The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring, versus 2 man-hours to make one SST ring. 1) How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $30 and on an SST ring is $40? A) 0 VIP and 24 SST B) 16 VIP and 8 SST C) 12 VIP and 12 SST D) 8 VIP and 16 SST 2) How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $30 and on an SST ring is $60? A) 16 VIP and 8 SST B) 12 VIP and 12 SST C) 8 VIP and 16 SST D) 0 VIP and 24 SST Introduce slack variables as necessary and write the initial simplex tableau for the problem. 3) Maximize z = 4x1 + 2x2 subject to: 2x1 + X2 s 28 3x1 + 5x2 s 77 X1 2 0, x2 2 0 A) X1 X2 $1 $2 Z B) X1 X2 $1 $2 Z N 1 1 0 0 |28 2 0 0 77 3 5 0 1 0 77 3 5 0 0 4 2 0 0 1 0 4 2 0 0 1 0] C) X1 X2 51 $2 Z D) X1 X2 S1 Z 2 1 1 0 0 28 2 1 1 0 0 77 3 5 0 0 77 3 5 0 0 28 -4 -2 0 0 1 0 -4 -2 0 0 0Find the pivot in the tableau. 4) x1 x2 x3 x4 x5 2. D 2 l 1 4 D 6 1 4 CI 1 2 D 10 A) 2 in row 2, column 5 B) 4 in row 2, column 2 C) 4 in row 1, column 5 D) 2 in row 1, column 2 Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0. 5) x1x2 x3 x4 x5 2 3 4 0 3 1 O 17 A)x1 0,x2 O,x3 25,)(4 O,x5 17,2 20 B)x1 25,)(2 O,x3 U,x4 0,x5 17,2 20 C)x1=25,x2=0,X3=l7,X4=0,x5=0,z=20 D)x1=0,x2=0,X3=17,x4=0,x5=25,z=20 Use the simplex method to solve the linear programming problem. 7) Maximize z = 3x1 + 2x2 subject to: 2x1 + 3x2 5 4 4x1 + 2x2 5 12 with x120,x220 A) Maximum is 6 when x1 : 2, x2 : 0 B) Maximum is 9 when x1 : 3, x2 : 0 C) Maximum is 12 when x1 = 2, x2 = 3 D) Maximum is 4 when x1 = 0, x2 = 2 8) Maximize z = 4x1 + 2x2 subject to: 2x1 + 3x2 5 6 x1 + 3x2 3 4 2x1 + 2x2 5 8 with X1 20,xz20 A) Maximum is 6 when x1 : 0, x2 : 3 B) Maximum is 8 when x1 : 0, x2 : 2 C) Maximum is 16 when x1 = 4, x2 = 0 D) Maximum is 12 when x1 = 3, X2 = 0 Answer questions 9, 10, 11, 12, and 13 with these given information. A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows: Department Production requirement Departmental capacity by product (hours) (Total hours) Assembling 30,000 Painting 38,000 Finishing 28,000 9) What are the coefficients of the objective function? A) 1, 2, 2 B) 2, 3, 1 C) 3, 6, 15 D) 2, 3, 2 10) What is the constraint for the assembling department? A) 2A + 3B + 2C 5 30,000 B) 2A + 3B + 28 5 38,000 C) 2A + 3B + 2C 5 28,000 D) 1A + 2B + 3C 3 30,000 11) What is the constraint for the painting department? A) 1A + 2B + 2C 2 38,000 B) 2A + 3B + 1C 2 28,000 C) 1A + 2B + 2C 5 38,000 D) 2A + SE + 2C 5 30,000 12) What is the constraint for the finishing department? A) 1A + 2B + 2C s 38,000 B) 2A + 3B + 2C > 30,000 C) 2A + 3B + 1C z 28,000 D) 2A + 3B + 1C s 28,000 13) What is the maximum profit in this model? A) $225,000 B) $75,000 C) $57,000 D) $95,000 Use duality to solve the problem. 14) Minimize w =5y1 + 2y2 subject to: y1 + y2 > 19.5 2y1 + y2 2 24 y1 2 0, y2 2 0 A) 54 B) 30.5 C) 31.5 D) 48 15) Minimize w = 4y1 + 2y2 subject to: 3y1 + 2y2 2 60 4y1 + y2 2 40 y1 2 0, y2 2 0 A) 64 B) 66 C) 48 D) 80 A toy making company has at least 300 squares of felt, 700 oz of stuffing, and 230 ft of trim to make dogs and dinosaurs. A dog uses 1 square of felt, 4 oz of stuffing, and 1 ft of trim. A dinosaur uses 2 squares of felt, 3 oz of stuffing, and 1 ft of trim. 16) It costs the company $1.28 to make each dog and $1.70 for each dinosaur. The company wants to minimize its costs. What are the coefficients of the objective function? A) 2, 3, 1 B) 300, 700, 230 C) 1, 4, 1 D) 1.28, 1.70Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4 units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C. 17) Fill #1 costs 11 cents, pill #2 costs 15 cents, and pill #3 costs 7 cents. Larry wants to minimize cost. What are the coefficients of the objective function? A) 4, 1, 10 B) 11, 15, 7 C) 10, 12, 20 D) 11, 4, 3 18) Pill #1 costs 9 cents, pill #2 costs 7 cents, and pill #3 costs 14 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin A? A) 9P1 + 7P2 + 14P3 2 10 B) 3P1 + 2P2 + P3 :12 (1)4132 + 5P3 2 20 D) 4P1 + P2 + 10193 2 10 19) Pill #1 costs 3 cents, pill #2 costs 6 cents, and pill #3 costs 14 cents. Larry wants to minimize cost. What is the constraint inequality for Vitamin B? A) 3P1 + 2P2 + P3 5 6 B) 3P2 + 6P3 + 14P3 2 12 C) 3P1 + 2P2 + P3 2 12 D) 4P1 + P2 + 10P3 2 10 20) Fill #1 costs 3 cents, pill #2 costs 9 cents, and pill #3 costs 2 cents. Larry wants to minimize cost. What is the constraint inequality for vitamin C? A)4P2+5P352 B)3P1+2P2+P3212 C) 4P1 + P2 + 10P3 2 10 D) 4P2 + 5P3 2 20 Solve the problem. 22) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A veterinarian wants to set up a special diet that will contain at least 500 units of vitamin 31 at least 800 units of vitamin 32 and at least 700 units of vitamin B6. She also wants to limit the diet to at most 300 total grams. There are three feed mixes available, mix P, mix Q, and mix R. A gram of mix P contains 3 units of vitamin B1, 5 units of vitamin 132, and 8 units of vitamin 136. A gram of mix Q contains 9 units of vitamin B1, 8 units of vitamin Bz, and 6 units of vitamin 36' A gram of mix R contains 7 units of vitamin Bl, 6 units of vitamin 32, and 9 units of vitamin B6. Mix P costs $0.10 per gram, mix Q costs $0.12 per gram, and mix R costs $0.21 per gram. How many grams of each mix should the veterinarian use to satisfy the requirements of the diet at minimal costs? (Let x1 equal the number of grams of mix P, x2 equal the number of grams of mix Q, and x3 equal the number of grams of mix R that are used in the diet). A) Minimize C = 0.10x1 0.12x2 0.21x3 B) Minimize C = 0.10x1 + 0.12x2 + 0.21X3 subject to subject to C) Minimize subject to 3x1 + 9x2 + 7x3 s 500 5x1 + 8x2 + 6x3 3 800 8x1 + 6x2 + 9x3 2 700 x1 +x2+x3 $300 x1, x2, x3 2 0 C : 0.10x1 0.12x2 0.21x3 3x1 + 9x2 + 7x3 s 500 5x1 + 8x2 + 6x3 5 800 8x1 + 6x2 + 9x3 2 700 x1 +x2+x3 S300 x1, x2, x3 2 0 D) Minimize subject to 3x1 + 9x2 + 7x3 2 500 5x1 + 8x2 + 6x3 2 800 8x1 + 6x2 + 9x3 2 700 x1+x2+x3 $300 x1, x2, x3 2 0 C : 0.10x1 0.12x2 + 0.21x3 3x1 + 9x2 + 7x3 5 500 5x1 + 8x2 + 6x3 2 800 8x1 + 6x2 + 9x3 2 700 x1+x2+x3 $300 x1, x2, x3 2 0 Construct a truth table for the proposition. 23) ~13 ~(~p A 8) Tell whether the statement is true or false. 24) 0 e E A) True B) False 25) {all odd integers greater than 3 and less than 5} = {1, 1, 3} A) True B) False Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. 26) C s D A) True B) False Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false. 27) {3, 0, 8} c A A) True B) False Provide an appropriate response. 28) One of the following is false; indicate by letter which one: A) 4 e {3, 4, 5, 6} B) 5 e {3, 4, 5, 6} C) 4 c {3, 4, 5, 6} D) {4} c {3, 4, 5, 6} 29) Which of the following is NOT a subset of the set {p, o, 7}? A) {p, o, 7} B) {0, 7} C) 7 D) (3 Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 30) In a marketing survey involving 1,000 randomly chosen people, it is found that 660 use brand P, 440 use brand Q, and 220 use both brands. How many people in the survey use brand P and not brand Q? Use the Venn diagram below to find the number of elements in the region. 31) n(A) A) 17 32) n(A u B) A) 14 33) n(A n B n C) A) 16 34) n((A u B) n C) A) 11 B) 12 B) 21 B) 18 B) 14 Q9 C) 29 C)8 C) 33 D)4 D) 11 D) 44 D) 15 Use the addition principle for counting to solve the problem. 35) If n(A) = 20, n(A u B) = 58, and n(A n B) = 16, find n(B). A) 53 B) 55 C) 58 D) 54 Pravide an appropriate response. 36) The access code to a house's security system consists of five digits. How many different codes are available if each digit can be repeated? A) 5 B) 100,000 C) 32 D) 3125 E) 45 Evaluate. 9! 37) 712! A) 72 B) 36 C) 9 D) 1 61 38) 4! 2! A) 15 B) 6 C) 1 D) 30 Determine whether the selection is a permutation, a combination, or neither. 39) Mary baked 3 pies: 1 for her father, 1 for her friend Joe, and 1 for her coworkers. A) permutation B) neither C) combination 40) An animal trainer selects 2 of the 5 baboons to showcase on the talk show. A) neither B) combination C) permutation