3. (
Use the graphical method for the following problem
): A technology company manufac-
tures televisions and soundbars. The following table gives the labor requirements and profits per
automobile:
Televisions
Soundbars
Assemble
10 labor-hours
8 labor-hours
Load and Transport
2 labor-hours
1 labor-hours
Profit
$150
$100
Each day, 360 labor-hours are available to assemble these products and 60 labor-hours are avail-
able to load and trasport them. The manufacturer wants to determine how many televisions and
soundbars to produce of each type in order to maximize profit under these constraints.
(a)
[5 points]
Define variables for modeling this scenario, and write down the objective function.

(b) Identify the constraints of the problem.

Assembly: 10
x
+ 8
y
≤
360
,
Load and Transport: 2
x
+
y
≤
60
,
We also have our physical constraints (no negative items can be produced):
x
≥
0
, y
≥
0
.
The complete system of constraints is given by
10
x
+ 8
y
≤
360
2
x
+
y
≤
60
x
≥
0
, y
≥
0
(c) List the corner points of the feasible region.

Thus the corner points are (0
,
0)
,
(0
,
45)
,
(20
,
20)
,
(30
,
0)
.
(d)
[5 points]
How many televisions and soundbars should be produced each day to maximize
the total profit?

4. A colleague approaches you with the following simplex tableau:
x
y
z
s
t
u
p
0
5
0
1
4
0
0
72
1
0
0
0
3
1
0
60
1
1
1
0
1
0
0
22
3
-2
0
0
-4
0
1
140
(a)
[10 points]
Your colleague does not remember which variables to list in the first column of the
tableau. Write down these variables, and give their values.