Date:_______________________________
Name:______________________________
Linear Programming: To Find the Maximum Profit
1. A shoe company sells both running shoes and dress shoes: The company wants to know how many of
each pair of shoes they should sell an hour to make the most profit, if the profit from one pair of running
shoes is $21 and the profit from one pair of dress shoes is $24. Find the maximum profit if the
constraints are shown in the following table:
Variable to
represent
Type of shoe
Amount of time to
make (Hours)
Amount of material
required
Dress shoe
2
1
Running shoe
1
2
6
(max time)
6
(max material)
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
6
y
Point Of Intersection: (
5
4
3
2
1
x
1
2
3
4
5
Find the Profit:
Profit equation:___________________
6
,
)
Date:_______________________________
Name:______________________________
2. The Precision Tool Company wants to make hammers and chisels. Each hammer needs 1 hour on
machine A and 2 hours on machine B. Each chisel needs 2 hours on machine A and 1 hour on machine
B. Neither machine can work more than 30 hours per week. The profit is $3 on each hammer and $2 on
each chisel. We eventually want to find how many of hammers and chisels should be made to get a
maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
(max time)
(max time)
hammer
chisel
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
30
y
Point Of Intersection: (
25
20
15
10
5
x
5
10
15
20
25
Find the Profit:
Profit equation:___________________
30
,
)
Date:_______________________________
Name:______________________________
3. A hat company sells both visors and baseball hats. The company wants to know how many of each type
of hat they should make a day to the most profit.. The profit on the visor is $4 and the profit on the
baseball hats is $5. Find the maximum profit if the constraints are shown in the following table:
Variable to
represent
Type of Hat
Amount of time to
make (Hours)
Amount of material
required
Visor
1
1
Baseball
1
3
4
(max time)
6
(max material)
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
6
y
Point Of Intersection: (
5
4
3
2
1
x
1
2
3
4
5
Find the Profit:
Profit equation:___________________
6
,
)
Date:_______________________________
Name:______________________________
4. The ABC Company wants to make jig-saws and drills. Each jig-saw needs 4 hours on machine A and 2
hours on machine B. Each drill needs 5 hours on machine A and 2 hours on machine B. Neither
machine can work more than 40 hours per week. The profit is $12 on each jig-saw and $8 on each drill.
We eventually want to find how many jig-saws and drills should be made to get maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
(max time)
(max time)
Jig-saw
drill
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
y
Point Of Intersection: (
x
5
10
15
20
Find the Profit:
Profit equation:___________________
25
30
,
)
Date:_______________________________
Name:______________________________
5. The Too-Comfy Furniture Store wants to make sofas and lazy-boy chairs. Each sofa needs 4 hours on
machine A and 4 hours on machine B. Each lazy-boy chair needs 2 hours on machine A and 4 hours on
machine B. Machine A is available for only 12 hours a day. The maximum daily time for machine B is
16 hours. The profit is $120 on each sofa and $110 on each lazy-boy chair. Find how many sofas and
lazy-boys should be made to get a maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
(max time)
(max time)
Sofa
Lazy-boy chair
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
5
y
Point Of Intersection: (
4
3
2
1
x
1
2
3
4
5
Find the Profit:
Profit equation:___________________
,
)
Date:_______________________________
Name:______________________________
6. The XYZ Sound Company makes CDs and DVDs. The two departments are data transfer and plastic
coating. A DVD takes 4 minutes in data transfer and 1 minute in plastic coating. Each CD takes 1
minute in data transfer and 1 minute in plastic coating. The total available time for data transfer is 400
minutes per day and the total time for plastic coating is 250 minutes a day. The profit on a DVD is $12
and the profit on a CD is $10. Find the number of each product to maximize profit. What is the
maximum profit?
Variable to
represent
Item
Data Transfer
Plastic Coating
(max time)
(max time)
DVD
CD
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
500
y
Point Of Intersection: (
450
400
350
300
250
200
150
100
50
x
50
100
Find the Profit:
Profit equation:___________________
150
,
)
Date:_______________________________
Name:______________________________
7. The ABC Tool Company makes monkey wrenches and adjustable wrenches. The wrenches go through
2 departments, moulding and forging. A monkey wrench spends 3 minutes in moulding and 2 minutes
in forging. An adjustable wrench spends 2 minutes in moulding and 4 minutes in forging. The total time
available for moulding is 600 minutes per day and for forging the maximum time is 800 minutes per
day. The profit on a monkey wrench is $25 and the profit on an adjustable wrench is $22. Find the
maximum profit. That is the number of each type of wrench necessary to make a maximum profit?
Variable to
represent
Tool
Moulding
Forging
(max time)
(max time)
Monkey wrench
Adjustable wrench
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
500
y
Point Of Intersection: (
400
300
200
100
x
100
200
300
400
500
Find the Profit:
Profit equation:___________________
,
)
Date:_______________________________
Name:______________________________
8. The Ace Manufacturing Company wants to make plates and cups. Each cup needs 3 hours on
machine A and 1 hour on machine B. Each plate needs 1 hour on machine A and 2 hours on
machine B. Neither machine can work more than 15 hours per day. The profit is $0.40 on each
cup and $0.25 on each plate. We eventually want to find how many of cups and plates should be
made to get maximum profit.
Variable to
represent
Total:
Inequations: _______________________
_______________________
Graph the inequations on the following graph:
Point Of Intersection: (
Find the Profit:
Profit equation:___________________
,
)
Date:_______________________________
Name:______________________________
ANSWERS
1. A shoe company sells both running shoes and dress shoes: The company wants to know how many of
each pair of shoes they should sell an hour to make the most profit, if the profit from one pair of running
shoes is $21 and the profit from one pair of dress shoes is $24. Find the maximum profit if the
constraints are shown in the following table:
Variable to
represent
Type of shoe
Amount of time to
make (Hours)
Amount of material
required
X
Dress shoe
2
1
Y
Running shoe
1
2
6
(max time)
6
(max material)
Total:
Inequations:
2 x  1y  6
x  2y  6
Graph the inequations on the following graph:
6
y
Point Of Intersection: ( 2 , 2 )
5
(0, 0)  $0
(0,3)  24(0)  21(3)  $63
(2, 2)  24(2)  21(2)  $90
(3, 0)  24(3)  21(0)  $72
4
3
2
1
x
1
2
3
4
5
6
Find the Profit:
Profit equation: P  24 x  21y
Make two pairs of each to earn maximum profit of $90
Date:_______________________________
Name:______________________________
2. The Precision Tool Company wants to make hammers and chisels. Each hammer needs 1 hour on
machine A and 2 hours on machine B. Each chisel needs 2 hours on machine A and 1 hour on machine
B. Neither machine can work more than 30 hours per week. The profit is $3 on each hammer and $2 on
each chisel. We eventually want to find how many of hammers and chisels should be made to get a
maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
x
hammer
1
2
y
chisel
2
1
30
(max time)
30
(max time)
Total:
Inequations:
x  2 y  30
2 x  y  30
Graph the inequations on the following graph:
30
y
Point Of Intersection: ( 10 , 10 )
25
(0, 0)  $0
(0,15)  3(0)  2(15)  $30
(10,10)  3(10)  2(10)  $50
(15, 0)  3(15)  2(0)  $45
20
15
10
5
x
5
10
15
Find the Profit:
Profit equation: P  3x  2 y
20
25
30
Date:_______________________________
Name:______________________________
3. A hat company sells both visors and baseball hats. The company wants to know how many of each type
of hat they should make a day to the most profit.. The profit on the visor is $4 and the profit on the
baseball hats is $5. Find the maximum profit if the constraints are shown in the following table:
Variable to
represent
Type of Hat
Amount of time to
make (Hours)
Amount of material
required
x
Visor
1
1
y
Baseball
1
3
4
(max time)
6
(max material)
Total:
Inequations:
x y4
x  3y  6
Graph the inequations on the following graph:
6
y
Point Of Intersection: ( 3 , 1 )
P  3x  2 y
5
(0,0)  $0
(0,2)  4(0)  5(2)  $10
(3,1)  4(3)  5(1)  $17
(4,0)  4(4)  5(0)  $16
4
3
2
1
x
1
2
3
4
5
6
Find the Profit:
Profit equation: P  4 x  5 y
Need to make 3 visors and 1 hat to get a maximum profit of $17
Date:_______________________________
Name:______________________________
4. The ABC Company wants to make jig-saws and drills. Each jig-saw needs 4 hours on machine A and 2
hours on machine B. Each drill needs 5 hours on machine A and 2 hours on machine B. Neither
machine can work more than 40 hours per week. The profit is $12 on each jig-saw and $8 on each drill.
We eventually want to find how many jig-saws and drills should be made to get maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
x
Jig-saw
4
2
y
drill
5
2
40
(max time)
40
(max time)
Total:
Inequations:
4 x  5 y  40
2 x  2 y  40
Graph the inequations on the following graph:
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
y
Point Of Intersection: (NONE)
(0, 0)  $0
(0,8)  12(0)  8(8)  $64
(10, 0)  12(10)  8(0)  $120
x
5
10
15
20
25
30
Find the Profit:
Profit equation: P  3x  2 y
Make 10 jig-saws and no drills to get a maximum profit of $120
Date:_______________________________
Name:______________________________
5. The Too-Comfy Furniture Store wants to make sofas and lazy-boy chairs. Each sofa needs 4 hours on
machine A and 4 hours on machine B. Each lazy-boy chair needs 2 hours on machine A and 4 hours on
machine B. Machine A is available for only 12 hours a day. The maximum daily time for machine B is
16 hours. The profit is $120 on each sofa and $110 on each lazy-boy chair. Find how many sofas and
lazy-boys should be made to get a maximum profit.
Variable to
represent
Tool
Amount of time on
machine A
Amount of time on
machine B
X
Sofa
4
4
Y
Lazy-boy chair
2
4
12
(max time)
16
(max time)
Total:
Inequations:
4 x  2 y  12
4 x  4 y  16
Graph the inequations on the following graph:
5
y
Point Of Intersection: (
,
)
4
(0, 0)  $0
(0, 4)  120(0)  110(4)  $440
(2, 2)  120(2)  110(2)  $460
(3, 0)  120(3)  110(0)  $360
3
2
1
x
1
2
3
4
5
Find the Profit:
Profit equation: P  120 x  110 y
Make 2 Sofas and 2 Lazy-boys to get a maximum profit of $460
Date:_______________________________
Name:______________________________
6. The XYZ Sound Company makes CDs and DVDs. The two departments are data transfer and plastic
coating. A DVD takes 4 minutes in data transfer and 1 minute in plastic coating. Each CD takes 1
minute in data transfer and 1 minute in plastic coating. The total available time for data transfer is 400
minutes per day and the total time for plastic coating is 250 minutes a day. The profit on a DVD is $25
and the profit on a CD is $7. Find the number of each product to maximize profit. What is the
maximum profit?
Variable to
represent
Item
Data Transfer
Plastic Coating
x
DVD
4
1
y
CD
1
1
400
(max time)
250
(max time)
Total:
Inequations:
4 x  y  400
x  y  250
Graph the inequations on the following graph:
Point Of Intersection: ( 50, 2000 )
500
y
(50,200)
(0,250)
(100,0)
450
400
350
300
250
200
150
100
50
x
50
100
150
Find the Profit:
Profit equation: P = 12x + 10y
Make 50 DVDs and 200 CDs to get a maximum profit of $2600
P = $2600
P = $2500
P = $1200
Date:_______________________________
Name:______________________________
7. The ABC Tool Company makes monkey wrenches and adjustable wrenches. The wrenches go through
2 departments, moulding and forging. A monkey wrench spends 3 minutes in moulding and 2 minutes
in forging. An adjustable wrench spends 2 minutes in moulding and 4 minutes in forging. The total time
available for moulding is 600 minutes per day and for forging the maximum time is 800 minutes per
day. The profit on a monkey wrench is $25 and the profit on an adjustable wrench is $22. Find the
maximum profit. That is the number of each type of wrench necessary to make a maximum profit?
Variable to
represent
Tool
Moulding
Forging
x
Monkey wrench
3
2
y
Adjustable wrench
2
4
600
(max time)
800
(max time)
Total:
Inequations:
3 x  2 y  600
2 x  4 y  800
Graph the inequations on the following graph:
500
y
450
Point Of Intersection: ( 100, 150 )
400
(0, 0)  $0
(0, 200)  25(0)  22(200)  $4400
(100,150)  25(100)  22(150)  $5800
(200, 0)  25(200)  22(0)  $5000
350
300
250
200
150
100
50
x
50 100 150 200 250 300 350 400 450 500
Find the Profit:
Profit equation: P  25 x  22 y
Make 100 monkey wrenches and 150 adjustable wrenches to get a maximum profit of $5800
Date:_______________________________
Name:______________________________
8. The Ace Manufacturing Company wants to make plates and cups. Each cup needs 3 hours on
machine A and 1 hour on machine B. Each plate needs 1 hour on machine A and 2 hours on
machine B. Neither machine can work more than 15 hours per day. The profit is $0.40 on each
cup and $0.25 on each plate. We eventually want to find how many of cups and plates should be
made to get maximum profit.
Variable to
represent
Time on Machine A Time on Machine B
x
Plates
1
2
y
Cups
3
1
15
15
Total:
Inequations: ___ x  3 y  15 __________
____ 2 x  y  15 _________
Graph the inequations on the following graph:
15
y
Point Of Intersection: ( 6
10
, 3 )
(0,5) P = $2
(6,3) P = $2.70
(7.5,0) P = $1.88
Make 6 plates and 3 cups for a max profit of $2.70
5
x
5
10
15
Find the Profit:
Profit equation:__ P  0.25 x  0.40 y ______