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.
Max has a bowl of 30 grapes of which 60% of which are red grapes
The cone is exactly half full of water by volume.how deep is the water in the cone
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.
What is formula to find perimeter of trapezoidal prism?
Answer:
7 duckpin utuyuuuu I have no idea where to watch the video is not a good day at work now and
How do I find the critical values and rejection region? and how is it graphed?
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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1. There were 45 more crossword puzzle books than jigsaw puzzles. 5/9 of the crossword puzzle books and 25% of the jigsaw puzzles were sold. Among the items sold, there were 91 more crossword puzzle books than jigsaw puzzles. How many crossword puzzle books and jigsaw puzzles remained in total?
2. In a warehouse, 60% of the table lamps were 30 more than 80% of the standing lamps. Given that there were a total of 155 table and standing lamps in the warehouse at first, how many table lamps were left if 30% of them were removed from the warehouse?
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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Can someone please help me. I need so much help with these two questions. Find all six trigonometric ratios for the triangle shown. Show all work.
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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(Scatterplots/Line
of Best Fit) Michael *
is a manufacturing analyst. He records
the number of water bottles produced
by five companies along with their
production costs in the table below.
Create a line of best fit and determine
the number of water bottles that can
be produced at a production cost of
$5,258.
Water Bottles Produced Production Cost
1,200
$885
1,825
$1,402
2,658
$1,868
4,022
$3,048
6,280
$4,582
Approximately 7,173 water bottles
can be produced
Approximately 70,181 water bottles
can be produced
Approximately 3,259 water bottles
can be produced
Approximately 3,854 water bottles
can be produced
7173 water bottles that can be produced at a production cost of $5,258.
To find the slope (m), we can use the formula:
m = (4582 - 885) / (6280 - 1200)
m = 3697 / 5080
m ≈ 0.7274
Now, let's find the y-intercept (b) using the equation:
b = y - mx
b = 885 - 0.7274 x 1200
b = 885 - 872.88
b ≈ 12.12
Therefore, the equation of line is
y = 0.7274x + 12.12
Now, the production cost of $5,258 means put y= 5258
5258 = 0.7274x + 12.12
5239.88 = 0.7274x
x= 7203
Thus, 7173 water bottles that can be produced at a production cost of $5,258.
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Example
8 ft
6 ft
10 ft
15 ft
1. Area of each base:
2. Lateral area:
0:00/2:48
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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What is the area of this figure?
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
A total of 240 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was two times the number of adult tickets sold. How many adult tickets were sold?
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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Termino en la sucesión 28 23 18 13
Hello !
28 23 18 13 8 ...
Para ir de izquierda a derecha, restamos 5 cada vez.
Lucy is riding on a bike course that is 88 miles long. So far, she has ridden 22 miles of the course. What percentage of the course has Lucy ridden so far?
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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How do I solve for x? please explain.
The value of x in the intersecting chords is determined as 5.
What is the value of x?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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Find the midpoint, M of AB A=(-3,-1) B=(-7,-7)
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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The average price of a movie ticket in 1990 was $2.97. Since then, the price has
increased by approximately 2.4% each year. Find the price of a ticket in 2014 using an
exponential function to model the situation. Round your answer to the nearest cent
(hundredth).
Britney got a summer job at a movie theater cleaning the aisles after each film. This Sunday, 8 movies
are scheduled to show, 5 of which feature a teenager as the one main protagonist.
If Britney is randomly assigned to clean up after 4 movies, what is the probability that all of them
feature a teenager as the one main protagonist?
Write your answer as a decimal rounded to four decimal places.
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 radius of a circle is 10 feet. what is the circumference Give the exact answer in simplest form
Answer: 62.83
Step-by-step explanation:
The formula for the circumference of a circle is C = 2 π r
So, let’s set up our equation.
2 •π •10 (your radius is 10!)
Now just multiply it together.
2 • π • 10 = 62.83 ft^squared
Hopefully this is what you were looking for. :)
Can yall help me with this im struggling
Check the picture below.
The radius of a circle is 6 meters. What is the area? Give the exact answer in simplest form
Answer:
Step-by-step explanation:
area of circle = pi*r^2
pi * 36 = 36pi = 113.04 which is about 113
Please solve the question
[tex]\frac{f(x)}{g(x)} =\frac{-10x}{x+3\\}\\ (\frac{f}{g} )(2)=\frac{-10(2)}{2+3}\\ \\\frac{-20}{5} =-4[/tex]
Answer is -4
Two lines intersect and create four angles one angle is 70 degrees the angle x and y are both linear angle pairs with 70 degrees the angles x is a vertical angle to 70 degrees find the degrees of the angles x and y and z.
Angle x is 70 degrees, angle y is 110 degrees, and angle z is also 110 degrees.
When two lines intersect, they form four angles known as vertical angles. Given that one angle is 70 degrees, we can determine the degrees of angles x, y, and z.
Vertical angles are congruent, meaning they have the same measure. Thus, angle x is also 70 degrees.
Since angles x and y are linear angle pairs with 70 degrees, their sum is 180 degrees. Therefore, angle y can be calculated as 180 - 70 = 110 degrees.
To find the measure of angle z, we can use the fact that the sum of angles in a straight line is 180 degrees. Angle x and angle z form a straight line, so their sum is 180 degrees. Angle x is 70 degrees, so angle z can be found by subtracting 70 from 180: 180 - 70 = 110 degrees.
In summary, angle x is 70 degrees, angle y is 110 degrees, and angle z is also 110 degrees.
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Find the missing angle of arc:
W
88°
●
?
X
176° is the measurement of the angle asked.
From the Theorem,
An angle at the circumference of a circle is half the angle at the center subtended by the same arc.
The desired angle will be equal to twice the angle WYX
Thus,
Angle made by the arc = 2 * 88
Angle made by the arc = 176°
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all formulars for solving Trigonometry
Answer:
These are all of the formulars for solving Trigonometry:
-sin θ = Opposite Side/Hypotenuse
-cos θ = Adjacent Side/Hypotenuse
-tan θ = Opposite Side/Adjacent Side
-sec θ = Hypotenuse/Adjacent Side
-cosec θ = Hypotenuse/Opposite Side
-cot θ = Adjacent Side/Opposite Side
Here is the visual edition below:
11 over 7=6 over v what is v
Answer:
42/11
Step-by-step explanation:
11/7 = 6/v
cross multiply:
11v = 6x7 = 42.
v = 42/11 ≈ 3.81
if possible please solve on paper so I can better understand thank you
Answer:
Step-by-step explanation:
10 : 25
then simplifies
2:5
Karla bought 3 packages of chicken. The total weight of the chicken in these packages is 7.52 pounds. This table shows the weight of the chicken in two packages
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:
good people please help!!!!!!! please!!!!!!
Clearer image, please!
help me please
with this
Please
Can someone answer this question
Answer:2nd one
Step-by-step explanation:
Find an equation for the slope of the graph of y=x²+2x at any point.
Answer:
dy/dx = 2x + 2
Step-by-step explanation:
the slope (gradient) is given by differentiating y = x² + 2x.
dy/dx = 2x + 2