MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 LINEAR PROGRAMMING (Applied Finite Mathematics, \"Linear Programming: A Geometric Approach\") For the following exercises, solve using the graphical method. Choose your variables, write the objective function and the constraints, graph the constraints, shade the feasibility region, label all corner points, and determine the solution that optimizes the objective function. 1. A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit? 2. Mr. Tran has $24,000 to invest, some in bonds and the rest in stocks. He has decided that the money invested in bonds must be at least twice as much as that in stocks. But the money invested in bonds must not be greater than $18,000. If the bonds earn 6%, and the stocks earn 8%, how much money should he invest in each to maximize profit? 3. A computer store sells two types of computers, desktops and laptops. The supplier demands that at least 150 computers be sold a month. In order to keep profits up, the number of desktops sold must be at least twice of laptops. The store pays its sales staff a $75 commission for each desk top, and a $50 commission for each lap top. How many of each type of computers must be sold to minimize commission to its sales people? What is the minimum commission? 4. Mr. Shoemacher has $20,000 to invest in two types of mutual funds, Coleman High-yield Fund, and Coleman Equity Fund. The High-yield fund gives an annual yield of 12%, while the Equity fund earns 8%. Mr. Shoemacher would like to invest at least $3000 in the High-yield fund and at least $4000 in the Equity fund. How much money should he invest in each to maximize his annual yield, and what is the maximum yield? 5. Dr. Lum teaches part-time at two different community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 5 classes per semester. For every class taught by him at Hilltop College, he needs to spend 3 hours per week preparing lessons and grading papers, and for each class at Serra College, he must do 4 hours of work per week. He has determined that he cannot spend more than 18 hours per week preparing lessons and grading papers. If he earns $4,000 per class at Hilltop College and $5,000 per class at Serra College, how many classes should he teach at each college to maximize his income, and what will be his income? 6. Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost? 1 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 7. Mr. Boutros wants to invest up to $20,000 in two stocks, Cal Computers and Texas Tools. The Cal Computers stock is expected to yield a 16% annual return, while the Texas Tools stock promises a 12% yield. Mr. Boutros would like to earn at least $2,880 this year. According to Value Line Magazine's safety index (1 highest to 5 lowest), Cal Computers has a safety number of 3 and Texas Tools has a safety number of 2. How much money should he invest in each to minimize the safety number? Note: A lower safety number means less risk. 8. A department store sells two types of televisions: Regular and Big Screen. The store can sell up to 90 sets a month. A Regular television requires 6 cubic feet of storage space, and a Big Screen television requires 18 cubic feet of space, and a maximum of 1080 cubic feet of storage space is available. The Regular and Big Screen televisions take up, respectively, 2 and 3 sales hours of labor, and a maximum of 198 hours of labor is available. If the profit made from each of these types is $60 and $80, respectively, how many of each type of television should be sold to maximize profit, and what is the maximum profit? 9. A small company manufactures two types of radios- regular and short-wave. The manufacturing of each radio requires two operations: Assembly and Finishing. The regular radios require 1 hour of Assembly and 3 hours of Finishing. The short-wave radios require 3 hours of Assembly and 1 hour of Finishing. The total work-hours available per week in the Assembly division is 60, and in the Finishing division, 60. If a profit of $50 is realized for every regular radio, and $75 for every short-wave radio, a. how many of each should be manufactured to maximize profit, and b. what is the maximum profit? 10. A company produces two types of shoes - casual, and athletic - at its two factories, Factory I and Factory II. Daily production of each factory for each type of shoe is listed below. Factory I Factory II Casual 100 200 Athletic 300 100 The company must produce at least 8000 pairs of casual shoes, and 9000 pairs of athletic shoes. If the cost of operating Factory I is $1500 per day and the cost of operating Factory II is $2000, a. how many days should each factory operate to complete the order at a minimum cost, and b. what is the minimum cost? 2 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 SETS AND COUNTING (Applied Finite Mathematics, \"Sets and Counting\") List the elements of the following sets: 11. Let Universal set=U={a,b,c,d,e,f,g,h,i,j}, V={a,e,i,f,h}, and W={a,c,e,g,i}. List the members of the following sets. V W VW V W' V' W' 12. Let Universal set=U={1,2,3,4,5,6,7,8,9,10}, A={1,2,3,4,5}, B={1,3,4,6}, and C={2,4,6}. List the members of the following sets. A B AC A B'C A' BC' Find the number of elements in the following sets. 13. In Mrs. Yamamoto's class of 35 students, 12 students are taking history, 18 are taking English, and 4 are taking both. Draw a Venn diagram and determine how many students are taking neither history nor English? 14. In the County of Santa Clara 700,000 people read the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and 100,000 read both newspapers. How many read either the Mercury News or the Examiner? 15. A survey of athletes revealed that for their minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. No athletes reported using neither. How many athletes were surveyed? 16. In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands? 3 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 17. This quarter, a survey of 100 students at De Anza College finds that 50 take math, 40 take English, and 30 take history. Of these 15 take English and math, 10 take English and history, 10 take math and history, and 5 take all three subjects. Draw a Venn diagram and determine the following. a. The number of students taking math but not the other two subjects. b. The number of students taking English or math but not history. c. The number of students taking none of these subjects. 18. In a survey of investors it was found that 100 invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and bonds, and 20 in all three. Determine the following. a. The number of investors that participated in the survey. b. How many invested in stocks or mutual funds but not in bonds? c. How many invested in exactly one type of investment? TREE DIAGRAMS AND THE MULTIPLICATION AXIOM Do the following problems using a tree diagram or the multiplication axiom. 19. In a city election, there are 2 candidates for mayor, and 3 for supervisor. Use a tree diagram to find the number of ways to fill the two offices. 20. Brown Home Construction offers a selection of 3 floor plans, 2 roof types, and 2 exterior wall types. Use a tree diagram to determine the number of possible homes available. 21. A Virginia license plate consists of three letters followed by four digits. How many such plates are possible? 22. How many different 4-letter radio station call letters can be made if the first letter must be K or W and none of the letters may be repeated? 23. How many seven-digit telephone numbers are possible if the first two digits cannot be ones or zeros? 24. How many 3-letter word sequences can be formed using the letters {a,b,c,d} if no letter is to be repeated? 25. A family has two children, use a tree diagram to determine all four possibilities. 4 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 26. A coin is tossed three times and the sequence of heads and tails is recorded. Use a tree diagram to determine the different possibilities. 27. A combination lock is opened by first turning to the left, then to the right, and then to the left again. If there are 30 digits on the dial, how many possible combinations are there? 28. How many different answers are possible for a multiple-choice test with 10 questions and five possible answers for each question? PERMUTATIONS Do the following problems using permutations. 29. How many three-letter words can be made using the letters {a,b,c,d,e} if no repetitions are allowed? 30. A grocery store has five checkout counters, and seven clerks. How many different ways can the clerks be assigned to the counters? 31. A group of fifteen people who are members of an investment club wish to choose a president, and a secretary. How many different ways can this be done? 32. In how many different ways can five people be seated in a row if two of them insist on sitting next to each other? 33. In how many different ways can five people be seated in a row if two of them insist on not sitting next to each other? 34. In how many ways can 3 English, 3 history, and 2 math books be set on a shelf, if they are grouped by subject? 35. You have 5 math books and 6 history books to put on a shelf with five slots. In how many ways can you put the books on the shelf if the first two slots are to be filled with the books of one subject and the next three slots are to be filled with the books of the other subject? COMBINATIONS Do the following problems using combinations. 36. How many different 5-player teams can be chosen from eight players? 37. In how many ways can a person choose to vote for three out of five candidates on a ballot for a school board election? 5 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 38. How many 13-card bridge hands can be chosen from a deck of cards? 39. There are twelve people at a party. If they all shake hands, how many different hand-shakes are there? 40. In how many ways can a student choose to do four questions out of five on a test? 41. There are five teams in a league. How many total league games are played if every team plays each other twice? 42. A team plays 15 games a season. How many ways are there to end up with 8 wins and 7 losses for the season (for example: win the first 8, lose the last 7; win 4 then lose 7 then win 4, etc.)? 43. In how many different ways can a 4-child family have 2 boys and 2 girls? 44. A coin is tossed five times. In how many ways can it fall three heads and two tails? COMBINATIONS INVOLVING SEVERAL SETS Following problems involve combinations from several different sets. 45. How many 5-people committees consisting of three boys and two girls can be chosen from a group of four boys and four girls? 46. How many 4-people committees chosen from four men and six women will have at least three men? 47. A batch contains 10 transistors of which three are defective. If three are chosen, in how many ways can one get two defective? 48. In how many ways can five counters labeled A, B, C, D and E at a store be staffed by two men and three women chosen from a group of four men and six women? 49. Three marbles are chosen from an urn that contains 5 red, 4 white, and 3 blue marbles. How many samples of the following type are possible? a. b. c. d. e. f. All three white. Two blue and one white. One of each color. All three of the same color. At least two red. None red. 6 MATH 106 Week 4 LEO Discussion Topic Exercises Version: 2015-1 50. Five coins are chosen from a bag that contains 4 dimes, 5 nickels, and 6 pennies. How many samples of five of the following type are possible? a. b. c. d. e. f. At least four nickels. No pennies. Five of a kind. Four of a kind. Two of one kind and two of another kind. Three of one kind and two of another kind. 51. Find the number of different ways to draw a 5-card hand from a deck to have the following combinations. a. b. c. d. e. f. Three face cards. A heart flush (all hearts). Two hearts and three diamonds. Two cards of one suit, and three of another suit. Two kings and three queens. Two cards of one value and three of another value. 7