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exam3a06S_sol - MATH23O Spring 2006 Exam 3a Name K A“ z/2...

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Unformatted text preview: MATH23O Spring 2006 Exam 3a Name: K A“ z /2 0 Section: Instructions: 1. Do not start until instructed to do so. 2. You may use a calculator and one 3x5 card (front and back) with notes, but nothing else. 3. SHOW ALL WORK to receive full credit. 4. Clearly indicate your answers. 5. The work you turn in must be your own. 1. 2 points Let A = [—36 1:]. For what value of kdoes A not have an inverse? MPG?) 2302 ~ (~ W = 3 b M :0 Por k ’1 ”‘5 3 2 x 0 2. 4points Let A: ,x= ,and b: . 5 4 y 1 2 —1 —5/2 3/2 524% ; 2(o\~I(I> : —) (,2va ”r %® 3/2. Use A'1 =[ J to solve the matrix equation Ax = b for x. 3. 10 points The Trimetal Mining Company operates three mines from which gold, silver. and copper are mined. The Alpha and Beta mines cost $1000 per day to operate and each yields 30 ounces of gold, 200 ounces of silver, and 400 pounds of copper each day. The Omega mine costs $1200 per day to operate and it yields 40 ounces of gold, 400 ounces of silver, and-300 pounds of copper each day. The company has a contract to supply at least 500 ounces of gold, 8000 ounces of silver, and 4000 pounds of copper. How many days should each mine be operated so that the contract can be filled at minimum cost? Set up this problem as a linear programming problem. Be sure to define all variables. DO NOT SOLVE. Main/112e, 55: loo mg + IOOQXL + Izoox; SJojecf {a 30x, +50XL+H©X3 Z. 5'00 all 200 x, + ZOGXL-l’woxj 2-?5000 BMW Lloo X1, +Lioo><Z +3c>oxg Z‘looo anger > > >0 Xi") ) XZ-O/ X5, Questions 4 — 5: Consider the following linear programming problem: Maximize z = 2x — y subject to y S 2 x S 2 x+y+120 x—yZO 4. 8 points Carefully graph the feasible set and find all corner points. : - X 'I J_ _ -l. X t ‘ 2 \/ ‘ 7. 5. 3 points Solve the linear programming problem. Cor/1M {Dc/NW Z 2 quim/ firm} (2;?) 2(2) ’63) : all 7 for (2/2) 2(L7—Z: 2.. X3?— 6. 4 points List the four assumptions implied when formulating a linear programming problem. /, 1mm/pmwor+-1M%~/ ’2, Anfiflzvi 9“ / 33-, OiWSibnf‘ZLy ll, G‘/'/'~im 71/ Questions 7 — 9: For each of the following augmented matrices. identify whether the corresponding system of equations (with variables x, y, and 2) has no solutions, exactly one solution, or infinitely many solutions. If there is exactly one solution, give the solution. If there are infinitely many solutions, describe them. 1 0 1/3 0 No twig/+2310 7. 2points 0 1 —4/3 0 0 0 0 1 1 0 0 2 Dearly 9136 ’ wit/hm 8. 2points 0 1 0 0 .. 0011 X‘Z/7*O/2‘/ 1 O —1 1 ~nf‘fl,n,k’}()\/ Ny/i/ #“/\/1L) ”/3; 9 2 't 0 1 1 1 / . poms .1. iv 2: 0 0 0 0 X- ~ t. A? *1 ‘ 0000 2 7/..«(M/ 10. 5 points Use the augmented matrix technique to find the solution to 5x+2y=16 6x+3y=21 é[s 2. My] #42:? 2/3 “0/5 4 3 2| 7 O 51; 9/; I O z x:z Questions 11 — 12: Consider the following system of equations. x, +x2 +3):4 =4 x, ~Jr3 —Jr4 =3 —5x1+2x2 —Jr3 =—1 11. 3 points Find the augmented matrix of the system. i I O 3 1+ io—i—I 3 —5 Z"! O “i 12. 5 points Use a matrix equation to represent the system. Clearly define a|| matrices/vectors you use. ...
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