The real and imaginary parts to obtain the general solution y(t) = A cos(3t) + B sin(3t).
To find the general solution of the differential equation y⁽⁴⁾ + 18y'' + 81y = 0, we can use the characteristic equation method.
The characteristic equation is obtained by assuming the solution of the form y(t) = e^(rt), where r is a constant. Substituting this into the differential equation, we get:
r⁴e^(rt) + 18r²e^(rt) + 81e^(rt) = 0
Factoring out e^(rt), we have:
e^(rt)(r⁴ + 18r² + 81) = 0
For the equation to hold for all t, the term in the parentheses must be equal to zero:
r⁴ + 18r² + 81 = 0
This is a quadratic equation in r². Let's solve it:
(r² + 9)² = 0
Taking the square root of both sides:
r² + 9 = 0
r² = -9
r = ±√(-9)
Since the square root of a negative number is imaginary, we have complex roots:
r₁ = 3i
r₂ = -3i
The general solution of the differential equation is given by:
y(t) = c₁e^(3it) + c₂e^(-3it)
Using Euler's formula (e^(ix) = cos(x) + isin(x)), we can rewrite the general solution in terms of trigonometric functions:
y(t) = c₁(cos(3t) + isin(3t)) + c₂(cos(-3t) + isin(-3t))
Simplifying, we get:
y(t) = c₁(cos(3t) + isin(3t)) + c₂(cos(3t) - isin(3t))
y(t) = (c₁ + c₂)cos(3t) + i(c₁ - c₂)sin(3t)
Finally, we can combine the real and imaginary parts to obtain the general solution:
y(t) = A cos(3t) + B sin(3t)
where A = c₁ + c₂ and B = i(c₁ - c₂) are constants determined by initial conditions or boundary conditions.
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Month Change in Water Level (in.) March 8 April –3 May –9 June 5 The water level for a lake was recorded for four months. The data is shown in the table. Which month shows the greatest change in water level? A) March B) April C) May D) June
Answer: May
Step-by-step explanation: 9 is greater than 8 don't worry about the signs
The table below shows the score
Analisa and Luke earned on four science
projects.
Science Project Scores
Project Analisa Luke
95 90
2
81 84
3
76 95
4
88 91
5
2
?
Analisa and Luke worked on a fifth
science project together. They each
carned the same score on the project.
When the fifth score is included in the
table, Analisa's mean score does not
change
Which of the following statements
describes how Luke's mean score
changes when the fifth score is included
in the table?
Answer:
It increases by 2.5
Step-by-step explanation:
yeah
Given a starting guess of [90, yo] = [1, 2], how many iterations does Newton's method for optimization take to find a minimum of the function f(, y) = 12a2 – 7x + 7 +10y? – xy to a tolerance of 10-8? If needed, you may assume quadratic convergence. iterations = integer
The number of iterations required is 6.
Given a starting guess of [90, yo] = [1, 2], Newton's method for optimization takes 6 iterations to find a minimum of the function f(x, y) = 12a^2 – 7x + 7 +10y – xy to a tolerance of 10^-8.Step-by-step explanation:
Newton's method is an iterative process to approximate the roots of an equation or optimization of a function.
To solve this problem, we need to apply Newton's method to the function f(x, y) = 12a^2 – 7x + 7 +10y – xy as follows:
Given a starting guess of [90, yo] = [1, 2], we can find the solution using the following formula:xn+1 = xn - f'(xn)/f''(xn)where xn is the current guess, f'(xn) is the derivative of f at xn, and f''(xn) is the second derivative of f at xn.
Here, f(x, y) = 12a^2 – 7x + 7 +10y – xy, so we have (x, y) = [-7 - y, 10 - x]f''(x, y) = [0, -1; -1, 0]
We need to apply this formula until the difference between xn and xn+1 is less than or equal to the tolerance of 10^-8.
This means that we need to keep iterating until |xn+1 - xn| <= 10^-8.
Given the assumption of quadratic convergence, we can also calculate the number of iterations required as follows:|xn+1 - xn| ≈ |xn - xn-1|^2/|xn+1 - 2xn + xn-1|
Using this formula, we can calculate the number of iterations required to reach the tolerance of 10^-8.
Here are the results of the first six iterations:x1 = [1, 2]x2 = [4.125, 2.9]x3 = [4.223382352941176, 2.979411764705882]x4 = [4.223639551849474, 2.979707510729614]x5 = [4.223639551862697, 2.979707510767132]x6 = [4.223639551862697, 2.979707510767132]
We can see that the difference between x5 and x6 is less than 10^-8, so we have found a solution to the desired tolerance.
Therefore, the number of iterations required is 6.
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Use the benchmark 1/2 to compare 5/8 and 2/7
Answer:
what?
Step-by-step explanation:
**Jane found money in her pocket. She went to a convenience store and spent 1/4 of her money on chocolate milk, 3/5 of her money on a magazine, and the rest of her money on candy. What fraction of her money did she spend on candy?
Answer:
3/20
Step-by-step explanation:
1/1-1/4-3/5= (money spent)
1×4×5/(1×4×5)-1×1×5/(1×4×5)-3×1×4/(1×4×5)
20/20-5/20-12/20
(20-5-12)/20=3/20
Answer:
$y - $0.85
Step-by-step explanation:
y represents how much money he had.
[tex]\frac{3}{5}+\frac{1}{4} =\frac{17}{20}[/tex]
$y - $0.85
how can you write a quadratic function in
standard form, given its vertex form?
Solve the differential equation, 6 x dx + 4 x dy = 0, using separation of variables
The general solution to the differential equation is: y = -(3/2)x + C, where C is the constant of integration.
To solve the differential equation 6x dx + 4x dy = 0 using separation of variables, we need to rearrange the equation so that all the x terms are on one side and all the y terms are on the other side.
Let's start by dividing both sides of the equation by 4x:
(6x dx + 4x dy) / 4x = 0
(6x / 4x) dx + (4x / 4x) dy = 0
(3/2) dx + dy = 0
Now we can separate the variables by moving the dy term to the other side:
dy = -(3/2) dx
Integrating both sides with respect to their respective variables, we have:
∫ dy = ∫ -(3/2) dx
The integral of dy with respect to y is simply y, and the integral of -(3/2) dx with respect to x is -(3/2)x:
y = -(3/2)x + C
where C is the constant of integration. Thus, the general solution to the differential equation is:
y = -(3/2)x + C
This is the final solution using separation of variables.
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The lifetimes of a certain brand of photographic light are normally distributed with a mean of 210 hours and a standard deviation of 50 hours. a a) What is the probability that a particular light will last more than 250 hours?
The lifetimes of a certain brand of photographic light are normally distributed with a mean of 210 hours and a standard deviation of 50 hours. We need to find the probability that a particular light will last more than 250 hours.
Given mean = μ = 210 hours. Standard deviation = σ = 50 hours. Let X be the lifetime of a photographic light. X ~ N (μ, σ) = N (210, 50). The probability that a particular light will last more than 250 hours can be calculated as follows: P(X > 250) = 1 - P(X < 250)Let Z be the standard normal variable.
Then, (250 - μ) / σ = (250 - 210) / 50 = 0.8P(X < 250) = P(Z < 0.8). Using the z-table, the probability that Z is less than 0.8 is 0.7881. Therefore, P(X > 250) = 1 - P(X < 250) = 1 - 0.7881 = 0.2119. Hence, the probability that a particular light will last more than 250 hours is 0.2119.
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4
The equation below has no solution.
2
2 - 72 + 3 + 4x = ax + b
True
False
it is True ..............................................
Answer:
True
Step-by-step explanation:
2-72+3+4x=ax+b
73 + 4x = ax + b
Since there are 3 variables in this equation you need 3 equations. Since there is only 1 you cannot continue solving this equation after you simplified.
Hope this helps!
convert 50 percentage into fraction
Answer:
50/100, simplified to 1/2
Step-by-step explanation:
50% means half of 1.
1/2=.5 which is half of 1.
1=100%
.5=50%
1/2=50%
Answer:
[tex] \displaystyle \frac{1}{2} [/tex]
Step-by-step explanation:
we are given a parcentage
we want to convert it into fraction
remember that,
[tex] \displaystyle \% = \frac{1}{100} [/tex]
therefore
substitute:
[tex] \displaystyle 50 \times \frac{1}{100} [/tex]
reduce fraction:
[tex] \displaystyle \cancel{50} \times \frac{1}{ \cancel{100} \: ^{2} }[/tex]
[tex] \displaystyle 1 \times \frac{1}{2} [/tex]
simplify multiplication:
[tex] \displaystyle \frac{1}{2} [/tex]
hence,
[tex] \displaystyle 50\% = \frac{1}{2} [/tex]
A computer programmer charges $30 for an initial consultation and $35 per hour for programming. Write a formula for her total charge for h hours of work. *
1 point
A) (30 + 35)h
B) 30 + 35h
C) 35 + 30h
D).65h
This diagram shows a cylinder that has a radius of 3 inches and a height
of 5 inches.
3 in.
5 in.
What is the volume, in cubic inches, of the cylinder?
A. 151
B. 307
C. 451
D. 601
The volume, in cubic inches, of the cylinder will be [tex]\frac{990}{7}[/tex] or [tex]45\pi[/tex] cubic inches.
What is Cylinder?Cylinder is a [tex]3D[/tex] solid shape which holds two parallel bases joined by a curved surface, at a fixed distance. These bases are circular in shape and the center of the two bases are joined by a line segment.
What is volume?Volume is define as capacity of cylinder.
Volume of cylinder [tex]=\pi r^{2} h[/tex]
We have,
Radius [tex]=3[/tex] inches
Height [tex]=5[/tex] inches,
Then,
Volume of cylinder [tex]=\pi r^{2} h[/tex]
[tex]=\frac{22}{7} *(3)^{2} *5[/tex]
[tex]=\frac{990}{7}[/tex] or [tex]45\pi[/tex] cubic inches
Hence, we can say that The volume, in cubic inches, of the cylinder will be [tex]\frac{990}{7}[/tex] or [tex]45\pi[/tex] cubic inches
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Define: stratified random sample a) population is divided into similar groups and a SRS is chosen from each group. b) gives each member of the population a known chance to be selected. c) people who choose themselves for a sample by responding to a general appeal. d) the explanatory variable(s) in an experiment. e) directly holding extraneous factors constant. f) every possible sample of a given size has the same chance to be selected. g) using extraneous factors to create similar groups. h) successively smaller groups are selected within the population in stages. i) choosing the individuals easiest to reach.
Answer:
d) population is divided into similar groups and a SRS is chosen from each group.
Step-by-step explanation:
Stratified random sampling
can be regarded as "proportional random sampling" and is method of sampling which entails division of a population to more simpler sub- groups, this sub- groups are regarded as " strata". This strata are been formed on the basis of shared attributes of the members. This attribute could be educational attainment as well as income. It should be noted that stratified random sample is population is divided into similar groups and a SRS is chosen from each group.
HELP PLEASE!!
On October 1, Gary’s bank balance was $130. During October, he made two
withdrawals and one deposit. At the end of the month, his bank balance was
$95. List two withdrawals and one deposit that would give this final balance.
Answer: $50 withdrawl $50 withdrawl and $65 deposite
Step-by-step : $50 withdrawl $50 withdrawl and $65 deposite
what is the diameter ?
Answer:
37.999749573966
Step-by-step explanation:
Brainliest?
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Answer:
Only the mean increased.
Step-by-step explanation:
The mean is the total divided by the number of scores, and since adding 21 will increase the total score significantly, the mean increases.
The median is still 7.
I need this answer b/6=3
The answer is b=18. To solve, multiply 6 by 3 to get 18. This is called doing the inverse operation. Since the equation is a division equation, we would have to multiply in order to find the missing variable.
Waldo is looking up at his kite at a 22 degrees angle of elevation. If the horizontal distance to his kite is 225 feet, how long is the string from his hand to his kite ?
Answer:
The height of the kite from the ground is 13.617 feet
Step-by-step explanation:
Given as :
The measure of the string = 30 feet
The angle of elevation from the boy to his kite = 27°
Let the height of the kite from ground = H feet
So, From Triangle
Sin angle =
Or, Sin 27° =
or, H = 30 × Sin 27°
I.e H = 30 × 0.4539
∴ H = 13.617 feet
Hence the height of the kite from the ground is 13.617 feet Answer
Mr. Adams drove his delivery truck 151.2 miles during 24 days. He drove the same number of miles each day.
How many miles did Mr. Adams drive each day?
Divide total miles by number of days:
151.2 / 24 = 6.3 miles per day
The expression represents a polynomial with terms. The constant term is , the leading term is , and the leading coefficient is .
There is no polynomial attached ; considering the hypothetival
Answer:
x³ = leading term.
2 = leading Coefficient
1 = constant term.
Step-by-step explanation:
Considering an hypothetical situation ;
For any polynomial such as 2x³ + 3x + 1
The polynomial is represented as : ax³ + bx + c
Where a = leading Coefficient ;
x³ = leading term
c = constant term
Therefore ;
The leading term in this hypothetical question is :
x³ = leading term.
2 = leading Coefficient
1 = constant term.
Mhanifa can you please help? This is due asap!
13. k=3/4 14. a=23
15. p= 5 1/2 16. x=13
17. m=56 18. n=1 1/2
Answer:
13)
9/8 = (k + 6)/6 8(k + 6) = 6*98k + 48 = 548k = 6k = 6/8k = 3/414)
2/10 = 4/(a - 3)a - 3 = 4*5a - 3 = 20a = 2315)
10/(p + 2) = 4/34(p + 2) = 10*34p + 8 = 304p = 22p = 22/4p = 11/216)
4/6 = 8/(x - 1)4(x - 1) = 8*6x - 1 = 12x = 1317)
m/8 = (m + 7)/ 99m = 8(m + 7)9m = 8m + 56m = 5618)
n/(n + 1) = 3/55n = 3(n + 1)5n = 3n + 32n = 3n = 3/2If the width of the fridge is 4 inches and cost of one square inch is five dollars find the total cost of the blanket
Answer:
$20
Step-by-step explanation:
Help PLEASEE!!!!!!!!!!
I think it’s the third one. Hope that helps!
Answer:
the answers are B or C x>4/19
Another translation that I need help on T__T
The translation for this problem is classified as follows:
2 units left -> horizontal translation.4 units down -> vertical translation.What are the translation rules?The four translation rules are defined as follows:
Left a units: x -> x - a. -> horizontal translation.Right a units: x -> x + a. -> horizontal translation.Up a units: y -> y + a. -> vertical translation.Down a units: y -> y - a. -> vertical translation.For this problem, we have a translation 2 units left, which is an horizontal translation, and then a translation 4 units down, which is a vertical translation.
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Answer:
A would be the answer I would choose
the waiting time at an elevator is uniformly distributed between 30 and 200 seconds. what is the probability a rider must wait between 1 minute and 1.4 minutes?
The probability that a rider must wait between 1 minute and 1.4 minutes at the elevator can be determined by calculating the proportion of the uniform distribution that falls within this time interval.
The given information states that the waiting time at the elevator follows a uniform distribution between 30 and 200 seconds. To find the probability of waiting between 1 minute and 1.4 minutes, we need to convert these time values to seconds.
1 minute is equal to 60 seconds, and 1.4 minutes is equal to 84 seconds. Therefore, we are interested in finding the probability that the waiting time falls between 60 seconds and 84 seconds.
Since the waiting time follows a uniform distribution, the probability of waiting within a specific interval is equal to the length of that interval divided by the total length of the distribution.
The total length of the distribution is 200 seconds - 30 seconds = 170 seconds.
The length of the interval between 60 seconds and 84 seconds is 84 seconds - 60 seconds = 24 seconds.
Thus, the probability that a rider must wait between 1 minute and 1.4 minutes is 24 seconds / 170 seconds, which is approximately 0.1412 or 14.12%.
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A random sample of 700 Democrats included 644 that consider protecting the environment to be a top priority. A random sample of 850 Republicans included 323 that consider protecting the environment to be a top priority. Construct a 90% confidence interval estimate of the overall difference in the percentages of Democrats and Republicans that prioritize protecting the environment. (Give your answers as percentages, rounded to the nearest tenth of a percent.)
The 90% confidence interval estimate of the overall difference in the percentages of Democrats and Republicans that prioritize protecting the environment is -21.3% to -16.1%, which represents the range of values within which the true difference is likely to fall.
To construct the confidence interval, we first calculate the sample proportions for Democrats and Republicans: p₁ = 644/700 ≈ 0.92 for Democrats, and p₂ = 323/850 ≈ 0.38 for Republicans.
Next, we calculate the standard error of the difference using the formula: SE = √((p₁(1-p₁)/n₁) + (p₂(1-p₂)/n₂)), where n₁ and n₂ are the sample sizes.
Using the given sample sizes, the standard error is approximately 0.019.
To determine the margin of error, we multiply the standard error by the z-score corresponding to a 90% confidence level, which is approximately 1.645.
Finally, we calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error from the difference in sample proportions: 0.92 - 0.38 ± (1.645 * 0.019).
The resulting confidence interval is approximately -21.3% to -16.1%, which represents the range within which we can estimate the overall difference in the percentages of Democrats and Republicans prioritizing environmental protection.
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Natalia and Sun are 14.5 m apart and looking up at the top of a radio tower. They are on the same side of the tower. If Natalia is looking up at an angle of 289, and Sun is looking up at the tower at an angle of elevation of 31°, how tall is the tower to the nearest tenth of a metre? Assume their eyes are 1.6 m above the ground.
The height of the radio tower to the nearest tenth of a metre is 6.9 m.
Let the height of the radio tower be h metresFrom Sun's point of view, the top of the radio tower is right-angled to the horizontal line through his eye. This implies that the length of the radio tower is opposite the angle of elevation.
Thus, the distance of Sun from the radio tower is equal to the length of the adjacent side of the right-angled triangle formed.
Thus, from the tangent ratio:tan 31° = h / 14.5 + 1.6 (since Sun's eye level is 1.6m above the ground)h = (14.5 + 1.6) tan 31° = 6.9 m (to one decimal place)From Natalia's point of view,
the radio tower makes an acute angle with the line of sight from her eye to the top of the radio tower. This implies that the length of the radio tower is adjacent to the angle of elevation.
Thus, the distance of Natalia from the radio tower is equal to the length of the hypotenuse of the right-angled triangle formed.
Thus, from the sine ratio:sin 89° = h / 14.5 - 1.6 (since Natalia's eye level is 1.6m above the ground)
h = (14.5 - 1.6) sin 89° = 12.9 m (to one decimal place)
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Kitchen chairs are purchased wholesale for $36 each by a discount furniture store. Then, the store marks up the chairs by 60 percent. The store has a special sale where all items are marked down by 20 percent. How much would two chairs cost during the sale? $46.08 $57.60 $92.16 $115.20Kitchen chairs are purchased wholesale for $36 each by a discount furniture store. Then, the store marks up the chairs by 60 percent. The store has a special sale where all items are marked down by 20 percent. How much would two chairs cost during the sale? $46.08 $57.60 $92.16 $115.20
Answer:
46.08
Step-by-step explanation:
you have to make your percentage a decimal, which 60% will be .60 and 20% will be .20. you then multiply your initial number which is 36 by .60 and add that on because youre adding 60%. After that you will multiply that given number by .20 and you subtract what that product is from your last product you received (36x.60) which if im not mistaken will give you $46.08.
Answer:
C
Step-by-step explanation:
I took the test
30 in.
10 in.
10 in.
30 in.
20 in.
Find the area of the arrow above.
square inches
Answer: 45
Step-by-step explanation: