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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

Minimize 16x + 42y subject to the constraints
Minimize 10x + 12y subject to the constraintsDetermine the dual problem of the given linear programming problem.
Maximize 80x + 70y + 120z subject to the constraintsDetermine the dual problem of the given linear programming problem.
Minimize 3x + 5y + z subject to the constraintsDetermine the dual problem of the given linear programming problem.
Maximize 2x - 3y + 4z - 5w subject to the constraintsDetermine the dual problem of the given linear programming problem.
1. The final simplex tableau for the linear programming problem of Exercise 1 is as follows. Give the solution to the problem and to its dual.2. The final simplex tableau for the dual of the linear programming problem of Exercise 2 is as follows. Give the solution to the problem and to its dual.
1. The final simplex tableau for the dual of the linear programming problem of Exercise 3 is as follows. Give the solution to the problem and to its dual.2. The final simplex tableau for the linear programming problem of Exercise 4 is as follows. Give the solution to the problem and to its dual.
What is the standard maximization form of a linear programming problem?
Explain how the dual problem can be used to decide whether to introduce a new product.
What is a slack variable? A group I variable? A group II variable?
Explain how to construct a simplex tableau corresponding to a linear programming problem in standard maximization form.
Explain how to convert a minimization problem to a maximization problem.
Explain how to obtain the matrix formulation of a linear programming problem.
1. Solve the problem geometrically. 2. By looking at your graph from part 1, can you determine the shadow price of cheddar? Jason's House of Cheese offers two cheese assortments for holiday gift giving. In his supply refrigerator, Jason has 3600 ounces of cheddar, 1498 ounces of Brie, and 2396
1. Solve the problem by the simplex method. The solution should be the same as in part 1. Verify your answer to part 2 by looking at your final tableau. 2. What are the shadow prices for Brie and Stilton? Jason's House of Cheese offers two cheese assortments for holiday gift giving. In his supply
What would the maximum revenue be if there were 3620 ounces of cheddar, 1500 ounces of Brie, and 2400 ounces of Stilton? Jason's House of Cheese offers two cheese assortments for holiday gift giving. In his supply refrigerator, Jason has 3600 ounces of cheddar, 1498 ounces of Brie, and 2396 ounces
Go back to the original problem, and state its dual problem. What information do the original slack variables u, y, and w give us about the dual problem? Determine the solution to the dual problem from your final tableau in part 3, and give an economic interpretation. Jason's House of Cheese offers
Maximize 3x + 4y subject to the constraintsUse the simplex method to solve the linear programming problem.
Maximize 3x + 4y + 5z + 4w subject to the constraintsUse the simplex method to solve the linear programming problem.
1. Determine the dual problem of the linear programming problem in Exercise 3.In exercise2. Determine the dual problem of the linear programming problem in Exercise 7. In exercise
1. The final simplex tableau for the linear programming problem of Exercise 3 is as follows. Give the solution to the problem and to its dual.2. The final simplex tableau for the dual of the linear programming problem of Exercise 7 is as follows. Give the solution to the problem and to its dual.
Consider the linear programming problem in Exercise 3. Identify the matrices A, B, C, X, and U and state the problem and its dual in terms of matrices.In exercise
Consider the linear programming problems in Exercise 7. Identify the matrices A, B, C, X, and U and state the problem and its dual in terms of matrices.In exercise
A lens manufacturer has the capability of making three types of lenses. Type A lenses require 4 minutes of grinding, 2 minutes of polishing, and 4 minutes of coating, and produce a profit of $12. Type B lenses require 2 minutes of grinding, 6 minutes of polishing, and 2 minutes of coating, and
A camp counselor wants to make a smoothie for a group of children using at most 75 pounds of fruit consisting of oranges, cherries, and blueberries. Each pound of oranges contains 230 calories, 3 mg of sodium, and 4 g of protein. Each pound of cherries contains 260 calories, 7 mg of sodium, and 5 g
Consider Exercise 42 of Section 3.3. (a) Solve the problem by the simplex method. (b) The Bluejay Lacrosse Stick Company is considering diversifying by also making tennis rackets. A tennis racket requires 1 labor-hour for cutting, 4 labor-hours for stringing, and 2 labor-hours for finishing. How
Maximize 2x + 5y subject to the constraintsUse the simplex method to solve the linear programming problem.
Consider the stereo store of Exercise 24 in Section 4.2. A fourth brand of stereo system has appeared on the market. Brand D takes up 3 cubic feet of storage space and generates a sales commission of $30. What profit would the store have to realize on the sale of each brand D stereo set in order to
Maximize 2x + 3y subject to the constraintsUse the simplex method to solve the linear programming problem.
Maximize 3x + 7y subject to the constraintsUse the simplex method to solve the linear programming problem.
Minimize x + y subject to the constraintsUse the simplex method to solve the linear programming problem.
Minimize 3x + 2y subject to the constraintsUse the simplex method to solve the linear programming problem.
Minimize 20x + 30y subject to the constraintsUse the simplex method to solve the linear programming problem.
Minimize 5x + 7y subject to the constraintsUse the simplex method to solve the linear programming problem.
Maximize 36x + 48y + 70z subject to the constraintsUse the simplex method to solve the linear programming problem.
Let U = {1, 2, 3, 4, 5, 6, 7}, S = {1, 2, 3, 4}, and T = {1, 3, 5, 7}. List the elements of the following sets: (a) S' (b) S ∪ T (c) S ∩ T (d) S' ∪ T
Refer to Table 3 on the next page. Let U = {all years from 1996 to 2015}, A = {all years during which the index declined during the first 5 business days}, and B = {all years during which the index declined for the entire year}. List the elements of the following sets: (a) A (b) B (c) A ∩ B (d)
Refer to Exercise 9. Describe in words the fact that S ∩ T' has two elements. The Standard and Poor's Index measures the price of a certain collection of 500 stocks. Table 3 on the next page compares the percentage change in the index during the first 5 business days of certain years with the
Refer to Exercise 10. Describe in words the fact that A' ∩ B has two elements. Refer to Table 3 on the next page. Let U = {all years from 1996 to 2015}, A = {all years during which the index declined during the first 5 business days}, and B = {all years during which the index declined for the
Let U = {a, b, c, d, e, f}, R = {a, b, c}, S = {a, b, d}, and T = {e, f}. List the elements of the following sets:(a) (R ˆª S)'(b) R ˆª S ˆª T(c) R ˆ© S ˆ© T(d) R ˆ© S ˆ© T(e) R' ˆ© S ˆ© T(f) S ˆª
Let U = {1, 2, 3, 4, 5}, R = {1, 3, 5}, S = {3, 4, 5}, and T = {2, 4}. List the elements of the following sets: (a) R ∩ S ∩ T (b) R ∩ S ∩ T' (c) R ∩ S ∩ T (d) R' ∩ T (e) R ∪ S (f) R' ∪ R (g) (S ∩ T)' (h) S' ∪ T
Simplify each given expression. 15. (S')' 16. S ∩ S' 17. S ∪ S 18. S ∩ Ø 19. T ∩ S ∩ T 20. S ∪ Ø
Let U = {1, 2, 3, 4, 5}, S = {1, 2, 3}, and T = 556. List the elements of the following sets: (a) S' (b) S ∪ T (c) S ∩ T (d) S' ∩ T
A large corporation classifies its many divisions by their performance in the preceding year. Let P = {divisions that made a profit}, L = {divisions that had an increase in labor costs}, and T = {divisions whose total revenue increased}. 1. {Divisions that had increases in labor costs or total
An automobile insurance company classifies applicants by their driving records for the previous three years. Let S = {applicants who have received speeding tickets}, A = {applicants who have caused accidents}, and D = {applicants who have been arrested for driving while intoxicated}. Describe the
Let U = {all letters of the alphabet}, R = {a, b, c}, S = {a, e, i, o, u}, and T = {x, y, z}. List the elements of the following sets: (a) R ∪ S (b) R ∩ S (c) S ∩ T (d) S' ∩ R
Let U = {people at Mount College}, A = {students at Mount College}, B = {teachers at Mount College}, C = {people at Mount College who are older than 35}, and D = {people at Mount College who are younger than 35}. 1. A ∩ D 2. B ∩ C 3. A ∩ B 4. B ∪ C 5. A ∪ C 6. (A ∩ D) 7. D' 8. D ∩ U
Let U = {a, b, c, d, e, f, g}, R = {a}, S = {a, b}, and T = {b, d, e, f, g}. List the elements of the following sets: (a) R ∪ S (b) R ∩ S (c) T' (d) T' ∪ S
Let U = {all people}, S = {people who like strawberry ice cream}, V = {people who like vanilla ice cream}, and C = {people who like chocolate ice cream}. 1. {People who don't like vanilla ice cream} 2. {People who like vanilla but not chocolate ice cream} 3. {People who like vanilla but not
Let U be the set of vertices in Fig. 1. Let R = {vertices (x, y) with x > 0}, S = {vertices (x, y) with y > 0}, and T = {vertices (x, y) with x ‰¤ y}. List the elements of the following sets:(a) R(b) S(c) T(d) R' ˆª S(e) R' ˆ© T(f) R ˆ© S ˆ©
Let U be the set of vertices in Fig. 2. Let R = {vertices (x, y) with x ‰¥ 150}, S = {vertices (x, y) with y ‰¤ 100}, and T = {vertices (x, y) with x + y ‰¤ 400}. List the elements of the following sets.(a) R(b) S(c) T(d) R ˆ© S'(e) R' ˆª T(f) R'
1. Ed's Cheese-steaks offers any combination of three toppings on his sandwiches: peppers, onions, and mushrooms. How many different ways can you order a sandwich from Ed? List them. 2. Amy ordered a baked potato at a restaurant. The server offered her butter, cheese, chives, and bacon as toppings.
1. List all subsets of the set {1, 2}. 2. List all subsets of the set {1, 2, 3, 4}.
1. Let S = {1, 3, 5, 7} and T = {2, 5, 7}. Give an example of a subset of T that is not a subset of S. 2. Suppose that S and T are subsets of the set U. Under what circumstance will S ∩ T = T? 3. Suppose that S and T are subsets of the set U. Under what circumstance will S ∪ T = T? 4. Find
Determine whether the statement is true or false. 1. 5 ∈ {3, 5, 7} 2. {1, 3} ⊆ {1, 2, 3} 3. {b} ⊆ {5b, c} 4. 0 ∈ {1, 2, 3} 5. 0 ∈ Ø 6. Ø ⊆ {a, b, c} 7. {b, c} ⊆ {b, c} 8. 1 ∉ {1}
Let U = {all college students}, F = 5all freshman college students), and B = {all college students who like basketball}. Describe the elements of the following sets: (a) F ∩ B (b) B' (c) F' ∪ B (d) F ∪ B
Let U = {all corporations}, S = {all corporations with headquarters in New York City}, and T = {all privately owned corporations}. Describe the elements of the following sets: (a) S' (b) T' (c) S ∩ T (d) S ∩ T'
The Standard and Poor's Index measures the price of a certain collection of 500 stocks. Table 3 on the next page compares the percentage change in the index during the first 5 business days of certain years with the percentage change for the entire year. Let U = {all years from 1996 to 2015}, S =
1. Find n (S ∪ T), given that n (S) = 4, n (T) = 4, and n (S ∩ T) = 2. 2. Find n (S ∪ T), given that n (S) = 17, n (T) = 13, and n (S ∩ T) = 0.
Suppose that all of the 1000 first-year students at a certain college are enrolled in a math or an English course. Suppose that 400 are taking both math and English and 600 are taking English. How many are taking a math course?
Of the 26 capital letters of the alphabet, 11 have vertical symmetry (for instance, A, M, and T), 9 have horizontal symmetry (such as B, C, and D), and 4 have both (H, I, O, X). How many letters have neither horizontal nor vertical symmetry?
A survey of employees in a certain company revealed that 250 people subscribe to a streaming video service, 75 subscribe to a streaming music service, and 25 subscribe to both. How many people subscribe to at least one of these services?
Motors Inc. manufactured 325 cars with navigation systems, 216 with push-button start, and 89 with both of these options. How many cars were manufactured with at least one of the two options?
A survey of 120 investors in stocks and bonds revealed that 90 investors owned stocks and 70 owned bonds. How many investors owned both stocks and bonds?
Draw a two-circle Venn diagram and shade the portion corresponding to the set. 1. S ∩ T' 2. S' ∩ T' 3. S' ∪ T 4. S' ∪ T'
Draw a three-circle Venn diagram and shade the portion corresponding to the set. 1. R ∩ S ∩ T' 2. R' ∩ S' ∩ T 3. R ∪ (S ∩ T) 4. R ∩ (S ∪ T)
Find n (S ∩ T), given that n (S) = 6, n (T) = 9, and n (S ∪ T ) = 15.
Use De Morgan's laws to simplify each given expression. 1. S' ∪ (S ∩ T)' 2. S ∩ (S ∪ T)' 3. (S' ∪ T)' 4. (S' ∩ T')' 5. T ∪ (S ∩ T)' 6. (S' ∩ T)' ∩ S
Find n (S ∩ T), given that n (S) = 4, n (T) = 12, and n (S ∪ T) = 15.
Give a set-theoretic expression that describes the shaded portion of each Venn diagram.1.2. 3. 4. 5. 6.
Find n (S), given that n (T) = 7, n (S ∩ T) = 5, and n (S ∪ T) = 10.
By drawing a Venn diagram, simplify each of the expressions to involve at most one union and the complement symbol applied only to R, S, and T. 1. (T ∩ S) ∪ (T ∩ R) ∪ (R ∩ S') ∪ (T ∩ R' ∩ S') 2. (R ∩ S) ∪ (S ∩ T) ∪ (R ∩ S' ∩ T') 3. ((R ∩ S') ∪ (S ∩ T') ∪ (T ∩
Assume that the universal set U is the set of all people living in the United States. Let A be the set of all U.S. citizens, let B be the set of all children under 5 years of age, let C be the set of children from 5 to 18 years of age, let D be the set of everyone over the age of 18, and let E be
1. Find n (T), given that n (S) = 14, n (S ∩ T) = 6, and n (S ∪ T) = 14. 2. If n (S) = n (S ∩ T), what can you conclude about S and T? 3. If n (T) = n (S ∪ T), what can you conclude about S and T?
Suppose that each of the 314 million adults in South America is fluent in Portuguese or Spanish. If 170 million are fluent in Portuguese and 155 million are fluent in Spanish, how many are fluent in both languages?
The Venn diagram in Fig. 10 classifies the 100 books in a family's library as hardback (H ), fiction (F), and children's (C). Exercises 1-10 refer to this Venn diagram.1. How many books are hardback fiction? 2. How many books are paperback fiction? 3. How many books are fiction? 4. How many books
Let R, S, and T be subsets of the universal set U. Draw an appropriate Venn diagram, and use the given data to determine the number of elements in each basic region. 1. n (U) = 17, n (S) = 12, n (T) = 7, n (S ∩ T ) = 5 2. n (U) = 20, n (S) = 11, n (T) = 7, n (S ∩ T ) = 7
A survey of 70 high school students revealed that 35 like rock music, 15 like hip-hop music, and 5 like both. How many of the students surveyed do not like either rock or hip-hop music?
A total of 900 Nobel Prizes had been awarded by 2015. Fourteen of the 112 prizes in literature were awarded to Scandinavians. Scandinavians received a total of 57 awards. How many Nobel Prizes outside of literature have been awarded to non-Scandinavians?
One of Shakespeare's sonnets has a verb in 11 of its 14 lines, an adjective in 9 lines, and both in 7 lines. How many lines have a verb but no adjective? An adjective but no verb? Neither an adjective nor a verb?
How many students correctly answered either the first or second question? The results from an exam taken by 150 students were as follows: 90 students correctly answered the first question, 71 students correctly answered the second question, 66 students correctly answered both questions.
How many students did not answer either of the two questions correctly? The results from an exam taken by 150 students were as follows: 90 students correctly answered the first question, 71 students correctly answered the second question, 66 students correctly answered both questions.
How many students answered either the first or the second question correctly, but not both? The results from an exam taken by 150 students were as follows: 90 students correctly answered the first question, 71 students correctly answered the second question, 66 students correctly answered both
How many students answered the second question correctly, but not the first? The results from an exam taken by 150 students were as follows: 90 students correctly answered the first question, 71 students correctly answered the second question, 66 students correctly answered both questions.
How many students missed the second question? The results from an exam taken by 150 students were as follows: 90 students correctly answered the first question, 71 students correctly answered the second question, 66 students correctly answered both questions.
Out of 35 students in a finite math class, 22 are male, 19 are business majors, 27 are first-year students, 14 are male business majors, 17 are male first-year students, 15 are first-year business majors, and 11 are male first-year business majors. (a) Use this data to complete a Venn diagram
A survey of 100 college faculty who exercise regularly found that 45 jog, 30 swim, 20 cycle, 6 jog and swim, 1 jogs and cycles, 5 swim and cycle, and 1 does all three. How many of the faculty members do not do any of these activities? How many just jog?
How many flags contain red, but not white or blue? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both
How many flags contain exactly one of the three colors? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both
How many flags contain none of the three colors? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both white
How many flags contain exactly two of the three colors? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both
How many flags contain red and white, but not blue? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both
How many flags contain red or white, but not both? The three most common colors in the 193 flags of the member nations of the United Nations are red, white, and blue: 52 flags contain all three colors 103 flags contain both red and white 66 flags contain both red and blue 73 flags contain both
How many people learned of the sale from newspapers or the Internet but not from both? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from the
How many people learned of the sale only from newspapers? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from the Internet and television 90 from
How many people learned of the sale from the Internet or television but not from newspapers? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from
How many people learned of the sale from at least two of the three media? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from the Internet and
How many people learned of the sale from exactly one of the three media? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from the Internet and
How many people learned of the sale from the Internet and television but not from newspapers? A merchant surveyed 400 people to determine from what source they found out about an upcoming sale. The results of the survey follow: 180 from the Internet 190 from television 190 from newspapers 80 from
Table 1 shows the number of students enrolled in each of three science courses at Gotham College. Although no students are enrolled in all three courses, 15 are enrolled in both chemistry and physics, 10 are enrolled in both physics and biology, and 5 are enrolled in both biology and chemistry. How

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