Textbook, Chapter 6 Review Exercises, page 331, Problem 20: Each week at a furni

Question

Textbook, Chapter 6 Review Exercises, page 331, Problem 20: Each week at a furniture company, 2000 work hours are available in the construction department, 1400 work hours in the painting department, and 1300 work hours available in the packing department. Producing a chair requires 2 hours of construction, 1 hour of painting, and 2 hours of packing. Producing a table requires 4 hours of construction, 3 hours of painting, and 3 hours of packing. Producing a chest requires 8 hours of construction, 6 hours of painting, and 4 hours for packing. If all available time is used in every department, how many of each item are produced each week?

Explanation / Answer

CHAPTER 6 REVIEW EXERCISES, PAGE 331, PROBLEM 20:     

ANSWER:

this isn't probability? (unless I'm missing something)

so we get 2000c, 1400p, 1300P. Let's use 2000x, 1400y, 1300z for the rest of this.
a=2x+1y+2z
b=4x+3y+3z
c=8x+6y+4z

2000=2a+4b+8c (rewrite the columns going down as = to it)
1400=1a+3b+6c
1300=2a+3b+4c

then we can just use a calculator or manipulate the rows. I get:
a = 200, b = 100, c = 150

available construction hours = 2000
available painting hours = 1400
available packaging hours = 1300

for chair 2C , 1P , 2Pa
for table 4C , 3p , 3Pa
for chest 8C , 6P , 4Pa

adding 14C , 10P, 9 Pa

Now see which is multiple of either 14 or 10 or 9 and gives minimum value

observing above we will see 1400 / 10 = 140 minimum
so we can produce 140 chiars, 140 tables and 140 chests.

with 40 construction and40 packaging hours left.   

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