Question

from my homework

a small manufacturing plant makes three types of inflatabl boats: one-person, two-person, and 4 person models. each boa requires the services of three departments as listed in the tabble. the cutting, assemply, and packaging depepartments have available a masimum of 380,330, and 120 labor-hours per week respectively.
table

collumns are one person, 2 person, 4 person respectivly
rows
cutting 0.5hours 1hour 1.5hour
assembly 0.6hours 0.9hour 1.2hour
packaging 0.2hour 0.3hour 0.5hour


A. how many boats of each type ust be produced each week for the plant to operate at full capacity?
B. How is the production schedule in part A affected if the packaging department is no longer used?
C. How is the production schedule in part A affected if the 4 person boat is no longer produced?
(use gauss-jordan elimination when possible and show all work and reasoning)

Answer 1

x=number of boats for one-person
y=number of boats for two-person
z=number of boats for four-person

We have the next system of equations.
0.5x+y+1.5z=380
0.6x+0.9y+1.2z=330
0.2x+0.3y+0.5z=120

we can solve this systema of equations by Gauss-Jordan elimination method.

0.5                  1                    1.5              380
0.6                  0.9                 1.2              330
0.2                  0.3                  0.5             120    



0.5                  1                    1.5              380
0                  -0.15                -0.3              -63         (0.5R₂-0.6R₁)
0                  -0.05                -0.05            -16         (0.5R₃-0.2R₁)


0.5                  1                    1.5                  380
0                  -0.15                -0.3                  -63     
0                       0                   0.0075          0.75          (0.15R₃-0.05R₂)         



0.5                  1                    1.5                  380 
0                  -0.15                -0.3                  -63     
0                       0                       1                 100           (R₃/0.0075)



0.5                  1                        0                  230         (R₁-1.5R₃)
0                  -0.15                     0                 -33           (R₂+0.3R₃)
0                       0                       1                 100
 



0.5                     1                        0                  230    
0                        1                        0                   220         (R₂/-0.15)
0                        0                        1                  100



1                        2                        0                    460          (2R₁)
0                        1                        0                    220       
0                        0                        1                     100



1                        0                        0                      20         (R₁-2R₂)
0                        1                        0                     220       
0                        0                        1                     100


Answer A)
20 boats for one-person
220 boats for two-person
100 boats for four-person.

B)We have 120 hour more of work.  
C) We have 100 boats manofactured less. We only manofacture 220 boats for two-person and 20 boats for one-person

Answer 2

Solutions 

To solve the problem lets use the variables "x,y, and z" 

x=number of boats for one-person
y=number of boats for two-person
z=number of boats for four-person

⇒ System of equations

0.5x+y+1.5z=380
0.6x+0.9y+1.2z=330
0.2x+0.3y+0.5z=120

To solve these equations we have to use the Gauss-Jordan elimination method.

$\left[\begin{array}{ccc}0.5&1&1.5\\0.6&0.9&1.2\\0.2&0.3&0.5\end{array}\right]$  


$\left[\begin{array}{ccc}0.5&1&1.5\\0&-0.15&-0.3\\0&-0.05&-16\end{array}\right]$ 

(0.5R₂-0.6R₁) 
(0.5R₃-0.2R₁)

$\left[\begin{array}{ccc}0.5&1&1.5\\0&-0.15&-0.3\\0&0&0.0075\end{array}\right]$ 

(0.15R₃-0.05R₂)    

$\left[\begin{array}{ccc}0.5&1&1.5\\0&-0.15&-0.3\\0&0&1\end{array}\right]$ 

(R₃/0.0075)

$\left[\begin{array}{ccc}0.5&1&0\\0&0.15&0\\0&0&1\end{array}\right]$ 

(R₁-1.5R₃) 
(R₂+0.3R₃)

$\left[\begin{array}{ccc}0.5&1&0\\0&1&0\\0&0&1\end{array}\right]$ 

 (R₂/-0.15)  

$\left[\begin{array}{ccc}1&2&0\\0&1&0\\0&0&1\end{array}\right]$ 

(2R₁)

$\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$ 

(R₁-2R₂) 



These numbers have to be added to each one 

1 - 380, 330, 120 
2 - 380, -63, -16
3 - 380, -63, 0.75
4- 380, -63, 100 
5 - 230, -33, 100 
6 - 230, 220, 100 
7 -  460, 220, 100 
8 - 20, 220, 100

Answer A) 

20 boats for one-person
220 boats for two-person
100 boats for four-person. 

B)We have 120 hour more of work.
   
C) We have 100 boats manufactured less. We only manufacture 220 boats for two-person and 20 boats for one-person