2
Mathematics 171A
The vertices are
A
(0
,
0)
,
B
(66
2
3
,
0)
,
C
(50
,
25)
, and
D
(0
,
50)
. We ﬁnd
P
(
A
) = 0
,
P
(
B
) =
200
,
P
(
C
) = 250
, and
P
(
D
) = 200
, where
P
denotes the proﬁt at the point
(
x,y
)
.
(b)
Which corner point represents the maximum proﬁt for each of the following proﬁt
formulas:
(i)
proﬁt = 3
x
+ 4
y
(the deﬁnition used in class); From part
(a)
, the point
C
gives
the maximum.
(ii)
proﬁt = 2
x
+ 5
y
;
In this case
P
(
A
) = 0
,
P
(
B
) = 133
1
3
,
P
(
C
) = 225
, and
P
(
D
) = 250
, whence
D
gives the maximum.
(iii)
proﬁt = 5
x
+ 3
y
.
For this part,
P
(
A
) = 0
,
P
(
B
) = 333
1
3
,
P
(
C
) = 325
, and
P
(
D
) = 150
, from which
we conclude that
B
is the maximizer with associated maximum
333
1
3
.
Exercise 1.2.
Consider the following six constraints in two variables: (1)
x
1
+2
x
2
≤
6; (2)
x
1

x
2
≥
2; (3)
x
2
≤
1; (4)
x
1

x
2
≤
4; (5)
x
1
≥
0; and (6)
x
2
≥
0.
(a)
Deﬁne the matrix
A
and vector
b
that express these constraints in the form
Ax
≥
b
.
Inequalities (1), (3), and (4) are multiplied by

1
giving
A
=

1

2
1

1
0

1

1
1
1
0
0
1
and
b
=

6
2

1

4
0
0
.
(b)
Draw the feasible region deﬁned by the six constraints. Outline the feasible region on
your graph and label each of the vertices.