. 7. (16 points) Find the general solution to the homogeneous system of DE:
x' = Ax where A = [ -11 -26 4 9
Hint: Write your answer y(t) in the form of eat [cos(bt) + sin(bt)].
Answers
The given homogeneous system of DE isx' = Ax where A = [ -11 -26 4 9].Step-by-step explanation to find the general solution to the homogeneous system of DE: To solve this we have to follow the given hint that is, write the answer y(t) in the form of eat [cos(bt) + sin(bt)].
The characteristic equation is given by |A- λI| = 0 Where A is the matrix, I is the identity matrix of order 2 and λ is an unknown constant.
|A- λI| = [ -11 -26 4 9 - λ]
= λ² + 2λ - 70
The roots of the characteristic equation can be found as follows:
λ² + 2λ - 70
= 0(λ + 10)(λ - 7)
= 0λ₁
= -10,
λ₂ = 7
Let's take the value of λ₁ = -10, For λ₁ = -10, the corresponding eigenvector v₁ can be found by solving the homogeneous system of equations
(A-λ₁I) v₁ = 0, that is [ -11 -26 4 9 - (-10)]
v₁ = 0 [ -11 -26 4 19]v₁ = 0
Then, we get a system of linear equations as follows:
-11x - 26y = 0,
4x + 19y = 0.
From the first equation, we get x = \(\frac{-2y}{5}\) Hence, the eigenvector corresponding to λ₁ = -10 isv₁ = [2, -5].
Let's take the value of λ₂ = 7.For λ₂ = 7, the corresponding eigenvector v₂ can be found by solving the homogeneous system of equations
(A-λ₂I) v₂ = 0, that is[ -11 -26 4 9 - 7]
v₂ = 0[ -11 -26 4 2]
v₂ = 0
Then, we get a system of linear equations as follows:
-11x - 26y = 0,
4x + 2y = 0.
From the first equation, we get x = - 2y/5
Hence, the eigenvector corresponding to λ₂ = 7 isv₂ = [5, -2].
Now, the general solution to the homogeneous system of DE can be written as follows:
\(y(t) = c_1 e^{-10t} \left( 2\cos(5t) - 5\sin(5t) \right) + c_2 e^{7t} \left( 5\cos(2t) + 2\sin(2t) \right)\)
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Related Questions
Which of the following sets represents the solution of the equation below?
-3/2x² = x+1
Answers
The solution set is: \($\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$\)
Option (D) is correct.
What is a quadratic equation?
A quadratic equation is a type of equation in algebra that can be written in the form of ax^2 + bx + c = 0, where x is the unknown variable, and a, b, and c are constants with a not equal to zero
The equation is:
\($-\frac{3}{2}x^2 = x + 1$\)
We can rewrite this equation as:
\($-\frac{3}{2}x^2 - x - 1 = 0$\)
To solve for x, we can use the quadratic formula:
\($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$\)
Where a = -3/2, b = -1, and c = -1. Substituting these values into the quadratic formula, we get:
\($x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-\frac{3}{2})(-1)}}{2(-\frac{3}{2})}$\)
Simplifying this expression, we get:
\($x = \frac{1 \pm \sqrt{5}}{3}$\)
Therefore, the solution set is:
\($\left{\frac{-1+\sqrt{5}}{3}, \frac{-1-\sqrt{5}}{3}\right}$\)
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Find the measurement of the angle in the given picture
Answers
Draw a A PQR if PQ = 6.5cm, m< PQR=75 °and m <PRQ=45° using ruler and compasses
only.
Answers
Answer:
Check your question well
Step-by-step explanation:
U didn't ask to draw angle p so how can we get the angle r
A pile of 100 $5coins is 150 mm high. One coin is______mm thick
Answers
Answer:
1.5mm
Step-by-step explanation:
100x = 150. x = height is one $5 coin
x = 150/100
x = 1.5
Solve each inequality. Check your solution. (x/3) + 5 ≥ 1/6
Answers
The inequality given by (x/3) + 5 ≥ 1/6 has the solution x ≥ -29/2 or [-29/2 , +∞).
Inequality refers to the relationship between two non-equal expressions. It can be denoted by > for greater than, < for less than, >/= for greater than and equal to, and </= for less than and equal to.
To solve an inequality is to find the values of x such that when we substitute that number for x we have a true statement.
Given the inequality (x/3) + 5 ≥ 1/6, subtract 5 from both sides.
(x/3) + 5 - 5 ≥ 1/6 - 5
x/3 ≥ -29/6
Multiply both sides by 3.
3(x/3) ≥ 3(-29/6)
x ≥ -29/2
To check, let x = -29/2
x = -29/2 ≥ -29/2
Substitute this value to the inequality.
(x/3) + 5 ≥ 1/6
(-29/2)/3 + 5 ≥ 1/6
1/6 ≥ 1/6
let x = -14,
x = -14 ≥ -29/2
Substitute this value to the inequality.
(x/3) + 5 ≥ 1/6
-14/3 + 5 ≥ 1/6
1/3 ≥ 1/6
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25. Find the number whose square is 3 less than 4
times the number.
Answers
Answer:
1 or 3
Step-by-step explanation:
hope this helps:)
Answer:
\(\huge\boxed{\sf x =\{3,1\}}\)
Step-by-step explanation:
Let the number be x
Given condition:
x² = 4x - 3
Subtract 4x to both sides and add 3 to both sides
x² - 4x + 3 = 0
Using mid-term break
x² - 3x - x + 3 = 0
Take common
x(x-3) - 1(x-3) = 0
Take (x-3) common
(x - 3)(x - 1) = 0
Using zero's method
Either,
x - 3 = 0 OR x - 1 = 0
x = 3 OR x = 1
\(\rule[225]{225}{2}\)
Question Progress
Homework Progress
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Here is a pattern made from sticks:
1
a) How many sticks would be in pattern number 6?
b)
How many sticks would be in pattern number n?
Answers
Answer:
a) 32
b) 2 +5n
Step-by-step explanation:
Given the stick patterns shown, with 7, 12, 17 sticks, you want to know (a) the number of sticks in pattern 6, and (b) the number of sticks in pattern n.
SequenceThe numbers of sticks in the patterns shown are ...
7, 12, 17
The differences between these numbers are ...
12 -7 = 5
17 -12 = 5
That is, the first pattern has 7 and the common difference between patterns is 5. This indicates the sequence is arithmetic.
b) General termThe general term of an arithmetic sequence with first term a1 and common difference d is ...
an = a1 +d(n -1)
an = 7 +5(n -1) = 7 +5n -5
an = 5n +2 . . . . . . sticks in pattern n
a) Sixth termThe term when n=6 is ...
a6 = 5·6 +2 = 32 . . . . . . sticks in pattern 6
__
Additional comment
We find it easier to determine sticks in pattern 6 by first determining the formula for pattern n.
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CASE :
A stationery shop has been running for 10 years and is located
near a campus where you study in Jakarta. You often stop by this
store to buy stationery, markers, rulers, and photocopiers for
co
Answers
The given paragraph states that a stationery shop has been running for 10 years near campus in Jakarta.
Students like the asker often visit this store to purchase stationery, markers, rulers, and photocopiers.
The word "been" is a past participle of the verb "be." It is used to show the completion of an action in the past.
In this case, it implies that the stationery shop has been operating for the past 10 years near the campus.
The sentence can be rewritten as follows to make use of the word "been":
"The stationery shop has been providing stationery, markers, rulers, and photocopiers to students near the campus for 10 years."
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what value of d makes the equation true
Answers
Answer:
16.8
Step-by-step explanation:
the one above would the y
x What does the figurative language in paragraph 2
help the reader understand about the pencil?
X
A The pencil is looking for its
eraser in the classroom
B The pencil longs for the day
when it gets another eraser
C The pencil is membering back
to when it had an eraser
D The pencil no longer has an
eraser on the end
Answers
The use of this figurative language helps to convey the theme of how memories can be fleeting and easily forgotten over time.
In paragraph 2, the author uses the figurative language of an "eraser on the end" to describe how time seems to fade away certain memories. The eraser is a symbol for forgetfulness, and the fact that it is on the end of something suggests that it is far away and difficult to reach.
This can be seen as a metaphor for how certain memories can become distant and hard to recall as time goes on.
Additionally, the eraser symbolizes how memories can be altered or erased entirely over time, much like how an eraser can remove pencil marks from paper.
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Charlie is making chocolate cake for his birthday party. He found a recipe that needs 2 cups of cocoa powder for every 3 cups of sugar. All of his friends are coming, so he uses 6 cups of cocoa powder in his cake. How many cups of sugar should Charlie use?
Answers
Answer:
9 cups of sugar
Step-by-step explanation:
you do 6/2=3 and then 3X3=9
Simplify The answer please
Answers
Answer:
-18/35
Step-by-step explanation:
If f(x)=x^5+, 5 then what is the remainder when f(x) is divided by x+2
Answers
Step-by-step explanation:
f(x)= Quotient × (x+2) + remainder
From this equation we obtain remainder by putting x=2 ( can you see why?)
Hence remainder is f(2)
Find the value of x. Round to
the nearest tenth.
Answers
Answer:
\(\displaystyle x \approx 38.2 ^{\circ}\)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtract Property of EqualityPre-Calculus
Law of Sines: \(\displaystyle \frac{sin(A)}{a} =\frac{sin(B)}{b} =\frac{sin(C)}{c}\)
A, B, C are angle measuresa, b, c are leg lengthsStep-by-step explanation:
Step 1: Identify
A = 27°, a = 11
B = x°, b = 15
Step 2: Solve for x
Substitute [LOS]: \(\displaystyle \frac{sin(27^{\circ})}{11} =\frac{sin(x^{\circ})}{15}\)Cross-multiply: \(\displaystyle 11sin(x^{\circ}) = 15sin(27^{\circ})\)Isolate x term: \(\displaystyle sin(x^{\circ}) = \frac{15sin(27^{\circ})}{11}\)Isolate x: \(\displaystyle x = sin^{-1}(\frac{15sin(27)}{11})\)Evaluate: \(\displaystyle x = 38.2488 ^{\circ}\)Round: \(\displaystyle x \approx 38.2 ^{\circ}\)Triangle ABC with angle ACB a right angle and CD perpendicular AB at D
Answers
We have proved that BC² ×AD = AC² ×BD using AAA congruence rule.
What is congruence rule?Triangles that are identical in size and shape are said to be congruent. This implies that the corresponding sides and angles are equal.
Without comparing each of the triangles' sides and angles, we can determine whether two triangles are congruent. The four rules that demonstrate triangle congruence will be discussed in this lesson. They are referred to as the SSS rule, SAS rule, ASA rule, and AAS rule.
We'll discuss the Hypotenuse Leg rule, a right triangle proof, in a later lesson. It is sufficient to demonstrate that the two triangles are congruent if only one of the rules is accurate.
Consider △ACD&△ABC
∠CAD=∠CAB
∠CDA=∠ACB
∴△ACD∼△ABC,{ By AA similarity criterion}
⟹ AC/AB = AD/AC
∴ AC ² =AB×AD
Similarly, △BCD∼△BAC { by AA similarity criterion}
⟹ BC/BA =BD/BC
∴BC² =BA×BD
∴ BC²/AC² = AB×BD/AB×AD
∴BC² ×AD = AC² ×BD
Thus, We have proved that BC² × AD = AC² × BD using AAA congruence rule.
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Complete question:
Triangle ABC is right angled at C and CD is perpendicular to AB. Prove that BC² ×AD = AC² ×BD
smith is a weld inspector at a shipyard. he knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. if smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find more than 25 substandard welds?
Answers
0.763
Probability is defined as the likelihood of an event occurring or the proportion of outcomes that match the criteria. The probability of finding more than 25 substandard welds is the probability that a weld will be substandard multiplied by the probability of finding more than 25 substandard welds, as stated in the question.What is the probability that Smith will find more than 25 substandard welds?We are given that Smith is a weld inspector who is well aware that 5% of all welds completed in the afternoon shift will be substandard. Smith has also examined 300 of the 7500 welds completed that shift. To determine the probability of locating more than 25 substandard welds, we will use the following formula: P(X > 25) = 1 - P(X ≤ 25)We'll now figure out the probability of locating 25 or fewer substandard welds using the formula below:P(X ≤ 25) = P(X = 0) + P(X = 1) + P(X = 2) + … + P(X = 25)Now we'll use the binomial probability formula to find the probability of finding each value, as shown below: P(X = k) = (n C k) pk (1 - p)n - k Where: k = 0, 1, 2,..., 25 (because we're looking for P(X ≤ 25))n = 300, p = 0.05. We can calculate this using the calculator: P(X ≤ 25) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 25) = 0.237Using this value, we can find P(X > 25):P(X > 25) = 1 - P(X ≤ 25) = 1 - 0.237 = 0.763. Therefore, the probability that Smith would locate more than 25 substandard welds is 0.763.
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is a rectangle. and . What is ?
A.
12
B.
6
C.
15
D.
24
Answers
Answer:
6
Step-by-step explanation:
diagonals bisect each other in rectangle
3x-6 = 2x
x=6
a manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 402 402 gram setting. is there sufficient evidence at the 0.05 0.05 level that the bags are underfilled or overfilled? assume the population is normally distributed. state the null and alternative hypotheses for the above scenario.
Answers
For the weight of bags 402 grams filled with chocolate chips the required null hypothesis and alternative hypothesis equal to H0: μ = 402 and Ha: μ ≠ 402.
Weight of the bag filled with chocolate chips made by machine works = 402 grams
The null hypothesis for the bag filled with machine work representing the statement has no effect.
And it represents that the mean weight of the bag filled with chocolate chips is 402grams.
Null hypothesis for the given scenario is equal to ,
H0: μ = 436
The alternative hypotheses for the above scenario of the bag isn't filled with chocolate chips with weight 402grams.
They are either under filled or that is less than 402grams or overfilled that is above 402grams.
Alternative hypothesis for the given scenario is equal to,
Ha: μ ≠ 402
Statistics is the study of surveys and research of the given numerically data.
Two different kinds of hypotheses we have,
A null hypothesis is one that holds true for the given scenario.
And an alternate hypothesis is another.
The null hypothesis is a default condition to represents nothing is happening.
Possibility of no relationship between two given measured groups.
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Scarlett is going to invest $7,900 and leave it in an account for 18 years. Assuming the interest is compounded continuously, what interest rate, to the nearest tenth of a percent, would be required in order for Scarlett to end up with $12,300?
Answers
Answer:
2.5%
Step-by-step explanation:
PLEASE HELP!
Priya is with her mother at the zoo’s gorilla exhibit. Priya is studying the chart displayed beside the gorilla exhibit, which shows the name and weight of each gorilla. Use the information in the chart to answer the questions in this activity.
Gorilla Weight Plugged into True or False?
5a + 8 = 318
(pounds)
Sophia 318
George 417
Nancy 310
Hank 136
Paula 62
Gavin 24
Part A
Priya’s favorite gorillas are a mother named Sophia and her baby. The zoo attendant said that Sophia weighs 8 pounds more than 5 times her baby’s weight. To determine the baby gorilla’s weight, Priya can use the equation 5a + 8 = 318, where a is the weight of the baby gorilla. Plug the weight of each gorilla into the equation to check which value makes the equation true.
Answers
Answer :
The value that makes the equation true is Paula 62
Step-by-step explanation:
Here, we want to plug the weight of each of the Gorilla to determine the weight that makes the equation true
all we have to do here is that in each of the cases, we substitute the value of a for the weight of each of the Gorilla
Sophia
a = 318
5a + 8
= 5(318) + 8
as we can see this is not equal to 318
George
For George, we substitute a for 417
That will be;
5(417) + 8
This is not 318
for Nancy
5(310) + 8
This is not equal to 318
for Hank
we have a as 136
5(136) + 8
This is not equal to 136
for Paula
5(62) + 8
= 310 + 8 = 318
we can see this is correct
for Gavin,
5(24) + 8
= 128 which is not equal to 318
The only equal value is that of Paula which is our answer
Answer:
Gorilla Weight
(pounds) Plugged into
5a + 8 = 318 True or False?
George 417
5(417) + 8 = 318
2,093 = 318
false
Nancy 310
5(310) + 8 = 318
1,558 = 318
false
Hank 136
5(136) + 8 = 318
688 = 318
false
Paula 62
5(62) + 8 = 318
318 = 318
true
Gavin 24
5(24) + 8 = 318
128 = 318
false
Step-by-step explanation:
In a school, three-quarters of students are involved in sports, half are involved in cultural activities, and one-eighth are involved in neither. Calculate the probability that a student is involved in
Answers
The probability that a student is involved in sports and cultural activities is 3/8
Calculate the probability that a student is involved in sports and cultural activitiesFrom the question, we have the following parameters that can be used in our computation:
P(Sport) = 3/4
P(Cultural) = 1/2
using the above as a guide, we have the following:
P(Both) = P(Sport) * P(Cultural)
Substitute the known values in the above equation, so, we have the following representation
P(Both) = 3/4 * 1/2
Evaluate
P(Both) = 3/8
Hence, the probability is 3/8
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Question
In a school, three-quarters of students are involved in sports, half are involved in cultural activities, and one-eighth are involved in neither.
Calculate the probability that a student is involved in sports and cultural activities
A store sells 6 bananas for $4. Determine how much it would cost to buy 15 bananas
Answers
Answer:
$10
Step-by-step explanation:
Since 6 cost four dollars 3 would cost 2 dollars multiply 3 to get 15 bananas and 2 to get $10
Answer:
$10
Step-by-step explanation:
Make proportion:
?= cost of bananas
6 - 4
15 - ?
?= 15×4÷6=10
Name the intersection of lines a and c
Answers
Hi there, your answer will be p because the line a goes up and meets With the straight line c and right as they meet you see p which is your answer.
hope this helps!!
Which ratio is less then 7 to 15 is it 9 to 15 2 to 5 3 to 5 24 to 45
Answers
Answer:
2:5 is less than 7:15
Step-by-step explanation:
7:15 = 7/15 = .47 = .5
the other given ratios are,
9:15 = 3:5 = 3/5 = .6
2:5 = 2/5 = .4
3:5 =3/5 = .6
24:45 = 8:15 = 8/15 = .5
thus, the ratio which is less than 7:15 is 2:5
write two inequalities to
compare -12 and -6
write two inequalities to compare 8 and -10
Answers
Answer:
First one: -12 < -6
-6 > -12
Second one:
-8 > -10
-10 < -8
Step-by-step explanation:
Well you just compare them with greater then or less then signs.
Evaluate the integral: ∫0 to π/8 8tan(2x)dx. Please show work to receive any credit. Give exact answer (NO decimal) and simplify the final answer to the simplest form.
Answers
The value of the integral ∫(0 to π/8) 8tan(2x) dx is 4 ln(cos(π/4)).
To evaluate the integral, we can use the substitution method. Let u = 2x, then du = 2 dx, and dx = du/2. Also, note that tan(2x) = sin(2x)/cos(2x). Substituting these values into the integral, we have:
∫(0 to π/8) 8tan(2x) dx
= ∫(0 to π/8) 8(sin(2x)/cos(2x)) dx
= 8 ∫(0 to π/8) (sin(u)/cos(u))(du/2)
= 4 ∫(0 to π/8) (sin(u)/cos(u)) du.
Now, we can simplify the integral by using the trigonometric identity sin(u)/cos(u) = tan(u). Therefore, the integral becomes:
= 4 ∫(0 to π/8) tan(u) du.
Integrating tan(u) with respect to u gives us -ln|cos(u)|. Applying the limits of integration, we have:
= 4 [-ln|cos(u)|] evaluated from 0 to π/8
= 4 [-ln|cos(π/8)| + ln|cos(0)|]
= 4 [-ln|cos(π/8)| + ln|1|]
= 4 [-ln|cos(π/8)| + 0]
= 4 ln|cos(π/8)|.
Since cos(π/8) = sin(π/4) = 1/√2, we have:
= 4 ln(1/√2)
= 4 ln(1) - 4 ln(√2)
= 4 (0) - 4 ln(√2)
= -4 ln(√2)
= -4 ln(2^(1/2))
= -4 (1/2) ln(2)
= -2 ln(2).
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Solve for the variables:
10x + 48 = 15x + 36
Answers
Answer:
x=2.4
Step-by-step explanation:
Step 1: move the 10x over by subtracting it from both sides: 48=5x+36
Step 2: move the 36 over by subtracting it from both sides: 12=5x
Step 3: divide by 5 on both sides: x=2.4
Hope that helps
Ryan works at a cell phone store Monday to Friday. He earns $80
per day plus $8
per phone sold. On Tuesday, Ryan makes at least $176.
Hurry up pls
Answers
Answer:
If you are asking how many phones he sold Tuesday he sold 12
Step-by-step explanation:
you subtract 80 from 176 then you get 96 and divide that by 8 to get 12 phones
why couldn't Pythagoras use the pythagorean theorem as we know it?
Answers
Pythagoras was an ancient Greek mathematician who founded the Pythagorean school of thought. The Pythagorean theorem is a fundamental concept in mathematics that is attributed to Pythagoras and his followers.
It states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. While this theorem is considered a cornerstone of mathematics today, it is important to understand that Pythagoras did not have access to the advanced mathematical tools and methods that we have today.
He had to rely on geometric constructions and reasoning to prove his theorem. Furthermore, Pythagoras believed that all numbers could be expressed as ratios of whole numbers, which is not always true in reality. Despite these limitations, the Pythagorean theorem has stood the test of time and continues to be a crucial tool in mathematics and other fields.
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Identify the transversal and classify the angle pair:
Answers
Answer:line l is the transversal because it intersect both sets of angles. and 7 and 11 are corresponding because they are in the same positions of the intersections.
Step-by-step explanation: usually a transversal is associated with parallel lines and corresponding angles are equal but this is not true with a triangle