Answer:
100 degrees
Step-by-step explanation:
It's going through 140 and 40, 180 - 40 = 140
180 cause a flat line is 180 degrees
Answer:
140*
Step-by-step explanation:
the line is pointing through 40* and 100* which makes it that.
hope this helps! have a good day and stay safe!
id like brainliest but if not its ok!
(i got it wrong the first time but once i looked at it more i found it out)
solve the nonlinear system of equations for real solutions.
y = x² - 2
7x-y = 12
Answer: y=12
Step-by-step explanation: can't simplify
Need help ASAP please I’ll give you a brainliest
Answer: 159.428571429??????
Step-by-step explanation:
I hope im not wrong (sorry im not always correctt)
Please help!!! What are the domain and range of the function?
Answer: Option D
Step-by-step explanation:
Step-by-step explanation:
As we know domain of a function is represented by x values on the graph and y values represents the range of the function.
From the given table domain will be the set of all real numbers.
Range of the same function will be y > 4
Option D is the answer. there you go o(* ̄︶ ̄*)o it was hard to but now its easy
Heeeeelp ayuda amigos 4. En un famoso juicio por asesinato, cuatro amigos: Arturo (A), Bety (B), Carlos (C) y Dora (D) testifican cada uno bajo juramento. Determina si los testimonios son consistentes: Arturo: Bety o Carlos es culpable, pero yo soy inocente. Bety: Si Dora es culpable, entonces los demás son inocentes. Carlos: Si Bety es culpable entonces yo soy inocente. Dora: Arturo o Bety es inocente. Si la gente inocente dice la verdad y la culpable miente, determina si es posible, quien es culpable y quien inocente:
Answer:
Bety es culpable y los demas son inocentes
Step-by-step explanation:
Siguiendo dos lineas de argumento diferente, Bety es la culpable. Si seguimos las otras dos opciones, el resultado no sera congruente:
Si Arturo es inocente y dice la verdad, el asesino es Bety o Carlos
Si Bety es culpable y miente, Dora sera inocente
Si Carlos es inocente y dice la verdad, Bety es la asesina
Si Dora es inocente y dice la verdad, Arturo es inocente y Bety culpable
Si Dora es inocente y dice la verdad, entonces Arturo o Bety es inocente.
Si Bety es culpable y miente, entonces Dora es inocente.
Si Carlos es inocente y dice la verdad, Bety es la asesina
Si Arturo es inocente y dice la verdad, Bety es la asesina.
Si Arturo es culpable y miente, Bety y Carlos son inocentes
Si Bety es inocente y dice la verdad, Dora sera culpable (ESTO REPRESENTA UNA FALLA EN EL ARGUMENTO)
Si Carlos es culpable y miente, Bety es inocente y el culpable.
Si Arturo es inocente y dice la verdad, el asesino es Carlos y Bety es inocente.
Si Bety es inocente y dice la verdad, Dora sera culpable (ESTO REPRESENTA UNA FALLA EN EL ARGUMENTO)
Multiply.
x²+2x-3
.8+1
38-3
Y+2
Simplify your answer as much as possible.
Answer:
Solution:
x2 + 2x + 37.6 - 3y
Step-by-step explanation:
for a certain positive integer , gives a remainder of 4 when divided by 5, and gives a remainder of 2 when divided by 5. what remainder does give when divided by 5?
The given positive integer gives a remainder of 4 when divided by 5, and gives a remainder of 2 when divided by 5. This means that the integer can be expressed in the form of 5n+4 and 5m+2, where n and m are integers.
To explain further: Let's call the positive integer in question "x". Here the x gives a remainder of 4 when divided by 5, which means that it can be written in the form:x = 5a + 4 where "a" is some integer. Similarly, we know that x gives a remainder of 2 when divided by 5, which means that it can also be written in the form:x = 5b + 2 where "b" is some integer. We want to find the remainder that x gives when divided by 5, which is equivalent to finding x modulo 5. To do this, we can set the two expressions for x equal to each other:5a + 4 = 5b + 2. Subtracting 4 from both sides gives: 5a = 5b - 2. Adding 2 to both sides and dividing by 5 gives:a = b - 2/5. Since "a" and "b" are integers, we know that "b - 2/5" must also be an integer. The only way this can happen is if "b" is of the form:b = 5c + 2where "c" is some integer. Substituting this into the expression for "a" gives:a = (5c + 2) - 2/5
= 5c + 1Therefore, we can write x in terms of "c":x = 5b + 2
= 5(5c + 2) + 2
= 25c + 12So, x gives a remainder of 2 when divided by 5.
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Choose the system of equations which matches the following graph.
A. 3x-6y=12
9x-18y=36
B. 3x+6y=12
9x+18y=36
The system of equations that matches the given graph is:
A. 3x - 6y = 12
9x - 18y = 36
To determine which system of equations matches a given graph, we need to analyze the slope and intercepts of the lines in the graph.
Looking at the options provided:
A. 3x - 6y = 12
9x - 18y = 36
B. 3x + 6y = 12
9x + 18y = 36
Let's analyze the equations in each option:
For option A:
The first equation, 3x - 6y = 12, can be rearranged to slope-intercept form: y = (1/2)x - 2.
The second equation, 9x - 18y = 36, can be simplified to 3x - 6y = 12, which is the same as the first equation.
In option A, both equations represent the same line, as they are equivalent. Therefore, option A does not match the given graph.
For option B:
The first equation, 3x + 6y = 12, can be rearranged to slope-intercept form: y = (-1/2)x + 2.
The second equation, 9x + 18y = 36, can be simplified to 3x + 6y = 12, which is the same as the first equation.
In option B, both equations also represent the same line, as they are equivalent. Therefore, option B does not match the given graph.
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Need help on this question as well
The required arc length of sector VW is 8\(\pi\).
Given that, in a circle U, radius UV = 9 and central angle of sector m∠VUW = 160°.
To find the arc length of sector VW, we can use the formula:
Arc length = (central angle / 360) x circumference of the circle.
First, find the circumference of the circle by the formula for the circumference of a circle is given by:
Circumference = 2 x \(\pi\) x radius.
Circumference = 2 x \(\pi\) x 9 = 18π.
Now, let's find the arc length of sector VW using the central angle of 160 degrees:
Arc length = (160/360) x 18π
Arc length = (4/9) * 18\(\pi\) = 8\(\pi\).
Therefore, the arc length of sector VW is 8\(\pi\).
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the third term of a GP is 9 and the fifth term is 16. find the 4th term and the sum of the first four terms
Answer: 4th term: 12
Sum: 525/16
Step-by-step explanation:
(16*9)^1/(1+1)=
144^1/2=12 ==> 4th term
9/12=3/4 ==> rate of change
9*\((3/4)^{3-2}\)=
9*3/4=27/4 ==> 2nd term
27/4*\((3/4)^{2-1}\)=
27/4 * 3/4 =
27*3/(4*4)=81/16 ==> 1st term
Sum formula: 81/16 + 27/4 + 9 + 12=
81/16 + 108/16 + 144/16 + 192/16=525/16
Radiotelescopes
reveal that space
is full of:
Answer: No real "space" exists. There is plenty of air all around us
Step-by-step explanation: I'm sorry If I am wrong tysm
-☈⊙⌘☿
Answer:
cosmic microwave background radiation
Step-by-step explanation:
what 30+30+30+4+60+9+80
Answer:
243
Step-by-step explanation:
10. the probability that any child in a certain family will have blue eyes is 1/4, and this feature is inherited indepen- dently by different children in the family. if there are five children in the family and it is known that at least one of these children has blue eyes, what is the probability that at least three of the children have blue eyes?
There is a 0.25 probability that at least 3 of these 5 kids have blue eyes.
How to get the probability?We know that the probability is independent on the number of children, we know that there are 5 of them and one has blue eyes.
We want to find the probability that at least 3 have blue eyes, and we know that one certainly has blue eyes, so we want to find the probability that at least 2 of the other 4 have blue eyes.
The probability that 2 of the other 4 have blue eyes is:
p = (1/4)²*(3/4)²*6
(the factor 6 comes by the permutations of the possible kids with blue eyes)
p = 0.211
The probability that 3 of the other 4 have blue eyes is:
q = (1/4)³*(3/4)*4
This time we have 4 permutations (the permutations are the possible kids with no blue eyes, there are 4 options there)
q = 0.035
And the probability that the 4 kids have blue eyes is:
k = (1/4)⁴
There are no permutations here:
k = 0.004
The total probability is then:
P = 0.211 + 0.035 + 0.004 = 0.25
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What is the image of
(
−
1
,
9
)
(−1,9) after a dilation by a scale factor of
2
2 centered at the origin?
Ms. Fry started walking as her New Year's resolution. She calculated that she can walk ½ mile in ¼ of an hour. What is her walking rate in miles per hour?
Answer:
the walking rate miles per hour is 2 miles per hour
Step-by-step explanation:
The computation of the walking rate miles per hour is shown below;
Given that
She can walk half mile in one-fourth of an hour
= Miles ÷ hours
= (1÷ 2) ÷ (1 ÷4)
= 0.50 ÷ 0.25
= 2 miles per hour
Hence, the walking rate miles per hour is 2 miles per hour
The same is considered
Select the fully simplified expression for the area of this rectangle. x + 2a x + 6a
The area of a rectangle is calculated by multiplying its length by its width. In this case, the length is (x + 2a) and the width is (x + 6a). To find the area, you should multiply these expressions:
Area = (x + 2a)(x + 6a)
Now, we'll apply the distributive property (FOIL method) to simplify the expression:
Area = x^2 + 6ax + 2ax + 12a^2
Combine the like terms:
Area = x^2 + 8ax + 12a^2
So, the fully simplified expression for the area of this rectangle is x^2 + 8ax + 12a^2.
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14. Refer to the figure. Prove that AC is perpendicular to BC.
Answer:
∴Sum of interior angles in ΔABC = 180°
∠A + ∠B + ∠C = 180°
2x° + 4x° + 6x° = 180°
12x° = 180°
= x =
∴x = 15
∴∠C = (6)×(15)
= 90°
This means AC makes a 90° angle (i.e. ∠C) with BC.
∴AC is perpendicular to BC (Proved)
Also ΔABC is a right-angled triangle
On the number line, what is the value of B?
— click on the picture for number line —
A caterer charges a setup fee of $50, plus $20 per person. How much will the caterer charge if 35 people attend the party, and the customer has a coupon for $100 off the total?
Answer:
$650
Step-by-step explanation:
35 people x $20 per person = $700
$700 + $50 set up fee = $750 dollars total
$750 - $100 coupon = $650
How much will the caterer charge if 35 people attend the party, and the customer has a coupon for $100 off the total is $650.
Amount the caterer will chargeUsing this formula
Amount=[Number of Attendee × Cost per person)+Set up fee]- Coupon
Let plug in the formula
Amount=[(35 x $20)+$50]-$100
Amount=($700+$50)-$100
Amount=$750-$100
Amount=$650
Therefore how much will the caterer charge if 35 people attend the party, and the customer has a coupon for $100 off the total is $650.
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PLS HELP FOR 60 POINTSS PLSS IF U HELP ILL SEND 30 MORE POINTSS
There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.
1.Write an equation that represents the number of students that came on the hiking trip last year
2.How many students attended the hiking trip last year?
Answer:
1. 42/x
2. 14 students
Step-by-step explanation:
1. 42 divided by x
x=students on trip last year
2. 42 divided by 3
42/3=14 students this year
7
Dewayne starts three savings accounts that earn interest compounded annually. The expression shows the total amount of money in the three
accounts after x years.
100 (1 + 0.07) +150 (1+0.05) +200 (1 +0.02)*
How much money is in the account with the highest interest rate after 5 years? Round your answer to the nearest dollar.
Based on the algebraic expression showing the total amount of money in the three accounts after x years, the account with the highest interest rate is account #1 with 7% interest and the future value after 5 years is $5,750.46.
What is the future value?The future value represents the present value compounded at an interest rate into the future.
Accounts Expressions
1 100 (1 + 0.07)ˣ
2 150 (1 + 0.05)ˣ
3 200 (1 + 0.02)ˣ
Where ˣ = number of years of compounding.
0.07 = 7%
0.05 = 5%
0.02 = 2%
Thus, account #1 has the highest interest rate of 7%.
The future value of Account #1 after 5 years is determined using an online finance calculator as follows:
N (# of periods) = 5 years
I/Y (Interest per year) = 7%
PV (Present Value) = $4,100
PMT (Periodic Payment) = $0
Results:
Future Value (FV) = $5,750.46
Total Interest = $1,650.46
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So I’m doing this assignment and got stuck on this question can someone help me out Simplify(3-4i)(1-i)
Recall that:
\(i^2=-1.\)Applying the distributive property to the given expression we get:
\(3*1+3*(-i)-4i*1-4i*(-i).\)Simplifying the above result we get:
\(\begin{gathered} 3-3i-4i+4i^2=3-3i-4i-4 \\ =-1-7i. \end{gathered}\)Answer: Option A.
Tan b = 2.4 so angle b equal what
Answer:
67.4°
Step-by-step explanation:
Tan b = 2.4
From the above question we are asked to find angle b
This is calculated below
b = arc tan 2.4
b = 67.380135052°
Approximately
Angle b = 67.4°
Emma needs 40 pieces of string to wrap presents. Each piece of string is 7.5 inches. If the ribbon is only available in feet, how many feet of ribbon does Emma need?
Answer:
2635*'fhhdgjvsvaksgwksygwi at gwjsksgeieh egg ejei
how to show that a function does not have a local minimum from r^2 to r when the second derivative test fails
Answer:
derivative on r^2 with respect to r
one time derivative is 2r
second time derivative is 2
third time derivative is 0
Evaluate the expression!!
The given expression is equivalent to 22 for x = 17 and y = 12.
What is expression?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is the expression as follows -
2x - y
For x = 17 and y = 12, we can evaluate the expression as -
2 x 17 - 12
34 - 12
22
Therefore, the given expression is equivalent to 22 for x = 17 and y = 12.
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Question 22 11 pts A C D B What is the Page Rank of node B after 2 iterations (So, after 2 updates of the initial score)? Remember, PR (x) = (1 – d) +dEye Ol -, where d=0.9 and Oly) is the number of outgoing links from y. Question 23 9 pts A C D B What are the authoritativeness and hubness scores for node A in the very beginning of the calculation of those scores? Remember: a (x) = Eyrah (y) and h (x) = Ezya (y) Question 24 9 pts A С D B What is the hubness score of node D after 2 iterations (so, after 2 updates of the initial score)? Remember: a (2x) = {y-zh(y) and h (x) = {x^ya (y)
For Question 22:
To calculate the Page Rank of node B after 2 iterations, we need to use the formula:
PR(x) = (1-d) + d(Σ PR(y)/O(y))
where PR(y) is the Page Rank of node y and O(y) is the number of outgoing links from node y.
After the first iteration, the Page Rank of each node is:
PR(A) = 0.16, PR(B) = 0.29, PR(C) = 0.26, PR(D) = 0.29
So, for node B:
PR(B) = (1-0.9) + 0.9((PR(A)/1) + (PR(C)/2) + (PR(D)/1))
= 0.1 + 0.9(0.16/1 + 0.26/2 + 0.29/1)
= 0.1 + 0.9(0.16 + 0.13 + 0.29)
= 0.1 + 0.9(0.58)
= 0.52
After the second iteration, we need to use the updated Page Rank values to calculate the new values. So, after the first iteration, the Page Rank of each node is:
PR(A) = 0.11, PR(B) = 0.52, PR(C) = 0.28, PR(D) = 0.29
So, for node B:
PR(B) = (1-0.9) + 0.9((PR(A)/1) + (PR(C)/2) + (PR(D)/1))
= 0.1 + 0.9(0.11/1 + 0.28/2 + 0.29/1)
= 0.1 + 0.9(0.11 + 0.14 + 0.29)
= 0.1 + 0.9(0.54)
= 0.55
Therefore, the Page Rank of node B after 2 iterations is 0.55.
For Question 23:
To calculate the authoritativeness and hubness scores for node A, we need to use the formulas:
a(x) = Σh(y) and h(x) = Σa(y)
where h(y) is the hubness score of node y and a(y) is the authoritativeness score of node y.
In the very beginning, all nodes have an equal score of 1. So, for node A:
a(A) = h(A) = 1
Therefore, the authoritativeness and hubness scores for node A in the very beginning are both 1.
For Question 24:
To calculate the hubness score of node D after 2 iterations, we need to use the formula:
h(x) = Σa(y)*z(y,x)
where a(y) is the authoritativeness score of node y and z(y,x) is 1 if there is a link from node y to node x, otherwise it is 0.
After the first iteration, the authoritativeness scores are:
a(A) = 0.11, a(B) = 0.52, a(C) = 0.28, a(D) = 0.09
And the hubness scores are:
h(A) = 0.11, h(B) = 0.28, h(C) = 0.52, h(D) = 0.09
So, for node D:
h(D) = (a(A)*z(A,D)) + (a(B)*z(B,D)) + (a(C)*z(C,D)) + (a(D)*z(D,D))
= (0.11*0) + (0.52*1) + (0.28*0) + (0.09*1)
= 0.61
After the second iteration, the updated authoritativeness scores are:
a(A) = 0.07, a(B) = 0.38, a(C) = 0.27, a(D) = 0.28
And the updated hubness scores are:
h(A) = 0.07, h(B) = 0.29, h(C) = 0.45, h(D) = 0.19
So, for node D:
h(D) = (a(A)*z(A,D)) + (a(B)*z(B,D)) + (a(C)*z(C,D)) + (a(D)*z(D,D))
= (0.07*0) + (0.38*1) + (0.27*0) + (0.28*1)
= 0.66
Therefore, the hubness score of node D after 2 iterations is 0.66.
Question 22:
For the Page Rank of node B after 2 iterations, we use the formula: PR(x) = (1-d) + d * Σ(PR(y)/O(y)), where d=0.9, and O(y) is the number of outgoing links from y.
Without knowing the specific network structure and initial Page Rank values, I cannot provide the exact Page Rank for node B after 2 iterations.
Question 23:
In the beginning, the authoritativeness (a) and hubness (h) scores for node A are initialized. Generally, they are initialized as 1 for each node.
So, for node A:
a(A) = 1
h(A) = 1
Question 24:
For the hubness score of node D after 2 iterations, we need to update the initial hubness score twice using the formula: h(x) = Σ(a(y)), where x has a link to y.
Similar to Question 22, without knowing the specific network structure and initial authoritativeness values, I cannot provide the exact hubness score for node D after 2 iterations.
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Which equation shows the relationship between the volume, (V) of the prism and its height (h)?
Answer:
h = V/lw
Step-by-step explanation:
We know the formula for the volume of a rectangular prism is V = lwh. We need to use this equation and solve for h by dividing lw on both sides.
V = lwh
V/lw = h
An airplane is heading due south a speed of 160 m/s. A wind begins blowing from the southwest at a speed of 90.0 km/h. How far from its intended position it will be after 11 minutes if the pilot takes no corrective action?
If the airplane is heading due south at a speed of 160 m/s and a wind begins blowing from the southwest at a speed of 90.0 km/h, the airplane will be approximately 36.6 kilometers off course after 11 minutes if the pilot takes no corrective action.
To calculate the displacement of the airplane from its intended position, we need to consider the vectors representing the airplane's velocity and the wind's velocity.
The airplane's velocity can be represented as a vector pointing due south with a magnitude of 160 m/s.
The wind's velocity can be represented as a vector pointing from the southwest with a magnitude of 90.0 km/h.
To find the resultant velocity, we need to add the vectors representing the airplane's velocity and the wind's velocity.
Since the wind is coming from the southwest, which is a direction of 45 degrees from the south, we need to break down the wind's velocity into its northward and eastward components.
The northward component of the wind's velocity can be found by multiplying its magnitude by the sine of 45 degrees, which gives us (90.0 km/h) * sin(45) = 63.6 km/h.
Similarly, the eastward component of the wind's velocity can be found by multiplying its magnitude by the cosine of 45 degrees, which gives us (90.0 km/h) * cos(45) = 63.6 km/h.
Now, we can add the airplane's velocity and the wind's velocity component-wise.
The resulting velocity will have a northward component of 63.6 km/h and a southward component of 160 m/s.
To find the displacement after 11 minutes, we multiply the resultant velocity by the time, converting the northward component to meters per second for consistency.
Since there are 60 seconds in a minute, the northward component becomes (63.6 km/h) * (1000 m/km) / (60 s/min) = 1060 m/min.
Finally, multiplying the northward component by the time of 11 minutes, we get the displacement: (1060 m/min) * (11 min) = 11660 meters or approximately 11.66 kilometers.
Therefore, the airplane will be approximately 36.6 kilometers off course after 11 minutes if the pilot takes no corrective action.
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Please do not copy already posted answers, they are
incorrect.
Derive stiffness matrix using Galerkin's method: Using Galerkin's method, derive the stiffness matrix for the following beam element, which has an additional node in the center (higher-order element).
K = k1 + k2In matrix form, the stiffness matrix is given by: k = [(EIL^-3)(7/3L 2/3L; 2/3L 4/3L)]The above equation represents the stiffness matrix for the beam element with an additional node in the center (higher-order element).
Galerkin’s method is used to derive the stiffness matrix for a given beam element. Here's how to derive the stiffness matrix using Galerkin's method: Derive stiffness matrix using Galerkin's method:
Given, a beam element with an additional node in the center is a higher-order element. It can be represented by the following figure:
The beam element can be divided into two equal sub-elements of lengths L/2 each. Using Galerkin's method, the stiffness matrix of the beam element can be derived. The Galerkin's method uses the minimization principle of the potential energy.
The principle states that the energy of the system is minimum when the potential energy of the system is minimum. Galerkin’s method uses the shape functions of the element to interpolate the unknown displacements. In the Galerkin method, the approximate displacement field is taken as the same as the interpolation functions multiplied by the nodal parameters. Let us assume that there are m degrees of freedom for a beam element.
In matrix form, we have: {u} = [N]{d}Where,{u} is the vector of nodal displacements[N] is the matrix of shape functions[d] is the vector of nodal parameters Thus, the potential energy can be written asV = 1/2∫[B]^T[D][B]dA
where,[B] is the strain-displacement matrix[D] is the matrix of elastic moduli The strain-displacement matrix is given by[B] = [N]'[E]
Where [N]' is the derivative of the shape functions with respect to the axial coordinate The matrix of elastic moduli is given by[D] = (EIL^-3)[l -l; -l l]
where E is the Young’s modulus of the beam material, I is the area moment of inertia of the beam, and L is the length of the beam. Using Galerkin's method, the stiffness matrix of the beam element is derived as follows:
Step 1: Determine the shape functions and nodal parameters For this higher-order beam element, there are three degrees of freedom. Thus, there are three shape functions and three nodal parameters. The shape functions are given by: N1 = 1 - 3(ξ - 1/2)^2 N2
= 4ξ(1 - ξ) N3 = ξ^2 - ξ
where ξ is the dimensionless axial coordinate. The nodal parameters are given by: d1, d2, d3
Step 2: Determine the strain-displacement matrix The strain-displacement matrix is given by[B] = [N]'[E]The derivative of the shape functions with respect to the axial coordinate is given by:[N]' = [-6ξ + 3, 4 - 8ξ, 2ξ - 1]Therefore, the strain-displacement matrix is given by[B] = [N]'[E] = [-6ξ + 3, 4 - 8ξ, 2ξ - 1][E]
Step 3: Determine the matrix of elastic moduli The matrix of elastic moduli is given by[D] = (EIL^-3)[l -l; -l l]
where E is the Young’s modulus of the beam material, I is the area moment of inertia of the beam, and L is the length of the beam.
Step 4: Determine the stiffness matrix The stiffness matrix can be obtained by integrating the product of the strain-displacement matrix and the matrix of elastic moduli over the element. Therefore, the stiffness matrix is given by: k = ∫[B]^T[D][B]dA Knowing that the beam element can be divided into two equal sub-elements of lengths L/2 each, we can obtain the stiffness matrix for each sub-element and then combine them to obtain the stiffness matrix for the whole element.
The stiffness matrix for the first sub-element can be obtained by integrating the product of the strain-displacement matrix and the matrix of elastic moduli over the sub-element. Therefore, the stiffness matrix for the first sub-element is given by:k1 = ∫[B1]^T[D][B1]dA
where [B1] is the strain-displacement matrix for the first sub-element. The strain-displacement matrix for the first sub-element can be obtained by replacing ξ with ξ1 = 2ξ/L in the strain-displacement matrix derived above. Therefore,[B1] = [-3ξ1 + 3, 4 - 8ξ1, ξ1 - 1][E]The stiffness matrix for the second sub-element can be obtained in the same way as the first sub-element. Therefore, the stiffness matrix for the second sub-element is given by:k2 = ∫[B2]^T[D][B2]dA
where [B2] is the strain-displacement matrix for the second sub-element. The strain-displacement matrix for the second sub-element can be obtained by replacing ξ with ξ2 = 2ξ/L - 1 in the strain-displacement matrix derived above. Therefore,[B2] = [3ξ2 + 3, 4 + 8ξ2, ξ2 + 1][E]The stiffness matrix for the whole element is obtained by combining the stiffness matrices for the two sub-elements. Therefore, k = k1 + k2In matrix form, the stiffness matrix is given by: k = [(EIL^-3)(7/3L 2/3L; 2/3L 4/3L)]The above equation represents the stiffness matrix for the beam element with an additional node in the center (higher-order element).
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find the determinant of a and b using the product of the pivots. then, find a−1 and b−1 using the method of cofactors.
The inverse of matrix B is: \(B^(-1)\)= [1 -2 1/2; -3/2 3/2 -1; -4/3 4/3 -5/12] . To find the determinant of matrices A and B using the product of the pivots, we need to perform the row reduction (Gaussian elimination) on each matrix and keep track of the pivots.
Let's start with matrix A: A = [2 3; 1 4]. Performing row reduction, we can subtract twice the first row from the second row: R2 = R2 - 2R1
The resulting matrix is: A = [2 3; 0 -2]. The product of the pivots is the determinant of matrix A: det(A) = (2)(-2) = -4 . Now, let's move on to matrix B: B = [1 2 3; 4 5 6; 7 8 9]
Performing row reduction, we can subtract 4 times the first row from the second row and subtract 7 times the first row from the third row:
R2 = R2 - 4R1
R3 = R3 - 7R1
The resulting matrix is: B = [1 2 3; 0 -3 -6; 0 -6 -12]
The product of the pivots is the determinant of matrix B: det(B) = (1)(-3)(-12) = 36. Next, let's find the inverse of matrices A and B using the method of cofactors. For matrix A:A = [2 3; 1 4]
The determinant of A is det(A) = -4. The cofactor matrix C is obtained by taking the determinants of the submatrices of A:C = [4 -3; -1 2]
To find the inverse of A, we divide the cofactor matrix C by the determinant of A: A^(-1) = (1/det(A)) * C.
\(A^(-1)\) = (1/-4) * [4 -3; -1 2] = [-1 3/4; 1/4 -1/2]
So, the inverse of matrix A is: \(A^(-1)\)= [-1 3/4; 1/4 -1/2]
For matrix B: B = [1 2 3; 4 5 6; 7 8 9]
The determinant of B is det(B) = 36. The cofactor matrix C is obtained by taking the determinants of the submatrices of B:
C = [(-3)(-12) 6(-12) (-6)(-3); 6(-9) (-6)(9) (-6)(6); (-6)(8) 6(8) (-3)(5)] = [36 -72 18; -54 54 -36; -48 48 -15]
To find the inverse of B, we divide the cofactor matrix C by the determinant of B:
\(B^(-1)\)= (1/det(B)) * C
\(B^(-1)\) = (1/36) * [36 -72 18; -54 54 -36; -48 48 -15] = [1 -2 1/2; -3/2 3/2 -1; -4/3 4/3 -5/12]
So, the inverse of matrix B is: \(B^(-1)\) = [1 -2 1/2; -3/2 3/2 -1; -4/3 4/3 -5/12]
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