To solve the initial-value problem, we find the complementary solution by solving the associated homogeneous equation, which yields yc(t) = C1e^(56.909t) + C2e^(-0.909t). The particular solution is found using the method of undetermined coefficients. The general solution is given by y(t) = yc(t) + yp(t), and the specific solution satisfying the initial conditions can be obtained by substituting the values and solving for the constants.
To solve the given initial-value problem, we will find the particular solution and the complementary solution.
1. Finding the complementary solution:
The homogeneous equation associated with the given initial-value problem is y" - 55y' + 9y' - 45y = 0. To find the complementary solution, we solve this homogeneous equation. The characteristic equation is obtained by substituting y(t) = e^(at) into the homogeneous equation:
(a^2 - 55a + 9) e^(at) - 45e^(at) = 0
Simplifying, we get:
a^2 - 55a + 9 - 45 = 0
a^2 - 55a - 36 = 0
Using the quadratic formula, we find two solutions for 'a': a1 ≈ 56.909 and a2 ≈ -0.909. Therefore, the complementary solution is given by:
yc(t) = C1e^(56.909t) + C2e^(-0.909t), where C1 and C2 are arbitrary constants.
2. Finding the particular solution:
To find the particular solution, we need to solve the non-homogeneous part of the equation, which is sec(3t). A particular solution can be found using the method of undetermined coefficients. We assume a particular solution of the form:
yp(t) = A sec(3t)
Differentiating twice and substituting into the non-homogeneous equation, we can solve for the constant A.
3. Solution of the initial-value problem:
Now we have the complementary solution yc(t) and the particular solution yp(t). The general solution of the initial-value problem is given by:
y(t) = yc(t) + yp(t) = C1e^(56.909t) + C2e^(-0.909t) + A sec(3t)
To find the specific solution that satisfies the initial conditions, substitute y(0) = 2, y'(0) = 0, and y''(0) = 33 into the above equation and solve for the constants C1, C2, and A.
Note: Please note that the provided solution is only a general outline of the process. Calculating the specific values of the constants and solving the initial-value problem would involve further calculations.
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A large box of cereal has 26.8 ounces costs $5.79. A small box of the same cereal has 19 ounces and costs 4.19. Which box of cereal is the better buy
Answer:
A small box of the same cereal has 19 ounces and costs 4.19.
Step-by-step explanation:
In the first case, you have $3.37 per 15.3 ounces
Thus take 3.37 / 15.3 and you get $0.2203 per ounce or 22 cents per ounce
In the second case, you have $4.59 per 20 ounces
Thus take 4.59 / 20 and you get $0.2295 per ounce or almost 23 cents per ounce
So the smaller box actually has a slightly cheaper cost per ounce associated with it.
algebra if x/y = 5/8 which of the following must be true?
can someone please help me pls
Answer:
The answer is 21 units²
Step-by-step explanation:
To find the area of a right triangle you use the formula A=ab/2
So
A=6×7/2 = 21
Can somebody help me with that ?
Answer:
a complementary angle adds up to 90 degrees.So 90 minus 64 and the measure of the other angle is 26
Step-by-step explanation:
Hope it helped ;)
i need help please explain to me the answer !
Answer:
If the triangle is similar then their ratio is also same
x = 8
6 4.8
X =8×6
4.8
x = 10 cm
hope this helps
good day mate
OMG PLS HELP WITH THIS IM PANICKING OMG I GOT A F IN MATH MY MOM JUST YELLED AT ME IM CRYING PLS HELP WITH THS- PLSS TELL ME WHAT TO DRAW
Answer:
The answer to this question is $14,916
Step-by-step explanation:
18,469 - 3,553 = 14916
As he already has money for the car.
A bar model is shown in the linked picture.
Prove or disprove the following statement: if A, B, and C are
sets with finite cardinalities that satisfy A ∩ B ∩ C = ∅, then |A
∪ B ∪ C| = |A| + |B| + |C|.
The statement |A ∪ B ∪ C| = |A| + |B| + |C| is true, which means the statement is proven.
Let A, B, and C be sets with finite cardinalities that satisfy A ∩ B ∩ C = ∅.
We have to prove that |A ∪ B ∪ C| = |A| + |B| + |C|.
The addition law of sets states that the cardinality of a union of two finite sets is equal to the sum of the cardinalities of the sets minus the cardinality of their intersection.
Thus, we get: |A ∪ B ∪ C| = |A ∪ B| + |C| − |(A ∩ B) ∪ C| = (|A| + |B| − |A ∩ B|) + |C| − |(A ∩ B) ∩ C| = |A| + |B| + |C| − |A ∩ B ∩ C|.Since A ∩ B ∩ C = ∅, we get |A ∪ B ∪ C| = |A| + |B| + |C|.
Thus, the statement is proven.
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Discrete math
Euclidean Algorithm (a) Find the Greatest Common Divisor of 27,720 and 58,212 (b) Find integers r and s so that 27720-5 + 58212.5 = GCD(27720, 58212) =
a) The GCD of 27,720 and 58,212 is 1.
b) Integers r = 3 and s = -1 satisfy the equation 27720r + 58212s = GCD(27720, 58212).
How to find the greatest common divisor (GCD) of 27,720 and 58,212?(a) To find the greatest common divisor (GCD) of 27,720 and 58,212, we can use the Euclidean Algorithm.
Divide 58,212 by 27,720.
58,212 ÷ 27,720 = 2 remainder 2
Divide the previous divisor (27,720) by the remainder (2).
27,720 ÷ 2 = 13,860
Divide the previous remainder (2) by the new remainder (2).
2 ÷ 2 = 1
Since the remainder is now 1, the GCD is found.
Therefore, the greatest common divisor (GCD) of 27,720 and 58,212 is 1.
How to find integers r and s such that 27720r + 58212s = GCD(27720, 58212)?(b) To find integers r and s such that 27720r + 58212s = GCD(27720, 58212), we can use the Extended Euclidean Algorithm.
Using the Euclidean Algorithm from part (a), we can backtrack to find the coefficients r and s:
2 = 27,720 - 13,860
Substituting the values:
2 = 27,720 - (58,212 - 2 * 27,720)
Simplifying:
2 = 3 * 27,720 - 58,212
Comparing with the form 27720r + 58212s = GCD(27720, 58212):
r = 3
s = -1
Therefore, we substitute r = 3 and s = -1 into the equation 27720r + 58212s, it will result in the greatest common divisor (GCD) of 27720 and 58212.
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PLEASE PLEASE PLEASE HELP
WITH 2D AND 3A ONLY, PLEASE SHOW YOUR WORKING
Answer:
2d) d = 60°
3a) x = 50°
Step-by-step explanation:
2d) both of the diagonals pass through the centre of the circle and divide the rectangle into 4 triangles,then it means that;
30 + d = 90°
d = 90 - 30
d = 60°
3a) since O is the centre of the circle, and we can see that the 2 chords directly above and below the centre are parallel to each other, it means that angle x = 50° since the two sides projecting from the centre to form x° and 50° are equal and in essence used to form isosceles triangle.
How many more gift card raffle tickets will
be entered than shopping spree raffle tickets
if there are 500 total raffle tickets?
Answer:
25 tickets
Step-by-step explanation:
If the answer is wrong, I don't know what to say because the answer I got is 25
The would be 25 more gift card raffle ticket than shopping spree raffle tickets if there are 500 total raffle tickets
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
The shopping spree raffle tickets is 35% while the gift card tickets is 40%. Hence for 500 tickets:
Difference = (40% - 35%) * 500 = 25 tickets
The would be 25 more gift card raffle ticket than shopping spree raffle tickets if there are 500 total raffle tickets
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In 20 seconds, a roller coaster goes up a 100-meter hill, then down 72 meters and then back up a 48-meter rise. what integer represents the vertical change from the start of the ride for the coaster after the 20 seconds
This is the answer
I hope if you get it correct or right
Good luck everyone
The total vertical change from the start of the ride for the roller coaster after the 20 seconds 76 m.
What is vertical change?The rise is the vertical difference between two points called vertical change.
Calculation for the vertical change:Step 1: Calculation for the the total rise when roller coaster goes up a 100 meter hill and the down 72 meter.Thus, the total vertical rise will be,
100 (upward) - 72 (downward) = 28(upward)
Step 2: Calculation for the total rise after the roller coaster again goes vertical to 48 meter.So, the total rise will become,
28(upward) + 48(upward) = 76(upward)
Therefore, the total vertical rise of the roller is 76 meter during the time of 20 seconds.
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Determine the surface area of a cereal box with these dimensions width: 3 inches length: 8 inches height: 12 inches
Estimate the area under the graph of f(x) =10 sqrt x from x = 0 to x = 4 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.)
(a) Use four approximating rectangles and right endpoints.
R4=______________________
(b) Use four approximating rectangles and left endpoints.
L4=_______________________
(a) R4= 61.4626 and (b) L4= 61.4626.
In order to estimate the area under the graph of f(x) = 10 sqrt x from x = 0 to x = 4 using four approximating rectangles and right endpoints, we need to use the formula:
Rn = Δx [f(x1) + f(x2) + f(x3) + ... + f(xn)], where Δx is the width of each rectangle and f(xi) is the height of the rectangle at the right endpoint of the ith interval.
Step 1: Calculation of ∆x.∆x = (4 - 0)/4 = 1
Step 2: Calculation of xi for i = 1, 2, 3 and 4.x1 = 1, x2 = 2, x3 = 3, x4 = 4
Step 3: Calculation of f(xi) for i = 1, 2, 3 and 4.f(x1) = 10√(1) = 10f(x2) = 10√(2) ≈ 14.1421f(x3) = 10√(3) ≈ 17.3205f(x4) = 10√(4) = 20
Step 4: Calculation of R4.R4= ∆x [f(x1) + f(x2) + f(x3) + f(x4)]R4= 1[10 + 14.1421 + 17.3205 + 20]= 61.4626Area ≈ 61.4626 square units.
Step 5: Calculation of L4.Ln = ∆x [f(x0) + f(x1) + f(x2) + ... + f(xn-1)]
Where x0 is the initial value.
Here, we can find the value of L4 by using the left endpoints.
So, x0 = 0L4 = ∆x [f(x0) + f(x1) + f(x2) + f(x3)]L4 = 1 [f(0) + f(1) + f(2) + f(3)]L4 = 1 [10 + 14.1421 + 17.3205 + 20]L4 = 61.4626
Therefore, (a) R4= 61.4626 and (b) L4= 61.4626.
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Help me please!!!!!!
can someone help plz ... mark brainliest ❤️
Answer:
Top:(4r⁹+3r³-5)
Bottom: r⁵
Step-by-step explanation:
please help me do it
10 points!! Please give me real help!!
Answer:
32
Step-by-step explanation:
Find a primitive root, for all positive integral m, modulo each integer below. (a) 7 (Hint: Using Corollary 5.15, find a common primitive root r modulo 7 and 72. The proof of Proposition 5.17 then guarantees that r is a primitive root modulo 7" for all positive integral m.) (b) 11m (c) 13m (d) 17m
To find a primitive root modulo a given integer, we can use Corollary 5.15 and Proposition 5.17. For (a) 7, a common primitive root r can be found by using these results. Similarly, for (b) 11m, (c) 13m, and (d) 17m, the same approach can be applied to find primitive roots modulo each integer.
(a) To find a primitive root modulo 7 and 72, we can use Corollary 5.15, which states that if r is a primitive root modulo n, then r is also a primitive root modulo any power of n. Since 7 is a prime number, we can easily find a primitive root modulo 7. Let's say r is a primitive root modulo 7. Using Proposition 5.17, which guarantees that a primitive root modulo a prime number remains a primitive root modulo any positive integral power of the prime, we can conclude that r is a primitive root modulo 7 for all positive integral m.
(b) Similarly, for 11m, we can find a primitive root modulo 11 and use Proposition 5.17 to prove that it is a primitive root modulo 11m for all positive integral m.
(c) The same approach can be applied to find a primitive root modulo 13m.
(d) Finally, for 17m, we can find a primitive root modulo 17 and use Proposition 5.17 to prove its primitiveness modulo 17m for all positive integral m.
By using Corollary 5.15 and Proposition 5.17, we can find a common primitive root for all the given integers modulo their corresponding powers.
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Find the surface area.
24 in.
40 in.
10 in.
26 in.
Answer:
24 IN.
Sorry for all caps
Step-by-step explanation:
Solve the inequality
*
2x^2+ 5x – 3 < 0.
Answer:
heres the answer I think
Step-by-step explanation:
-3 < x < 1/2
Method begins with a matrix or table describing the characteristics of a target population? a. probability b. snowball c. quota d. simple random sampling
The method that begins with a matrix or table describing the characteristics of a target population is quota sampling.
Quota sampling is a non-probability sampling method in which the researcher first creates a matrix or table describing the characteristics of the target population. Once the matrix is created, the researcher then selects a certain number of people from each category in the matrix until the desired sample size is reached.
For example, if the researcher is interested in studying the political views of college students, they might create a matrix with the following categories:
Gender: Male, Female
Age: 18-21, 22-25, 26-30
Political affiliation: Democrat, Republican, Independent
Once the matrix is created, the researcher might then select 100 students from each category, for a total sample size of 300.
Quota sampling is a quick and easy way to obtain a sample of a population. However, it is not a probability sampling method, which means that the sample may not be representative of the population. This is because the researcher is not randomly selecting participants from the population. Instead, they are selecting participants based on their characteristics.
As a result, quota sampling can lead to bias in the results of a study. For example, if the researcher selects more students from one political affiliation than another, the results of the study may be biased in favor of that political affiliation.
Overall, quota sampling is a useful tool for researchers who need to obtain a sample of a population quickly and easily. However, it is important to be aware of the limitations of this method and to take steps to minimize bias in the results of the study.
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A researcher checks different times that students take to complete an IQ test. The Samples are from different states. Please check if their means are significantly different or the difference is just by chance. Significance level is 5%. The premise is that the ANOVA assumption of normality and homogeneity of variance has been satisfied.
If we fail to reject the null hypothesis, we cannot conclude that the means are significantly different.
To check whether the means of the samples from different states are significantly different or if the difference is just by chance, we can use the one-way analysis of variance (ANOVA).
The premise is that the ANOVA assumption of normality and homogeneity of variance has been satisfied.
Here, the null hypothesis is that the means of all the samples are equal while the alternative hypothesis is that at least one mean is different.
The significance level is 5%.
To perform the ANOVA test, we can follow these steps:
Step 1: State the null and alternative hypotheses
H0: μ1 = μ2 = μ3 = ... = μk (where k is the number of groups)
Ha: At least one mean is different
Step 2: Set the significance level (α)α = 0.05
Step 3: Calculate the test statistic
The test statistic used in ANOVA is the F-statistic, which is the ratio of the between-group variance to the within-group variance.
If the F-statistic is large and the p-value is less than the significance level, we can reject the null hypothesis and conclude that at least one mean is different. If the F-statistic is small and the p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that the means are not significantly different.
Step 4: Calculate the p-value
Using the F-statistic and the degrees of freedom for the between-group and within-group variance, we can calculate the p-value. If the p-value is less than the significance level, we can reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis.
Step 5: State the conclusion
Based on the p-value, we can either reject or fail to reject the null hypothesis. If we reject the null hypothesis, we can conclude that at least one mean is different. If we fail to reject the null hypothesis, we cannot conclude that the means are significantly different.
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Dwayne buys ingredients to make a cake. He buys 1 1/2 pounds of flour, 12 ounces of coconut, and 1 1/4 pounds of sugar. What is the total weight of the ingredients Dwayne bought?
Answer:
56 oz or 3 1/2 lbs
Step-by-step explanation:
1 1/4 + 1 1/2 = 2 3/4lbs = 44 oz
44 oz + 12 oz = 56 oz
Answer:
81/4
Step-by-step explanation:
the answer is not sure
on the 3-days of their vacation the Davis family traveled 417 mi 45.3 mi and 366.9 mi how far did they travel all together
Answer:
829.2 miles
Step-by-step explanation:
To find how far they traveled in total, add together all of the distances:
417 + 45.3 + 366.9
= 829.2
So, all together, they traveled 829.2 miles
If the longest side of a triangle is 12 inches, what must be true about the lengths of the remaining sides?
What is the sum of 10/12and 11/12? Make sure to show all work to receive full credit!
Answer:
[tex]1\frac{3}{4}[/tex]
Step-by-step explanation:
[tex]\frac{10}{12} +\frac{11}{12} \\\\\frac{21}{12} \\\\1\frac{9}{12} \\\\1\frac{3}{4}[/tex]
Can someone help me pls I’m stuck on both problems
Answer:
171
Step-by-step explanation:
6x6x4.75 is 171
can someone help me I’m stuck on this and can’t figure it out
Answer:
20%
Step-by-step explanation:
Answer:
20%
Step-by-step explanation:
19.99 - 3.998(20%) = 15.992 which when rounded is 15.99
Please help me with the
Answer: i think its 10
Step-by-step explanation:
Test for equality of population means against the alternative that the means are different assuming normality, choosing ? 5% and using two samples of sizes 12 and 18, with mean 10 and 14, respectively, and equal standard deviation 3.
The p-value (0.0208) is less than the significance level (0.05), we reject the null hypothesis. This indicates that there is sufficient evidence to suggest the case.
Null hypothesis (H0): The population means are equal, μ1 = μ2.
Alternative hypothesis (Ha): The population means are different, μ1 ≠ μ2.
Select the significance level (α): In this case, α = 0.05.
Now, The test statistic for a two-sample t-test is given by:
t = (sample mean 1 - sample mean 2) / √((s1² / n1) + (s2² / n2))
where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Here X₁ = 10, X₂ = 14, s= 3, n₁ =12, n₂ = 18
So, t= (10-14) / √(3²/12) + (3²/18) et:
t= (-4)/ √(3/4 + 1/2)
t= -4 / (5/4)
t= -2.7931
Now, degrees of freedom (df):
= (9/12 + 9/18)² / (((9/12)² / 11) + ((9/18)² / 17))
= 25.164
So, the p value will be 0.0208.
As, p-value is less than α (0.0208 < 0.05), we reject the null hypothesis.
Since the p-value (0.0208) is less than the significance level (0.05), we reject the null hypothesis. This indicates that there is sufficient evidence to suggest the case.
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