find the general solution of the given higher-order differential equation. 16 d 4y dx4 16 d2y dx2 4y
The general solution of the given higher-order differential equation is y(x) = A * e^(-x/2) * cos((x/√2)) + B * e^(-x/2) * sin((x/√2)).
To find the general solution of the given higher-order differential equation, first, let's rewrite the equation in a more standard form:
16y'''' + 16y'' + 4y = 0
Now, we'll solve this fourth-order linear homogeneous differential equation using the characteristic equation method.
1. Write the characteristic equation:
16r^4 + 16r^2 + 4 = 0
2. Divide the equation by 4 to simplify:
4r^4 + 4r^2 + 1 = 0
3. Apply a substitution: let t = r^2, then the equation becomes:
4t^2 + 4t + 1 = 0
4. Solve the quadratic equation for t using the quadratic formula:
t = (-B ± √(B^2 - 4AC)) / 2A
t = (-4 ± √(4^2 - 4(4)(1))) / (2 * 4)
t = (-4 ± √(16 - 16)) / 8
t = -4/8 = -1/2
5. Substitute back r^2 for t:
r^2 = -1/2
6. Solve for r:
r = ±√(-1/2)
r = ±(1/√2)i
7. Since the roots are complex conjugates, the general solution can be written as:
y(x) = A * e^(-x/2) * cos((x/√2)) + B * e^(-x/2) * sin((x/√2))
where A and B are arbitrary constants determined by the initial conditions of the problem.
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Solve −9=2/7x+5
x=________
Due today please help
The area of the triangles are:
Triangle 1: 144√3 unit²
Triangle 2: 64√3 unit²
How to find the area of the triangle?
The area of a triangle is given by the formula:
A = 1/2 * b * h
where b is the base and h is height of the triangle
1st triangle:
The vertical line divides the base into two right triangles.
Considering the right triangle to the right. Let L represent the length of the base. Thus, we can say:
tan 60° = (12√3)/L
√3 = (12√3)/L
(√3)L = 12√3
L = 12
Since b = 2* L
b = 24 units
A = 1/2 * 24 * 12√3
A = 144√3 unit²
2nd triangle:
cos 60° = L/16
0.5 = L/16
L = 0.5 * 16
L = 8 units
b = 2 * 8 = 16 units
h = √(16² - 8²) (Pythagoras)
h = 8√3 units
A = 1/2 * 16 * 8√3
A = 64√3 unit²
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The experimental probability of choosing the name Maya is ___.
The experimental probability of choosing the name Maya is [tex]23/118[/tex].
and theoretical probability is of choosing name maya is [tex]1/5[/tex]
What is Probability?Probability is a measure of how likely an event is to occur. Many events are impossible to forecast with absolute accuracy. We can only anticipate the possibility of an event occurring, i.e. how probable they are to occur, using it. Probability can range from 0 to 1, with 0 indicating an impossible event and 1 indicating a certain event. The probability of all events in a sample space equals one.
According to the given information:
The experimental probability = [tex]\frac{Number of times an event occur}{ Total number of trials}[/tex]
Here total trials = 118,
The number of times the name maya has been chosen =23 times.
Therefore, experimental probability = [tex]\frac{23}{118}[/tex].
Theoretical probability = [tex]\frac{Number of favourable outcome}{Number of possible otucome}[/tex]
favourable outcome = 1 (Maya)
Possibleble outcome = 5 (Total no of Names in a hat)
And the theoretical possibility of the name maya has been chosen is [tex]1/5[/tex]
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an artist needs to enlarge a statue by making each linear dimension 75% greater. if n cm3 of clay was used to make the original sculpture, how many cubic centimeters of clay are needed to make the larger sculpture?
To make the larger sculpture with each linear dimension 75% greater, the artist needs to use 265.625% or 3.65625 times more clay than the original sculpture.
Let's assume that the original sculpture is a cube with side length x cm. Therefore, its volume would be x³ cm³.
To make the larger sculpture, each linear dimension needs to be increased by 75%, which means that each side length will become 1.75x cm. Thus, the new volume of the sculpture will be (1.75x)³ = 5.378125 x³ cm³.
To find the amount of clay required to make the larger sculpture, we need to find the ratio of the new volume to the original volume (5.378125 x³ cm³) / (x³ cm³) = 5.378125
This means that the artist needs to use 5.378125 times more clay than the original amount used to make the larger sculpture.
Therefore, to find the actual amount of clay needed, we multiply the original amount of clay by 5.378125: n cm³ x 5.378125 = 5.378125n cm³
So, the artist needs to use 3.65625 or 265.625% times more clay than the original sculpture to make the larger sculpture with each linear dimension 75% greater.
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There are 101 girls and 105 boys in Year 11 in a school. One girl is going to be chosen to be Head Girl and a different girl to be the Deputy Head Girl. One boy is going to be chosen to be Head Boy and a different boy to be the Deputy Head boy. How many different possible selections could be made?
110,664,000 possible selections for Head Girl, Deputy Head Girl, Head Boy, and Deputy Head Boy from a group of 101 girls and 105 boys.
How to calculate different possible selections could be made?
To find the number of possible selections that can be made, we need to multiply the number of choices available for each position.
For the Head Girl position, there are 101 girls to choose from. After one girl is selected for Head Girl, there are 100 girls remaining to choose from for the Deputy Head Girl position. So there are a total of 101 x 100 = 10,100 possible combinations of girls that can be chosen for these two positions.
Similarly, there are 105 boys to choose from for Head Boy, and 104 boys remaining to choose from for Deputy Head Boy. So there are a total of 105 x 104 = 10,920 possible combinations of boys that can be chosen for these two positions.
To find the total number of possible selections, we multiply the number of combinations for the girls by the number of combinations for the boys:
10,100 x 10,920 = 110,664,000
Therefore, there are 110,664,000 different possible selections that could be made.
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Please Help!
Which line of music shows a translation?
Answer:
3 one
Step-by-step explanation:
dy/dx=(x+1)(y+1) with initial condition of y(4)=3
The particular solution to the differential equation dy/da = (a+1)(y+1) with the initial condition y(4) = 3 is:
[tex]y = 4e^{\frac{1}{2}a^2 + a + 10} - 1[/tex]
What is differential equation?
Differential equations are used to model a wide range of phenomena in science, engineering, and mathematics, including physics, chemistry, biology, economics, and engineering.
Integration is a mathematical concept and technique used to find the integral of a function. The integral of a function is a measure of the area under the curve of the function, between two specified points on the x-axis.
To find the particular solution to the differential equation
[tex]\frac{dy}{da} = (a+1)(y+1)[/tex]
with the initial condition y(4) = 3, we can use separation of variables:
[tex]\frac{dy}{y+1} = (a+1) da[/tex]
Integrating both sides:
[tex]\int \frac{dy}{y+1} = \int (a+1) da\ln |y+1| = \frac{1}{2}a^2 + a + C_1, \text{ where } C_1 \text{ is an arbitrary constant of integration}[/tex]
Solving for y, we get:
[tex]y+1 = Ce^{\frac{1}{2}a^2 + a}[/tex]
where C is another arbitrary constant of integration. We can solve for C using the initial condition y(4) = 3:
[tex]3+1 = Ce^{\frac{1}{2}(4^2) + 4}C = 4e^{10}[/tex]
Substituting back, we get:
[tex]y+1 = 4e^{\frac{1}{2}a^2 + a + 10}y = 4e^{\frac{1}{2}a^2 + a + 10} - 1[/tex]
Therefore, the particular solution to the differential equation dy/da = (a+1)(y+1) with the initial condition y(4) = 3 is:
[tex]y = 4e^{\frac{1}{2}a^2 + a + 10} - 1[/tex]
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The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Then answer and explanation is in the doc below.
Express as a product
[tex]1-4sin^{2} a[/tex]
Thus, the expression [1 - 4sin²a] written in the product of its values is:
(1 + 2sina)(1 - 2sina).
Explain about the algebraic properties?We can solve mathematical equations thanks to algebra's inherent characteristics. Take note that both addition and multiplication adhere to these characteristics.
One is the multiplicative identity of real numbers. Any real number will return to its original value after being multiplied by one.Zero is the additive identity for adding real numbers.The multiplicative identity is 1, whereas the additive identity is 0.This Order of Operations is a set of rules that enables mathematicians to solve issues uniformly. In this order:
Do anything indicated by a parenthesis first.Exponents: All exponents (powers) must then be evaluated.Division and multiplication are both performed simultaneously from left to right.Addition as well as subtraction are also performed simultaneously, from left to right.using the algebraic formula:
(a² - b²) = (a + b)(a - b)
Given expression:
= 1 - 4sin²a
This can be written as:
= 1² - (2sina)²
Now, using identity:
= (1 + 2sina)(1 - 2sina)
Thus, the expression [1 - 4sin²a] written in the product of its values is:
(1 + 2sina)(1 - 2sina).
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Complete question-
Express the given as the product of both values.
1 - 4sin²a
the data are the number of machines in a gym. you sample five gyms. one gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. what type of data is this? group of answer choices qualitative data
As per the number of gym machines, the type of data is b. quantitative discrete data.
A discrete quantitative variable has a distinct quantitative meaning but can only take particular integer values rather than any value within a period. The number of needle sticks, the number of births, and the number of hospitalisations are a few examples of discrete numeric factors.
Similarly to that, the information given in the question refers to the quantity of gym equipment, which qualifies as quantitative data because it is a numerical assessment. Furthermore, rather than representing a continuous range of values, the data is discrete, which means that it reflects a limited collection of whole integers such as 10, 12, 15, 20, and 22. These numbers are the total number of gym machines that are available.
Complete Question:
the data are the number of machines in a gym. you sample five gyms. one gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. what type of data is this?
a. Qualitative data
b. Quantitative discrete data
c. Discreet data
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help asap will give brainliest!!!
Answer:
15.625 inches^3
Step-by-step explanation:
The volume of a cube is a simple calculation:
V=a^3
"a" is any edge of the cube.
Therefore, it would be (2.5^3), which would be 15.625 inches^3
You have $50 to buy T-shirts. You can buy 3 T-shirts for $24. Do you have enough money to buy 7 T-shirts? Justify your answer.
No , because 7 t shirts will cost 56 $ in linear equation.
What in mathematics is a linear equation?
An algebraic equation with simply a constant and a first-order (linear) component, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
You have $50 to buy T-shirts.
You can buy 3 T-shirts for $24.
You can buy 1 T-shirts for = 24/3 = 8
then 3 + 3 = 6 = 24 + 24 = 48 + 8 = $56
so, you don't buy 7 t - shirts because you have only $56 .
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A rectangular prism can hold 600 in³. Which of the boxes described below could NOT be this rectangular prism?
Group of answer choices
A. 5 in by 20 in by 6 in
B. 15 in by 4 in by 10 in
C. 2 in by 15 in by 20 in
D. 20 in by 30 in by 10 in
Answer:
D. 20 in by 30 in by 10 in
Step-by-step explanation:
The rest of them would equal 600 in³ but answer choice D when multiplied together equaled 6,000 in³.
Is -20 a reasonable solution to -6+2log(-5x)=-2?
A) Yes, -20 is a reasonable solution because at x=-20 the logarithm is defined and the equation is true.
B) No, -20 is NOT a reasonable solution because logarithmic equations cannot have negative solutions.
C) No, -20 is NOT a reasonable solution because at x=-20, the logarithm is not defined.
asking again but this time its worth 20pts
PLEASE HELP!
option (c) -No, 20 is NOT a reasonable solution because at x=-20,
what is linear equation?an equation in which highest power of variable is always 1.
by solving the equation -6+2log(-5x)=-2 :
-6 + 2log(-5x) = -2
2log(-5x) = 4
log(-5x) = 2
-5x = 10^2
-5x = 100
x = -20
So, we found that x = -20 satisfies the equation. However, we need to check whether it is a reasonable solution or not.
Option C is correct: -20 is NOT a reasonable solution because at x = -20, the logarithm is not defined. The logarithm function is defined only for positive real numbers.
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Regenerate response
The perimeter of a rectangle is 110, and the width is 4 times
the length. What is the width?
O 33
O 44
O 66
O 55
Answer:
44
Step-by-step explanation:
come on now.
the perimeter is
2×length + 2×width = 110
width = 4×length
so, we use the second equation in the first :
2×length + 2×(4×length) = 110
2×length + 8×length = 110
10×length = 110
length = 110/10 = 11
width = 4×length = 4×11 = 44
Which ratio could be used to determine the scale factor?
Step-by-step explanation:
6.3 goes to 2.52
6.3 : 2.52 = 2.5 :1 = 5 : 2
PLS HELP !! WILL GIVE BRAINLIEST
The area of a circle is 16m square inches. What is the diameter of the circle?
Answer:
The answer to your problem is, 4.51 or in option terms A. 4
Step-by-step explanation:
A=π [tex]R^{2}[/tex]
d = 2 r
We would need to solve for D
d = 2 [tex]\sqrt{\frac{A}{tt} } = 2 x \sqrt{\frac{16}{tt} }-around- 4.51352-or-4.51[/tex]
Thus the answer to your problem is, 4.51 or A. 4
Find the surface area of the following figure in terms of pi.
Answer:
A
Step-by-step explanation:
Given:
r (radius) = 2 in
h (height) = 8 in
Find:
A (surface area) - ?
.
First, we have to find the lateral area of the cylinder:
[tex]a(lateral) = 2\pi \times r \times h = 2 \times π\times 2 \times 8 = 32\pi \: {in}^{2} [/tex]
Now, let's find the area of both bases (since a cylinder has 2 identical bases, we have to multiply the area of one base by 2):
[tex]a(2 \: bases) = 2 \times \pi {r}^{2} = 2 \times π\times {2}^{2} = 8\pi \: {in}^{2} [/tex]
In order to find the surface area, we have to add both lateral and bases' areas together:
[tex]a(surface) =32\pi + 8\pi = 40\pi \: {in}^{2} [/tex]
Help me please this is trigonometry and I don’t understand
Answer:
x ≈ 8.0
Step-by-step explanation:
Using the tangent of the angle is equal to the opposite over the adjacent
tan(37º) = 6/x
x = 6/tan(37º)
x=7.96226892972
at a office supply store, the ratio of computers to printers is 7:5 . if the office supply store receives a shipment with 28 computers , how many printers can be expected in the shipments?
Answer:
20
Step-by-step explanation:
7 computers every 5 printers
7/5 = 28/x
Cross multiply
7x = 5*28
7x = 140
x = 20
Or in words,
7 times 4 is 28
5 times 4 is 20, which is your answer
Kelvin has a bag containing 5 red marbles, 2 blue marbles, 4 green marbles, and 1 white marbles. He randomly picks one marble from the bag. What is the theoretical probability that he will select a red marble
Thus, the theoretical probability that Kelvin will select a red marble is 5/12.
Define about the theoretical probability:Probability that is established using logic is known as theoretical probability.
Experimental probability is calculated based on the outcomes of numerous iterations of an experiment.
A probability is a number that falls between (and includes) 0 and 1.
If the likelihood of a situation E is represented by P(E), then:
In the event that E is an impossibility, P(E) = 0.When and only when E is a specific event, then P(E) = 1.Given:
Total marbles in bag = 12
5 red marbles, 2 blue marbles, 4 green marbles, and 1 white marbles.theoretical probability P(E) = favourable outcome / total outcome
P(red marble) = 5/12
Thus, the theoretical probability that Kelvin will select a red marble is 5/12.
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please help me !! i’m struggling with this
Answer:
Step-by-step explanation:
3 divided by 3/4 can be changes to multiplication
We can solve the equation like 3*4/3
3 divided by 3/4 is 4
What is the answer:
6x-12=
Answer:
x=2
Step-by-step explanation:
Answer: 6x-12= -72
Step-by-step explanation: 12 is a bigger number than 6 and a negative times a positive is always a negative. 6x12 is already 72 but we have to keep the negative so it becomes -72.
Or if its another equation involving x then it's going to be 2. ;)
PLS Give answer for 7. 8. 9.
Answer:
7. shifted right 8 units
8. shifted left 4 and up 12 units
9. shifted up 8 units
Step-by-step explanation:
You want to compare the given functions with p(x) = a(x -h)² +k and tell the value and effect of a, h, and k on the vertex.
TransformationsIn the equation for p(x) above, the value 'a' represents a vertical scale factor. It expands the graph vertically by the factor 'a'. If a < 0, the graph is reflected across the x-axis.
The values 'h' and 'k' in the equation for p(x) represent a right shift of the graph by 'h' units and a shift up by 'k' units. That is, the graph is translated by (h, k).
7. g(x)We have h = 8. The graph is shifted right 8 units.
8. h(x)We have (h, k) = (-4, 12). The graph is shifted left 4 units and up 12 units.
9. f(x)The values of the transformation parameters are a = -1/2 and k = 8. The vertical compression and reflection do not affect the vertex. The vertex is shifted up 8 units.
Cual es la mitad de 432
Answer: 216
Step-by-step explanation:
Brainliest pls:)
HS HONORS GEOMETRY!!!
PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST
In ALMN, m/L= 40° and m/M = 41°. Which statement about the sides of ALMN
must be true?
OMN > LM > NL
OLM > NL > MN
ONLMN > LM
ONL> LM > MN
OMN > NL > LM
OLM > MN > NL
We know that the sum of the angles of a triangle is 180 degrees. So, we can find the value of angle LNM as follows:
m/L + m/M + m/N = 180 degrees
Substituting the given values, we get:
40 + 41 + m/N = 180
m/N = 99
Now, we can see that angle LNM is greater than both angles L and M. So, we can conclude that:
OMN > NL > LM
Helllo pleas answer all of them with work !
Vertical angles are a pair of opposite angles formed by two intersecting lines. Vertical angles are congruent, meaning they have the same measure.
For example, if two lines intersect and form four angles, the two angles that are opposite each other are vertical angles.
Adjacent angles are two angles that have a common vertex and a common side but do not overlap.
In other words, they share a ray or line segment, but they don't share any interior points.
Adjacent angles may or may not be congruent, depending on their measurement.
For example, if two angles are next to each other and share a vertex and a side, they are adjacent angles.
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PLEASE HELP ME PLEASE IM BEGGING PLEASE ANSWER CORRECTLY!!!!
The equation of the line parallel to the line shown in the graph passing through the point (-2, 3) is y = (2/3)x + (13/3) and the equation of the line perpendicular to the line shown in the graph passing through the point (-2, 3) is y = (-3/2)x.
What is slope?
The slope of a line passing through two points [tex](x_1, y_1) \: and \: (x_2, y_2)[/tex] is given by slope = [tex]\frac{(y_2 - y_1)}{(x_2 - x_1)}[/tex]
Given line passes through (0,-6) and (9,0)
To find the equation of a line parallel to a given line, we need to use the fact that parallel lines have the same slope.
The slope of the line passing through (0,-6) and (9,0) can be found using the slope formula
slope = (change in y) / (change in x)
= (0 - (-6)) / (9 - 0)
= 6 / 9
= 2/3
Therefore, any line parallel to this line will also have a slope of 2/3.
We can now use the point-slope form of a line to find the equation of the line passing through (-2,3) with a slope of 2/3:
y - y1 = m(x - x1) (point-slope form)
where m is the slope, and (x1,y1) is a point on the line.
Substituting the values, we get:
y - 3 = (2/3)(x - (-2))
y - 3 = (2/3)(x + 2)
y - 3 = (2/3)x + (4/3)
y = (2/3)x + (4/3) + 3
y = (2/3)x + (13/3)
Therefore, the equation of the line parallel to the line passing through (0,-6) and (9,0) shown in the graph passing through the point (-2, 3) is y = (2/3)x + (13/3).
Using the points (0,-6) and (9,0), we can find the slope of the original line:
slope = (0 - (-6)) / (9 - 0) = 6/9 = 2/3
The slope of any line perpendicular to this line will be the negative reciprocal of this slope. So, the slope of the perpendicular line will be:
perpendicular slope = -1 / (2/3) = -3/2
Now we have the slope of the perpendicular line, and we also have a point it passes through: (-2, 3). We can use the point-slope form of a line to write the equation of the perpendicular line: y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is the point it passes through. Plugging in the values we have, we get:
y - 3 = (-3/2)(x - (-2))
Simplifying:
y - 3 = (-3/2)x - 3
y = (-3/2)x + 0
So the equation of the line perpendicular to the line passing through (0,-6) and (9,0), and passing through (-2, 3), is y = (-3/2)x.
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