College Mathematics: Learning Worksheets Chapter 4
135
7. 1
2
37 65
21 3
x
x
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
⎡
⎢
⎣
Dadbc
=−
1
⎡
⎢
⎢
⎣
1
db
Aca
⎣
−
−
⎡
⎤
=
⎢
⎡
⎢
⎣
8. 1
2
25 31
23 25
x
x
(2)(3) ( 5)( 2)
610
4
D
D
D
1
35
44
1
11
22
1
35
1
22
4
db
D
A
A
35
44
11
31 8
xAB
College Mathematics: Learning Worksheets Chapter 4
136
Write the following systems as a matrix equation and solve by using inverses.
9. 121
32
x
xk
x
12
12
a) 13 5
b
) 5 3
kk
kk
==
==
1
12 125 1
xk
−−−−
⎡⎤ ⎡⎤
⎡⎤ ⎡⎤⎡⎤⎡⎤
c) 11
1
12 12 10 10
xk
AB
−−−−−
⎡⎤ ⎡⎤
⎡⎤ ⎡⎤⎡⎤⎡⎤
== = =
College Mathematics: Learning Worksheets Chapter 4
137
23
x
xk
x
12
12
a) 6 12
b
) 3 3
kk
kk
==
==−
11
(2)(1) ( 3)(1)
2(3)
5
D
D
D
1
1
13
1
12
5
0.2 0.6
0.2 0.4
D
A
A
a) 11
1
0.2 0.6 0.2 0.6 6 8.4
xk
AB
1
0.2 0.6 0.2 0.6 3 1.2
xk
c) 11
1
0.2 0.6 0.2 0.6 9 1.8
xk
AB
College Mathematics: Learning Worksheets Chapter 4
138
11. A dog food manufacturer makes two types of dog food: Sparky’s Special Selection and
Fido’s Favorite Food. Each type of food uses two types of additives: Additive #1 and
Additive #2. Each bag of Sparky’s Special Selection has 5 units of Additive #1 and 3 units
a) Additive #1: 160 units Additive #2: 180 units
b) Additive #1: 141 units Additive #2: 107 units
c) Additive #1: 130 units Additive #2: 176 units
Let f be the number of bags of Fido’s Favorite and s be the number of bags of Sparky’s
Special. Based on the amount of available additives, we would have the following:
1
2
52
34
sfa
sfa
52
A
Dadbc
1
1
db
Aca
a)
21 21
77 77
1
1
160 20
a
sAB a
−⎡⎤⎡⎤
−−
⎡⎤
⎡⎤ ⎡ ⎤ ⎡ ⎤
⎢⎥⎢⎥
== = =
b)
21 21
77 77
1
1
141 25
a
sAB a
−⎡⎤⎡⎤
−−
⎡⎤
⎡⎤ ⎡ ⎤ ⎡ ⎤
⎢⎥⎢⎥
== = =
⎢⎥
⎢⎥ ⎢ ⎥ ⎢ ⎥
College Mathematics: Learning Worksheets Chapter 4
139
21 21
77 77
1
1
130 12
a
sAB a
−⎡⎤⎡⎤
−−
⎡⎤
⎡⎤ ⎡ ⎤ ⎡ ⎤
12. Farmer Phil needs to treat a field with potash and nitrogen. He can use Grow Rite brand
and Great Green brand. Grow Rite has 3 pounds of potash and 5 pounds of nitrogen per bag.
Great Green has 1 pound of potash and 2 pounds of nitrogen per bag. How many bags of
each brand should he use on his field if he needs exactly the following amounts?
a) Potash: 1320 pounds Nitrogen: 2240 pounds
b) Potash: 1250 pounds Nitrogen: 2150 pounds
c) Potash: 750 pounds Nitrogen: 1350 pounds
Let G
Gbe the number of bags of Great Green brand and
R
Gbe the number of bags Grow
Rite brand. Based on the amount of available treatments, we would have the following:
RG
GG p
College Mathematics: Learning Worksheets Chapter 4
140
b) 12 1 2 1 1250 350
R
Gp
AB
c) 12 1 2 1 750 150
R
Gp
AB
College Mathematics: Learning Worksheets Chapter 4
141
Name ________________________________ Date ______________ Class ____________
Goal: To solve application problems using the Leontief input–output analysis
1. An economy is based on two industrial sectors, electricity and oil. Production of a dollar’s
worth of electricity requires an input of $0.20 from electricity and $0.40 from oil. Production
of a dollar’s worth of oil requires an input of $0.30 from electricity and $0.10 from oil. Find
the output from each sector that is needed to satisfy a final demand of $20 billion for
electricity and $16 billion for oil.
a) Define the variables and find the variable matrix X.
b) Find the technology matrix M.
Section 4-7 Leontief Input–Output Analysis
Solution to a Two-Industry Input–Output Problem:
Given two industries, 1
Cand 2,Cwith defined technology (M), output (X), and final
demand (D) matrices as follows:
x
⎢
⎣
12
CC
x
d
⎡
⎤
where
ij
ais the input required from i
Cto produce a dollar’s worth of output for
College Mathematics: Learning Worksheets Chapter 4
142
x
x
2
x
x
EO
c) Based on the given information, 20
16
E
DO
0.3 0.9
⎡⎤⎡ ⎤
⎢⎥
−
⎣⎦
⎡
⎢
⎣
Dadbc
=−
1
1
()
db
IM ca
⎣
−
−
⎡
⎤
−=
⎢
⎥
e) 1
()
XIM D
−
=− ⋅
42
133
20
x
⎡⎤
⎡⎤ ⎡⎤
g) To meet the final demand, the output from electricity must be $37.33 billion and the
output from oil must be $34 billion.
143
2. An economy is based on two industrial sectors, coal and steel. Production of a dollar’s
worth of coal requires an input of $0.40 from coal and $0.20 from steel. Production of a
dollar’s worth of steel requires an input of $0.40 from coal and $0.20 from steel. Find the
output from each sector that is needed to satisfy a final demand of $25 billion for coal and
$22 billion for steel.
a) Let 1
x
be the output from coal and 2
x
be the output from steel. 1
2
x
X
x
CS
c) Based on the given information, 25
22
C
DS
0.4 0.8
Dadbc
1
1
()
db
IM ca
e) 1
()
X
IM D
f) 1
20.5 25
x
g) To meet the final demand, the output from coal must be $61 billion and the output
from steel must be $58 billion.
3. An economy is based on two industrial sectors, transportation and agriculture. Production
of a dollar’s worth of transportation requires an input of $0.30 from transportation and $0.11
from agriculture. Production of a dollar’s worth of agriculture requires an input of $0.50
from transportation and $0.35 from agriculture. Find the output from each sector that is
needed to satisfy a final demand of $40 billion for transportation and $25 billion for
agriculture.
a) Let 1
x
be the output from transportation. and 2
x
be the output from agriculture.
x
x
b) Based on the given information, 0.3 0.11
TA
T
c) Based on the given information, 40
25
T
DA
d) 10 0.30.11
IM
(0.7)(0.65) ( 0.11)( 0.5)
0.455 0.055
0.4
D
D
D
1
1
1
0.65 0.11
1
() 0.5 0.7
0.4
1.625 0.275
()1.25 1.75
db
D
IM
IM
e) 1
()
XIM D
f) 1
1.625 0.275 40
x
College Mathematics: Learning Worksheets Chapter 4
4. An economy is based on two industrial sectors, iron and steel. Production of a dollar’s
worth of iron requires an input of $0.20 from iron and $0.25 from steel. Production of a
dollar’s worth of steel requires an input of $0.40 from iron and $0.25 from steel. Find the
output from each sector that is needed to satisfy a final demand of $27 million for iron and
$23 million for steel.
a) Let 1
x
be the output from iron and 2
x
be the output from steel. 1
x
X
x
b) Based on the given information, 0.2 0.25
IS
I
I
d) 10 0.20.25
IM
Dadbc
1
1
()
db
IM ca
e) 1
()
XIM D
College Mathematics: Learning Worksheets Chapter 4
146