exam3_solutions - Math 2602 Exam#3 Fall 2008 Name...

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Unformatted text preview: Math 2602 Exam #3 Fall 2008 Name: GTid (9xxxxxxxx): Instructor: Stephen J. Young There are 4 questions on this exam on 4 pages (not counting this coverpage). 0 Be sure to fully explain your answers, as answers that are not accompanied by explana- tions/ work may receive no credit. 0 Place your name and problem number on each solution sheet. The exams will be separated to be graded. Anyone turning in a solution sheet without a name will receive stern looks and runs the risk of the scores not being accurately totalled. 9 You are to complete this exam completely alone, without the aid of calculators, cellular telephones, personal digital assistants, or any other mechanical or digital calculating device. By signing on the line below, you agree to abide by the Georgia Tech Honor Code, the principles of which are embodied by the Challenge Statement: I commit to uphold the ideals of honor and integrity by refusing to betray the trust bestowed upon me as a member of the Georgia Tech community. Student signature: ------ Math 2602 Exam #3 25 November 2008 1. (5 points) Translate the following linear program into standard form and find the optimal solution using the simplex algorithm. max éfik subject to —2£L‘1 + $2 S 2 $1 + $2 S 6 {1:1 S 4 1131,.1‘2 2 0 m SanCl'A VEVM «a? ‘l’lhs LP :3 Max 379w: 6i. “2x, txz +31 9‘: z xii/ix; SD $2133 2 6 7165 yields cm (halal 6%fo VlaUem; dfi "‘3‘l0006 ~21\o 2‘3 3: i * . O ““ C) C) 1 4“;le O l i O 2 it) 5%“ "-9 O l O l‘l Z 1 C3 0 l “l 3r“) ’ “It, oel‘w’vl Salo‘lnm iS (Ll/21?,OIO\ wilt Value, I“! Page 1 of 4 N ame:._M—v Points earned: Math 2602 Exam #3 25 November 2008 3. (5 points) The following Markov Chain has a unique limiting distribution, find it and explain why it is unique. 5W llfi 7‘03"“?04’ 69' AZ ““5415, m. Mi; [L743 cjsitnéugm cs mice film. l Wémrlogélacev one. “#‘Wl Wkth‘odb/Mwl J 1/; “2/310 allw[i3 1 2 1(1th .c:>€-rg 5‘0] m3 ’4‘va- ‘W‘S‘ 2% 1:0 c) S’ f’o 5 Ys—l 5/3‘6 L7 \‘3 lo 73, 2/3 ”‘1 £0. ‘7 <1le 4L waning“ fir AL! 611:] WWIVZVES 11034 MWLAMM 3,4— [7/10] own Page 3 of 4 Namezm_—_._ Points earned: CLQ‘JMIWI KWMWWMMWNW-wwmmnmmmmwarm mw%mMWmmmmmmzwmwmm.WWW».W»,.. Math 2602 Exam #3 25 November 2008 2. (5 points) Write an linear program to solve the following problem and translate it into canonical form. Be sure to explain What each variable in your initial formulation represents. A burger joint wants to create a new healthier burger by mixing beef, chicken, and a soy vegetable mixture. In each gram of beef there are 2.6 calories, 0.2 grams of fat, 1 mg of cholesterol, and 1 mg of sodium. Each gram of chicken has 1.5 calories, 0.07 grams of fat, 0.8 mg of cholesterol, and 0.8 mg of sodium. The vegetable mixture contains 2.0 calories, 0.] grams of fat, no cholesterol, and 4 mg of sodium. The resulting burger must weight at least 150 grams, have no more than 375 calories, no more than 10 grams of fat, no more than 50 mg of cholesterol, and no more than 400 mg of sodium. The beef costs half a cent per gram, the chicken costs .07 cents per gram, and the vegetable mixture costs a cent per gram. Find the mixture that satisfies all the requirements and is the cheapest. 2th +l.§'ulv £373" .2), r.6‘7e*‘|lv é 1'0 is +— Xc, Q So \3 4— 3L Viv 5- [1/50 L, + c +v' 250 Mic/v20 Page 2 of 4 Name: __l§£>l_— ( 5.1M («59% (elmleéud\ (’rolum\ lo 9+ .ggc 4/1", 4;. ‘7/00 (with Points earned: « i E 6 r § § i a i .; E § § 3 «Mariammm«.x»m..xm.uwvm.mmMaw- Math 2602 Exam #3 25 November 2008 4. (5 points) Thoroughly explain how you would solve the following standard form linear program. Do NOT solve the linear program. max 4:5+5y+3z—28-—t subjectto 3m+2y-z—s:5 x+7z+53+t=—23 —y+3z+6s—t=—4 —16w+3y——4z+6s+7t==—14 $,y,z,s,t20. Sm ‘lewe, t$ M— le lawns we at» WWW! Umdwug O‘I/C‘Zflg, gxl'z‘y'l- ”$47215?— “- ‘X ¥ 7% “%¢S‘$%% M2“- *ZE “Ly ¥3Z— %S "H: +0“; :3"): Hex 23y ”£623ng =‘V7 E X‘,’ asfiflw diva ‘26 SOlistdl' 4° VQLLK dwel‘rwls 63;: 931%; $37 Lp 7W,m¢g% “Clim'fl‘z‘fi‘qya WAR sup/lap oe/amwlfl. (/0105; (Aug/05’“? Kg {(2, a? BMW [mg ml ~Hs o n 18 n 600049 aswfnwlllogflnawlmm A393“ (@153 wally , MAM“ MN m 15mm rib/KL owfiwl LP is ('k‘fiv‘skL/egu Ta [Jr ‘5 .Qméré, 4&1;er fwilmafl‘feugé» stwublo KW berm/552.5 (lerm’lvélecwl) Max #34. v/vL-J— aw "LL-€55 :7» I Y M W W‘Ls‘m‘fiz WSét/W'Vfléus‘n Malena m 6, fix raw fiswflqc aJZMflkoc/x “/{cs Sim/Ar 674%“ ch] Page 4 of 4 Name:_..__.._._____.._ Points earned: % 2) x a ,Seéxaéiz, iiiziasagsée3?. :2:1.E§32§§i.z.x§.5: it,gzzgnytxflsge. _ngukisié5x:5§§:§5§§§§§pg?_ ,A 7,,‘4 ,,,,,,, A”, ,,,,,,A,% ,, . , A _ h » , . 7 g , ...
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