Answer:
36893/12, de gramo de cobre y plata para las artesanías.
También pudiste haber encontrado el resultado escrito en forma de número mixto que en este caso sería 3074 y 5/12
La respuesta es 3 mil 74 gramos y cinco doceavos.
En este ejercicio utilizaremos los conocimientos de estadística para calcular cuantas piezas se necesitarán para hacer el conteo, de esta forma encontramos que:
tres mil setenta y cuatro gramos y cinco doceavos.
Para iniciar el cálculo tendremos que:
\(36893/12 = 5/12\)
También pudiste haber encontrado el resultado escrito en forma de número mixto que en este caso sería:
tres mil setenta y cuatro gramos y cinco doceavos.
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EASY MATH PLEASE HELP!! : )
I WILL GIVE BRAINLIEST!
QUESTION: find the area of the shaded part.
120
10 x 8 = 80
8 x 5 = 40
80 + 40 = 120
Answer:
is it 40
Step-by-step explanation:
6×10=60. 4×5=20
60-20=40
I NEED THIS FOR ONCE
Answer:
correct frequency is A
Step-by-step explanation:
There is a frequency table and no. of vegtables eating for 36 days so we write fill the table as
0 1×1 7×2 6×3 8×4 6×5 2×6 5×So the correct frequency is A which fulfill every frequency of no.
A ski instructor recorded the amount of snow that fell each Saturday during ski season. He displayed the data in a line plot.
What is the range in this set of data?
the range in this set of data is 25 inches. To find the range of snowfall amounts, we need to look at the line plot and identify the highest and lowest amounts of snowfall that were recorded.
The range in a set of data is the difference between the maximum and minimum values in the set.
To find the range of snowfall amounts, we need to look at the line plot and identify the highest and lowest amounts of snowfall that were recorded.
Let's say that the highest amount recorded was 30 inches and the lowest amount recorded was 5 inches. Then the range would be:
range = highest amount - lowest amount
range = 30 - 5
range = 25
Therefore, the range in this set of data is 25 inches.
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Construct a counterexample by model for the following invalid argument, and then explain exactly how your counterexample proves that the argument is invalid.
¼ of the hairs in Plato’s beard are gray. ¾ of the hairs in Aristotle’s beard are gray. So, Aristotle has a greater number of gray hairs in his beard than Plato does.
The given argument is invalid. It says that if one person has more gray hair in their beard than another person, then they have a higher percentage of gray hair in their beard.
The argument is invalid because it is possible for one person to have a higher percentage of gray hair in their beard but a lower number of gray hairs in total.One possible counterexample by model is:Suppose Plato has 200 hairs in his beard and 50 of them are gray. So, 1/4 of his hairs are gray. Now, let's suppose Aristotle has 400 hairs in his beard, and 300 of them are gray. So, 3/4 of his hairs are gray. Even though Plato has fewer gray hairs than Aristotle, his percentage of gray hairs is more significant (1/4) than Aristotle's (3/4).
Therefore, the given argument is invalid since Aristotle's beard contains more gray hairs in quantity, but Plato has a higher percentage of gray hair in his beard.
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PLZZ HELP
When dividing fractions, the ________ term becomes the reciprocal.
Neither
Second
Both
First
Calculate the perimeter and the area of the figure use 3.14
Answer:
Step-by-step explanation:
13.
P= 19.42mm
A= 18.32mm²
Answer:
Half of 6 is 3(new radius)
2 x 3.14 x 3 = 18.84, 18.84 divided by 2 = 9.42
2 + 2 + 2 = 6
6 - 2 = 4
Add lengths:
9.42 + 6 + 4 = 19.42 mm is your perimeter
2 x 2 = 4
3^3 = 9
9 x 3.14 = 28.26, 28.26 divided by 2 (semi circle) = 14.13
Add areas:
4 + 14.13 = 18.13 mm2 is your area
For f(x) = 5x+1, find f(x_ when x = 2 and when x = 3.
Answer:
f(2) = 11
f(3) = 16
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Function NotationStep-by-step explanation:
Step 1: Define
f(x) = 5x + 1
x = 2
x = 3
Step 2: Evaluate
x = 2
Substitute: f(2) = 5(2) + 1Multiply: f(2) = 10 + 1Add: f(2) = 11x = 3
Substitute: f(3) = 5(3) + 1Multiply: f(3) = 15 + 1Add: f(3) = 16The formula for has the same numerator as the formula for but contains an extra factor of:_____.
The formula for nCr contains the same numerator as the formula for nPr but contains an extra factor of r! in the denominator.
What are factorials?In mathematics, the factorial of a non-negative integer n, represented by n! exists as the product of all positive integers less than or equal to n. The factorial of n even equals the product of n with the next less factorial:
Combinations exist as a method to estimate the total outcomes of an event where the order of the outcomes does not matter.
nCr = n! / r! \(*\) (n - r)!
where n denotes the total number of items, and r denotes the number of items being chosen at a time.
The term permutation directs to a mathematical analysis of the number of ways a particular set can be arranged. Put simply, a permutation exists as a term that defines the number of ways things can be collected or arranged.
nPr = n! / (n - r)!
Therefore, the correct answer is r!.
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The complete question is:
The formula for nCr has the same numerator as the formula for nPr but contains an extra factor of _____ in the denominator.
Simplify the following
-5 +6-8
Answer: -7
Step-by-step explanation:
Answer:
-5+6-8
-5+6=1
1-8=-7
because if u own someone five dollars and you have six dollars you can pay the person and it will remain one dollar. And the one dollar you own another person eight dollars you can give the person the one dollar for the mean time while it will be remaining seven dollars for you to pay the person
Write an example from daily life that uses each type of real number.
rational numbers
Write an example from daily life that uses rational numbers .
Real numbers that may be expressed as p/q, where p, q are integers, and q 0, are rational numbers. The money we have is a rational number. If we spend some sum out of it, it is subtraction of rational number.The running race involves rational numbers if you're an athlete.Rational numbers include the following:the distance runthe time needed to complete the distance,the number of competitors placing first, second, or thirdthe number of heartbeats you take per minute.We use fractions to represent taxes.If you split a pizza or any other food.When you finish a piece of your work, you indicate that you have done half, or 50%.Loan and mortgage interest rates.Savings account interest.To learn more about rational numbers, refer:
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A tailor spends $63.75 to buy 7.5 yards of satin. What is the cost of the satin fabric per yard?
Answer:
12.75
Step-by-step explanation:
use a truth table to determine whether the symbolic form of the argument is valid or invalid ~p -> q
The symbolic argument ~p -> q is valid.
To determine the validity of the argument ~p -> q using a truth table, we need to consider all possible combinations of truth values for p and q and evaluate the truth value of the implication ~p -> q.
The symbolic form of the argument is ~p -> q, which can also be written as ¬p → q.
A truth table for this argument would have columns for p, ~p, q, and ~p -> q.
Let's construct the truth table:
| p | ~p | q | ~p -> q |
| True | False | True | True |
| True | False | False | False |
| False | True | True | True |
| False | True | False | True |
In the truth table, we consider all possible combinations of true (T) and false (F) for p and q. For ~p, we negate the value of p.
For the implication ~p -> q, it is true (T) if either ~p is false (F) or q is true (T). In all other cases, it is false (F).
Looking at the truth table, we can see that in all rows where ~p -> q is true (T), the corresponding conclusion q is true (T). Therefore, the argument ~p -> q is valid because whenever ~p is true (F), the conclusion q is also true (T).
In summary, the symbolic argument ~p -> q is valid.
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a delivery person uses service elevator to bring boxes of books up to an office. the delivery peron weights 190lb and each box of books weighs 50lb . the maximum capacity of the elevator is 1430 lb how many boxes of books can the delivery person bring up at one time
The delivery person can bring up 28 boxes of books at one time because the maximum capacity of the elevator is 1430 pounds and each box of books weighs 50 pounds.
The delivery person weighs 190 pounds, and each box of books weighs 50 pounds. So, the maximum weight of the boxes of books that the delivery person can bring up at one time is 1430 - 190 = 1240 pounds.
Since each box of books weighs 50 pounds, the delivery person can bring up 1240 / 50 = 24.8 boxes of books at one time.
However, we cannot bring up a fraction of a box of books. So, the delivery person can bring up 24 boxes of books at one time.
The delivery person weighs 190 pounds.Each box of books weighs 50 pounds.The maximum capacity of the elevator is 1430 pounds.The maximum weight of the boxes of books that the delivery person can bring up at one time is 1430 - 190 = 1240 pounds.Since each box of books weighs 50 pounds, the delivery person can bring up 1240 / 50 = 24.8 boxes of books at one time.However, we cannot bring up a fraction of a box of books. So, the delivery person can bring up 24 boxes of books at one time.
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Um- Please help me, Thanks in advance
(1) Combining fractions is just a matter of writing each individual fraction in terms of a common denominator. The least common multiple of 6, 4, and 8 is
LCM(6, 4, 8) = 24
since 6×4 = 4×6 = 8×3 = 24. So
2/6 = (2×4)/(6×4) = 8/24
1 1/4 = 4/4 + 1/4 = 5/4 = (5×6)/(4×6) = 30/24
13 4/8 = 104/8 + 4/8 = 108/8 = (108×3)/(8×3) = 324/24
Then
2/6 + 1 1/4 + 13 4/8 = 8/24 + 30/24 + 324/24 = 362/24 = 181/12
(2) We have
LCM(12, 9) = 36
since 12×3 = 9×4 = 36. Then
121 2/12 = 1452/12 + 2/12 = 1454/12 = (1454×3)/(12×3) = 4362/36
781 6/9 = 7029/9 + 6/9 = 7035/9 = 2345/3
(4x^2y)(8x^5y^3 find product of monomials
The product of the monomials is 32x^7y4
What is a monomial?A monomial is said to be a product of powers of variables with a non-negative integer exponents, or a product of variables.
Monomials include numbers, variables or multiple numbers or variables multiplied together.
Given the monomials;
(4x²y)(8x^5y³)
Finding the product is simply done by multiplying the two expressions, we have
The product of the numbers = 4 × 8 = 32
The product of the 'x' variables = x^2 + 5 = x^7
The product of the 'y' variables = y^1 + 3 = y^4
32x^7y4
Thus, the product of the monomials is 32x^7y4
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Please help I am in trouble
Answer:
y decreases by 4 as x increases by 2
or
there is a -2 change between the relationship of x and y
I hope I understood the question correctly
can i get some help on this question
please help me solve this HCF
Answer:
(x+y)
Step-by-step explanation:
\( {x}^{2} - {y}^{2} = (x + y)(x - y)\)
Thus, the HCF is (x+y).
Answer:
(x + y)
Step-by-step explanation:
Factorizing both terms :
x² - y² = (x + y)(x - y)(x + y)² = (x + y)(x + y)The HCF of the terms is (x + y).
You may need to use the appropriate technology to answer this question.
The Consumer Reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains. † One of the variables in the study is meal price, the average amount paid per person for dinner and drinks, minus the tip. Suppose a reporter for a local newspaper thought that it would be of interest to her readers to conduct a similar study for restaurants located in her city. The reporter selected a sample of 8 seafood restaurants, 8 Italian restaurants, and 8 steakhouses. The following data show the meal prices ($) obtained for the 24 restaurants sampled.
Italian Seafood Steakhouse
$12 $17 $23
13 18 18
16 18 24
16 27 25
18 23 21
19 15 23
18 20 28
24 14 30
Use α = 0. 05 to test whether there is a significant difference among the mean meal price for the three types of restaurants.
State the null and alternative hypotheses.
H0: μItalian = μSeafood = μSteakhouse
Ha: Not all the population means are equal.
H0: μItalian ≠ μSeafood ≠ μSteakhouse
Ha: μItalian = μSeafood = μSteakhouse
H0: At least two of the population means are equal.
Ha: At least two of the population means are different.
H0: Not all the population means are equal.
Ha: μItalian = μSeafood = μSteakhouse
H0: μItalian = μSeafood = μSteakhouse
Ha: μItalian ≠ μSeafood ≠ μSteakhouse
Find the value of the test statistic. (Round your answer to two decimal places. )
test statistic=
Find the p-value. (Round your answer to four decimal places. )
p-value =
What is your conclusion?
Do not reject H0. There is not sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
Reject H0. There is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
Do not reject H0. There is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
Reject H0. There is not sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants
A) The stated the null and alternative hypotheses are defined as
H₀ : μItalian = μSeafood = μsteakhouse
Hₐ : Not all the population means are equal. So, correct answer is option (a).
B) The value of the test statistic is equals to the 6.70. The p-value is equals to the 0.0056.
C) Conclusion : The null hypothesis H0 is rejected, so there is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
We have a consumer reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains. The above table data show the meal prices ($) obtained for the 24 restaurants sampled ( 8 seafood restaurants, 8 Italian restaurants, and 8 steakhouses). So, sample size , n = 24
Significance level = 0.05
A) For Hypothesis testing, the null and alternative hypothesis are defined as
H₀ : μItalian = μSeafood = μsteakhouse
Hₐ : Not all the population means are equal. So, correct answer is option (a).
B) We have more than two groups for analysis so we use ANOVA test to compare the means among three or more groups. Using the Excel the above second table is made which represents the mean, standard deviations and standard error for data. It also contains ANOVA table. So,
Mean value of Italian = 17
Mean value of Seafood = 19
Mean value of steakhouse = 24
Uding all the above details or Excel command, The value of the F test- statistic value = 6.70
Using the distribution table, the p-value for test-static value 6.70 is 0.0056.
C)Now, we see P-value = 0.0056 < 0.05, which implies the null hypothesis is rejected. Thus, There is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
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Complete question:
You may need to use the appropriate technology to answer this question. The Consumer Reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains. † One of the variables in the study is meal price, the average amount paid per person for dinner and drinks, minus the tip. Suppose a reporter for a local newspaper thought that it would be of interest to her readers to conduct a similar study for restaurants located in her city. The reporter selected a sample of 8 seafood restaurants, 8 Italian restaurants, and 8 steakhouses. The following data show the meal prices ($) obtained for the 24 restaurants sampled.
Italian Seafood Steakhouse
$12 $17 $23
13 18 18
16 18 24
16 27 25
18 23 21
19 15 23
18 20 28
24 14 30
Use α = 0. 05 to test whether there is a significant difference among the mean meal price for the three types of restaurants. State the null and alternative hypotheses.
a) H0: μItalian = μSeafood = μSteakhouse
Ha: Not all the population means are equal.
b)H0: μItalian ≠ μSeafood ≠ μSteakhouse
Ha: μItalian = μSeafood = μSteakhouse
c) H0: At least two of the population means are equal.
Ha: At least two of the population means are different.
d)H0: Not all the population means are equal.
Ha: μItalian = μSeafood = μSteakhouse
e) H0: μItalian = μSeafood = μSteakhouse
Ha: μItalian ≠ μSeafood ≠ μSteakhouse
B) Find the value of the test statistic. (Round your answer to two decimal places. )
test statistic=___
Find the p-value. (Round your answer to four decimal places. )p-value =____
V)What is your conclusion?
a)Do not reject H0. There is not sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
b)Reject H0. There is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
c)Do not reject H0. There is sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants.
d)Reject H0. There is not sufficient evidence to conclude that the mean meal prices are not all the same for the three types of restaurants
given the numbers 0.29 , 0.816¯¯¯ , 2.515115111…, 2.63¯¯¯ , 0.125, and 0.418302…, select all of the rational numbers.
The rational numbers in the given list are 0.29, 0.816¯¯¯, 2.63¯¯¯, and 0.125.
A rational number is any number that can be expressed as a ratio of two integers (where the denominator is not equal to zero).
Looking at the given numbers:
- 0.29 can be expressed as 29/100, so it is rational.
- 0.816¯¯¯ can be expressed as 816/999, so it is rational.
- 2.515115111… and 0.418302… are both irrational since they are non-repeating and non-terminating decimals.
- 2.63¯¯¯ can be expressed as 263/99, so it is rational.
- 0.125 can be expressed as 1/8, so it is rational.
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1. explain the The Butler–Volmer (BV)
2. Equation example of BV equation with values
3. a report on BV
1. The Butler-Volmer equation is an empirical equation used to describe electrochemical reaction kinetics at the electrode-electrolyte interface. 2. An example equation using the Butler-Volmer equation with values would depend on the specific electrochemical system and reaction being studied.
3. A report on the Butler-Volmer equation would typically involve an analysis of electrochemical reactions.
1. The Butler-Volmer equation is an empirical equation used to describe the kinetics of electrochemical reactions occurring at an electrode-electrolyte interface. It relates the rate of electrochemical reactions to the electrode potential and the concentrations of reactants in the electrolyte. The equation considers both the forward and backward reaction rates, taking into account the activation energy and the transfer of charge between the electrode and the electrolyte.
2. The general form of the Butler-Volmer equation is given as:
i = i₀[exp((αₐFη)/(RT)) - exp((-αᵦFη)/(RT))]
where:
i is the current density,
i₀ is the exchange current density,
αₐ and αᵦ are the anodic and cathodic charge transfer coefficients, respectively,
F is the Faraday's constant,
η is the overpotential (the difference between the electrode potential and the thermodynamic equilibrium potential),
R is the gas constant,
T is the temperature.
An example equation using the Butler-Volmer equation with values would depend on the specific electrochemical system and reaction being studied.
3. A report on the Butler-Volmer equation would typically involve an analysis of electrochemical reactions and their kinetics at the electrode-electrolyte interface. The report may include a theoretical background on the Butler-Volmer equation, its derivation, and its applications. It would also discuss experimental methods used to determine the parameters in the equation, such as the exchange current density and charge transfer coefficients. The report may present experimental data, discuss the limitations and assumptions of the equation, and compare the results with theoretical predictions.
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Help MEEEEE PLEASEEE
Answer:
below.
Step-by-step explanation:
2. 4.
3. 180 3. Internal angles between parallel lines.
4. 6.
5. 20 160.
Find the distance between each pair of parallel lines with the given equations Y=7. Y = -1
The distance between the two parallel lines with the equations y = 7 and
y = -1 is 8 units
What is the distance of a line between 2 points?
The distance of a line between 2 points is always positive and given by the formula
Let the first point be A ( x₁ , y₁ ) and the second point be B ( x₂ , y₂ )
The distance between A and B is D , and the distance D is
\(Distance D = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} }\)
Given data ,
Let the equation of line be y = 7 and y = -1
Now , the slope of both the lines is 0
Slope m = 0
And the y intercept of the equation 1 or line 1 is = 7
So , the point is ( 0 , 7 )
And the y intercept of the equation 2 or line 2 is = -1
So, the point is ( 0 , -1 )
Now , the distance between the points ( 0 , 7 ) and ( 0 , -1 ) is given by the equation ,
\(Distance D = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2} }\)
Substituting the values in the equation , we get
Distance D = √ ( 7 - ( - 1 ) )²
Distance D = √ ( 8 )²
Distance D = 8 units
Therefore , the value of D is 8 units
Hence , The distance between the two parallel lines with the equations y = 7 and y = -1 is 8 units
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pease help me i really need this
Answer:
2:140
1:70
Step-by-step explanation:
:)
PLEASEEE I NEED MORE ANSWERS
Answer:
3 loaves of bread
Step-by-step explanation:
2/3*3=6/3= 2 cups of oil
A rectangular storage container without a lid is to have a volume of 10 m3. the length of its base is twice the width. material for the base costs $15 per square meter. material for the sides costs $9 per square meter. let w denote the width of the base. Find a function in the variable w giving the cost C in dollars) of constructing the box.
The function in variable w giving the cost C (in dollars) of constructing the box is C(w) = 30w² + 270/w. The result is obtained by using the formula of volume and area of the box.
How to determine the function?We have a rectangular storage container without a lid.
Volume, V = 10 m³Length, l = 2wWidth, w = wBase costs $15/m²Sides costs $9/m²The formula of volume of the box is
V = l × w × h
Where
l = lengthw = widthh = heightSo, the height is
10 = 2w × w × h
10 = 2w² × h
h = 10/2w²
h = 5/w²
To find the total cost, calculate the area of base and sides of the box!
See the picture in the attachment!
The base area is
A₁ = 2w × w = 2w² m²
The sides area is
A₂ = 2(2wh + wh)
A₂ = 2(3wh)
A₂ = 6wh
A₂ = 6w(5/w²)
A₂ = 30/w m²
The total cost is
C = $15(2w²) + $9(30/w)
C = $30w² + $270/w
The function of the total cost is
C(w) = 30w² + 270/w
Hence, the function of constructing the box is C(w) = 30w² + 270/w.
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A man wants to sell his house. He priced it at $150,000 the first day. The second day he reduced the price by 12%. What was the price of the house after this reduction
Answer:
$132,000
Step-by-step explanation:
Subtract 12% from 100%: 88%. Now convert 88% into a decimal fraction: 0.88. Finally, multiply $150,000 by (0.88). Answer: $132,000
the parking meter changes 18$ every 6 hours. Based on this rate, fill in the table
Answer:
1 hour=18
2 hour=32
3 hour=54
4 hour=70
5 hour=88
so on so forth
Step-by-step explanation: Take however many hours you have and multiply it by 18. Don't forget to add the dollar sign!
I need help desperately! Could someone explain this math concept for me please?
The inequality that represents the range of k so that inequality 1 has real solution is -3 < k < 5
How to determine the range?Inequality 1
x^2 + (k + 1)x + k + 4 > 0
The discriminant is calculated using
D = b^2 - 4ac
So, we have:
D = (k + 1)^2 - 4 * 1 * (k + 4)
Evaluate
D = k^2 + 2k + 1 - 4k - 16
This gives
D = k^2- 2k - 15
Factorize the equation
D = (k+ 3)(k -5)
Solve for k
k = -3 and k = 5
Express the k values as inequalities
-3 < k < 5
Inequality 2
The inequality cannot be read, because of the image quality
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(0)
Question 2 Consider the dynamic system described by Equation Q2. 85.16400 083.3770 0046.999 ⃗+ 0.079400 00.7030 001.07 ×10⃗+ 0.013600 03.1390 005.124 ×10⃗= 0 0 0 Equation Q2 (a) Calculate the spectral matrix, the undamped natural frequencies and damping ratios of the system in Equation Q2. Identify its fundamental frequency. (b) The following mode shape vectors have been used to diagonalise the equations of motion of the dynamical system presented in Equation Q2: f1 = [0.8076 1.0000 0.8039]T; f2 = [-0.9694 -0.1620 1.0000]T and f3 = [-0.5342 1.0000 -0.3523]T. Calculate the respective matrix of mass normalised mode shapes. (c) Using the mode superposition method, calculate the response of the system for the first physical coordinate y1 assuming the following initial conditions expressed in terms of the modal coordinates: the initial modal displacements are [0 0.5 0]T m and the initial modal velocities are [0 -3 0]T m/s.
The first physical coordinate y1 can be expressed as y1 = [1 0 0]Y, & The mass-normalised mode shapes can be normalising the mode shape vectors f1, f2, and f3.
Part (a)
In Equation Q2, the spectral matrix, undamped natural frequencies, damping ratios, and fundamental frequency need to be calculated.
The mass matrix is given by [85.16400 083.3770 0046.999; 0.079400 00.7030 001.07 × 10; 0.013600 03.1390 005.124 × 10].
The stiffness matrix is given by [0.16400 00.3770 000.999; 0.079400 00.7030 001.07 × 10; 0.013600 03.1390 005.124 × 10].
The damping matrix is given by [0 0 0; 0 0 0; 0 0 0].The undamped natural frequencies, damping ratios, and fundamental frequency for the system in Equation Q2 can be calculated from the spectral matrix.
The characteristic equation can be written as det(K-mω^2M)=0.where K is the stiffness matrix, M is the mass matrix, ω is the angular frequency, and m is the mass-normalised mode shape.
The roots of this equation are the undamped natural frequencies, and the damping ratios can be calculated from the undamped natural frequencies and mode shapes.
The mass-normalised mode shapes can be calculated by normalising the mode shape vectors f1, f2, and f3.
Part (b)
The mass-normalised mode shapes can be calculated using the mode shape vectors f1, f2, and f3.Part (c)The response of the system for the first physical coordinate y1 can be calculated using the mode superposition method. The initial modal displacements and velocities are given in terms of the modal coordinates.
The response is then calculated using the equation y(t)= Σ ai φi(t), where ai are the modal amplitudes, and φi(t) are the modal shapes given by the mode shape vectors f1, f2, and f3.
The first physical coordinate y1 can be expressed as y1 = [1 0 0]Y, where Y is the vector of physical coordinates. The modal amplitudes can be calculated from the initial modal displacements and velocities.
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