Lembar Kerja Mahasiswa Kalkulus dan Aljabar Linier (4 sks)

Jurusan Sistem Informasi FTIF – ITS 2013 Materi : SPL sd Invers Matriks Kelas : D Sifat : Take Home

Nama : NRP :  

 1. After  taking  nutrition  class,  a  big  Sabrina’s  ® Mac  and  Cheese  fan  decides  to  improve  the  

levels  of  protein  and  fiber  in  her  favourite  lunch  by  adding  broccoli  and  canned  chicken.  The  nutritional  information  for  the  foods  referred  to  are  given  in  the  table.  

 If  she  wants  to  limit  her  lunch  to  400  calories  but  get  30  g  of  protein  and  10  g  of  fiber,  what  proportions  of  servings  of  Mac  and  Cheese,  broccoli,  and  chicken  should  she  use  ?    She  found  that  there  was  too  much  broccoli  in  the  proportion  from  previous  plan,  so  she  decided  to  switch  from  classical  Mac  and  Cheese  to  Sabrina’s  Whole  Wheat  Shells  and  White  Cheddar.  What  proportions  of  servings  of  each  food  should  she  use  to  meet  the  same  goals  as  in  previous  plan  ?    (Hint   :   Set   up  matrix   and   vector   that   gives   the   amount   of   calories,   protein,   and   fiber  contained  in  a  mixture  of  servings)    

2. Three neighbors have backyard vegetable gardens. Neighbor A grows tomatoes, neighbor B grows corn, and neighbor C grows lettuce. They agree to divide their crops among themselves as follows: A gets ½ of the tomatoes, 1/3 of the corn, and ½ of the lettuce. B gets 1/3 of the tomatoes, 1/3 of the corn, and ¼ of the lettuce. C gets 1/6 of the tomatoes, 1/3 of the corn, and ½ of the lettuce. What prices should the neighbors assign to their respective crops if the equilibrium condition of a closed economy is to be satisfied, and if the lowest-priced crop is to have a price of $100?   Equilibrium prices : total outputs of various sectors in such a way that the income of each sector exactly balances its expenses    (Hint  :  Row  reduce  the  augmented  matrix)    

3. Urban  planners  and  traffic  engineers  monitor  the  pattern  of  traffic  flow  in  a  grid  of  city  streets.   Electrical   engineers   calculate   current   flow   through   electrical   circuits.   An  economists   analyse   the   distribution   of   products   from   manufacturers   to   consumers  through  a  network  of  wholesalers  and  retailers.    A  basic  assumption  of  network  flow  is  that   the   total   flow   into   the  network  equals   the   total   flow  out  of   the  network  and   the  total  flow  into  a  junction  equals  the  total  flow  out  the  junction.  The  network  in  Figure  shows  the  traffic  flow  in  downtown  England  during  a  typical  early  afternoon.  Determine  the  general  flow  pattern  for  the  network.  Assuming  that  the  flow  must  be  in  the  directions  indicated,  find  the  minimum  flows  in  the  branches  denoted  by  x2,  x3,  x4,  and  x5.    (Hint  :  Write  equations  that  describe  the  flow  and  then  find  the  general  solution  of  the  system.)  

   

4. Write   a   matrix   equation   that   determines   the   loop   currents   !     No   need   to   find   the  solution.  A   voltage   source   such   as   battery   forces   a   current   of   electrons   to   flow   through   the  network.  When  the  current  passes  through  a  resistor  (such  as  a  lightbulb),  some  of  the  voltage  is  used  up  by  Ohm’s  law  :  V  =  RI  (V  :  voltage  in  volts,  R  :  resistance  in  ohms,  I  :  current  flow  in  amperes).  Current  flow  in  a  loop  is  governed  by  the  Kirchhoff’s  Voltage  Law  :    The  algebraic  sum  of  the  loop  of  the  RI  voltage  drops  in  one  direction  around  a  loop  equals  the  algebraic  sum  of  the  voltage  sources  in  the  same  direction  around  the  loop.  

   

5. Chemical  equations  describe  the  quantities  of  subtances  consumed  and  produced  by  chemical  reactions.  For  instance,  Alka-­‐Seltzer  contains  sodium  bicarbonate  (NaHCO3)  and  Citric  Acid  (H3C6H5O7).  When  a  tablet  is  dissolved  in  water,  the  following  reaction  produces  sodium  citrate,  water,  and  carbon  dioxide  (gas)  :  

 To  balance  this  equation,  a  chemist  must  find  whole  numbers  x1,…,xn  such  that  the  total  numbers  of  atoms  on  the  left  match  the  corresponding  numbers  of  atoms  on  the  right  (because  atoms  are  neither  destroyed  nor  created  in  the  reaction).    Balance  the  chemical  equations  above  using  the  vector  equation  approach  !