Suppose R and R’ are 2 × 3 row-reduced echelon matrices and that the systems RX = 0 and R’X = 0 have exactly the same solutions. Prove that R = R’

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Algebra

1.We say that two matrices A and B are row-equivalent if one can be obtained from the other by a finite sequence of elementary row operation. Show that if A and B are row-equivalent matrices, then the homogeneous systems of linear equations Ax = 0 and Bx = 0 have exactly same solutions.

  1. Suppose R and R’ are 2 × 3 row-reduced echelon matrices and that the systems RX = 0 and R’X = 0 have exactly the same solutions. Prove that R = R’
  2. Suppose A is a 2 × 1 matrix and that B is a 1 × 2 matrix. Prove that C = AB is not invertible.
  3. For each matrix use elementary row operations to discover whether it is invertible, and to find the inverse in case it is.
  4. a) 2 5 −1

4 −1 2

6 4 1

  1. b) 1 −1 2

3 2 4

0 1 −2

  1. c) 1 2 3 4

0 2 3 4

0 0 3 4

0 0 0 4

 

 

 

List the operations of relational algebra and the purpose of each. What is union compatibility? Why do the UNION, INTERSECTION, and DIFFERENCE operations require that the relations on which they are applied be union compatible?

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The Domain Relational Calculus

8.1. List the operations of relational algebra and the purpose of each.

8.2. What is union compatibility? Why do the UNION, INTERSECTION, and DIFFERENCE operations require that the relations on which they are applied be union compatible?

8.3. Discuss some types of queries for which renaming of attributes is necessary in order to specify the query unambiguously.

8.4. Discuss the various types of inner join operations. Why is theta join required?

8.5. What role does the concept of foreign key play when specifying the most common types of meaningful join operations?

8.6. What is the FUNCTION operation? For what is it used?

8.7. How are the OUTER JOIN operations different from the INNER JOIN operations? How is the OUTER UNION operation different from UNION?

8.8. In what sense does relational calculus differ from relational algebra, and in what sense are they similar?

8.9. How does tuple relational calculus differ from domain relational calculus?

8.10. Discuss the meanings of the existential quantifier (∃) and the universal quantifier (∀).

8.11. Define the following terms with respect to the tuple calculus: tuple variable, range relation, atom, formula, and expression.

8.12. Define the following terms with respect to the domain calculus: domain variable, range relation, atom, formula, and expression.

8.13. What is meant by a safe expression in relational calculus?

8.14. When is a query language called relationally com

 

What is the payback period for the ion exchange unit? Based on your economic analysis, would you recommend that the ion exchange unit should be installed to the metal plater? Why? Discuss how two other factors besides economics might influence the decision.

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Engineering Economics and Pollution Prevention

You are working for a factory and have been asked to assess the economic impacts of implementing a new pollution prevention program at the factory. The factory managers required you to design the program so that the production from the factory is not reduced by the pollution prevention program. Currently, the factory has an annual operation cost of $480,000/yr. It currently costs the factory $75,000 per year to dispose of its waste. The plan that you developed will have an initial equipment cost of $100,000. However, the result of your plan will reduce the annual operating cost to $400,000/yr and the waste disposal cost to $30,000/yr.

  • Assuming a constant discount rate of 7 percent, how long will it take the factory to break even on their investment in your pollution prevention program?

A metal plating operator is considering installing an ion exchange unit to recover and reuse metals currently lost in the rinse waters from the plating line. The company presently pays a sewer usage fee of $4.00 per kg of metal sent to the sewage treatment plant. The metal plater is currently discharging 1,000 kg/yr of metal to the sewer. The metal costs the company $120 per kilogram to purchase. The ion exchanger will recover 98 percent of this metal, and all of this recovered metal can be reused in the metal plating operation. The ion exchange unit will cost $50,000 to purchase and install and $12,000 per year to operate.

  • What is the net cost or benefit of the project, assuming a discount rate of 10 percent over 10 years (based on an expected equipment life of 10 years)?
  • What is the payback period for the ion exchange unit?
  • Based on your economic analysis, would you recommend that the ion exchange unit should be installed to the metal plater? Why?
  • Discuss how two other factors besides economics might influence the decision.

 

Based on your research in Part (a) and what you learned in class, decide on a type of ballot and voting method you would choose to have for the presidential election. Make sure to explain your choices.

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Methods Voting

There are many types of ballots used in real-life elections, ranging from the simple (winner only) to the exotic (each voter has a fixed number of points to divide among the candidates any way he or she sees fit).

(a) Research other types of ballots; how, where, and when they are used; and what are the arguments for and against their use.
(b) Based on your research in Part (a) and what you learned in class, decide on a type of ballot and voting method you would choose to have for the presidential election. Make sure to explain your choices.

Post a message in the application discussion forum for this unit. In your message, describe the problem and how you solved it. Use the equation editor as necessary to show the mathematical operations.

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DISCUSSION QUESTIONS

Find a problem in your life that you can solve using the consumer math that you learned in this unit. Most of us have home, car, or education loans for which we sometimes need to calculate payments or time required to repay. Many also have savings accounts or other investments and want to calculate growth in value over a period of time.

Solve the problem:

Use the consumer math learned in this unit to solve your problem. The better you use the appropriate mathematics to correctly solve the problem, the more points you will earn.

Present the problem and your solution to the others in the class:

Post a message in the application discussion forum for this unit. In your message, describe the problem and how you solved it. Use the equation editor as necessary to show the mathematical operations. The better you communicate, the more points you will earn. If you enjoy and know how to use multimedia, such as video, audio, and graphics, you may use those as well, but this is not required.

View and respond to the applications submitted by your classmates.

Pick two of your classmates’ applications that were particularly helpful to you. Write a response to each, explaining in a paragraph or two why their applications helped you better understand the mathematics for this unit or better understand how the mathematics for this unit could be used outside of class.

 

What’s are the common factors of 36, 72 and 18?

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Find the common factors

What’s are the common factors of 36, 72 and 18?

 

How do demographers model world population? Is this different from how they model, say, the population of the United States? How does this process compare with that used by biologists in determining the size of a future salmon spawn?

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Population Growth Model

How do demographers model world population? Is this different from how they model, say, the population of the United States? How does this process compare with that used by biologists in determining the size of a future salmon spawn? Compare and contrast the process demographers use to model human population growth with that which biologists use to model animal populations.
Real life example, connection and conclusion?

In an oil rig a thermodynamic system, K = A , where R, K and A are constants Find the stationary point of the function y = x2 − 2x + 3 and hence determine the nature of this point.

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Calculus

Q1. a. Find the eigen values of the following matrix and discuss the applications of eigen values in engineering disciplines. (8 marks)
b. Temperature of a disk brake plate at any point (x, y) varies is represented by the T(x,y)=100/(1+x3+y3 ) where T measure in °C and x, y in meters. Find the rate of change of temperature with respect to x direction and y direction and also the rate at a point (2,1). (12 marks)

Q2. a. In an automobile testing the relationship between the displacement s, velocity v and acceleration a of a piston is given by the following set of linear simultaneous equations:
Use Gauss-Jordon elimination method to determine the values of s, v and a. (15 marks)
b. The results obtained during helical spring loading test are as follows:
Force (Newton) Time (Seconds)
11.4 0.56
18.7 0.35
11.7 0.55
12.3 0.52
14.7 0.43
18.8 0.34
19.6 0.31
⦁ Determine the equation of the regression line of time on force.
⦁ Find the equation for the regression line of force on time.
⦁ Draw the scatter diagram. (10 marks)

Q3 a. In an oil rig a thermodynamic system, K = A , where R, K and A are constants Find the stationary point of the function y = x2 − 2x + 3 and hence determine the nature of this point. (14 marks)

Q4. a. Solve the linear equation using MATLAB
5x = 3 y – 2 z + 10
8 y + 4 z = 3 x + 20
2 x + 4 y – 9 z = 0
(5 marks)
b. Consider the two matrices A= and B= using MATLAB, determine the following
⦁ A + B
⦁ AB
⦁ A2
⦁ AT
⦁ B-1
⦁ BT AT
⦁ A2 + B 2 + AB
⦁ Determinant of AB (20 marks)

Carter Manning has a weekly adjusted gross income of ​$986​, is​ single, and claims one withholding allowance. Find the federal tax withholding to be deducted from his weekly paycheck using the percentage method tables.

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Discussion Essay

(There are a total of 8 questions, some having one – two parts, please answer wherever there is ???)

A man earns ​$16.20 per hour with time and a half for overtime and he worked 48 hours during a recent week. Find his gross pay for the week.

The​ man’s gross pay for the week was ​$???

​(Round to the nearest cent as​ needed.)

 

A man is paid on a​ salary-plus-commission basis. He receives ​$277 weekly in salary and a commission based on 6​% of all weekly sales over ​$2,030. If he sold ​$7,822 in merchandise in one​ week, find his gross earnings for the week.

The​ man’s gross earnings for the week were $???

 

Khalid Khouri is​ married, has a gross weekly salary of ​$706 ​(all of which is​ taxable), and claims two withholding allowances. Use the tax tables to find the federal tax withholding to be deducted from his weekly salary.

The withholding tax is ​$???

 

Carter Manning has a weekly adjusted gross income of ​$986​, is​ single, and claims one withholding allowance. Find the federal tax withholding to be deducted from his weekly paycheck using the percentage method tables.

The federal tax withholding to be deducted from his salary is ​$

 

Dr. Josef Young earns an adjusted gross weekly income of ​$2,418. How much Social Security tax should be withheld the first week of the​ year? How much Medicare tax should be​ withheld? The Social Security tax rate is​ 6.2% from earnings to be taxed to a maximum annual wage of​ $118,500. The Medicare tax rate is​ 1.45% from all earnings to a maximum annual wage of​ $200,000.

The Social Security tax is ​$???

​(Type an integer or a decimal. Round to the nearest cent as​ needed.)

The Medicare tax is ​$???

 

John Waits owns a small shop with four employees. For one payroll period the total withholding tax for all employees was ​$1,550. The total​ employees’ Social Security tax was ​$175 and the total​ employer’s Social Security tax was ​$245. The total​ employees’ Medicare tax was ​$119 How much must John deposit as the​ employer’s share of Social Security tax and Medicare​ tax? What is the total tax that must be​ deposited?

John must deposit ​$??? as the​ employer’s share of Social Security tax and Medicare tax.

The total tax that must be deposited is ​$???

 

A loan is made on January 18 and has a due date of October 16 during a leap year. Find the exact time of the loan.

???days

 

Find the discount and proceeds on a ​$3,260 ​face-value note for nine months if the discount rate is 9.3​%. ​(Use the​ banker’s rule.)

The discount is ​$???

​(Round to the nearest cent as​ needed.)

The proceeds are ​$???

​(Round to the nearest cent as​ needed.)

Create an algorithm named forward, that will advance ONE value through a sequence of numbers 1, 2, 3 … MAX_NUMBER. In other words, when passed a value of 3 in the parameter current_number, it simply returns a 4.

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CMPSCI 235 – Algorithm Project

You must design 3 algorithms, and provide both a flow chart and pseudo code for the algorithms.

Algorithm descriptions:

Given an integer parameter named current_number and two constant global variables:

  • const int MIN_NUMBER = 1;
  • const int MAX_NUMBER = 8;

Create an algorithm named forward, that will advance ONE value through a sequence of numbers 1, 2, 3 … MAX_NUMBER. In other words, when passed a value of 3 in the parameter current_number, it simply returns a 4.

However, when MAX_NUMBER is reached the algorithm should wrap around back and return MIN_NUMBER. The algorithm will NEVER return a value larger than MAX_NUMBER.

Create an algorithm named backward, that will move through a sequence of numbers … 3, 2, MIN_NUMBER. In other words, when passed a value of 6 in the parameter current_number, it simply returns a 5.

When MIN_NUMBER is reached the algorithm should STOP and return the value MIN_NUMBER. This algorithm will NEVER wrap around.

Create an algorithm named createFileName, that takes a number as input, current_number, and builds and returns a string like “pictureX.gif”, where X is the value in the input parameter.

This should fit on 1 sheet of paper. Place the 3 flowcharts  on one side of the paper and the matching pseudo-code next to it or on the other side.