Step-by-step explanation:
in mathematics, a basic algebra corporation is there any one of the traditional operation of arithmetic, which are addition, subtraction, multiply, division, rising to an integer power, and taking root (fractional power ).
Use variation of parameters to find a general solution to the differential equation given that the functions y1 and y2 are linearly independent solutions to the corresponding homogeneous equation for t > 0. ty" + (2t - 1 )y' - 2y = 6t^2 e^-2t​; y1 = 22t −​1, y2 = e^-2t
Answer:
[tex]y_g(t) = c_1*( 2t - 1 ) + c_2*e^(^-^2^t^) - e^(^-^2^t^)* [ t^3 + \frac{3}{4}t^2 + \frac{3}{4}t ][/tex]
Step-by-step explanation:
Solution:-
- Given is the 2nd order linear ODE as follows:
[tex]ty'' + ( 2t - 1 )*y' - 2y = 6t^2 . e^(^-^2^t^)[/tex]
- The complementary two independent solution to the homogeneous 2nd order linear ODE are given as follows:
[tex]y_1(t) = 2t - 1\\\\y_2 (t ) = e^-^2^t[/tex]
- The particular solution ( yp ) to the non-homogeneous 2nd order linear ODE is expressed as:
[tex]y_p(t) = u_1(t)*y_1(t) + u_2(t)*y_2(t)[/tex]
Where,
[tex]u_1(t) , u_2(t)[/tex] are linearly independent functions of parameter ( t )
- To determine [ [tex]u_1(t) , u_2(t)[/tex] ], we will employ the use of wronskian ( W ).
- The functions [[tex]u_1(t) , u_2(t)[/tex] ] are defined as:
[tex]u_1(t) = - \int {\frac{F(t). y_2(t)}{W [ y_1(t) , y_2(t) ]} } \, dt \\\\u_2(t) = \int {\frac{F(t). y_1(t)}{W [ y_1(t) , y_2(t) ]} } \, dt \\[/tex]
Where,
F(t): Non-homogeneous part of the ODE
W [ y1(t) , y2(t) ]: the wronskian of independent complementary solutions
- To compute the wronskian W [ y1(t) , y2(t) ] we will follow the procedure to find the determinant of the matrix below:
[tex]W [ y_1 ( t ) , y_2(t) ] = | \left[\begin{array}{cc}y_1(t)&y_2(t)\\y'_1(t)&y'_2(t)\end{array}\right] |[/tex]
[tex]W [ (2t-1) , (e^-^2^t) ] = | \left[\begin{array}{cc}2t - 1&e^-^2^t\\2&-2e^-^2^t\end{array}\right] |\\\\W [ (2t-1) , (e^-^2^t) ]= [ (2t - 1 ) * (-2e^-^2^t) - ( e^-^2^t ) * (2 ) ]\\\\W [ (2t-1) , (e^-^2^t) ] = [ -4t*e^-^2^t ]\\[/tex]
- Now we will evaluate function. Using the relation given for u1(t) we have:
[tex]u_1 (t ) = - \int {\frac{6t^2*e^(^-^2^t^) . ( e^-^2^t)}{-4t*e^(^-^2^t^)} } \, dt\\\\u_1 (t ) = \frac{3}{2} \int [ t*e^(^-^2^t^) ] \, dt\\\\u_1 (t ) = \frac{3}{2}* [ ( -\frac{1}{2} t*e^(^-^2^t^) - \int {( -\frac{1}{2}*e^(^-^2^t^) )} \, dt] \\\\u_1 (t ) = -e^(^-^2^t^)* [ ( \frac{3}{4} t + \frac{3}{8} )] \\\\[/tex]
- Similarly for the function u2(t):
[tex]u_2 (t ) = \int {\frac{6t^2*e^(^-^2^t^) . ( 2t-1)}{-4t*e^(^-^2^t^)} } \, dt\\\\u_2 (t ) = -\frac{3}{2} \int [2t^2 -t ] \, dt\\\\u_2 (t ) = -\frac{3}{2}* [\frac{2}{3}t^3 - \frac{1}{2}t^2 ] \\\\u_2 (t ) = t^2 [\frac{3}{4} - t ][/tex]
- We can now express the particular solution ( yp ) in the form expressed initially:
[tex]y_p(t) = -e^(^-^2^t^)* [\frac{3}{2}t^2 + \frac{3}{4}t - \frac{3}{8} ] + e^(^-^2^t^)*[\frac{3}{4}t^2 - t^3 ]\\\\y_p(t) = -e^(^-^2^t^)* [t^3 + \frac{3}{4}t^2 + \frac{3}{4}t - \frac{3}{8} ] \\[/tex]
Where the term: 3/8 e^(-2t) is common to both complementary and particular solution; hence, dependent term is excluded from general solution.
- The general solution is the superposition of complementary and particular solution as follows:
[tex]y_g(t) = y_c(t) + y_p(t)\\\\y_g(t) = c_1*( 2t - 1 ) + c_2*e^(^-^2^t^) - e^(^-^2^t^)* [ t^3 + \frac{3}{4}t^2 + \frac{3}{4}t ][/tex]
We are interested in finding an estimator for Var (Xi), and propose to use V=-n (1-Xn). 0/2 puntos (calificable) Now, we are interested in the bias of V. Compute: E [V]-Var (Xi)-[n Using this, find an unbiased estimator V for p (1 - p) if n22. rite barX_n for Л n . 72 1--X 7t
Here is the full question .
We are interested in finding an estimator for [tex]Var (X_i )[/tex] and propose to use :
[tex]\hat {V} = \bar {X}_n (1- \bar {X} )_n[/tex]
Now; we are interested in the basis of [tex]\hat V[/tex]
Compute :
[tex]E \ \ [ \bar V] - Var (X_i) =[/tex]
Using this; find an unbiased estimator [tex][ \bar V][/tex] for [tex]p(1-p) \ if \ n \geq 2[/tex]
Write [tex]bar \ x{_n} \ for \ X_n[/tex]
Answer:
Step-by-step explanation:
[tex]\bar X_n = \dfrac{1}{n} {\sum ^n _ {i=1} } \\ \\ E(X_i) = - \dfrac{1}{n=1} \sum p \dfrac{1}{n}*np = \mathbf{p}[/tex]
[tex]V(\bar X_n) = V ( \dfrac{1}{n_{i=1} } \sum ^n \ X_i )} = \dfrac{1}{n^2} \sum ^n_{i=1} Var (X_i) \\ \\ = \dfrac{1}{n^2} \ \sum ^n _{i=1} p(1-p) \\ \\ = \dfrac{1}{n^2}*np(1-p) \\ \\ = \dfrac{p(1-p)}{n}[/tex]
[tex]E( \bar X^2 _ n) = Var (\bar X_n) + [E(\bar X_n)]^2 \\ \\ = \dfrac{p(1-p)}{n}+ p \\ \\ = p^2 + \dfrac{p(1-p)}{n} \\ \\ \\ \hat V = \bar X_n (1- \bar X_n ) = \bar X_n - \bar X_n ^2 \\ \\ E [ \hat V] = E [ \bar X_n - \bar X_n^2] \\ \\ = E[\bar X_n ] - E [\bar X^2_n] \\ \\ = p-(p^2 + \dfrac{p(1-p)}{n}) \\ \\ = p-p^2 -\dfrac{p(1-p)}{n}[/tex]
[tex]=p(1-p)[1-\dfrac{1}{n}] = p(1-p)\dfrac{n-1}{n}[/tex]
[tex]Bias \ (\bar V ) = E ( \hat V) - Var (X_i) \\ \\ = p(1-p) [1-\dfrac{1}{n}] - p(1-p) \\ \\ - \dfrac{p(1-p)}{n}[/tex]
Thus; we have:
[tex]E [\hat V] = p(1-p ) \dfrac{n-1}{n}[/tex]
[tex]E [\dfrac{n}{n-1} \ \ \bar V] = p(1 -p)[/tex]
[tex]E [\dfrac{n}{n-1} \ \ \bar X_n (1- \bar X_n )] = p (1-p)[/tex]
Therefore;
[tex]\hat V ' = \dfrac{n}{n-1} \bar X_n (1- \bar X_n)[/tex]
[tex]\mathbf{ \hat V ' = \dfrac{n \bar X_n (1- \bar X_n)} {n-1}}[/tex]
Please answer this correctly
4yd^2
2yd^2
6yd^2
Step-by-step explanation:
Rule: height x base/2
The area of the big triangle=2 x 4/2 = 4 yd^2
The area of the small one= 2 x2/2 =2yd^2
The total area of the trapezoid is the sum of these areas= 2+4=6yd^2
Please answer this question I give brainliest thank you! Number 9
Answer:
B
The mode is 11 and 3
The Median is 10
The mean is 12
Which set of ordered pairs does NOT represent a function ?
Answer:
The answer is C.
Step-by-step explanation:
For a function, we do a vertical line test. If there is more than one point in one single x-position, it is not a function. Example, the ordered pairs (1, 1) and (1, 2) do NOT describe a function because there are more than one point on x=1.
Scientists think that robots will play a crucial role in factories in the next several decades. Suppose that in an experiment to determine whether the use of robots to weave computer cables is feasible, a robot was used to assemble 507 cables. The cables were examined and there were 9 defectives. If human assemblers have a defect rate of 0.035 (3.5%), does this data support the hypothesis that the proportion of defectives is lower for robots than humans
Answer:
The data support the hypothesis that the proportion of defectives is lower for robots than humans.
Step-by-step explanation:
To know if the proportion of defectives is lower for robots than humans so as to prove if the hypothesis is true.
From the data given:
Total number of cables a robot assembled = 507
Defectives = 9
To get the defect rate = the number of defects divided by the total number of cables, multiplied by 100.
Defect rate = (9 / 507) x 100 = 0.01775 x 100
Defect rate for the robot = 1.775%
From the question, a robot was used and the defect rate after the calculation is 1.775%. While for humans, the defect rate is 3.5%. This implies, if humans were used to assembling the same 507 cables, there will be 17.745 defectives.
x / 507 = 3.5%
x (defectives) = 17.745
Therefore, the data support the hypothesis that the proportion of defectives is lower for robots than humans.
graph the equation in a coordinate plane, x+2y=4
Hope this helps!
Stay safe, have a good day :D
What are two possible measures of the angle below?
The smaller angle, inside the bold lines, is -90 degrees.
The larger angle, outside the bold lines, is 270 degrees.
Angles can be measured in increments between -90° and 630°.
What angles are created when two lines cross one other?Two straight lines are considered to be intersecting if they come together at the same point. The intersection of two lines is known as the junction point. When two lines intersect, four angles are produced. The sum of the four angles is always 360 degrees.
Two straight lines that cross one other and produce right angles are called perpendicular lines. There are four right angles created when two perpendicular lines cross.
There are two types of angle connections produced when lines intersect:
Congruent opposite angles
Nearby angles are helpful
The information is
Let O (0, 0) be the origin, where the y and x axes must connect.
Thus, the angles on the four quadrants of the axis are produced.
The fourth quadrant's crossing lines create an angle that is given by For, the anticlockwise measure, A = -90°.
B = 360n - 180° for the clockwise measure, and n = 3 in this case.
Hence, when we simplify, we obtain
The second angle has a length of = 630°.
As a result, the angles are 90° and 630°.
Go here to find out more about angles created by intersecting lines.
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Express the function G in the form f∘g. (Enter your answers as a comma-separated list. Use non-identity functions for
f(x) and g(x).)
Answer:
i dont really know what it is
Find the function value. cos150°
Answer:
[tex]cos150 = -\frac{\sqrt{3} }{2}[/tex]
Step-by-step explanation:
Recall the unit circle. At 150 deg, the point value is (-sqrt3/2, 1/2)
Remember the cosine is always the x-value, and sine is always the y-value.
This means that cosine will be -sqrt3/2.
Really in need of help :( please !
Answer:
A
Step-by-step explanation:
(2,1) - only one in the table. Remember (x,y)
There are 225 students at March middle school. On Friday, 135 students wore spirit shirts. What percent of the students did Not wear spirit shirts on Friday?
Answer:
40%
Step-by-step explanation:
To find the answer to this, you first can find out what percentage of students did wear spirit shirts. To do this you can divide 135 by 225 to give you 0.6. To convert the decimal into a percentage you can simply multiply by 100, giving you 60%. Then to find the percentage of students that did not wear spirit shirts, you can subtract 60 from 100, giving you 40%.
Two students, A and B, are working independently on homework (not necessarily for the same class). Student A takes X = Exp(1) hours to finish his or her homework, while B takes Y = Exp(2) hours. (a) Find the CDF of X/Y , the ratio of their problem-solving times. (b) Find the probability that A finishes his or her homework before B does.
Answer:
a) The CDF of X/Y is calculated as:
[tex]F_{z} (\zeta) = \frac{\zeta}{\zeta + 2}[/tex] for [tex]0 < \zeta < \infty[/tex]
[tex]F_{z} (\zeta) = 0[/tex] for [tex]\zeta \leq 0[/tex]
Note: Z = X/Y
b) Probability that A finishes before B = 1/3
Step-by-step explanation:
For clarity and easiness of expression, this solution is handwritten and attached as a file. Check the complete solution in the attached file.
If you put all of the bean sprouts together end to end what
would be the total length of all the objects?
inches
Answer:
5.625 = 45/8 = 5 5/8
Step-by-step explanation:
Find the area of a rectangle with base 4ft and height 4 in.
Hey there!
Answer:
A = 192 in²
Step-by-step explanation:
Base = 4 ft
Height = 4 in
Convert the length of the base to inches:
4 ft = 4 ×12 = 48 in
Formula for the area of a rectangle: A = l × w (in this instance, 'l' is the height)
4 × 48 = 192 in²
Answer:
A =192 in^2
Step-by-step explanation:
Change the 4 ft to inches
4 * 12 = 48 inches
A = l*w
A = 48*4
A =192 in^2
Antonio burns 75 calories for every 15 minutes
Answer is 5 calories/min
75 divided by 15 is 5
Answer:five cal per min.15 in five groups equals ''75''
Determine whether the following procedure is a binomial experiment.
If it is not, explain why. Drawing 5 marbles from a bag with 10 red, 8 green and 12 yellow marbles without replacement and finding out how many of these five are green.
a. Yes, this is a binomial experiment.
b. No, the outcomes cannot be classified into two categories.
c. No, the trials are not independent
Answer:
C. The trails are not independent.
The probability of drawing one marble will not be independent of others thus option (c) is correct.
What is probability?The probability of an event occurring is defined by probability.
Probability is also called chance because if you flip a coin then the probability of coming head and tail is nothing but chances that either head will appear or not.
As per the given,
Drawing 5 marbles from a bag with 10 red, 8 green, and 12 yellow marbles without replacement.
In without replacement, the remaining balls in each draw will go to be decreased thus they will be dependent events so binomial distribution will not be applied.
Hence "One marble's likelihood of being drawn won't be independent of the other marbles".
For more information about the probability,
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Find the value of y when equals zero. -7x+3y=30
Answer:
x = -30/7
Step-by-step explanation:
-7x+3y=30
Let y=0
-7x +0 = 30
Divide by -7
-7x /-7 = 30/-7
x = -30/7
Answer:
[tex]-\frac{30}{7}[/tex]
Step-by-step explanation:
y equals zero => y = 0
-7x+3y=30
-7x +3.0 = 30
-7x + 0 = 30
-7x = 30
-7x/-7 = 30/-7
x = -30/7
Hope this helps ^-^
What preserves a shapes orientation?
a. Vertical translation
b. Reflection across the shapes base
c. Rotation about its center
Answer:
a.vertical translation
I don’t know how to do this can someone help?
Answer:
67
Step-by-step explanation:
Using triangle property
127+x=180
x=53
53+60+y=180
113+y=108
y=67
Statistics show that about 42% of Americans voted in the previous national election. If three Americans are randomly selected, what is the probability that none of them voted in the last election
Answer:
19.51% probability that none of them voted in the last election
Step-by-step explanation:
For each American, there are only two possible outcomes. Either they voted in the previous national election, or they did not. The probability of an American voting in the previous election is independent of other Americans. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
42% of Americans voted in the previous national election.
This means that [tex]p = 0.42[/tex]
Three Americans are randomly selected
This means that [tex]n = 3[/tex]
What is the probability that none of them voted in the last election
This is P(X = 0).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{3,0}.(0.42)^{0}.(0.58)^{3} = 0.1951[/tex]
19.51% probability that none of them voted in the last election
Solve for y=x squared -18 solve for x
Step-by-step explanation:
[tex]y = {x}^{2} - 18 \\ y + 18 = {x}^{2} \\ square \: root \: both \: sides \: \\ \sqrt{y + 18} = \sqrt{ {x}^{2} } [/tex]
[tex]x = \sqrt{y + 18} [/tex]
Answer:
√y + 18 = x
Step-by-step explanation:
Let us solve it now.
y = x² - 18
Take -18 to the left side
y + 18 = x²
Now remove the square of x
√y + 18 = x
Which shows one way the equation can be represented in words?
Z-6=1.4
Answer:
6 less than a number is equal to 1 and 4 tenths
Step-by-step explanation:
What’s the correct answer for this question?
Answer: Choice C
Step-by-step explanation:
Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true.
3/10≠3/5*1/4
so event A and B are not independent.
Which of the following linear equations has the steepest slope?
A. Y = -2x +11
B. y=+x+4
C. y - x +7
D. y - 7+2
Answer:
A --- unless d is supposed to be "y= -7x + 2"
Step-by-step explanation:
The slope is m in y=mx + b
So:
a. y= -2x + 11 slope= -2
b. y= x + 4 slope= 1
c. y= -x + 7 slope= -1
d. y= -7 + 2 (I don's see an x but if there were an x I assume that the slope would equal -7)
The higher the m value, the steeper the slope because it is m/1
So, -2/1 is steeper than 1/1 or -1/1
On a residential single lane road there was a wreck that backed up traffic for 5 miles. 80% of the traffic consists of cars and 20% of the traffic consists of trucks. The average distance between vehicles is 3 feet. The average length of a car is 13.5 feet and the average length of a truck is 20 feet. Estimate how many vehicles are stuck in the traffic jam. (Hint: There are 5280 feet in 1 mile.) A. 853 vehicles B. 1510 vehicles C. 2103 vehicles D. 2320 vehicles
Answer: b) 1510 vehicles
Step-by-step explanation:
Total: 5 miles x 5280 ft per mile = 26,400
Cars: 80% of vehicles are cars with a length of 13.5 = 0.8(13.5)v = 10.8v
Trucks: 20% of vehicles are trucks with length of 20 = 0.2(20)v = 4v
Between: Distance between two vehicles is 3: (3/2)v = 1.5v
Total = Cars + Trucks + Between
26,400 = 10.8v + 4v + 1.5v
26,400 = 16.3v
1619.6 = v
the closest number of all of the options is (b) 1510
Find the point of diminishing returns (x comma y )for the function R(x), where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).
Complete Question
The complete question is shown on the first uploaded image
Answer:
The point of diminishing returns (x , y ) is (11, 21462)
Step-by-step explanation:
From the question we are told that
The function is [tex]R(x) = 10,000 -x^3 - 33x^2 + 800x , \ \ 0 \le x \le 20[/tex]
Here R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).
Now differentiating R(x) we have
[tex]R'(x) = -3x^2 +66x + 800[/tex]
Finding the second derivative of R(x)
[tex]R''(x) = -6x +66[/tex]
at inflection point [tex]R''(x) = 0[/tex]
So [tex]-6x +66 = 0[/tex]
=> [tex]x= 11[/tex]
substituting value of x into R(x)
[tex]R(x) = 10,000 -(11)^3 - 33(11)^2 + 800(11) ,[/tex]
[tex]R(x) = 21462[/tex]
Now the point of diminishing returns (x , y ) i.e (x , R(x) ) is
(11, 21462)
y
The figure shows A XYZ. XW is the angle
bisector of ZYXZ.
8
6.5
W
What is W Z?
Enter
your answer in the box. Do not round
your answer.
x
Z
6
units
Basic
Answer:
3.84 units
Step-by-step explanation:
By the properties of angle bisectors, ...
WZ/ZX = WY/YX
Solving for WY, we have ...
WY = (YX)(WZ)/(ZX) = (6.5/6)(WZ)
The length YZ is ...
YZ = 8 = WY +WZ
8 = (6.5/6)(WZ) +WZ = 12.5/6(WZ) . . . . substitute for WY
WZ = 8(6/12.5) . . . . multiply by 6/12.5
WZ = 3.84
Answer:
The correct answer is indeed 3.84 units
Step-by-step explanation:
I just took the test and got it correct hope this helps ☺
For a hyperbolic mirror the two foci are 42 cm apart. The distance of the vertex from one focus is 6 cm and from the other focus is 36 cm. Position a coordinate system with the origin at the center of the hyperbola and with the foci on the y-axis. Find the equation of the hyperbola.
Answer:
[tex]\dfrac{y^2}{225} -\dfrac{x^2}{216}=1[/tex]
Step-by-step explanation:
For a hyperbolic mirror the two foci are 42 cm apart.
The distance between the foci = 2c.
Therefore:
2c=42c=21The distance of the vertex from one focus = 6 cm
The distance of the vertex from the other focus = 36 cm
2a=36-6=30
a=15Now:
[tex]c^2=a^2+b^2\\21^2=15^2+b^2\\b^2=21^2-15^2\\b^2=216\\b=6\sqrt{6}[/tex]
If the transverse axis lies on the y-axis, and the hyperbola is centered at the origin. Then the hyperbola has an equation of the form:
[tex]\dfrac{y^2}{a^2} -\dfrac{x^2}{b^2}=1[/tex]
Therefore, the equation of the hyperbola is:
[tex]\dfrac{y^2}{225} -\dfrac{x^2}{216}=1[/tex]
Help asap giving branlist!!
Answer:
option 3
Step-by-step explanation:
x = 2 is a vertical line with an x-intercept of (2, 0) so the answer is Option 3.
Answer:
Option 3
Step-by-step explanation:
The value of x will always be 2. Y can be anything it wants to be and x will still be 2 no matter what, You could pick multiple points on the line for each graph, and only Option 3 will have x always being 2.