Problem #1. The following matrices are augmented matrices of linear systems. How many solutions has each system? Motivate your answer.

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1 Exam #4 covers the material about systems of linear equations and matrices (CH , PART II); systems of linear inequalities in two variables (geometric approach) and linear programming (CH , PART II); sets theory, principles of counting, permutations, and combinations (CH , PART II). Problem #1. The following matrices are augmented matrices of linear systems. How many solutions has each system? Motivate your answer A = B = C = D = F =

2 Problem #2. Suppose that the supply and demand for a commodity are p = 0.001q Supply equation p = 0.001q Demand equation where p is the price in dollars and q is a quantity. Find the equilibrium price and equilibrium quantity graphically. Label axes and lines. Indicate viewing window. Round the price to the nearest dollar. Problem #3. Solve the following linear systems using Gauss-Jordan elimination. Clearly indicate all row operations you are using. a) b) 2x 4y+ 12z = 20 2x+ y+ z = 6 x+ 3y+ 5z = 15 x+ y+ z = 5 3x 7y+ 7z = 5 x+ 2y+ 3z = 10 c) d) 2

3 x+ 2y 3z = 1 2x y+ z = 3 x+ 3y 2z = 4 x+ 2y 4z = 4 2x y+ z = 1 x+ 3y 2z = 1 Problem #4. Cost of production. A contractor builds three kinds of houses, models I, models II, and models III. Matrix M presents amounts of exterior materials concrete (in cubic yards), lumber (in units of 1000 board feet), brick (in thousands), and shingles (in units of ft ) planned for each model. Cost (in dollars) of the materials (per unit) is given by matrix C. M = C = Use multiplication of matrices to get the total cost of the exterior materials for all models of houses. Which model is the cheaper? Problem #5. Multiply the following matrices A= B =

4 Problem #6. If matrices A and B have dimensions 3 5and 4 3 respectively, what are dimensions of the matrix C = B A? Is it possible to multiply A B? Problem #7. Mary has in the retirement portfolio in the Vanguard with 400 shares of VSC found, 350 shares of VASIX found, and 500 shares of VAAPX found. The closing prices of these stocks for one week are given in the following table. VSC VASIX VAAPX Monday Tuesday Wednesday Thursday Friday Use matrix multiplication to find the portfolio values for each day of this week. Show all details of your computation. Indicate the days with the largest and smallest price of this portfolio. 4

5 Problem #8. Business lease. A corporation wants to lease a fleet of 12 airplanes with a combined carrying capacity of 320 passengers. The three available types of planes carry 10, 20, and 40 passengers, respectively. How many of each type of planes should be leased? Write your answers in the table provided. Type of airplanes 10 passengers capacity 20 passengers capacity 40 passengers capacity Number of airplanes Number of airplanes Number of airplanes Number of airplanes 5

6 Problem #9. a) Maximize P = 4x+ 7 y subject to x+ y 11 3x+ y 5 x 0, y 0 Instructions: Graph all straight lines using x - and y - intercepts. Label coordinates of all x - and y - intercepts. Shade the feasible region. Find and state all corner points of feasible region algebraically. Specify why the optimization problem has solution(s). x 0 y 0 x 0 y 0 Corner points P = 4x+ 7y Answer: P = at max 6

7 b) Maximize P = 10x+ 5y subject to x+ y 11 3x+ y 5 x 0, y 0 Instructions: Graph all straight lines using x - and y - intercepts. Label coordinates of all x - and y - intercepts. Shade the feasible region. Find and state all corner points of feasible region algebraically. Specify why the optimization problem x 0 y 0 x 0 y 0 Corner points P = 10x+ 5y Answer: P = at max 7

8 Problem #10. Infotron Inc. makes electronic hockey and soccer games. Each hockey game requires 2 labor-hours of assembly and 2 laborhours of testing. Each soccer game requires 3 labor-hours of assembly and 1 labor-hours of testing. Each day there are 42 labor-hours available for assembly and 26 labor-hours available for testing. The company makes a profit of $10 on each hockey game and $8 profit on each soccer game. Construct a mathematical model in the form of a linear programming problem. DO NOT SOLVE THE LINEAR PROGRAMMING PROBLEM! 1. Introduce decision variables. 2. Determine the objective and the objective function. 3. Write all necessary constraints. 8

9 Problem #11. (Application of Venn diagram). A survey of 60 freshman business students at a large university produces the following results: 19 students read Business Week, 18 read The Wall Street Journal, 50 read Fortune, 14 read Business Week and The Wall Street Journal, 11 read The Wall Street Journal and Fortune, 13 read Business Week and Fortune, 9 read all three. Fill out the Venn diagram and use it to answer following questions. a) How many students read no publications? b) How many read only Fortune c) How many read at least one source? d) How many read at least two sources? e) How many read exactly one source? f) How many read exactly two sources? 9

10 Problem #12. (Application of Venn diagram). A survey yields the following information about the musical preferences of students: 30 like classical, 24 like country, 31 like jazz, 9 like country and classical, 12 like country and jazz, 10 like classical and jazz, 4 like all three, 6 like none of the three. Draw a diagram that shows this breakdown of musical tastes. a) Determine the total number of students interviewed. b) How many students like only one kind of music? c) How many students like at least two types of music? d) Exactly two types? e) If A is the set of all students that like at least one of listed types of music, what is the complement to such set, A? Give a word description of A. 10

11 Permutations and Combinations. Problem #13.How many arrangements of three people seated along one side of a table are possible if there are 8 people to select from? Problem #14.In how many ways can five essays be ranked in a contest? Problem #15. Six runners are competing in a 100-meter race. In how many ways can runners finish in first, second, and third place? Problem #16.How many three-digit numbers can be formed using the digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}if a) Repetitions are not allowed? b) B) Repetitions are allowed? 11

12 Problem #17. From a committee of 20 people, in how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position? Problem #18. Three women are selected from the audience of a style show to receive a purse, a pair of gloves, and a scarf. If 30 women are present, in how many different ways may the gifts be given? Problem #19. In how many ways the committee of 3 people can be selected from a department with 20 employees? (Compare with the Problem #17). Problem #20. The Beta Club has 14 male and 16 female members. A committee composed of 3 men and 3 women is formed. In how many ways can this be done? 12