The volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis is π cubic units.
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 2 - (1/2)x, y = 0, x = 1, and x = 2 about the x-axis, the method of cylindrical shells.
The volume V can be calculated using the following formula:
V = ∫(2πx × h) dx
where h represents the height of the cylindrical shell at each value of x.
The height h can be determined as the difference between the y-values of the curves y = 2 - (1/2)x and y = 0.
The integral,
V = ∫(2πx × h) dx
= ∫(2πx ×(2 - (1/2)x)) dx
= 2π ∫(2x - (1/2)x²) dx
= 2π [(x²) - (1/6)(x³)] evaluated from 1 to 2
Evaluating the definite integral,
V = 2π [(2²) - (1/6)(2³)] - 2π [(1²) - (1/6)(1³)]
= 2π [4 - (1/6)(8)] - 2π [1 - (1/6)(1)]
= 2π [4 - (4/6)] - 2π [1 - (1/6)]
= 2π [4 - (2/3)] - 2π [1 - (1/6)]
= 2π [4/3] - 2π [5/6]
= (8π/3) - (5π/3)
= (3π/3)
= π
To know more about volume here
https://brainly.com/question/28058531
#SPJ4
A compact disc is designed to last an average of 4 years with a standard deviation of 0.8 years. What is the probability that a CD will last less than 3 years?
A- 1.11%, B - 10.56%, C - 86.65%, D - 100%
Answer:
the answer is B.........................
Two mechanics were working on your car. One can complete the given job
in six hours, but the new guy takes eight hours. They worked together for
the first two hours, but then the first guy left to help another mechanic on
a different job. How long will it take the new guy to finish your car?
PLS EXPLAIN YOUR WORK AND SHOW YOUR STEPS CLEARLY
It will take the new guy approximately 3.33 hours to finish your car.To find out how long it will take the new mechanic to finish the car, we need to consider their individual rates of work.
Let's denote the first mechanic's rate as "M1" (job per hour) and the new guy's rate as "M2" (job per hour).
We are given that M1 can complete the job in six hours, so his rate of work is 1/6 of the job per hour. Similarly, the new guy takes eight hours to complete the job, so his rate of work is 1/8 of the job per hour.
When they worked together for the first two hours, they combined their rates of work. So in the first two hours, they completed (2/6 + 2/8) of the job.
Now, the first mechanic leaves and only the new guy is left to finish the remaining portion of the job. Let's denote the time it takes for the new guy to complete the remaining portion as "T."
Since we know the portion of the job completed in the first two hours, we can set up the equation: (2/6 + 2/8) + (T/8) = 1.
Simplifying the equation, we have (8/24 + 6/24) + (T/8) = 1.
Combining the fractions, we get 14/24 + (T/8) = 1.
Subtracting 14/24 from both sides, we have T/8 = 10/24.
Simplifying further, we find T = (10/24) * 8 = 3.33 hours.
For more such questions on rates
https://brainly.com/question/25720319
#SPJ8
Given the data 21, 13, 13, 37, 13, 23, 25, 15: What is the outlier in the data? What is the mean with the outlier? What is the mean without the outlier? A. 13; 21; 17.6 B. 37; 20; 17.6 C. 37; 17.6; 20 D. 13; 17.6; 21
Consider f: R2 + R² given by (x,y) → (3x + 2y, 7.x + 5y). . Is f injective? Is f surjective? • Does f have an inverse and if yes what is it?
To determine if f has an inverse, we need to check if it is bijective (both injective and surjective). Since f is not injective, it cannot have an inverse.
To determine whether the function f: R^2 -> R^2 given by (x, y) -> (3x + 2y, 7x + 5y) is injective (one-to-one) and surjective (onto), we need to analyze its properties.
Injectivity:
A function f: A -> B is injective if for every pair of distinct elements in A, their corresponding images in B are also distinct. In other words, if f(x) = f(y), then x = y.
Let's consider two distinct points in R^2: (x1, y1) and (x2, y2). If f(x1, y1) = f(x2, y2), we have:
(3x1 + 2y1, 7x1 + 5y1) = (3x2 + 2y2, 7x2 + 5y2)
By comparing the corresponding components, we get the following system of equations:
3x1 + 2y1 = 3x2 + 2y2 (Equation 1)
7x1 + 5y1 = 7x2 + 5y2 (Equation 2)
We can subtract Equation 1 from Equation 2 to eliminate the variables:
7x1 + 5y1 - (3x1 + 2y1) = 7x2 + 5y2 - (3x2 + 2y2)
4x1 + 3y1 = 4x2 + 3y2
Rearranging the equation gives:
4x1 - 4x2 = 3y2 - 3y1
4(x1 - x2) = 3(y2 - y1)
Since this equation holds for any values of x1, x2, y1, and y2, it implies that the difference in the x-coordinates must be the same as the difference in the y-coordinates for any two distinct points. However, this is not always true, indicating that f is not injective. Therefore, the function is not one-to-one.
Surjectivity:
A function f: A -> B is surjective if every element in the codomain B has a preimage in the domain A. In other words, for every element b in B, there exists an element a in A such that f(a) = b.
To determine surjectivity, we need to find whether there exists an (x, y) in R^2 such that f(x, y) = (a, b) for any (a, b) in R^2.
Let's take an arbitrary point (a, b) in R^2. We need to find an (x, y) such that:
(3x + 2y, 7x + 5y) = (a, b)
This gives us a system of equations:
3x + 2y = a (Equation 3)
7x + 5y = b (Equation 4)
We have two equations with two variables, which can be solved to find the values of x and y. Since the system has a unique solution for every (a, b) in R^2, we can conclude that f is surjective.
Inverse:
To determine if f has an inverse, we need to check if it is bijective (both injective and surjective). Since f is not injective, it cannot have an inverse.
In summary:
The function f: R^2 -> R^2 given by (x, y) -> (3x + 2y, 7x.
Visit to know more about Injective:-
brainly.com/question/5614233
#SPJ11
Perform the following test of hypothesis. H0: μ = 285, H1: μ < 285, n = 55, x = 266.89, s = H0 is
By Performing the following test of hypothesis, H0 is rejected.
To perform the test of hypothesis, we compare the sample mean (x) to the hypothesized population mean (μ) and consider the sample size (n) and sample standard deviation (s).
Given:
H0: μ = 285 (null hypothesis)
H1: μ < 285 (alternative hypothesis)
n = 55 (sample size)
x = 266.89 (sample mean)
s = ?
To determine whether to reject or fail to reject the null hypothesis, we calculate the test statistic and compare it to the critical value or p-value.
Since the standard deviation (s) is not given, we cannot directly calculate the test statistic. Without the value of s, we cannot proceed with the hypothesis test. Please provide the value of s to continue with the calculation and draw a conclusion.
The test of hypothesis cannot be performed without the value of the sample standard deviation (s). Please provide the necessary information to proceed with the calculation and draw a conclusion.
To know more about test of hypothesis, visit
https://brainly.com/question/15980493
#SPJ11
A quantity with an initial value of 400 grows exponentially at a rate such that the quantity doubles every 8 days. What is the value of the quantity after 29 days, to the nearest hundredth?
Click the location of the point in the coordinate plane with the coordinates (–4, 3).
Answer:
I've put the red dot where location is.
Step-by-step explanation:
solve for x round to your nearest tenth
If x=6, what is the smallest natural number y that makes x2+y2 rational?
Given: If x=6, what is the smallest natural number y that makes x2+y2 rational .
To Find: Smallest value if y so that the expression is rational .
Solution : On substituting x = 6 , we have
[tex]=> x^2+y^2= 6^2+y^2\\\\ x^2+y^2= 36+y^2[/tex]
Now for the whole expression has minimum value y^2 should be minimum , also y should be natural no. The set of Natural no. is ,
[tex]=> N=\{ 1,2,3,4,... ..,\infty\}[/tex]
Therefore the smallest Natural no. is 1 . Therefore minimum value of y is 1.
The minimum value of y is 1.
Observation from two random and independent samples, drawn from population 1 and 2are given below. Use the Wilcoxon rank sum test to determine whether population 1 is shifted to the left of population 2 Sample 1 33 61 20 19 40 Sample 2 26 36 65 25 35 (1) State the null and alternative hypotheses to be tested.
The null hypothesis will be rejected if the test statistic is smaller than the critical value at a given significance level. The following hypotheses to be tested are:
H0: Population 1 = Population 2
H1: Population 1 < Population 2
Null hypothesis: Population 1 and Population 2 are not significantly different in their distributions of observations.
Alternative hypothesis: Population 1 is shifted to the left of Population 2 in their distributions of observations. This is a one-tailed test. Thus, the null hypothesis will be rejected if the test statistic is smaller than the critical value at a given significance level.
Therefore, the following hypotheses are to be tested:
H0: Population 1 = Population 2
H1: Population 1 < Population 2
The null hypothesis is a fundamental concept in statistical hypothesis testing. It is a statement that assumes there is no significant relationship between two variables or no difference between two groups being compared. The null hypothesis is often denoted as H0.
In simpler terms, the null hypothesis suggests that any observed differences or relationships in a study are due to random chance or sampling error rather than a genuine effect. It serves as a basis for comparison against an alternative hypothesis, which proposes a specific relationship or difference.
To conduct a hypothesis test, researchers typically formulate a null hypothesis and an alternative hypothesis. The alternative hypothesis (denoted as Ha or H1) represents the claim they want to support or prove. The null hypothesis, on the other hand, assumes that the alternative hypothesis is false or not valid.
To know more about null hypothesis, visit the link : https://brainly.com/question/4436370
#SPJ11
If using the method of completing the square to solve the quadratic equation x^2-3x+20=0x 2 −3x+20=0, which number would have to be added to "complete the square"?
Answer:
The polynomial must be added by 2.25 on each side of the equation.
Step-by-step explanation:
Let [tex]x^{2}-3\cdot x +20 = 0[/tex], we need to add each side by 2.25 to complete the square, that is, expanding the polynomial so that factor of a perfect square trinomial can be done:
1) [tex]x^{2}-3\cdot x +20 = 0[/tex] Given
2) [tex](x^{2}-3\cdot x +20) +2.25 = 2.25 + 0[/tex] Compatibility with addition/Commutative property
3) [tex](x^{2}-3\cdot x +2.25) +20 = 2.25[/tex] Commutative, associative and modulative properties.
4) [tex](x-1.5)^{2} +20 = 2.25[/tex] Perfect square trinomial/Result
Answer:
(x−21)^2=81/4
A study conducted by an airline showed that a random sample of 120 of its passengers arriving at Kennedy Airport on flights from Europe took an average of 24.15 minutes with a standard deviation of 3.29 minutes to claim their baggage and clear customs. What can the airline say with 95% confidence about the maximum error, if it uses x = 24.15 minutes as an estimate of the true average time that one of its passengers arriving at Kennedy Airport on a flight from Europe requires to claim his ticket? baggage and pass customs.?
The airline can say with 95% confidence that the maximum error in using x = 24.15 minutes as an estimate of the true average time is 24.736.
How do we solve confidence interval for the True average time?To calculate the confidence interval, we used the following formula:
CI = x ± z × SE.
CI is the confidence interval
x is the sample mean
z is the z-score for the desired confidence level (in this case, 95%)
SE is the standard error of the mean
The z-score for a 95% confidence interval is 1.95. ⇒ 0.05/2 = 0.025 = 1.95.
the sample standard deviation is 3.29 and the sample size is 120.
Therefore
CI = 24.15 + 1.95 × 3.29 / √(120) = 24.736
Find more exercises on confidence level;
https://brainly.com/question/22851322
#SPJ1
Identify the Type I and Type 2 error for the following claim
(a) The average time that customers wait on hold for customer wrvice at a certain company in less than 5 minutes
(b) The number of salespeople employed at a car lot in linearly correlated with the annual profit of the lot.
(a) Type I error: Rejecting the claim , Type II error: Failing to reject the claim. (b) Type I error: Rejecting the claim, Type II error: Failing to reject the claim.
For claim (a):
Type I error: Rejecting the claim that the average time customers wait on hold is less than 5 minutes when, in fact, it is true. This means concluding that the average wait time is longer than 5 minutes when it is not.
Type II error: Failing to reject the claim that the average time customers wait on hold is less than 5 minutes when, in fact, it is false. This means concluding that the average wait time is less than 5 minutes when it is actually longer.
For claim (b):
Type I error: Rejecting the claim that the number of salespeople employed at a car lot is linearly correlated with the annual profit of the lot when, in fact, it is true. This means concluding that there is no linear correlation between the number of salespeople and annual profit when there actually is.
Type II error: Failing to reject the claim that the number of salespeople employed at a car lot is linearly correlated with the annual profit of the lot when, in fact, it is false. This means concluding that there is a linear correlation between the number of salespeople and annual profit when there actually isn't.
Learn more about correlation here:
https://brainly.com/question/16206377
#SPJ11
Sonia has two packages of hamburger meat. The first package weighs 1.76 pounds and the second package weighs 2.29 pounds. She mixes the two packages together and forms hamburgers that weigh 0.25 pounds each. What is the greatest number of 0.25- pound hamburgers Sonia can make using the hamburger meat she has?
16 because o.25 times 16 = 4
1.76+2.29=4.05 so 16
Answer:
16.
Step-by-step explanation:
0.25 × 16 = 4
1.76 + 2.29 = 4.05
so the answer is 16
hope this helps!
~mina-san
A window is designed as shown below. Find the value of x
Answer:
x=20
Step-by-step explanation:
We know that a line is 180 degrees, and the base of the cemi-circle is a line. So, we would have:
x+7x+x=180
9x=180
x=20
Hope this helps!!
Answer:
x = 20
Step-by-step explanation:
x + 7x + x = 180
9x = 180
x = 20
Given a directed graph as depicted in Figure Q6. Figure Q6 (a) List the ordered pairs of the relation, R. (b) Give the matrix of the relation, MR. (c) Give the in-degree and out-degree of each vertex.
The task involves analyzing a directed graph and performing several operations related to relations and degrees of vertices. We need to list the ordered pairs of the relation, find the matrix of the relation, and determine the in-degree and out-degree of each vertex in the graph.
(a) To list the ordered pairs of the relation, R, we examine the directed edges in the graph. For each edge, we write down the corresponding ordered pair. For example, if there is an edge from vertex A to vertex B, we write (A, B). By listing all the directed edges in the graph, we obtain the ordered pairs of the relation, R.
(b) To find the matrix of the relation, MR, we use the vertices of the graph as rows and columns. If there is a directed edge from vertex i to vertex j, we place a 1 in the (i, j) entry of the matrix; otherwise, we place a 0. By examining the directed edges in the graph and filling in the matrix accordingly, we obtain the matrix of the relation, MR.
(c) To determine the in-degree and out-degree of each vertex, we count the number of incoming and outgoing edges for each vertex, respectively. The in-degree of a vertex represents the number of edges pointing towards it, while the out-degree represents the number of edges originating from it. By counting the incoming and outgoing edges for each vertex in the graph, we can determine their respective in-degrees and out-degrees.
Performing these operations will provide the necessary information about the relation and degrees of the vertices in the given directed graph.
Learn more about matrix here:
https://brainly.com/question/28180105
#SPJ11
Rectangle ABCD is congruent to rectangle A″B″C″D″ . Which sequence of transformations could have been used to transform rectangle ABCD to produce rectangle A″B″C″D″ ? Rectangle ABCD was reflected across the y-axis and then across the x-axis. Rectangle ABCD was translated 8 units left and then 7 units down. Rectangle ABCD was translated 2 units left and then 3 units down. Rectangle ABCD was rotated 180° around the origin and then translated 7 units down. A coordinate graph with rectangle A B C D and rectangle A double prime B double primes C double prime and D double prime. Rectangle A B C D has points at A begin ordered pair 2 comma 5 end ordered pair, B begin ordered pair 2 comma 2 end ordered pair, C begin ordered pair 6 comma 2 end ordered pair, D begin ordered pair 6 comma 4 end ordered pair. Rectangle A double prime B double prime C double prime D double prime has points at A double prime begin ordered pair negative 6 comma negative 3 end ordered pair, B double prime begin ordered pair negative 6 comma negative 5 end ordered pair, C double prime begin ordered pair negative 2 comma negative 5 end ordered pair, D double prime begin ordered pair negative 2 comma negative 3 end ordered pair
If f(z) = 2³ (z - 2)² (z+5)³ (z + 1)³(z − 1)4³ evaluate ƒ'(z) |z| =3 f(z)
ƒ'(z)|z|=3 f(z) = -20160The function is given as f(z) = 2³ (z - 2)² (z+5)³ (z + 1)³(z − 1)4³ and we need to evaluate ƒ'(z) |z| =3 f(z).
The value of f'(z) is found by differentiating f(z) with respect to z. Using the product rule of differentiation, we have;ƒ(z) = 2³ (z - 2)² (z+5)³ (z + 1)³(z − 1)4³Now, ƒ'(z) = [2³ * 2(z - 2) * (z+5)³ (z + 1)³(z − 1)4³] + [2³ (z - 2)² * 3(z+5)² (z + 1)³(z − 1)4³] + [2³ (z - 2)² (z+5)³ * 3(z + 1)² (z − 1)4³] + [2³ (z - 2)² (z+5)³ (z + 1)³ * 4(z − 1)³]Now, substitute |z| = 3 and evaluate.ƒ'(z)|z|=3 f(z) = -20160Thus, the value of ƒ'(z)|z|=3 f(z) is -20160. The derivative of the given function is calculated using the product rule of differentiation. The result is then substituted with |z| = 3 and evaluated.
Know more about product rule of differentiation here:
https://brainly.com/question/28993079
#SPJ11
Gigi's Family left their house and drove 14 miles south to a gas station and then 48 miles east to a waterpark. How much shorter with their trip to the waterpark have been if they hadn't stopped at the gas station and had driven along the diagonal path instead?
Answer:
If they hadn't stopped at the gas station and had driven along the diagonal path, their trip would have been 12 miles shorter.
Step-by-step explanation:
Given that Gigi's Family left their house and drove 14 miles south to a gas station and then 48 miles east to a waterpark, to determine how much shorter would their trip to the waterpark have been if they hadn't stopped at the gas station and had driven along the diagonal path instead the following calculation must be performed, using the Pythagorean theorem:
14 ^ 2 + 48 ^ 2 = X ^ 2
196 + 2,304 = X ^ 2
√ 2500 = X
50 = X
14 + 48 - 50 = X
62 - 50 = X
Thus, if they hadn't stopped at the gas station and had driven along the diagonal path, their trip would have been 12 miles shorter.
Question 8 of 10
Which of the following are remote interior angles of Z1? Check all that apply.
ole
O A. 25
B. 21
O c. 26
O D. 24
D E. 23
O F. 22
Answer:
D and E
Step-by-step explanation:
remote interior angles are not adjacent to a given angle. or basically 1 is our exterior angle, so the opposites of 1 and the remote interior angle.
The remote interior angles of Z1 are ∠4 and ∠3. The correct options are D and E.
What are remote interior angle?A triangle's remote interior angles are the two angles that are not adjacent to a given exterior angle. An exterior angle of a triangle is formed by extending one of the triangle's sides.
The following theorem can be used to calculate the remote interior angles of a triangle:
A triangle's remote interior angles are equal in size to the exterior angle that is not adjacent to them.
In other words, if we know the measure of one of a triangle's exterior angles, we can find the measure of the distant interior angles by subtracting that measure from 180 degrees.
Thus, the correct options are D and E.
For more details regarding angles, visit:
https://brainly.com/question/28451077
#SPJ7
A computer tablet is 0.24 meter long. How long is the computer tablet in centimeters?
Answer:
24 cm long
Step-by-step explanation:
What is the quotient of 1/6 divided by 5?
Answer:
0.03333333333 = 33/1000 = 3.3%
Step-by-step explanation:
Answer:
30
Step-by-step explanation:
Factor the expression (s):
7x+63
12x+15
36z+60
Part of the graph of the function f(x) = (x – 1)(x + 7) is shown below.
Which statements about the function are true? Select three options.
The vertex of the function is at (–4,–15).
The vertex of the function is at (–3,–16).
The graph is increasing on the interval x > –3.
The graph is positive only on the intervals where x < –7 and where
x > 1.
The graph is negative on the interval x < –4.
Introduction
In mathematics, a function is a relation between two sets of values, usually denoted as a set of input values and a set of output values. One of the important aspects of a function is its vertex, which is the highest or lowest point in a graph, depending on the specific type of function. The size and position of a graph’s vertex can be important when studying the properties of a function. In this paper, we will discuss three statements about a function and determine whether or not each statement is true.
Statement 1: The vertex of the function is at (–4,–15).
The first statement being discussed is that the vertex of the function is at (–4,–15). This statement is true. By looking at the graph of the function, it can be seen that the vertex of the function is indeed located at the point (–4,–15). At this point, the graph reaches its highest or lowest point.
Statement 2: The vertex of the function is at (–3,–16).
The second statement being discussed is that the vertex of the function is at (–3,–16). Unfortunately, this statement is false. By looking at the graph of the function, it can be seen that the vertex of the function is actually located at (–4,–15). The vertex is not located at (–3,–16).
Statement 3: The graph is increasing on the interval x > –3.
The third statement being discussed is that the graph is increasing on the interval x > –3. This statement is true. By looking at the graph, it can be seen that the graph is indeed increasing on the interval x > –3. On this interval, the y-values increase as the x-values increase.
Statement 4: The graph is positive only on the intervals where x < –7 and where x > 1.
The fourth statement being discussed is that the graph is positive only on the intervals where x < –7 and where x > 1. This statement is true. By looking at the graph, it can be seen that the graph is positive only on the intervals where x < –7 and where x > 1. On these intervals, the y-values are greater than 0.
Statement 5: The graph is negative on the interval x < –4.
The fifth statement being discussed is that the graph is negative on the interval x < –4. This statement is also true. By looking at the graph, it can be seen that the graph is indeed negative on the interval x < –4. On this interval, the y-values are less than 0.
Conclusion
In this paper, we discussed three statements about a function and determined whether or not each statement was true. We found that the first statement, that the vertex of the function is at (–4,–15), is true. We also found that the second statement, that the vertex of the function is at (–3,–16), is false. Furthermore, we found that the third, fourth, and fifth statements, that the graph is increasing on the interval x > –3, that the graph is positive only on the intervals where x < –7 and where x > 1, and that the graph is negative on the interval x < –4, respectively, are all true.
IVE BEEN STUCK ON THIS FOR AN HOUR PLEASE HELP WHAT IS 9.85714285714 X 7
Answer:
69
Step-by-step explanation:
lol
PLEASE HELP WITH THIS QUESTION!!
NO LINKS PLEASE
Answer:
Step-by-step explanation:
g(x)=[x-4] h(x)= [x] -5
h= -5
g=[x-4]
x x
g=4
Find the general solution to the differential equation (1+x)dy-2ydx=0
Tthe general solution to the differential equation is given by y = (1+x)^2C, where C is any real number.
The given differential equation is (1+x)dy - 2ydx = 0. To find the general solution, we can rearrange the equation as dy/dx = 2y/(1+x) and separate the variables, yielding (1/y)dy = (2/(1+x))dx. Integrating both sides gives us ln|y| = 2ln|1+x| + C, where C is the constant of integration. Simplifying further, we get ln|y| = ln|(1+x)^2| + C, which can be rewritten as ln|y| = ln|((1+x)^2e^C)|. By taking the exponential of both sides, we obtain y = (1+x)^2e^C, where C is an arbitrary constant.
In this differential equation, we initially rearrange it and separate the variables to obtain dy/dx = 2y/(1+x). Then, we integrate both sides, resulting in ln|y| = 2ln|1+x| + C. We simplify further by exponentiating both sides, which leads to y = (1+x)^2e^C. The constant of integration, C, is absorbed into a new constant, let's say C' = e^C. Therefore, the general solution to the differential equation is y = (1+x)^2C', where C' represents any real number.
This solution represents a family of curves that satisfy the original differential equation, and different values of C' will give different curves within this family.
Learn more equation here:
https://brainly.com/question/29657983
#SPJ11
PLEASE I NEED HELP
uhh whats 1 + 2
I don't get it. I have a feeling its 12 though.
your answer should be 3 :)
What are the solutions to the following system of equations?
y = 3x - 7
5x - y = 11
A(2, -1)
B (3, 4)
C (-3, 3)
D (-6, 1)
Answer:
A (2,-1)
Step-by-step explanation:
-1 = 3(2) -7
5(2) - (-1) = 11
Solve the system dxdt= ⎡⎣⎢⎢ 3 9 ⎤⎦⎥⎥ -1 -3 x with x(0)= ⎡⎣⎢⎢ 2 ⎤⎦⎥⎥ 4. Give your solution in real form
The solution to the system of differential equations dx/dt = [[3, 9], [-1, -3]]x with x(0) = [[2], [4]] is x = [[6cos(2t)], [2cos(2t)]].
To solve the system of differential equations dx/dt = [[3, 9], [-1, -3]]x with x(0) = [[2], [4]], we can use the eigenvalue method. The matrix [[3, 9], [-1, -3]] has eigenvalues λ₁ = 2 and λ₂ = -2, with corresponding eigenvectors v₁ = [[3], [1]] and v₂ = [[3], [-1]].
Let's denote x = [[x₁], [x₂]]. Using the eigenvectors, we can write x as a linear combination of the eigenvectors: [tex]\[x = c_1 \mathbf{v}_1 e^{\lambda_1 t} + c_2 \mathbf{v}_2 e^{\lambda_2 t}\][/tex], where c₁ and c₂ are constants to be determined.
Using the given initial condition x(0) = [[2], [4]], we have:
[[2], [4]] = c₁[[3], [1]] + c₂[[3], [-1]]
Solving this system of equations, we find c₁ = 2 and c₂ = 0.
Thus, the solution to the system of differential equations is:
[tex]\[x = 2 \begin{bmatrix} 3 \\ 1 \end{bmatrix} e^{2t}\][/tex]
In real form, we can expand the exponential term using Euler's formula: e^(2t) = cos(2t) + i sin(2t). So the solution becomes:
x = [[6cos(2t)], [2cos(2t)]]
In real form, the solution is x₁ = 6cos(2t) and x₂ = 2cos(2t).
For more such questions on differential equations
https://brainly.com/question/2273154
#SPJ8