MATH312-Extralistofexercises
         1. As responsible for urban planning at the town council, you have to completely reinstall all the
           electricity,gasandwaterlinesinthecity. Reinstallingonemileofelectricitylinecosts$12(million),
           takes10hoursandrequires13workers. Reinstallingonemileofgaslinecosts$7(million),takes4
           hoursandrequires7workers. Reinstallingonemileofwaterlinecosts$4(million),takes6hours
           andrequires6workers. Ifyouhaveamonthlybudgetof$195(million),162hourspermonthand
           ateamof216workers,howmanymilesofeachlinecanyoureinstallinamonth?
      Solution. Thevariables x, y and z will represent, respectively,
                             x= milesofelectricitylineinstalledinamonth.
                             y = milesofgaslineinstalledinamonth.
                             z = milesofwaterlineinstalledinamonth.
      Thenumberofmilesofelectricity line times the cost of one mile, plus the number of miles of gas line
      times the cost of one mile, plus the number of miles of water line times the cost of one mile has to be
      equaltothe$195(million)thatIhaveasbudget. Thisgivestheequation
                                         12x+7y+4z=195.
      If we do the same for hours of work, we get the equation 10x+4y +6z = 162; similarly, for number of
      workersneededgivestheequation13x+7y+6z=216.Thuswehavetosolvethesystem
                      12x+7y+4z=195                          12 7 4 195 
                      10x+4y+6z=162        whosematrixis     10 4 6 162 
                      13x+7y+6z=216                           13  7  6  216
      Let’s solve the systemusingGauss-Jordanelimination. Ateachstepthepivotwillbethecirclednumber.
      Forinstance,forthefirststepthenumber 12 ,inthetopleftcornerofthematrix,willbethepivot.
              12    7  4  195  R3−R1  12  7  4  195  R1↔R3  1  0  2   21 R2−10R1
                              −−−−−→  10  4  6  162  −−−−→  10 4  6  162  −−−−→
              10    4  6  162          1   0  2  21          12  7  4  195 R3−12R1
                13   7  6  216
               12   7   4   195 2R2−R3 12   7    4   195 R3−7R2 12  7  4   195 
               0    4   -14 -48  −−−−→  0   1   -8   -39  −−−−→  0  1  -8  -39 
               0    7   -20 -57        0    7   -20  -57         0   0  36  216
      Thustheinitialsystemofequationsisequivalent(inotherwords,ithasthesamesolutions)tothesystem
                                         12x+7y+4z=195
                                               y−8z=−39
                                                 36z=216
      Fromthebottomequationweget
                                     z = 216 =6milesofwaterline.
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                                                 1
       Replacingthisinthemiddleequationweget
                                 y =−39+8z=−39+48=9milesofgasline.
       Finally, replacing the values of y and z in the top equation, we get
                       x=195−7y−4z =195−63−24=108=9milesofelectricityline.
                               12            12        12
         2. Yourunadeliverycompany,deliveringinthreedifferentareasofManhattan,A,BandC.Inaver-
            age, a trip to the area A takes 4 hours, 5 gallons of fuel and you deliver 3 tons of goods. A trip to
            areaBtakes6hours,4gallonsoffuelandyoudeliver1tonofgoods. Finally,atriptoareaCtakes3
            hours,2gallonsoffuelandyoudeliver3tonsofgoods. everydayyourcompanydeliversthrough
            all 24 hours, and your budget allows to spend 16 gallons in total, to deliver 24 tons of goods. How
            manytripstoeachareacanyoudoeveryday?
       Solution. Thevariables x, y and z will represent, respectively,
                                            x= tripstoareaA.
                                            y = trips to area B.
                                            z = trips to area C.
       Theequationsaresetupastheywereinthepreviousexercise. Thatis, withrespect to time, we have 4
       hoursforeachtriptoA,6hourspertriptoBand3hourspertriptoC,andthetotalmustbeequalto24,
       sowegettheequation
                                             4x+6y+3z=24.
       Doingthesameforgallonsoffuel,wegettheequation5x+4y+2z=16;anddoingthesamefortonsof
       goodsdeliveredyieldstheequation3x+y+3z=24. Thesystemtosolveis
                         4x+6y+3z=24                            4 6 3 24 
                         5x+4y+2z=16         whosematrixis      5 4 2 16 
                         3x+y+3z=24                              3   1  3  24
       Againlet’ssolveitbyGauss-Jordanelimination.
        4    6  3  24         1    5  0   0         1    5    0  0          1   5   0   0 
                       R1−R3                 R2−5R1                 3R3−2R2
        5    4  2  16  −−−−→  5    4  2  16  −−−−→  0   -21   2  16  −−−−→  0  -21  2  16 
          3   1  3  24            3   1  3  24   R3−3R1  0  -14   3  24          0   0   5  40
       Sotheoriginalsystemisequivalenttothesystem
                                             x+5y       =0
                                             −21y+2z=16
                                                    5z =40
       Wegetthenz=8tripstoareaC.Replacinginthemiddleequationweget−21y =16−16=0,so y =0
       trips to area B, and then we see from the top equation, x+5y =0,that x =0tripstoareaA.
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         3. Youwanttorunfourdifferentexperimentsinalab. Thefirstexperimentwillrequire,persample,2
            hoursoflabwork,2hoursofprocessingtime,3labassistantsand$5,000. Asampleofthesecond
            experimentrequires2hoursoflabwork,noprocessingtime,3labassistantsand$1,000. Thethird
            experiment requires, per sample, 2 hours of lab work, 3 hours of processing time, 3 assistants,
            and $7,000. Finally, each sample of the fourth experiment takes 1 hour of lab work, 2 hours of
            processingtime,1labassistantand$5,000.
            Perday,yourscheduleandbudgetallowyoutospend3hoursonlabwork,4hoursonprocessing
            time, the use of 4 lab assistants and $11,000. How many samples of each experiment can you do
            perday?
       Solution. Let x, y, z and t denote the following
                                   x= numberofsamplesofexperiment1.
                                   y = numberofsamplesofexperiment2.
                                   z = numberofsamplesofexperiment3.
                                   t = numberofsamplesofexperiment4.
       Thesystemtosolveisthenthefollowing
        2x+2y+2z+t=3                                             2     2     2      1     3    
       
       
        2x     +3z+2t=4                                          2     0     3      2     4    
                                                inmatrixform                                    
        3x+3y+3z+t=4                                             3     3     3      1     4    
                                                                                               
        5000x+1000y+7000z+5000t=11000                           5000  1000   7000  5000  11000
       Onceagain,let’suseGauss-Jordanelimination.
          2      2     2     1      3           2 2 2 1        3         1    3  0  -1   0 
          2      0     3     2      4     R3−R2  2  0  3   2   4  R1↔R3  2    2  2   1   3 
                                          −−−−→                  −−−−→                      
                                                1 3 0 -1 0                                  
          3      3     3     1      4    R4/1000  5 1  7   5  11         2    0  3   2   4 
          5000  1000   7000  5000  11000                                       5   1  7   5  11
                      1   3    0  -1   0         1   3  0   -1  0        1   3   0  -1  0 
         R2−2R1,R3−2R1 0  -4   2   3   3 R4−2R3 0 -4 2      3   3  R2↔R4  0  -2  1  2   3 
           −−−−−−−−→                     −−−−→                    −−−−→                   
                                                 0 -6 3 4 4              0 -6 3 4 4 
            R4−5R1    0   -6   3   4   4         0 -2 1 2 3              0 -4 2 3 3 
                       0   -14  7  10  11
                                1   3   0   -1   0         1   3  0  -1   0 
                         R3−3R2  0  -2  1   2    3 R3−2R4 0 -2 1      2   3 
                         −−−−→                     −−−−→                    
                                0   0   0   -2  -5         0   0  0   0   1 
                         R4−2R2  0  0   0   -1  -3         0   0  0  -1  -3 
       The third row of the last matrix corresponds 0 = 1, which means that the system does not have any
       solutions.
         4. You are designing a new CPU combining four different microchips. Each of the first type of mi-
            crochip performs 2,000 operations per second, costs $3,000, takes 4 hours to be installed and
            requires 3 engineers to be set up. A microchip of the second type performs 2,000 operations per
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             second, costs $1,000, takes 4 hours to be installed, and requires 3 engineers to be set up. Each of
             the third type performs 2,000 operations per second, costs $4,000, takes 4 hours to be installed,
             andrequires 3 engineers to be set up. Finally, a microchip of the fourth type performs 1,000 op-
             erations per second, costs $3,000, takes 2 hours to be installed, and requires 2 operators to be set
             up.
             If youwantyourCPUtoperform3,000operationspersecond,youhave$10,000tospend,6hours
             and14engineersworkinginthelab,howmanychipsofeachtypedoyouneed?
       Solution. Thevariables x, y, z and t denote
       x= numberofchipsoftype1.y= numberofchipsoftype2.z= numberofchipsoftype3.t = numberofchipsoftype4.
       Thesystemthatwehavetosolveis
                                    2000x+2000y+2000z+1000t=3000
                                   
                                   
                                    3000x+1000y+4000z+3000t=10000
                                    4x+4y+4z+2t=6
                                   
                                    3x+3y+3z+2t=14
       AndagainwesolveitusingGauss-Jordanelimination.
          2000   2000   2000  1000    3000          2    2   2  1   3          2 2 2 1 3 
          3000   1000   4000  3000   10000 R1/1000   3   1   4  3   10  R4−R1  3  1   4  3   10 
                                             −−−−→                     −−−−→                    
            4      4     4      2      6                                       4 4 4 2 6 
            3      3     3      2      14   R2/1000  4   4   4  2   6          1 1 1 1 11 
                                                        3   3   3  2   14
               1    1   1  1  11               1    1  1    1    11          1   1  1    1    11 
        R1↔R4  3    1   4  3  10 R2−3R1,R3−4R1 0   -2  1    0    -23 R3−2R4 0   -2  1    0    -23 
        −−−−→                      −−−−−−−→                          −−−−→                        
                                               0    0  0    -2   -38         0   0  0    0     0 
               4    4   4  2   6     R4−2R1                                                       
                 2   2   2  1   3                  0   0  0    -1   -19           0   0  0    -1   -19
       Sincealltheentriesarezerointhethirdrow,weknowthatthesystemhasinfinitelymanysolutions. The
       original systemisequivalentto          
                                               x+y+z+t=11
                                               −2y+z     =−23
                                                         −t =19
       Fromthebottomequationweseethat t =19microchipsoftype4. Nextwehavetochooseoneofthe
       variables to become a parameter to express the solutions of the system. We have to be careful at this
       point because we cannot choose just any variable: the variables x and t cannot become parameters,
       sincethesecondequationdoesnotcontainanyofthesetwovariables! Thuswehavetochoosebetween
       y andz. Let’schoose y. Thisway,
                                        z =−23+2y microchipsoftype3.
       Thetopequationthengives
                     x=11−y−z−t=11−y−(−23+2y)−19=15−3ymicrochipsoftype1.
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