The general solution of the given system is:
x(t) = 5c_1 * e^(10t) + 5c_2 * e^(10t)x'(t) = 2c_1 * e^(10t) + 2c_2 * e^(10t)To find the general solution of the given system, let's represent the system as a matrix equation:
X' = AX
where X is a vector representing the variables x and x', and A is the coefficient matrix:
A = [[20, -25], [4, 0]]
To find the general solution, we need to find the eigenvalues and eigenvectors of matrix A. Let's proceed with the calculation:
First, we find the eigenvalues by solving the characteristic equation:
|A - λI| = 0
where I is the identity matrix. In this case, we have:
|20-λ, -25| |4, -λ| = 0
|4, -λ|
Expanding the determinant, we get:
(20-λ)(-λ) - (-25)(4) = 0
λ^2 - 20λ + 100 = 0
Solving this quadratic equation, we find two eigenvalues:
λ_1 = 10
λ_2 = 10
Since both eigenvalues are equal, we have repeated eigenvalues. To find the corresponding eigenvectors, we solve the following equations for each eigenvalue:
(A - λI)v = 0
For λ = 10, we have:
(20-10)v_1 -25v_2 = 0
4v_1 - 10v_2 = 0
Simplifying, we find:
2v_1 - 5v_2 = 0
v_1 = (5/2)v_2
We can choose v_2 = 2 as a free parameter, which gives v_1 = 5.
Therefore, the eigenvector corresponding to λ = 10 is:
v_1 = 5
v_2 = 2
To find the general solution, we can write:
X(t) = c_1 * e^(λ_1t) * v_1 + c_2 * e^(λ_2t) * v_2
Substituting the values:
X(t) = c_1 * e^(10t) * [5, 2] + c_2 * e^(10t) * [5, 2]
So, the general solution of the given system is:
x(t) = 5c_1 * e^(10t) + 5c_2 * e^(10t)
x'(t) = 2c_1 * e^(10t) + 2c_2 * e^(10t)
where c_1 and c_2 are arbitrary constants.
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4. The area of a rectangle is x² + 5x - 24. Which of the following could be one of the dimensions?
O(x-8)
O(x-6)
O(x+8)
O(x-4)
Answer:
the correct answer is c (x+8)
Step-by-step explanation:
x+8=0
=x= -8
put x= -8 in the given polynomial
(-8)²+5(-8)-24
= 64-40-24
=24-24= 0
Consider triangle EFG.
F
8
10
12
E
What is the approximate measure of angle G?
G
The approximate measure of angle G is 41.4°. The correct option is C. 41.4°
Law of cosinesFrom the question, we are to determine the measure of angle G
From the law of cosines, we can write that
[tex]cos\ G =\frac{e^{2}+f^{2}-g^{2} }{2ef}[/tex]
In the diagram,
e = 12
f = 10
g = 8
Putting the values into the equation
[tex]cos\ G =\frac{12^{2}+10^{2}-8^{2} }{2\times 12 \times 10}[/tex]
[tex]cos\ G =\frac{144+100-64 }{240}[/tex]
[tex]cos\ G =\frac{180}{240}[/tex]
cosG = 0.75
G = cos⁻¹(0.75)
G = 41.4°
Hence, the approximate measure of angle G is 41.4°. The correct option is C. 41.4°
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Describe how the graph of the function g(x) = -5x²-2 is related to the graph of
the function f(x) = 5x².
(A) reflection of f(x) = 5x² across the x-axis and translated left 2 units
(B) translation of f(x) = 5x² down 2 units
(C) reflection of f(x) = 5x² across the x-axis and translated down 2 units
(D) reflection of f(x) = 5x² right 2 units
Answer:
(C) reflection of f(x) = 5x² across the x-axis and translated down 2 units
Explain and demonstrate the multiplication of complex conjugates using 2+3i and its complex conjugate. Be sure to discuss all steps of the process, including the solution and to which number set it belongs.
The product of a complex number and its conjugate is (a + i b) · (a - i b), where a and b are real numbers, and the result for the complex number 2 + i 3 is 13.
What is the multiplication of a complex number and its conjugateLet be a complex number a + i b, whose conjugate is a - i b. Where a and b are real numbers. The product of these two numbers is:
(a + i b) · (a - i b)
Then, we proceed to obtain the result by some algebraic handling:
a · (a + i b) + (- i b) · (a + i b)
a² + i a · b - i a · b - i² b²
a² - i² b²
a² + b²
If we know that a = 2 and b = 3, then the product of the complex number and its conjugate is:
[tex]z\cdot \bar z = 2^{2} + 3^{2}[/tex]
[tex]z\cdot \bar z = 13[/tex]
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Barack is travelling to dubai. he wants to bring $350 cad with him. the conversion rate from cad to uae dirham is 2.94. how much money, in dirham, will barack be able to bring with him?
Barack be able to bring 1029 dirham with him.
What is currency conversion rate?The amount of one currency needed to buy items in another is shown by conversion rates.
Currency rates and spot prices on the foreign exchange market are comparable to conversion rates.
In relation to supply and demand, conversion rates are impacted.
Governments and central banks implement policies in response to supply and demand dynamics that affect the conversion rate.
According to the question,
The conversion rate from cad to uae dirham is 2.94
Money, in dirham, Barack would be able to bring with him,
=(350*2.94) dirham
=1029 dirham
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Question 5 of 10
Suppose f(x) = x² and g(x)= (1x)
graph of g(x) with the graph of f(x)?
Which statement best compares the
A. The graph of g(x) is the graph of f(x) horizontally stretched by a
factor of 4.
B. The graph of g(x) is the graph of f(x) vertically stretched by a
factor of 4.
C. The graph of g(x) is the graph of f(x) shifted units right.
D. The graph of g(x) is the graph of f(x) horizontally compressed by a
factor of 4.
Answer: A
Step-by-step explanation:
See attached image.
The table shows how far a distance runner has traveled since the race began. what is her average rate of change, in miles per hour, during the interval 0.75 to 1.00 hours?
The average rate of change will be 5 miles per hour from 0.75 to 1 hour.
The average rate of change is the ratio of change in the y-value or the function value by the change in the x-value.
So according to the question,
the interval between 0.75 to 1 hour is =1-0.75=0.25 hour.
At 0.75 hours elapsed, the distance traveled is 3.5 miles
at 1 hour elapsed, the distance traveled is 4.75miles.
during the interval, 0.75 hours to 1-hour distance travelled=4.75-3.5=1.25 miles.
the average rate of change will be = change in distance traveled during the interval/interval time.
=(4.75-3.5)/(1-0.75)=1.25/0.25=5 miles per hour
Therefore the average rate of change will be 5 miles per hour from 0.75 to 1 hour.
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Answer:
B
Step-by-step explanation:
If you don’t want to read
30) A student has not been to class and must pass a 20 question true/false exam to continue in the course. The student will guess on every question and must get at least 14 correct to pass the test. What is the probability that the student will pass the test
The probability that the student will pass the test is 0.057
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the discrete probability distribution of a random variable which takes the value 1 with probability [tex]p[/tex] and the value 0 with probability [tex]{\displaystyle q=1-p}[/tex]. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q.
The student will pass if he at least answers 14 questions correctly .
This can be written as a Bernoulli distribution function [tex]P(Q=i)= 20C_{i} (\frac{1}{2} )^i(1-\frac{1}{2} )^(20-i)[/tex]
P(Q≥ 14) = P(Q=14) + P(Q=15) + P(Q=16) +P(Q=17) + P(Q=18)+ P(Q=19) + P(Q=20)
P(Q=14) = 20[tex]C_{14}[/tex][tex](1/2)^{20}[/tex]
P(Q=15) = 20[tex]C_{15}[/tex][tex](1/2)^{20}[/tex]
P(Q=16) = 20[tex]C_{16}[/tex][tex](1/2)^{20}[/tex]
P(Q=17) = 20[tex]C_{17}[/tex][tex](1/2)^{20}[/tex]
P(Q=18) = 20[tex]C_{18}[/tex][tex](1/2)^{20}[/tex]
P(Q=19) = 20[tex]C_{19}[/tex][tex](1/2)^{20}[/tex]
P(Q=20) = [tex]20C_{20}(1/2)^{20}[/tex]
On summing up all we get,
P(Q≥ 14) = (38760 + 15504 + 4845+ 1140 + 190 + 20 + 1)/2^(20)
= 15115/262144=0.057
Thus the probability that the student will pass the test is 0.057
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Please help!!! Answer asap!!!
Other than translation, what transformation is used to create this frieze pattern?
180° rotation
horizontal reflection
glide reflection
vertical reflection
Answer:
Horizontal reflection
Step-by-step explanation:
a math chart that was labeled had these pattern examples with the the translation and transformation.
Experiments done without replacement will result in:
When sampling is finished without replacement, each member of a population could also be chosen just once. during this case, the chances for the second pick are full of the results of the primary pick. The events are considered to be dependent or not independent.
lets understand with the help of an Example
You have a good, well-shuffled deck of 52 cards. It consists of 4 suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit.
Sampling without replacement:
Suppose you decide three cards without replacement. the primary card you choose out of the 52 cards is that the K of hearts. you place this card aside and pick the second card from the 51 cards remaining within the deck. it's the three of diamonds. you place this card aside and pick the third card from the remaining 50 cards within the deck. The third card is that the J of spades. Your picks are . Because you have got picked the cards without replacement, you can not pick the identical card twice.
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Solve the equation
3(2x + 2) = 3x - 15
• Using distributive property
[tex]6x + 6 = 3x - 15[/tex]
• Solving the like terms
[tex]6x - 3x = - 15 - 6[/tex]
[tex]3x = - 2 1[/tex]
[tex]x = - \dfrac{21}{3} [/tex]
[tex]x = - 7[/tex]
Hope it helps you any confusions you may ask!
Please help I NEED IT ASAP
Answer:
she rides her bike at a rate of 8mph during the second part of the trip.
Step-by-step explanation:
for the first part of the trip, she rides 18 miles in 2 hours which means that she bikes at a rate of 9 miles per hour.
subtracting 18 from 58 gives us the distance she biked during the second part of her trip (which is 40 miles). subtracting 2 from 7 gives us that she biked for 5 more hours. Putting this together means she biked 40 miles in 5 hours.
Now divide miles by hours [tex]\frac{40mi}{5hrs} =8mph[/tex] to get your average rate of change for the second part of the trip.
A friend asked lara to help her divide 456.3 by 100. what should lara tell her? a. move the decimal point three places to the right. b. move the decimal point two places to the right. c. move the decimal point three places to the left. d. move the decimal point two places to the left.
Answer:
Step-by-step explanation:
The correct answer is D. If we are multiplying by 10 we would move the decimal place one place to the right to show that the number is now 10 times bigger. If we multiply by one hundred. We are getting 100 times bigger, so we would move the decimal place 2 places to the right.
The opposite is true if we are dividing. Instead of moving the decimal to the right to show that the number is getting bigger, we move to the left to show that the number is getting smaller. If we are getting smaller by 10, we would move the decimal one place to the left. In this case, we are getting 100 times smaller so we move the decimal two places to the left.
The length of the rectangle is 4 times its width. If the
erimeter of the rectangle is 80 m, find the length and
readth of rectangle.
Answer:
width is 8, length is 32
Step-by-step explanation:
* = multiply or times
Let the width be x, then the length would be 4x
Perimeter = 2*length+2*width
80 = 2*4x+2*x
80 = 8x+2x
80 = 10x
8 = x (width)
32 = 4x (length)
Which of these statements is true for f(x) = 2.3*?
OA. The y-intercept is (0, 1).
OB. The domain is x > 0.
OC. The y-intercept is (0, 2).
OD. It is always decreasing.
The true statement for f(x) = 2(3)^x is the y-intercept is (0, 2).
How to determine the true statement?The function is given as:
f(x) = 2(3)^x
Set x = 0 to determine the y-intercept
f(0) = 2(3)^0
Evaluate
f(0) = 2
This means that the y-intercept of f(x) = 2(3)^x is (0,2)
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"What is the y-intercept of the line that has a slope of- 4 and passes through the point (A)- 10 (B)-6 (C) 8 (D) 10
The y-intercept which is the point at which x = 0 is; A: -10
How to find the y-intercept?The general form of an equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept
We are given the slope;
m = -4
Now, the line passes through the point 0, -10. Thus;
(y - y1)/(x - x1) = -4
(y + 10)/(x - 0) = -4
y + 10 = -4x
y = -4x - 10
Thus, the y-intercept which is the point at which x = 0 is; y = -10
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i don’t know this answer help pls
Answer:
c
Step-by-step explanation:
Starting price is $5.50. And for every mile it costs $1.50.
Write an equation of a line that passes through (3, -1) and is parallel and perpendicular to y = 6x - 4
Here,
y=mx+c
the slope of the line 6x+4y= -16 is -20.
if the two lines are perpendicular, the slope of 1st line * slope of second line=-1
the slope of the 2nd line is -1/2
y-y1=m(x-x1)
y-13=-1/2(x- -18)
when you solve, you get the equation of the line as 2y+x=50
In basic geometry, if two geometry objects intersect at right angles (90 degrees or π / 2 radians), they are vertical.
If two lines intersect at right angles, the line is perpendicular to another line. Explicitly (1) if two lines intersect, the first line is perpendicular to the second line. (2) At the intersection, the straight line angle on one side of the first line is cut by the second line into two congruent angles. Verticality can be shown as symmetric. That is, if the first line is perpendicular to the second line, then the second line is also perpendicular to the first line. Therefore, two straight lines can be said to be perpendicular (to each other) without specifying the order.
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Given the functions:
f(x) = 6x + 11 and g(x) = x + 6,
which of the following represents f[g(x)] correctly?
A. F[g(x)] = 6x + 47
B. F[g(x)] = 6x + 17
C. F[g(x)] = 6x^2 +47
D. F[g(x)] = 6x^2 + 17
Answer:
f[g(x)] = 6x + 47
Step-by-step explanation:
The expression f[g(x)] represents a composite function. You need to plug the function g(x) into the "x" value of the f(x) function. Then, you can simplify.
f(x) = 6x + 11 g(x) = x + 6
f[g(x)] <----- Composition expression
f[g(x)] = 6(x + 6) + 11 <----- Plug g(x) into x
f[g(x)] = 6x + 36 + 11 <----- Multiply terms in parentheses by 6
f[g(x)] = 6x + 47 <----- Add 36 and 11
omg please help me!!!
area = pi * r^2
here, the radius is 2 as its the diameter/2.
so the area is 4pi.
circumference = 2*pi*r.
so the circumference is 4pi as well.
PLEASE ASAP: The tennis club has 5 new members each member buys a racet and a tube of balls. If 6 racket cost $186.00 and 6 tubes of balls cost $27.00 how much do the 5 people pay for rackets and balls?
Answer:
Total cost is $177.50.
Step-by-step explanation:
Rackets:
Divide the cost for racket by 6 to find the cost per racket.
186÷6=31
Multiply 31 by 5 to find the cost for 5 rackets.
31×5=155
$155
Tubes of balls:
Divide 27 by 6 to find the cost per a tube of balls.
27÷6=4.50
Multiply 4.50 by 5 to find the cost for 5 tubes of balls.
4.50×5=22.50
$22.50
Add the amounts for rackets and tubes of balls to find the total cost.
155+22.50=177.50
The total cost is $177.50.
Hope this helps!
Data definition involves describing the properties of the data that go into each database table, specifically the ____ or columns that make up the database.
Data definition involves describing the properties of the data that go into each database table, specifically the fields or columns that make up the database.
Database. a collection of related information organized for rapid search and retrieval.Record. a collection of fields that appear as a row in a database or table.The term DATABASE describes a collection of data organized in a manner that allows access, retrieval, and use of that data.What is a field record table and query?
Field: A field refers to an area within a record which is reserved for a specific piece of data. Eg. Employee ID. Table: Table is the collection of records of specific types. E.g. Employee table is a collection of record related to all the employees.Queries can perform many different functions in a database. Their most common function is to retrieve specific data from the tables.Learn more about database
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PLEASE HELP ASAP
Given a graph for the transformation of f(x) in the format g(x) = f(kx), determine the k value.
The value k needed for the transformation of f(x) to g(x) = f(k · x) is equal to 3.056.
How to find the find the dilation factor
In this problem we have the following relationship bewteen the two quadratic equations: g(x) = f(k · x), which means that for all y the following relationship between f(x) and g(x):
[tex]k = \frac{x_{g}}{x_{f}}[/tex]
Let suppose that y = 3, then [tex]x_{f} = - 5.5[/tex] and [tex]x_{g} = - 1.8[/tex], then the value k is:
k = (- 5.5)/(- 1.8)
k = 3.056
The value k needed for the transformation of f(x) to g(x) = f(k · x) is equal to 3.056.
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An arborist examined trees in an orchard to see if they were infected with a virus. 324
out of 540 trees were infected with the virus. What percentage of the trees were infected?
Write your answer using a percent sign (%).
Answer:
60%
Step-by-step explanation:
324/540 x100=60%
Np!
solve using substitution:
4x- 2y = 6 and x + y = 6
solve using elimination:
4x- 2y = 6 and x + y = 6
Answer:
Step-by-step explanation:
Substitution is a method that rewrites one of the equations in terms of one variable, and substitutes that in the other equations.
We can easily rewrite x + y = 6 by subtracting y from both sides, giving us
[tex]x = -y+6[/tex]
By putting this equation in place of x in the first equation, we can solve for y:
[tex]4(-y+6)-2y=6[/tex]
[tex]-4y+24-2y=6[/tex] [Distributing 4 into the parentheses]
[tex]-6y+24=6[/tex] [Combining like terms]
[tex]-6y = -18[/tex] [Subtracting 24 from both sides]
[tex]y=3[/tex] [Dividing both sides by -6]
Since we solved for y, we can again substitute by putting it into the second equation.
[tex]x+3=6[/tex]
[tex]x=3[/tex] [Subtracting 3 from both sides]
The solution for this system of equations is (3,3).
Elimination is another method of solving systems of equations. In this method, we multiply one equation such that when both equations are combined, only one variable is left.
We can first multiply both sides of the second equation by 2.
[tex]2x+2y=12[/tex]
Now, let's add both equations to cancel out y.
[tex]4x-2y=6[/tex]
[tex]2x+2y=12[/tex]
[tex]6x=18[/tex]
Now, we can divide 6 from both sides to get the value of x.
[tex]\frac{6x}{6} = \frac{18}{6}[/tex]
[tex]x=3[/tex]
We can now substitute this value into one of the equations to get the value of y. Here, I used the second equation.
[tex]3+y=6[/tex]
[tex]y=3[/tex]
The solution to our system of equations is (3,3).
there are five chemistry instructors and six physics instructors at a college. if a committee of four instructors is selected,
The probability of at least one of them being a physics instructor is 120 ways to form the committee.
Probability of Independent Events: The 'At Least One' Rule?Sometimes, simply the occurrence of the event at least once matters when computing independent events. The "At Least One" Rule is used to describe this. The complement of the probability of the event never occurring will be used to calculate the probability of the event occurring at least once.
There are five chemistry instructors and six physics instructors at a college. If a committee of four instructors is selected.
Starting with 6, choose two because we require at least two professors. Let's move on to the remaining 8 individuals (there were initially 10, but now there are just 8): and select 3.
6 select 2; 8 select 3
15×56 =840
We then chose 5 individuals from a group of 10, at least 2 of whom were lecturers.
(9) = 20 options for selecting the professors. 3
(3) = 6 options for selecting the professors.
(9) (¹) = 20 × 6 = 120 ways.
Therefore, 120 ways to form the committee.
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Complete question:
There are five chemistry instructors and six physics instructors at a college. If a committee of four instructors is selected, find the probability of at least one of them being a physics instructor?
Please help me!! due today. Find the volume of the right triangular prism !:)
Answer: 115.5 cubic m
Step-by-step explanation:
[tex]V=\frac{1}{2}(7)(11)(3)=115.5[/tex]
If the parabola of the form y = a(x – h)2 + k is always shifted horizontally h units and vertically k units, then its vertex is always
(–h, –k)
(h, k)
(–h, k)
(h, –k)
The vertex of the transformed parabola with the equation [tex]y=a(x-h)^2+k[/tex] is (h, k).
What are the transformation rules for horizontal and vertical shifts?The transformation rules are:
Translation: (x, y) → (x + a, y) or (x, y) → (x - a, y) (Horizontal shift)
Translation: (x, y) → (x , y + b) or (x, y) → (x , y - b) (Vertical shift)
Calculation:It is given that, the transformed parabola is [tex]y=a(x-h)^2+k[/tex]
This parabola has vertex at (h, k).
Since this equation is obtained by shifting h units horizontally and k units vertically.
So, the original vertex is at (h - h, k - k) = (0, 0)
Thus, the equation of the actual parabola is [tex]y=ax^2[/tex]
Therefore, the given transformed equation of parabola has vertex (h, k).
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Answer:
B] (h,k)
The next answer is Vertex (-5, -4)
☆
EDGE2022; Good luck :3!!!!
The planner needs a clearance of 9 feet under the arch has the planner met the minimum
Yes, the planner has met the minimum height, as the minimum height was 9 feet and the maximum height of the arc is 9.6 feet
How to explain the information?As we can see that at the mid value, for x=4, the value of f(x)=9.4
It means that the maximum height occurs at mid of the arc where it is height is 9.6 feet above the ground.
It means that the planner has met the minimum height, as the minimum height was 9 feet and the maximum height of arc is 9.6 feet
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If the original recipe made 14 pancakes and the cafeteria plans to charge $.50 per pancake, how much money will they make if they sell all of the
If they sell all the pancakes, they will win $7
how much money will they make if they sell all of the pancackes?
We know that they can make a batch of 14 pancakes, such that each one can be sold for $0.50
Then the total amount that they get if they sell all the pancakes is given by the product:
P = 14*$0.50 = $7
If they sell all the pancakes, they will win $7
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