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Unformatted text preview: Exam Name___________________________________ Questions 1 – 25 are worth 3 pts each for a complete solution. (TOTAL 75 pts) (Formulas, work, or detailed explanation required.) Question 26 – 30, worth 5 pts each for a complete solution, (TOTAL 25 pts) (Formulas, work required.) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the values of the variables in the matrix. 1) 1) _______ + = A) = 1, y = 11, a = 3, z = 9, m = 0 x B) = 2, y = 11, a = 1/ 3, z = 2/ 9, m = 0 x D) = 1, y = 11, a = 13/ 3, z = 2/ 9, m = 0 x C) = 1, y = 11, a = 13/ 3, z = 2/ 9, m = 0 x Perform the indicated operation where possible. 2) 2) _______ + B) C) D) ot possible N A) Provide an appropriate response. 3) ive the dimensions of the following matrix. G 3) _______ A) × 3 4 Solve the problem. 4) Let A = A) B) × 4 4 C) × 3 3 D) × 4 3 4) _______ . Find 4A. B) C) D) The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. 5) is 3 × 1, and B is 1 × 4. A 5) _______ A) B is 3 × 4; BA does not exist. A B) B is 3 × 3; BA is 1 × 1. A C) B is 1 × 1; BA is 3 × 3. A D) B does not exist; BA is 1 × 1. A 6) is 1 × 4, and B is 1 × 4. A 6) _______ A) B is 4 × 1; BA is 1 × 4. A B) B is 1 × 4; BA is 4 × 1. A C) B is 1 × 1; BA is 4 × 4. A D) B does not exist; BA does not exist. A Find the matrix product, if possible. 7) B) A) 8) A) B) 7) _______ C) D) oes not exist D 8) _______ C) D) oes not exist D Solve the problem. 9) company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the A amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers X and Y. Multiply the matrices. 9) _______ A) B) C) D) Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) 10) ______ 10) and A) o N B) es Y Solve the system of equations by using the inverse of the coefficient matrix. 11) 3x + y = 7 7x + 2y = 17 A) o inverse, no solution for system N B) 3, 2) ( C) 2, 3) ( D) 2, 3) ( 11) ______ 12) x + y + z = 4 x  y + 5z = 18 5x + y + z = 12 USE EXCEL or Graphing calculator A) o inverse, no solution for system N B) 5, 3, 4) ( C) 5, 4, 3) ( D) 4, 3, 5) ( 12) ______ Solve the matrix equation AX = B for X by finding , given A and B as follows. Use a graphing calculator/ EXCEL to obtain your answer, rounding all numbers to four decimal places. 13) ______ 13) A = A) , B = B) C) D) Convert the inequality into a linear equation by adding a slack variable. 14) 1 + 4X2 + 6X3 ≤ 101 X A) 1 + 4 X2 + 6 X3 + S1 ≤ 101 X B) 1 + 4 X2 + 6 X3 + S1 = 101 X C) 1 + 4 X2 + 6 X3 + S1 ≥ 101 X D) 1 + 4 X2 + 6 X3 + S1 + 101 = 0 X Use slack variables to convert the constraints into linear equations. 15) aximize z = 2 X1 + 8 X2 M subject to: X1 + 2 X2 ≤ 15 8 X1 + 2 X2 ≤ 25 with: X1 ≥ 0, X2 ≥ 0 A) X1 + 2 X2 + S1 = 15 B) X1 + 2v = S1 + 15 8 X1 + 2 X2 + S1 = 25 8 X1 + 2 X2 = S2 + 25 C) X1 + 2 X2 + S1 = 15 D) X1 + 2 X2 + S1 ≤ 15 8 X1 + 2 X2 + S2 = 25 8 X1 + 2 X2 + S2 ≤ 25 Write the solutions that can be read from the simplex tableau. 16) 14) ______ 15) ______ 16) ______ A) 1, X2, S1= 0, X5= 23, S2= 12, z = 19 X C) 1, X2, S1= 0, X3= 12, S2= 23, z = 19 X B) 1, X2, S1= 0, X1= 23, S2= 12, z = 19 X D) 1, X2, S1= 0, X3= 23, S2= 12, z = 19 X Pivot once about the circled element in the simplex tableau, and read the solution from the result. 17) 17) ______ A) 1= 48, S2= 16, z = 48; X2, X3, S1 = 0 X B) 1= 24, S2= 16, z = 24; X2, X3, S1 = 0 X C) 1= 48, S2= 16, z = 48; X2, X3, S1= 0 X D) 1= 24, S2= 16, z = 24; X2, X3, S2= 0 X Introduce slack variables as necessary, and write the initial simplex tableau for the problem. 18) ind X1 ≥ 0 and X2 ≥ 0 such that F 2 X1 + 5 X2 ≤ 9 3 X1 + 3 X2 ≤ 2 and z = 4 X1 + X2 is maximized. 18) ______ A) B) C) D) The initial tableau of a linear programming problem is given. Use the simplex method to solve the problem. 19) 19) ______ A) aximum at 18 for X2 = 8, X3= 2 M C) aximum at 9 for X1 = 8, X2= 2 M B) aximum at 32 for X2 = 8, S1= 2 M D) aximum at 36 for X2 = 2, S1= 8 M Use the simplex method to solve the linear programming problem. (You may use the SOLVER=show worksheet) 20) aximize z = 5 X1 + 3 X2 M 20) ______ subject to: 2 X1 + 4 X2 ≤ 13 X1+ 2 X2 ≤ 6 with X1≥ 0, X2≥ 0 A) aximum is 18 when X1 = 0, X2= 6 M B) aximum is 30 when X1 = 6, X2= 0 M C) aximum is 9 when X1 = 0, X2= 3 M D) aximum is 32.5 when X1 = 6.5, X2= 0 M Find the transpose of the matrix. 21) 21) ______ A) B) C) D) State the dual problem. UseY1, Y2, Y3and Y4 as the variables. Given: Y1≥ 0, Y2≥ 0, Y3≥ 0, and Y4 22) inimize M w = 11 X1 + 7 X2 + 4 X3 + 5X4 subject to: 11 X1 + 5 X2 + 7 X3 + 4 X4 ≥ 25 16 X1 + 5 X2 + 11 X3 + 7 X4 ≥ 30 X1≥ 0, X2≥ 0, X3≥ 0, X4≥ 0 A) aximize M z = 25Y1 + 30Y2 B) aximize M z = 25 Y1  30 Y2 subject to: 11 Y1 + 16Y2 ≥ 11 subject to: 11 Y1 + 16 Y2 ≤ 11 5 Y1 + 5 Y2 ≥ 7 5 Y1 + 5 Y2 ≤ 7 7 Y1 + 11 Y2 ≥ 4 7 Y1 + 11 Y2 ≤ 4 4 Y1 + 7 Y2 ≥ 5 4 Y1 + 7 Y2 ≤ 5 C) aximize M z = 25 Y1 + 30 Y2 D) aximize M z = 25 Y1 + 30 Y2 subject to: 11 Y1 + 16 Y2 ≤ 11 subject to: 11 Y1 + 16 Y2 ≤ 11 5 Y1 + 5 Y2 ≤ 7 5 Y1 + 5 Y2 ≤ 7 7 Y1 + 11 Y2 ≤ 4 7 Y1 + 11 Y2 ≤ 4 4 Y1 + 7 Y2 ≤ 5 4 Y1 + 7 Y2 ≤ 5 Rewrite the system of inequalities, adding slack variables or subtracting surplus variables as needed. 23) X1 + 5 X2 ≤ 5 2 X1 + 3 X2 ≥ 7 A) X1 + 5 X2  S1 = 5 2 B) X1 + 5 X2 + S1 = 5 2 X1 + 3 X2 + S2 = 7 X1 + 3 X2  S2 = 7 C) X1 + 5 X2 + S1 = 5 2 D) X1 + 5 X2 + S1 = 7 2 X1 + 3 X2  S1 = 7 X1  3 X2  S2 = 5 Rewrite the objective function into a maximization function. 24) inimize w = 2Y1 + 4 Y2+ 3Y3 M subject to: Y1+ Y2 ≥ 10 2 Y1 + 3 Y2 + Y3 ≥ 27 Y1+ 2 Y2 + Y3 ≥ 15 Y1≥ 0, Y2≥ 0, Y3≥ 0 A) aximize z = 2 X1 + 4 X2 – 3X3 M B) aximize z = 2 X1  3 X2  X3 ≤ 27 M D) aximize z =  X1  X2 ≤ 10 M C) aximize z = 2 X1  4 X2  3 X3 M 22) ______ 23) ______ 24) ______ Each 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 of A, 1 of B, and 5 of C. 25) #1 costs 9 cents, pill #2 costs 8 cents, and pill #3 costs Larry wants to minimize cost. 25) ______ Pill What are the coefficients of the objective function? A) 0, 12, 20 1 B) , 8, 10 9 C) , 4, 3 9 D) , 1, 10 4 Use a graphing calculator/ EXCEL to find the matrix product and/or sum, if possible. 26) ind A(B + C). F A = B = C = Solve the problem algebraically or Use EXCEL Solver. 27) basketball fieldhouse seats 15,000. Courtside seats cost $ 8, endzone seats cost $ 7, and balcony A seats cost The total revenue for a sellout is $ 88,000. If half the courtside seats, half the balcony seats, and all the endzone seats are sold; then the total revenue is $ 51,000. How many of each type of seat are there? 28) A bakery makes sweet rolls and donuts. A batch of sweet rolls requires of flour, 1 dozen eggs, and 2 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex of sugar. A batch of donuts requires tableau to maximize profit if :. The bakery has 680 lb of flour, 800 dozen eggs, 600 lb of sugar. The profit on a batch of sweet rolls is and on a batch of donuts is Use the simplex method to solve the linear programming problem. or EXCEL Solver 29) inimize w = 5Y1 + 2Y2 M subject to: Y1+ Y2 ≥ 19.5 2Y1 + Y2 ≥ 24 Y1≥ 0, Y2≥ 0 Solve the problem. (Use Simplex method or EXCEL Solver 30) n appliance store sells two types of refrigerators. Each CoolIt refrigerator sells for $ 640 and A each Polar sells for $ 740. Up to 330 refrigerators can be stored in the warehouse and new refrigerators are delivered only once a month. It is known that customers will buy at least 80 CoolIts and at least 100 Polars each month. How many of each brand should the store stock and sell each month to maximize revenue ...
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 Spring '10

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