155 - x"

So, Number of table lamps = (60/100) x = 0.6x

and, Number of standing lamps

= (80/100) (155 - x)

= 0.8(155 - x)

= 124 - 0.8x

Therefore, the number of table lamps left will be:

Remaining table lamps = (70/100) * 0.6x = 0.42x

To find the value of x, we can set up the equation:

0.42x = x - 30

Simplifying the equation:

x - 0.42x = 30

0.58x = 30

x = 30 / 0.58

x ≈ 51.72

So, the number of table lamps left is 0.42 x 51.72 ≈ 21.68.

Thus, 22 table lamps were left in the warehouse.

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The solution is: 0.003 area of of the room's area is, covered by one sheet of newspaper.

Here, we have,

given that,

60 cm * 50 cm what is the area covered by one sheet of newspaper if the area of the room is 79.2 m square​.

we know, that,

area of one sheet of newspaper = 60 cm * 50 cm

                                                      = 3000 cm^2

we know that,

100 cm = 1 m

so, 3000 cm^2 = 0.3 m^2

now, we have,

the area of the room is 79.2 m square​.

so, we get,

the area covered by one sheet of newspaper = 0.3/ 79.2

                                                                            = 0.003

Hence, The solution is: 0.003 area of of the room's area is, covered by one sheet of newspaper.

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The number of adult tickets sold were 80.

Let's denote the number of adult tickets sold as A and the number of student tickets sold as S.

The number of student tickets sold was two times the number of adult tickets sold:

S = 2A

We also know that the total number of tickets sold was 240:

A + S = 240

Substituting the value of S from the first equation into the second equation, we have:

A + 2A = 240

Combining like terms:

3A = 240

Divide both sides of the equation by 3:

A = 80

Therefore, 80 adult tickets were sold.

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Answer:

dy/dx = 2x + 2

Step-by-step explanation:

the slope (gradient) is given by differentiating y = x² + 2x.

dy/dx = 2x + 2