SOLUTION: A furniture company makes dining room furniture. A buffet requires 40 hours for construction 10 hours for finishing. A chair requires 20 hours for construction and 20 hours for fin

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Question 1203703: A furniture company makes dining room furniture. A buffet requires 40 hours for construction 10 hours for finishing. A chair requires 20 hours for construction and 20 hours for finishing. A table requires 40 hours for construction and 70 hours for finishing. The construction department has 500 hours of labor and the finishing department has 200 hours of labor available each week. How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
A furniture company makes dining room furniture.
A buffet requires 40 hours for construction 10 hours for finishing.
A chair requires 20 hours for construction and 20 hours for finishing.
A table requires 40 hours for construction and 70 hours for finishing.
The construction department has 500 hours of labor
and the finishing department has 200 hours of labor available each week.
How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?
~~~~~~~~~~~~~~~~~~ 

X buffets; Y chairs; Z tables.

Write equations for full capacity as you read the problem

    40X + 20Y + 40Z = 500  hours   (1)   (the construction department time)

    10X + 20Y + 70Z = 200  hours   (2)   (the finishing department time)


Simplify equations by dividing each term by 10

    4X + 2Y + 4Z = 50     (1')  

     X + 2Y + 7Z = 20     (2')  


From equation (2'), Z may have only three possible values:      Z = 0,  or  1,  or  2.


If Z= 0, then equations (1'), (2') take the form

    4X + 2Y = 50
     X + 2Y = 20

and have a UNIQUE solution X= 10, Y= 5.   (You can use the Elimination method to get this solution).


If Z= 1, then equations (1'), (2') take the form

    4X + 2Y = 46
     X + 2Y = 13

and have a UNIQUE solution X= 11, Y= 1.  (You can use the Elimination method to get this solution).


If Z= 2, then equations (1'), (2') take the form

    4X + 2Y = 42
     X + 2Y =  6

and have NO a solution in integer non-negative numbers.


Thus the problem has two possible solutions:  (X,Y,Z) = (10,5,0)  and/or  (X,Y,Z) = (11,1,1).


There are no other solutions.

Solved.



Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

B, C, and T must be non-negative integers:

system%2840B%2B20C%2B40T=500%2C10B%2B20C%2B70T=200%29

Subtract the 2nd eq. from the 1st.

matrix%283%2C1%2C%0D%0A30B-30T=300%2C%0D%0A++B-T=10%2C%0D%0AB=10%2BT%29

system%2840B%2B20C%2B40T=500%2C10B%2B20C%2B70T=200%29

Eliminate the constant, multiply 1st eq. by 2, and 2nd eq. by 5

system%2880B%2B40C%2B80T=1000%2C50B%2B100C%2B350T=1000%29



Answer: 10 buffets, 5 chairs, and no tables  OR
        11 buffets, 1 chair,  and  1 table.

Edwin