The general solution to the initial value problem is:
y(t) = y_c(t) + y_p(t)
= 8 cos(√7t) + cos(√7t)
To solve the initial value problem using the method of undetermined coefficients, we assume a particular solution of the form:
y_p(t) = A sin(√7t) + B cos(√7t)
where A and B are constants to be determined.
Taking the first and second derivatives of y_p(t), we have:
y'_p(t) = A√7 cos(√7t) - B√7 sin(√7t)
y''_p(t) = -A(√7)^2 sin(√7t) - B(√7)^2 cos(√7t)
Substituting these derivatives back into the differential equation, we get:
(-A(√7)^2 sin(√7t) - B(√7)^2 cos(√7t)) + 7(A sin(√7t) + B cos(√7t)) = 7 sin(√7t)
Simplifying, we have:
(-A(√7)^2 + 7A) sin(√7t) + (-B(√7)^2 + 7B) cos(√7t) = 7 sin(√7t)
To satisfy this equation for all t, the coefficients of sin(√7t) and cos(√7t) must be equal to the corresponding coefficients on the right side.
Therefore, we have the following system of equations:
-A(√7)^2 + 7A = 0 (coefficient of sin(√7t))
-B(√7)^2 + 7B = 7 (coefficient of cos(√7t))
Solving these equations, we find:
A = 0
B = 1
So, the particular solution is:
y_p(t) = cos(√7t)
To find the general solution, we need to find the complementary solution by solving the homogeneous equation:
y'' + 7y = 0
The characteristic equation is:
r^2 + 7 = 0
Solving this quadratic equation, we find two complex roots:
r₁ = √7i
r₂ = -√7i
The complementary solution is then:
y_c(t) = C₁ cos(√7t) + C₂ sin(√7t)
where C₁ and C₂ are constants to be determined.
Using the initial conditions y(0) = 8 and y'(0) = √7, we can substitute these values into the general solution and solve for C₁ and C₂.
y(0) = C₁ cos(0) + C₂ sin(0) = C₁ = 8
y'(0) = -C₁√7 sin(0) + C₂√7 cos(0) = C₂√7 = √7
Therefore, C₁ = 8 and C₂ = 1.
The general solution to the initial value problem is:
y(t) = y_c(t) + y_p(t)
= 8 cos(√7t) + cos(√7t)
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Need this answer as soon as possible
Answer:
x=22
Step-by-step explanation:
Rule states diagonals = same so 6x-36 = 96 in which x=22 cause 6(22)=132
132-36 = 96
Find the area of each rhombus. Write your answer as an integer or a simplified radical
Step-by-step explanation:
area of rhombus =1/2×d1×d2
area of rhombus =1/2×9cm×5cm
area of rhombus =22.5cm²
area of rhombus = 1/2×d1×d2
area of rhombus =1/2×8in×17in
area of rhombus= 68in²
Find mXY
X
23°
N
I
plz help
Answer:
arc XY = 46°
Step-by-step explanation:
The inscribed angle XZY is half the measure of its intercepted arc , then
arc XY = 2 × 23° = 46°
Consider the function y = 8x + 3 between the limits of x = 2 and x = 8.
a) Find the arclength L of this curve:
L: ___________ Round your answer to 3 significant figures.
b) Find the area of the surface of revolution, A, that is obtained when the curve isrotated by 2π radians about the x-axis.
Do not include the surface areas of the disks that are formed at x = 2 and x = 8.
A = ___________ Round your answer to 3 significant figures.
a) We have the function given by; y = 8x + 3We need to find the arclength of the curve between the limits of x = 2 and x = 8.The arclength L of the curve is given by; L = ∫(2,8) sqrt(1 + f'(x)²)dx Here, f(x) = 8x + 3 Differentiate f(x) with respect to x;f'(x) = 8Now, substitute f'(x) in the above equation; L = ∫(2,8) sqrt(1 + 8²)dx L = ∫(2,8) sqrt(65)dxL = sqrt(65)∫(2,8)dxL = sqrt(65) [x]₂⁸L = sqrt(65) [8 - 2]L = 6sqrt(65)Therefore, the arclength L of this curve is 6sqrt(65).
b) We are given the function y = 8x + 3We need to rotate this curve by 2π radians about the x-axis to get the required surface of revolution. The formula for the surface area of the surface of revolution generated by revolving the curve y = f(x) between x = a and x = b about the x-axis is given by;A = ∫(a,b) 2πf(x) sqrt(1 + f'(x)²)dx Here, f(x) = 8x + 3f'(x) = 8We know that the limits of integration are from x = 2 to x = 8.
Substitute the values in the above equation; A = ∫(2,8) 2π(8x + 3) sqrt(1 + 8²)dxA = 16π ∫(2,8) (8x + 3) sqrt(65)dxA = 16π [∫(2,8) (8x sqrt(65))dx + ∫(2,8) (3 sqrt(65))dx]A = 16π [2/3(8sqrt(65))² - 2/3(2sqrt(65))² + 3sqrt(65)(8 - 2)]A = 16π [2/3(8sqrt(65))² - 2/3(2sqrt(65))² + 3sqrt(65)(6)]A = 192πsqrt(65)
Therefore, the area of the surface of revolution, A that is obtained when the curve is rotated by 2π radians about the x-axis is 192πsqrt(65) square units.
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Find the flux of the vector field F across the surface S in the indicated direction.
F = x i + y j + z 2 k; S is portion of the cone z = 2 square root of x^2+y^2 between z = 2 and z = 4; direction is outward
The flux of the vector field F across the surface S in the indicated direction is 8π/3.
So, the flux of the given vector field F across the surface S can be calculated by the surface integral as follows:
Φ = ∫∫S F · dS = ∫∫S (xi + yj + z2k) · n(x, y, z) dS= ∫∫S (2x/z + 2y/z + z2(-1/2)) dS= ∫∫S (2x + 2y) / z dS= ∫0²∫2π 2rcosθ / z √(r² + z²) dr dθ= 8π/3.
The flux of the vector field F across the surface S in the indicated direction is 8π/3.
Given, vector field F = xi + yj + z2k,
S is the portion of the cone z = 2√(x² + y²) between z = 2 and z = 4 and the direction is outward.
The flux of the vector field F is given by the surface integral:Φ = ∫∫S F · dS .
Here, dS is the outward pointing unit normal vector of the surface S. Hence the flux Φ will be positive if F points outward, otherwise negative. The surface S can be parameterized as r(x, y, z) = xi + yj + zk, where z varies from 2 to 4 and (x² + y²) = (z²/4).
Then, the unit normal vector to the surface is given by n(x, y, z) = (2x/z)i + (2y/z)j - k/2.
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The degree of precision of a quadrature formula whose error term is 24 f'"'() is: 5 4 3 2
The degree of precision of the quadrature formula with an error term of 24 f‴() is 2.
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The degree of precision of a quadrature formula represents the highest power of x that the formula can integrate exactly. In this case, the error term of the formula is given as 24 f‴(), where f‴() denotes the third derivative of the function f(x). The degree of precision is determined by the highest power of x that appears in the error term.
In a quadrature formula, the error term typically has the form K * h^p, where K is a constant, h is the step size, and p is the degree of precision. In this case, the error term is 24 f‴(). We can see that there is no dependence on the step size h, which implies that h^p = h^0 = 1. Therefore, the highest power of x in the error term is determined by the highest power of x that appears in f‴().
Since the error term is 24 f‴(), it indicates that the highest power of x in f‴() is 1. Thus, the degree of precision of the quadrature formula is 2, as the highest power of x in the error term is two degrees less than the highest power of x that the formula can integrate exactly.
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in class of 40 pupils,20% are absent one day. (a) how many pupils are absent?
Answer: 8 pupils are absent
Step-by-step explanation:
20 percent * 40 =
(20* 40)/100 =
(800)100 =
=8
How many yards are equivalent to 38 feet? Sh
Answer:
16 2/3 yards
Step-by-step explanation:
Hope this helps and have a wonderful day!!!!
calculate the velocity of a stream if a drop of food coloring is timed traveling 32 feet in 15 seconds.
The velocity of the stream is approximately 2.13 feet per second.
To calculate the velocity of the stream, we can use the formula:
Velocity = Distance / Time
Given that the drop of food coloring travels 32 feet in 15 seconds, we can plug in these values into the formula:
Velocity = 32 feet / 15 seconds
To find the velocity, we divide 32 by 15:
Velocity ≈ 2.13 feet per second
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Imagine drawing a figure with the following conditions:
VX
A quadrilateral with at least two right angles. Is the figure
described unique? Explain why or why not.
Answer:
with atleast two right angles in rectangle and it do not describe unique because it is simple
please help i’ll give brainliest
Answer:
A insects and plants
Step-by-step explanation:
the decaying things are used as fertilizer for plants to grow and insects to eat
A 1-g antibiotic vial states "Reconstitute with 3.4 mL of sterile water for a final volume of 4 ml. * What is the powder volume in the vial?
A. 3.4 mL
B. 0.6 mL
C. 4 mL
D. 4.6 mL
The correct answer is option B. 0.6 mL which is the powder volume in the vial.
To determine that 0.6 mL of powder volume in the vial, we need to subtract the volume of the sterile water used for reconstitution from the final volume.
The vial states that it needs to be reconstituted with 3.4 mL of sterile water for a final volume of 4 mL. This means that 3.4 mL of sterile water will be added to the vial to make a total volume of 4 mL.
To find the powder volume, we subtract the volume of the sterile water (3.4 mL) from the final volume (4 mL):
Powder volume = Final volume - Volume of sterile water
Powder volume = 4 mL - 3.4 mL
Powder volume = 0.6 mL
Therefore, the powder volume in the vial is 0.6 mL.
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Lisa can mow the lawn in 3 hours. If Rhianna helps her with another mower, the lawn can be mowed in 2 hours. How long would it take Rhianna if she worked alone? PLEAsE HELPPPP
Answer:
5 hours or 300 mins
Step-by-step explanation:
Find the tangents of the acute angles in the right triangle. Write each answer as a fraction,
45
tan R-
tan S -
Answer:
Following are the solution to these question:
Step-by-step explanation:
Please find the complete question in the attachment file.
Formula:
[tex]\to \tan \theta= \frac{Perpendicular}{Base}[/tex]
[tex]\to \tan R = \frac{45}{28} \\\\ \to \tan S = \frac{28}{45}[/tex]
find the value of two numbers if their sum is 39 and their difference is 1
1/4x + 6 = 1/2 (x+4)
Answer:
Solution x=16
Step-by-step explanation:
solving for x, please help
Answer:
x = 5,-2
Step-by-step explanation:
Multiply the whole equation by 3(x+1) to get rid of the denominator.
[tex] \large{ \frac{x + 2}{3} \times 3(x + 1) = \frac{2(x + 2)}{x + 1} \times 3(x + 1)} \\ \large{(x + 2)(x + 1) = 2(x + 2) \times 3} \\ \large{ {x}^{2} + 3x + 2 = 6(x + 2)} \\ \large{ {x}^{2} + 3x + 2 = 6x + 12}[/tex]
Then we solve the equation.
[tex] \large{ {x}^{2} + 3x + 2 - 6x - 12 = 0} \\ \large{ {x}^{2} - 3x - 10 = 0} \\ \large{(x - 5)(x + 2) = 0} \\ \large{x = 5, - 2}[/tex]
Since we are also solving rational equation, make sure that x ≠ -1 for this problem. Because if x = -1, the denominator is 0 which is undefined. Since our solutions are x = 5,-2 and not -1. Therefore the answer is x = 5,-2
The work of a student trying to solve the equation 2(4x − 3) = 9 + 2x + 6 is shown below: Step 1: 2(4x − 3) = 9 + 2x + 6 Step 2: 8x − 3 = 15 + 2x Step 3: 8x − 2x = 15 + 3 Step 4: 6x = 18 Step 5: x = 3 In which step did the student first make an error and what is the correct step? (4 points) a Step 2: 8x − 3 = 15 + 2x b Step 2: 8x − 6 = 15 + 2x c Step 3: 8x + 2x = 15 + 2 d Step 3: 8x − 2x = 15 − 2
Answer:
Step-by-step explanation:
2(4x - 3) = 9 + 2x + 6 Combine the like terms on the right
2(4x - 3) = 2x + 15 The distributive property gets rid of the brackets
8x - 6 = 2x + 14 Add 6 to both sides
6 6
8x = 2x + 20 Subtract 2x from both sides
-2x -2x
6x = 20
Step 2 is the error. It is a very common error. You multiply 2 and - 3 together. as well as 2 and 4 together. If you are going to forget something that will be it.
Solve this please!!!!
Answer:
The volume of the rectangular prism is 336
The volume of the triangular prism is 24.875
Add them together, 360.875. You can round if you want
Find the equilibrium vector for the transition matrix 0.47 0.19 0.34 0 0.45 0.55 0 0 1 The equilibrium vector is (Type an integer or decimaldor each matrix element)
The equilibrium vector for the given transition matrix is approximately (0.359, 0.359, 0.284).
To find the equilibrium vector, we need to solve the equation [tex]T * v = v[/tex], where T is the transition matrix and v is the equilibrium vector.
Let's denote the equilibrium vector as (x, y, z). Setting up the equation, we have:
[tex]0.47x + 0.19y + 0.34z = x\\0.45x + 0.55y + 0z = y\\0x + 0y + 1z = z[/tex]
Simplifying the equations, we get:
[tex]0.46x - 0.19y - 0.34z = 0\\-0.45x + 0.45y = 0\\0x + 0y + 1z = z[/tex]
From the second equation, we can see that x = y. Substituting x = y in the first equation, we have:
[tex]0.46x - 0.19x - 0.34z = 0\\0.27x - 0.34z = 0[/tex]
Simplifying further, we get:
[tex]0.27x = 0.34z\\x = (0.34/0.27)z\\x = 1.259z[/tex]
Since the equilibrium vector must sum to 1, we have:
[tex]x + y + z = 1\\1.259z + 1.259z + z = 1\\3.518z = 1\\z - 0.284[/tex]
Substituting the value of z back into x, we get:
[tex]x = 1.259 * 0.284=0.359[/tex]
Therefore, the equilibrium vector is approximately (0.359, 0.359, 0.284).
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Theoretically, if a month is chosen 300 times,
how many times would you expect a month
that starts with the letter J?
Answer:
75 times.
Step-by-step explanation:
Well there are 12 months and 3 of them start with the letter J, so theoretically, 25% of the months chosen will start with the letter J because [tex]\frac{3}{12}=\frac{1}{4}[/tex], which is 25%. 25% of 300 (or [tex]\frac{300}{4}[/tex], for those who like fractions) is 75, so theoretically, we can expect a month that starts with the letter J 75 times.
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On a coordinate plane, a curved line with a minimum value of (1.5, negative 1) and a maximum value of (negative 1.5, 13), crosses the x-axis at (negative 3, 0), (1, 0), and (2, 0), and crosses the y-axis at (0, 6).
Which lists all of the x-intercepts of the graphed function?
(0,6)
(1,0)(2,0)
(1,0)(2,0) and (-3,0)
(1,0)(2,0)(-3,0) and (0,6)
Answer:
c
Step-by-step explanation:
In the diagram, mBDA = 150°. Find mBDC.
Answer:
a. 70°
Step-by-step explanation:
m∠BDC + m∠CDA = m∠BDA
-3x + 34 - 2x + 56 = 150
-5x + 90 = 150
-5x = 60
x = -12
m∠BDC = -3x + 34 = -3(-12) + 34
= 36 + 34 = 70°
Five more then six times a number is 17.what is the number
Answer:
2
Step-by-step explanation:
You set up and equation:
5+6x = 17
You solve it and get 2.
Step-by-step explanation:
17-5 = 12
Zoe works in a bakery. She uses 250 milliliters of milk to make a loaf of bread. How many liters of milk will she need to make 15 loaves of bread?
Answer: 3.75 liters
Step-by-step explanation:
Zoe uses 250 milliliters of milk to make a loaf of bread.
If she needed to make 15 loaves therefore, she would do the following:
= 250 * 15 loaves
= 3,750 milliliters of milk
Then convert the above quantity to liters.
1 Liter = 1,000 milliliters.
3,750 milliters to liters is:
= 3,750 / 1,000
= 3.75 liters
11 inches in miles?
24.5 km in miles?
Answer:
0.000173611 miles is 11 inches
15.22359 miles is 24.5km
Step-by-step explanation:
. Amy Company sold merchandise of $8,000 to Tory Turnbull with terms 2/10, n/30. Amy Company recorded this transaction using the gross method. If Tory Turnbull paid for all the merchandize within the discount period, the journal entry that Amy Company will make to record the collection of cash would include a: a. Credit to Sales Discount of $160 b. Credit to Account receivable $7,840 c. Debit to Sales Discount of $160 d. Credit to Cash of $160 Select-
The journal entry that Amy Company will make to record the collection of cash from Tory Turnbull, who paid within the discount period, would include a credit to Cash for $160. Therefore, option d is the correct answer.
This is because Tory Turnbull will pay $8,000 - $160 (2% of $8,000) to avail the discount. The Sales Discount account is not involved in the journal entry as the discount was taken by the customer, not given by Amy Company.
Therefore, the correct answer is d. Credit to Cash of $160. This entry reflects the cash received by Amy Company and the reduction in the Accounts Receivable balance for the amount paid by the customer.
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Is this a Function or not a Function?
Help
Answer:
yes
Step-by-step explanation:
no
Sara had 27 peaches and 11 pears left at her roadside fruit stand. She went to the orchard and picked more peaches to stock up the stand. There are now 63 peaches at the stand, how many did she pick
Answer:
36
Step-by-step explanation:
sara had 27 peaches when she left the stand. when she returns the total number of peaches is 63. So to find the answer you take 63 and subtract it by 27 to get your answer.
A random sample of 16 size A batteries for toys yield a mean of 3.29 hours with standard deviation, 1.4 hours.
(a) Find the critical value, t∗, for a 99% Confidence interval.
(b) Find the margin of error for a 99% Confidence interval.
a. Critical value, t∗, for a 99% Confidence Interval, n=16 is given by t∗=2.921.
b. The margin of error is 0.97.
a) Critical Value, t∗ for a 99% Confidence Interval: Critical value, t∗, for a 99% Confidence Interval, n=16 is given by t∗=2.921.
Note: t-table, with 15 degrees of freedom is used to determine the critical value for the given confidence interval.
b) Margin of Error for a 99% Confidence Interval:Margin of error for a 99% confidence interval, n=16 is given by E = t∗× s/√n, where s is the standard deviation.
E = 2.921 × 1.4/√16 = 0.97 (rounded off to two decimal places).
Hence, the margin of error is 0.97.
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Find the critical value, t∗, for a 99% Confidence interval.
The sample size is 16.The degree of freedom (df) = n - 1 = 16 - 1 = 15.
From t-table for 15 degrees of freedom (df) and 99% level of confidence (α = 0.01) (two-tail) t* = 2.947.a) t* = 2.947.
b) Find the margin of error for a 99% Confidence interval.
The margin of error (E) for a 99% confidence interval is given by:
E = t* × s / √n
Where s is the sample standard deviation.
s = 1.4 (given)E = 2.947 × 1.4 / √16E = 0.7285 or ≈ 0.73
Therefore, the margin of error for a 99% confidence interval is approximately 0.73.
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