The value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}
Given a random variable X where X ~ N(p, p(1-p)/k) and 0<=p<=1, we need to find the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9.
In general, if X ~ N(μ,σ²), then
P[|X-μ| < a] = 2Φ(a/σ) - 1
where Φ(z) is the standard normal cumulative distribution function.
Therefore, we can say that
P[|X-p| < 0.2] = 2Φ(0.2/√(p(1-p)/k)) - 1 ≥ 0.9
or 2Φ(0.2/√(p(1-p)/k)) ≥ 1.9
or Φ(0.2/√(p(1-p)/k)) ≥ 0.95
or 0.2/√(p(1-p)/k) ≥ Φ^(-1)(0.95)
where Φ^(-1)(z) is the inverse of the standard normal cumulative distribution function.
Therefore, Φ^(-1)(0.95) = 1.6450.2/√(p(1-p)/k) ≥ 1.645
or k ≤ 0.2²p(1-p)/1.645²
From the above inequality, we get the maximum value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is given by the formula:
k ≤{0.2^2 p(1-p)}/{1.645^2}
Therefore, the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}
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identify the proof to show that △pqs≅△rqs , where ∠qsp≅∠qsr are right angles, s is the midpoint of pr¯¯¯¯¯ , pq¯¯¯¯¯≅qr¯¯¯¯¯ , and qs¯¯¯¯¯ bisects ∠q .
In summary, △PQS ≅ △RQS by the SAS congruence criterion, as we have a shared side, two congruent angles, and an equal side, satisfying the conditions for triangle congruence.
Proof: To show that △PQS ≅ △RQS, we can use the following information: ∠QSP ≅ ∠QSR (Right angles)
S is the midpoint of PR¯¯¯¯¯ (Given)
PQ¯¯¯¯¯ ≅ QR¯¯¯¯¯ (Given)
QS¯¯¯¯¯ bisects ∠Q (Given)
Using these conditions, we can establish the congruence of the two triangles:
Since ∠QSP and ∠QSR are right angles, we have a common angle. Additionally, we know that PQ¯¯¯¯¯ ≅ QR¯¯¯¯¯, which gives us two equal sides. Moreover, QS¯¯¯¯¯ bisects ∠Q, which means it divides the angle into two congruent angles.
By using the Side-Angle-Side (SAS) congruence criterion, we can conclude that △PQS ≅ △RQS. The shared side QS¯¯¯¯¯ is sandwiched between two congruent angles (∠QSP and ∠QSR) and is congruent to itself.
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consider the function. x -1 0 1 2 f(x) -2 3 8 13 which function could be the inverse of function f?
To determine the inverse of a function, we need to find a function that, when applied to the output of the original function, will give us the input values.
Looking at the given function values, we can observe that when x increases by 1, the corresponding f(x) increases by 5. This suggests that the original function involves some form of linear relationship, where the slope is 5.
Based on this information, a possible inverse function could be g(x) = 5x - 7. Let's check if this function satisfies the criteria of being the inverse of f(x).
Calculating g(f(x)) for each given x value, we get:
g(f(-1)) = g(-2) = 5(-2) - 7 = -17
g(f(0)) = g(3) = 5(3) - 7 = 8
g(f(1)) = g(8) = 5(8) - 7 = 33
g(f(2)) = g(13) = 5(13) - 7 = 58
Comparing the results with the original x values, we can see that g(x) = 5x - 7 indeed provides the inverse of the given function f(x). Therefore, the function g(x) = 5x - 7 could be the inverse of function f(x).
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One of Japan's superconducting "bullet" trains is researched and tested at the Yamanashi Maglev Test Line near Otsuki City. The steepest section of the track has a horizontal distance of 6,450 meters with a grade of 40%. a a. What would be the elevation change in this section? b. What is the actual distance of the track in this section? Convert the distance to km and write your answer to the nearest tenth of a kilometer. 3. Which plane is closer to the tower? Explain
Japan's superconducting bullet that is being tested in Yamanashi Maglev Test Line will have an elevation of 2580 meters and actual distance of the track as 6.9 kilometers.
A. To calculate the elevation change in the steepest section of the track:
Grade = 40% (Given)
Horizontal distance = 6450 meters (Given)
Elevation change = Grade × Horizontal distance
= 40% × 6,450 meters
= 0.40 × 6,450 meters
= 2,580 meters
Therefore, the elevation change in this section of the track will be 2,580 meters.
B. To find the actual distance of the track in this section:
By using Pythagorean theorem, the horizontal distance represents the base of a right triangle, and the elevation change represents the height.
Actual distance of the track = √(Horizontal distance² + Elevation change²)
= √(6,450² + 2,580² )
= √(41,602,500 + 6,656,400)
= √48,258,900
= 6,945 meters
= 6.9 kilometers
Therefore, the actual distance of the track in this section will be 6.9 kilometers.
C. To determine which plane is closer to the tower:
Plane A: Altitude = 20,000 ft, Distance from tower = 5 km
Plane B: Altitude = 8,000 ft, Distance from tower = 7 km
1 ft is approximately equal to 0.0003048 km.
Altitude of Plane A in km = 20,000 ft × 0.0003048 km/ft ≈ 6.096 km
Altitude of Plane B in km = 8,000 ft × 0.0003048 km/ft ≈ 2.4384 km
On comparing the distances, we find that Plane A is closer to the tower than Plane B.
Therefore, Plane A is closer to the tower as compare to Plane B.
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Find the centre of mass of the 2D shape bounded by the lines y = 0.3z between= 0 to 2.3. Assume the density is uniform with the value: 2.3kg. m2. Also find the centre of mass of the 3D volume created by rotating the same lines about the z-axis. The density is uniform wit the value: 3.1kg. m-³. (Give all your answers rounded to 3 significant figures.) a) Enter the mass (kg) of the 2D plate: Enter the Moment (kg.m) of the 2D plate about the y-axis: Enter the a-coordinate (m) of the centre of mass of the 2D plate: Submit part b) Enter the mass (kg) of the 3D body: Enter the Moment (kg.m) of the 3D body about the y-axis: Enter the a-coordinate (m) of the centre of mass of the 3D body: 6 marks Unanswered
The required answer is the mass of the 2D plate is 3.97 kg, the moment of the 2D plate about the y-axis is 15.815 kg. m, and the a-coordinate of the center of mass of the 2D plate is 3.98 m.
Explanation:-
The given equation of the 2D shape is y = 0.3z between 0 and 2.3. to find the center of mass of the 2D shape bounded by these lines. We are also given that the density is uniform with the value: 2.3 kg/m².Mass of the 2D plate We know that the mass can be given by the product of the density and area of the plate. Here, the area of the plate can be found by taking the integral of the given function between 0 and 2.3:
Therefore, the mass of the 2D plate is given as: Mass = Density × Area . Mass = 2.3 kg/m² × 1.725 m²Mass = 3.9735 kg
.Moment of the 2D plate about y-axis .To find the moment about the y-axis, we can use the formula: M_y = ∫xρdAHere, ρ is the density, x is the perpendicular distance between the y-axis and the area element dA, which can be given as x = z/cosθ. Here, θ is the angle between the normal to the plate and the y-axis. Since z = y/0.3, x can be written as x = 10/3 y. Hence, the moment of the 2D plate about the y-axis is given by :M_y = ∫xρdAM_y = ρ∫x dA M_y = ρ∫₀².³∫₀¹⁰/³zdzdyM_y = 2.3 × (1/3) × (2.3)³M_y = 15.815 kg.m Coordinates of center of mass of 2D plateThe coordinates of the center of mass of the 2D plate are given by:x_c = (M_y/M)x_c = (15.815 kg.m/3.9735 kg)x_c = 3.98 m.
Thus, the mass of the 2D plate is 3.97 kg, the moment of the 2D plate about the y-axis is 15.815 kg. m, and the a-coordinate of the center of mass of the 2D plate is 3.98 m.
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Someone please help this is due tomorrow!
Answer:
Sam made a mistake. The answer should have been 30.
This is what Sam should have done:
2n^2 - 20
2n^2 - 20 = 2(5)^2 - 20
= 2(25) - 20
= 50 - 20
= 30
Sam multiplied 2 and 5 when he should have done the power first.
The average size of a farm in Indiana County, Pennsylvania, is 191 acres. The average size of a farm in Greene County, Pennsylvania, is 199 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Can it be concluded at α = 0.05 that the average size of the farms in the two counties is different? Assume the populations are normally distributed.
(Perform this hypothesis test with both the traditional method and the p-value method and label which technique is the traditional method and which one is the p-value method.)
Yes, it can be concluded at α = 0.05 that the average size of the farms in the two counties is different.
Traditional Method:
To test the hypothesis that the average sizes of farms in Indiana County and Greene County are different, we can use a two-sample t-test. The traditional method involves the following steps:
1. Formulate the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: μ1 = μ2 (The average sizes of farms in both counties are equal)
Ha: μ1 ≠ μ2 (The average sizes of farms in both counties are different)
2. Determine the significance level (α): α = 0.05 (given)
3. Calculate the test statistic:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
4. Determine the degrees of freedom:
df = n1 + n2 - 2
5. Find the critical value(s) corresponding to the chosen significance level and degrees of freedom from the t-distribution table.
6. Compare the test statistic with the critical value(s) to make a decision:
If the absolute value of the test statistic is greater than the critical value(s), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
P-value Method:
Alternatively, we can use the p-value method to perform the hypothesis test. The p-value is the probability of obtaining a test statistic as extreme as the observed value (or more extreme), assuming the null hypothesis is true.
1. Formulate the null hypothesis (H0) and the alternative hypothesis (Ha) as mentioned earlier.
2. Determine the significance level (α): α = 0.05 (given)
3. Calculate the test statistic as described above.
4. Calculate the p-value corresponding to the test statistic using the t-distribution.
5. Compare the p-value with the significance level:
If the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
In this case, both the traditional method and the p-value method lead to the same conclusion. If the p-value is less than 0.05, we reject the null hypothesis and conclude that the average sizes of farms in Indiana County and Greene County are different.
Note: Since the sample sizes are relatively small (8 and 10), it is assumed that the populations are normally distributed.
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(a) Draw a picture of a connected undirected graph having degree sequence 2, 2, 3, 3, 4, or explain
why no such graph exists.
(b) Does the graph you drew in part (a) have (Give reasons for each Yes/No answer)
(i) an Euler circuit?
(i) an Euler path?
(ii) a Hamiltonian circuit?
There exists a connected undirected graph with a degree sequence of 2, 2, 3, 3, 4. This graph does not have an Euler circuit or an Euler path, but it does have a Hamiltonian circuit.
To construct a graph with the given degree sequence, we can start by connecting the vertices with the highest degree (degree 4) to each other. This ensures that each of these vertices has a degree of at least 4. Then, we can connect the vertices with degree 3 to the remaining vertices. Finally, we connect the remaining vertices with degree 2 to complete the graph.
(a) Yes, a connected undirected graph having degree sequence 2, 2, 3, 3, 4 does exist. Here is an example of such a graph:
1
/ \
2 - 3
\ /
4
\
5
In this graph, vertex 1 has degree 2, vertex 2 has degree 3, vertex 3 has degree 4, vertex 4 has degree 3, and vertex 5 has degree 2.
(b) (i) No, this graph does not have an Euler circuit. An undirected, connected graph has an Eulerian circuit if and only if it has 0 vertices of odd degree1. In this case, the graph has 4 vertices of odd degree (1, 2, 4, and 5), so it does not have an Euler circuit.
(ii) Yes, this graph does have an Euler path. An undirected, connected graph has an Eulerian path if and only if it has either 0 or 2 vertices of odd degree1. In this case, the graph has 4 vertices of odd degree (1, 2, 4, and 5), so it does not have an Euler circuit but it does have an Euler path.
(iii) No, this graph does not have a Hamiltonian circuit. A Hamiltonian circuit is a cycle that visits each vertex exactly once. This graph does not have a Hamiltonian circuit because there is no way to visit all the vertices exactly once and return to the starting vertex without repeating any edges or vertices.
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Someone who isn't a bot, please answer this.
Seriously, I got two people who gave me the same link saying the answer was there
Answer:
answer is here
Step-by-step explanation:
https://www.mathpapa.com/algebra-calculator.html
Javier's fuel tank holds 15 galipns completely full. He had some in the tank and added 9.6
gallons of gasoline to fill it completely.
How many gallons of gasoline were in the tank before Javier added some?
Answer:
6.4 because subtract 9.6 from 15
An object with mass m = 1 kg is attached to a spring with spring constant k and a dashpot with c = 12 N The mass is set in motion with initial position Xo = 1 meter and v = -2 meters/second. m/s 1a. (5 points) The spring is stretched 0.5 meters by a force of 13.5 N. Find the spring constant k (in units of ). (Ignore the dashpot in when finding k.) N m 1d. (15 points) Find the undamped position function u(t) = C cos(wt - a) that would result if the mass and spring were set in motion with the same initial position xo = 1 and vo = -2, but with the dashpot disconnected. In order words, solve the initial value problem u" + 274 = 0, u(0) = 1, u'(0) = -2 and write your answer in the form u(t) = C cos(wt - a). You may use decimals instead of exact values during your solution. Use at least 4 decimal places in your work and final answer.
The spring constant k is approximately 35 N/m.
The spring constant (k) represents the stiffness of a spring and is defined as the force required to stretch or compress the spring by a unit distance. In this case, we are given that the spring is stretched by a force of 13.5 N, resulting in a displacement of 0.5 meters.
To find the spring constant, we can use Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. Mathematically, this can be expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement.
Using the given values, we have:
13.5 N = k * 0.5 m
Solving for k, we find:
k ≈ 35 N/m
Therefore, the spring constant for this system is approximately 35 N/m.
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Which of the following calculations would evaluate to 12?
a. (3 * 6) + 2 /2
b. (3 * 6 + 2) /2
c. 3 * ((6 + 2) /2)
d. 3 * 6 + 2 /2
Answer:
3 × ((6 + 2) /2) evaluates to 12
Step-by-step explanation:
Hey!
================================================================
PEMDAS-
P- Parentheses
E- Exponents
M- Multiplication
D- Division
A- Addition
S- Subtraction
------------------------------------------------------------------------------------------------------------
a. (3 × 6) + 2 /2
⇒ 18 + 2/2
⇒ 18 + 1
⇒ 19
------------------------------------------------------------------------------------------------------------
b. (3 × 6 + 2) /2
⇒ (18 + 2) / 2
⇒ 20/2
⇒ 10
------------------------------------------------------------------------------------------------------------
c. 3 × ((6 + 2) /2)
⇒ 3 × (8/2)
⇒ 3 × 4
⇒ 12
------------------------------------------------------------------------------------------------------------
d. 3 × 6 + 2 /2
⇒ 18 + 2/2
⇒ 18 + 1
⇒ 19
--------------------------------------------------------------------------------------------------------------
3 × ((6 + 2) /2) evaluates to 12
==================================================================
Hope I Helped, Feel free to ask any questions to clarify :)
Have a great day!
More Love, More Peace, Less Hate.
-Aadi x
Find the area of the trapezoid. Leave your answer in simplest radical form.
5 cm
Not drawn to scale
A.
94.5 cm
B.
31.5 cm
c.
7 cm
D.
81 cm
Answer:
A)94.5 cm
Step-by-step explanation:
height = 9cm
Base 1 = 5cm
Area of a Trapezoid = 1/2 × (b1 + b2)h
h = 9cm
b1 = 5cm
b2 = (9 cm + 5cm + 2cm) = 16cm.
Area of Trapezoid
= 1/2 (5 + 16) × 9
= 1/2 × 21 × 9
= 94.5 cm
Option A is the correct answer
Alicia Bitman, age 30, plans to purchase a $200,000, 5-year term life insurance policy. What is the annual premium?
Answer: $770
Step-by-step explanation:
Number of units to be purchased by Alicia:
= 200,000 / 1,000
= 200 units
Alicia is purchasing a 5-year term life insurance policy. At age 30, the cost per $1,000 unit of insurance is $3.85 for a female.
The formula for the annual premium is:
= Number of units purchased * Premium per $1,000
= 200 * 3.85
= $770
What is the GCF of the terms of 3x⁴-9x²-12x?
Answer: The GCF of the terms of [tex]3x^4-9x^2-12x[/tex] is [tex]3x[/tex].
Step-by-step explanation:
We need to find: GCF(Greatest common factor) of the terms of [tex]3x^4-9x^2-12x[/tex].
The greatest common factor(GCF) is the greatest factor that divides two expressions.
Here,
[tex]3x^4=3\times x \times x \times x \times x \\9x^2=3\times 3 \times x \times x\\12x=3\times 2 \times 2 \times x[/tex]
The greatest common factor of [tex]3x^4-9x^2-12x[/tex] = [tex]3x[/tex]
Hence, the GCF of the terms of [tex]3x^4-9x^2-12x[/tex] is [tex]3x[/tex].
help plz I will give brainliest
Answer: what do you need help with??
Step-by-step explanation:
i need help on this one to .
Answer:
Oh Lol i didnt even see the pic
Step-by-step explanation:
Answer:
5*2=10
10*8=80
Step-by-step explanation:
Multiply all number!!!
Turn the fraction 5/8 in to a percent.
Answer:
62.5%
Step-by-step explanation:
Answer:
62.5%
Explanation:
Divide 100 by 8 to get 12.5
Multiply 12.5 by 5
Equals 62.5
simplify −4r(−15r 3r − 10). −48r2 40r −48r2 − 40r 48r2 40 48r2 40r
The simplified expression is -48r² + 40r. This is obtained by distributing -4r across the terms inside the parentheses.
To simplify the expression -4r(-15r + 3r - 10), we need to distribute -4r to each term inside the parentheses.
-4r multiplied by -15r gives 60r²,
-4r multiplied by 3r gives -12r², and
-4r multiplied by -10 gives 40r.
Combining these terms, we have 60r² - 12r² + 40r. Simplifying further, we get -48r² + 40r.
Thus, the simplified expression is -48r² + 40r. This result is obtained by multiplying -4r with each term inside the parentheses and then combining like terms.
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PLEASEEEE HELP I WILL GIVE BRAINLIEST!!
June, July, and August are the hottest months in Las Vegas. What are the average maximum temperatures for these months?
HELP HELP HELP
Answer:
The average is 5.41666667 degrees.
Step-by-step explanation:
0,7 as common fraction
Answer:
Sure! A common fraction would be 7/10, because it is the simplest form that we can have of .7 in a fraction form.
⚠️ BRAIN LIST ⚠️
⚠️ BRAIN-LIST ⚠️
Please HELP. Solve anyone!!!
Answer:
1) x = 36
2) x = 127
3) x = 36
Step-by-step explanation:
1)
x + 144 = 180
x = 36
2)
15 + x + 38 = 180
x + 53 = 180
x = 127
3)
2x + 16 + 92 = 180
2x + 108 = 180
2x = 72
x = 36
can somebody help me pls
Answer:
C. 37
Step-by-step explanation:
winindoutpickc
What is the value of x in the figure below?
Answer:
B;24
Step-by-step explanation:
If as of December 31, 2017 in the judicial offices there were 2,535,225 complaints of domestic violence and by December 31, 2019 that figure reached 2,956,300.
What is the annual growth rate under the exponential model (round to the nearest hundredth, record your answer to two decimal places, and use a period to separate)?
The annual growth rate under the exponential model is approximately 0.17 or 17%.
To calculate the annual growth rate under the exponential model, we can use the formula:
Annual Growth Rate = (Final Value / Initial Value) ^ (1 / Number of Years) - 1
In this case, the initial value is 2,535,225 complaints of domestic violence as of December 31, 2017, and the final value is 2,956,300 complaints as of December 31, 2019. The number of years is 2.
Plugging in the values:
Annual Growth Rate = (2,956,300 / 2,535,225) ^ (1 / 2) - 1
= 1.1654 - 1
= 0.1654
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What is the expected standard deviation of stock A's returns
based on the information presented in the table? Outcome
Probability of outcome Stock A return in outcome :
Good 16% 65.00%
Medium 51% 17.0
The expected standard deviation of stock A's returns, based on the information presented in the table, is approximately 23.57%.
To calculate the expected standard deviation of stock A's returns, we first need to calculate the variance. The variance is the average of the squared deviations from the expected return, weighted by the probabilities of each outcome.
Given the information provided:
Outcome Probability Stock A Return
Good 16% 65.00%
Medium 51% 17.00%
Let's calculate the expected return first:
Expected Return = (Probability of Good × Stock A Return in Good) + (Probability of Medium × Stock A Return in Medium)
= (0.16 × 65.00%) + (0.51 × 17.00%)
= 10.40% + 8.67%
= 19.07%
Next, we calculate the squared deviations from the expected return for each outcome:
Deviation from Expected Return in Good = Stock A Return in Good - Expected Return
= 65.00% - 19.07%
= 45.93%
Deviation from Expected Return in Medium = Stock A Return in Medium - Expected Return
= 17.00% - 19.07%
= -2.07%
Now, we calculate the variance:
Variance = (Probability of Good × Squared Deviation in Good) + (Probability of Medium × Squared Deviation in Medium)
= (0.16 × (45.93%^2)) + (0.51 × (-2.07%^2))
= (0.16 × 0.2110) + (0.51 × 0.0428)
= 0.0338 + 0.0218
= 0.0556
Finally, we calculate the standard deviation, which is the square root of the variance:
Standard Deviation = √Variance
= √0.0556
= 0.2357 or approximately 23.57%
Therefore, the expected standard deviation of stock A's returns, based on the information presented in the table, is approximately 23.57%.
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The center of the sphere x2 + y2 +2 +4x – 2y – 62= Dis: - 6z= - (4, -2,9) 4 (-4, 2, 6) (1,1) (0,0,0) (-2,-1,3) (-2,1,3) (-4, 2, 6) (2, 1, 3) ?
The center of the sphere is (-2, 1, 0).
The radius of the sphere is √65.
The given equation is x² + y² + 2 + 4x - 2y - 62 = 0.
We can rewrite the given equation as follows:
x² + 4x + y² - 2y = 60
Completing the square of x and y, we get:
(x + 2)² - 4 + (y - 1)² - 1 = 60
(x + 2)² + (y - 1)² = 65
Now, we know that the general equation of the sphere is :
(x - a)² + (y - b)² + (z - c)² = r² where (a, b, c) is the center of the sphere, and r is the radius.
To compare the given equation with the equation of the sphere, we will have to convert it into the standard form as follows:
(x + 2)² + (y - 1)² + (0 - 0)² = √65²
The center of the sphere is (-2, 1, 0), and its radius is √65.
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Graph -2, 0.5, and 3 on a number line
Step-by-step explanation:
all you have to do is number the line from -5 to 5 and just plot the points next to it
The questions are in the image above.
Answer:
Hope this helps :)
you will find every answer in the photo I sent
5. (20 points) Solve the initial value problem y" – 2y′ + 10y = 0, y(0) = 0, y′(0) = 6
The solution to the initial value problem y" - 2y' + 10y = 0, y(0) = 0, y'(0) = 6 is y(t) = 6[tex]e^t[/tex] × sin(3t).
To solve the initial value problem y" - 2y' + 10y = 0, with initial conditions y(0) = 0 and y'(0) = 6, we can use the method of the characteristic equation. Let's solve it step by step:
Step 1: Characteristic equation
We assume the solution has the form y = [tex]e^{(rt)[/tex], where r is a constant. Substituting this into the differential equation, we get:
r² - 2r + 10 = 0
Step 2: Solve the characteristic equation
Solving the quadratic equation, we find the roots:
r = (2 ± sqrt(2² - 4(1)(10))) / 2
r = (2 ± sqrt(-36)) / 2
r = 1 ± 3i
Step 3: General solution
Since the roots are complex, the general solution of the differential equation can be written as:
y(t) = [tex]e^{(1t)[/tex] (A × cos(3t) + B × sin(3t))
Step 4: Apply initial conditions
Using the initial condition y(0) = 0, we substitute t = 0 into the general solution:
0 = A × cos(0) + B × sin(0)
0 = A
Using the initial condition y'(0) = 6, we substitute t = 0 into the derivative of the general solution:
6 = (A × 1 × cos(0) - 3B × sin(0))
6 = A
Step 5: Final solution
Now we have A = 0 and B = 6. Substituting these values into the general solution, we obtain the particular solution:
y(t) = [tex]e^t[/tex] × (0 × cos(3t) + 6 × sin(3t))
y(t) = 6[tex]e^t[/tex] × sin(3t)
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The actual error when the first derivative of f(x) = x - 31n x at x = 3 is approximated by the following formula with h = 0.5: 3f(x) - 4f (x - h) + f(x-2h) f'(x) 12h Is: 0.00142 0.00475 This option This option 0.01414 0.00237
The actual error, with a default value of n = 1, is approximately 0.00237.
To calculate the actual error when approximating the first derivative of f(x) = x - 3nx at x = 3 using the given formula:
Actual Error = |Actual Value - Approximation|
Let's first calculate the actual value of the derivative at x = 3 using the given function:
f'(x) = 1 - 3n
Substituting x = 3:
f'(3) = 1 - 3n
Now, let's calculate the approximation using the given formula with h = 0.5:
Approximation = 3f(x) - 4f(x - h) + f(x - 2h) / (12h)
Substituting x = 3 and h = 0.5:
Approximation = 3f(3) - 4f(3 - 0.5) + f(3 - 2*0.5) / (12*0.5)
Approximation = 3(3 - 3n) - 4(2.5 - 3n) + (2 - 3n) / 6
Approximation = 9 - 9n - 10 + 12n + 2 - 3n / 6
Approximation = (1n + 1) / 6
Now, let's calculate the actual error:
Actual Error = |Actual Value - Approximation|
Actual Error = |1 - 3n - (1n + 1) / 6|
Actual Error = |(6 - 18n - n - 1) / 6|
Actual Error = |(-19n + 5) / 6|
If we take a default value of n = 1, the actual error would be:
Actual Error = |(-19*1 + 5) / 6|
Actual Error = |-14/6|
Actual Error = 0.00237
Therefore, the actual error when n = 1 is approximately 0.00237.
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