Vector Worksheets With Answers - Math 125, Exam 3 Page 6
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195. (10 pts) A company produces stereos, CD players, and speakers at three different fac-
tories. At Factory 1, the daily 8 hour workday output is 63 stereos, 51 CD players, and 36
speakers. At Factory 2, the daily output is 31 stereos, 32 CD players, and 24 speakers. At
Factory 3, the daily output is 40 stereos, 60 CD players, and 48 speakers. The company
would like to close Factory 2. Is there some combination of outputs from the other two
factories that will equal the output of Factory 2? If so, what is that combination?
Solution: Let F denote the output of the i-th factory. Therefore, F
= (63, 51, 36), F
=
1
2
(31, 32, 24), and F
= (40, 60, 48) To determine if the output of Factory 2 can be replace by
3
the other two factories, we need to find the dependency equation for the three vectors – if
the dependency equation exists. To find the dependency equation we solve the augmented
matrix [ F
F
F
0]. Moving [ F
F
F
0] into reduced row echelon form, we get
1
2
3
1
2
3
1 0
4/3
0
.
0 1
4
0
0 0
0
0
4
If we view this table as a dependency table, we see that F
=
F
+ 4 F
. Solving for F
,
3
1
2
2
3
1
1
we get the equation F
=
F
+
F
.
2
1
3
3
4
We could also have found the dependency equation from the looking at the 0 as a linear
combination of the F ’s. Since the F
column does not have a leading one, we let c
= t.
3
3
4
Accordingly, c
=
t and c
=
4t. Therefore,
1
2
3
4
0 =
t F
+ 4t F
+ t F
.
1
2
3
3
To find the dependency equation, we let t = 1. This yields the dependency equation
4
0 =
F
4 F
+ F
.
1
2
3
3
1
1
Solving this dependency equation for F
, we again get F
=
F
+
F
.
2
2
1
3
3
4
The conclusion is: Everyday, we can replace the output of Factory 2 by running Factory
1 and extra one-third of a workday and Factory 3 and extra one-fourth of a workday.
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