SPRING 2014 MATH 1324 REVIEW EXAM 3_

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1 SPRING 214 MATH 1324 REVIEW EXAM 3_ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the constraints into linear equations by using slack variables. 1) Maximize z = 2x1 + 8x2 Subject to: x1 + 3x2 15 5x1 + 9x2 25 x1, x2 A) x1 + 3x2 + x3 = 15 5x1 + 9x2 + x4 = 25 C) x1 + 3x2 + x3 15 5x1 + 9x2 + x4 25 B) x1 + 3x2 = x x1 + 9x2 = x D) x1 + 3x2 + x3 = 15 5x1 + 9x2 + x3 = 25 1) 2) Maximize z = x1 + 2x2 + 3x3 Subject to: x1 + 9x2 + 3x3 4 4x1 + x2 + 3x3 5 x1, x2, x3 A) x1 + 9x2 + 3x3 + x4 = 4 4x1 + x2 + 3x3 + x4 = 5 B) x1 + 9x2 + 3x3 + x4 = 4 4x1 + x2 + 3x3 + x5 = 5 C) x1 + 9x2 + 3x3 = x x1 + x2 +3x3 = x ) Introduce slack variables as necessary and write the initial simplex tableau for the problem. 3) Maximize z = 4x1 + x2 subject to: 2x1 + 5x2 1 3x1 + 3x2 12 x1, x2 3) A) x1 x2 x3 x4 z C) x1 x2 x3 x4 z B) x1 x2 x3 x4 z D) x1 x2 x3 x4 z

2 4) Maximize z = 4x1 + 2x2 subject to: 2x1 + x2 3 3x1 + 5x2 66 x1, x2 4) A) x1 x2 x3 x4 z C) x1 x2 x3 x4 z B) x1 x2 x3 x4 z D) x1 x2 x3 x4 z Find the pivot in the tableau. 5) 5) A) 2 in row 1, column 1 B) 4 in row 1, column 3 C) 2 in row 2, column 1 D) 4 in row 2, column 2 6) 6) A) 1 in row 2, column 5 B) 4 in row 3, column 1 C) 3 in row 2, column 3 D) 2 in row 1, column 2 7) 7) A) 1 in row 3, column 1 B) 1 in row 2, column 5 C) 9 in row 3, column 3 D) 4 in row 1, column 2 2

3 Use the indicated entry as the pivot and perform the pivoting once. 8) 8) C) D) 9) 9) C) D) Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to. 1) x1 x2 x3 x4 x5 z A) x1 = 21, x2 =, x3 = 12, x4 =, x5 =, z = 18 B) x1 =, x2 =, x3 = 12, x4 =, x5 = 21, z = 18 C) x1 = 21, x2 =, x3 =, x4 =, x5 = 12, z = 18 D) x1 =, x2 =, x3 = 21, x4 =, x5 = 12, z = 18 1) 3

4 11) x1 x2 x3 x4 x5 z ) A) x1 = 3, x2 = 27, x3 = 3, x4 = 12, x5 = 12, z = 17 B) x1 = 2, x2 =, x3 =, x4 = 3, x5 =, z = 1 C) x1 =, x2 = 27, x3 = 3, x4 =, x5 = 12, z = D) x1 =, x2 = 27, x3 = 3, x4 =, x5 = 12, z = 17 Use the simplex method to solve the linear programming problem. 12) Maximize z = 5x1 + 3x2 subject to: 2x1 + 4x2 13 x1 + 2x2 6 with x1, x2 A) Maximum is 9 when x1 =, x2 = 3 B) Maximum is 32.5 when x1 = 6.5, x2 = C) Maximum is 18 when x1 =, x2 = 6 D) Maximum is 3 when x1 = 6, x2 = 12) A bakery makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, 1 dozen eggs, and 2 lb of sugar. A batch of donuts requires 5 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex tableau to maximize profit. 13) The bakery has 48 lb of flour, 64 dozen eggs, 64 lb of sugar. The profit on a batch of sweet 13) rolls is $72. and on a batch of donuts is $123.. A) B) C) D)

5 14) The bakery has 4 lb of flour, 23 dozen eggs, 4 lb of sugar. The profit on a batch of sweet rolls is $11. and on a batch of donuts is $1.. 14) A) B) C) D) 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 by product (hours) Departmental capacity (Total hours) A B C Assembling , Painting , Finishing , 15) What are the coefficients of the objective function? 15) A) 1, 2, 2 B) 2, 3, 1 C) 2, 3, 2 D) 3, 6, 15 16) What is the constraint for the assembling department? A) 2A + 3B + 2C 3, B) 1A + 2B + 3C 3, C) 2A + 3B + 2C 28, D) 2A + 3B + 2B 38, 16) Find the transpose of the matrix. 17) A) B) C) D) ) 5

6 18) A) B) C) D) ) 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 by product (hours) Departmental capacity (Total hours) A B C Assembling , Painting , Finishing , 19) What is the constraint for the painting department? 19) A) 1A + 2B + 2C 38, B) 1A + 2B + 2C 38, C) 2A + 3B + 2C 3, D) 2A + 3B + 1C 28, State the dual problem. Use y1, y2, and y3 as the variables. Given: y1, y 2, and y 3. 2) Minimize w = 6x1 + 3x2 subject to: 3x1 + 2x2 25 2x1 + 5x2 36 x1, x2 A) Maximize z = 36y1 + 25y2 subject to: 2y1 + 3y2 6 5y1 + 2y2 3 C) Maximize z = 36y1 + 25y2 subject to: 2y1 + 3y2 6 5y1 + 2y2 3 B) Maximize z = 25y1 + 36y2 subject to: 3y1 + 2y2 6 2y1 + 5y2 3 D) Maximize z = 25y1 + 36y2 subject to: 3y1 + 2y2 6 2y1 + 5y2 3 2) 6

7 21) Minimize w = 2x1 + 3x2 + x3 subject to: x1 + 3x2 + 2x3 43 2x1 + 4x2 + 3x3 6 x1, x2, x3 A) Maximize z = 6y1 + 43y2 subject to: 2y1 + y2 2 4y1 + 3y2 3 3y1 + 2y2 1 C) Maximize z = 43y1 + 6y2 subject to: y1 + 2y2 2 3y1 + 4y2 3 2y1 + 3y2 1 B) Maximize z = 6y1 + 43y2 subject to: 2y1 + y2 2 4y1 + 3y2 3 3y1 + 2y2 1 D) Maximize z = 43y1 + 6y2 subject to: y1 + 2y2 2 3y1 + 4y2 3 2y1 + 3y2 1 21) Use duality to solve the problem. 22) Minimize w = 5y1 + 2y2 subject to: y1 + y y1 + y2 24 y1, y2 A) 3.5 B) 54 C) 48 D) ) 23) Minimize w = 4y1 + 4y2 subject to: 5y1 + 1y2 1 1y1 + 2y2 15 y1, y2 A) 4 B) 6 C) 1 D) 2 23) TRUE/FALSE. Write ʹTʹ if the statement is true and ʹFʹ if the statement is false. Tell whether the statement is true or false. 24) {5, 8, 13} = {, 5, 8, 13} 24) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Insert ʺ ʺ or ʺ ʺ in the blank to make the statement true. 25) {2, 4, 6} {1, 2, 3, 4, 6} A) B) 25) 26) {16, 29, 34} {11, 29, 34, 44} A) B) 26) 27) A) B) 27) Find the number of subsets of the set. 28) {mom, dad, son, daughter} A) 8 B) 12 C) 16 D) 14 28) 7

8 29) {1, 2, 3,..., 9} A) 16 B) 58 C) 124 D) ) Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. 3) A Bʹ 3) A) {r, s, t, u, v, w, x, z} B) {u, w} C) {q, s, t, u, v, w, x, y} D) {t, v, x} 31) (A B)ʹ A) {r, s, t, u, v, w, x, z} B) {s, u, w} C) {r, t, v, x} D) {t, v, x} 31) 32) A (B C) A) {q, r, w, y, z} B) {q, y, z} C) {q, s, u, w, y, z} D) {q, w, y} 32) 33) B (A C) A) {q, w, y} B) {q, s, u, w, y, z} C) {q, s, y, z} D) {q, r, w, y, z} 33) Shade the Venn diagram to represent the set. 34) Aʹ Bʹ 34) 8

9 35) Cʹ (A B) 35) 36) (Aʹ B) C 36) 9

10 37) B (A Cʹ) 37) Use the union rule to answer the question. 38) If n(a) = 5, n(b) = 11, and n(a B) = 3; what is n(a B)? A) 16 B) 12 C) 13 D) 14 38) 39) If n(a) = 4, n(b) = 117, and n(a B) = 137; what is n(a B)? A) 1 B) 22 C) 2 D) 6 39) Use a Venn diagram to answer the question. 4) A survey of 26 families showed that 99 had a dog; 76 had a cat; 34 had a dog and a cat; 98 had neither a cat nor a dog, and in addition did not have a parakeet; 8 had a cat, a dog, and a parakeet. How many had a parakeet only? A) 21 B) 26 C) 36 D) 31 4) 41) A survey of a group of 115 tourists was taken in St. Louis. The survey showed the following: 63 of the tourists plan to visit Gateway Arch; 49 plan to visit the zoo; 1 plan to visit the Art Museum and the zoo, but not the Gateway Arch; 14 plan to visit the Art Museum and the Gateway Arch, but not the zoo; 19 plan to visit the Gateway Arch and the zoo, but not the Art Museum; 7 plan to visit the Art Museum, the zoo, and the Gateway Arch; 15 plan to visit none of the three places. How many plan to visit the Art Museum only? A) 36 B) 59 C) 14 D) 1 41) 1

11 Answer Key Testname: SPRING 214 MATH 1324 EXAM 3 1) A 2) B 3) D 4) A 5) D 6) C 7) C 8) A 9) A 1) D 11) D 12) D 13) B 14) C 15) D 16) A 17) B 18) B 19) B 2) D 21) D 22) C 23) A 24) FALSE 25) B 26) A 27) B 28) C 29) D 3) B 31) C 32) C 33) C 34) B 35) A 36) A 37) A 38) C 39) C 4) A 41) C 11