1. A snack bar cooks and sells hamburgers and hot dogs during football games. To stay in business, it must sell
at least 10 hamburgers but cannot cook more than 40 hamburgers. It must also sell at least 30 hotdogs but
cannot cook more than 70. The snack bar cannot cook more than 90 items total. Write a system of linear
inequalities to represent the constraints in the problem and then graph the feasible region.
y
let x = number of hamburgers sold per game
y = number of hot dogs sold per game
hamburgers
10 < x < 40
hot dogs
30 < y < 70
total items
cooked
x + y < 90
common sense
x > 0, y > 0
x-int (90,0)
y-int (0,90)
A
B
C
D
E
(10,70)
(20,70)
(40,50)
(40,30)
(10,30)
75
A
B
C
50
E
25
D
0
x
0
25
50
75
2. It takes a tailoring firm 2 hours of cutting and 4 hours of sewing to make a knit suit. To make a worsted suit
it takes 4 hours of cutting and 2 hours of sewing. At most, 20 hours per day are available for cutting, and at
most 16 hours per day are available for sewing. Write a system of linear inequalities to represent the constraints
y
in the problem and then graph the feasible region.
let
x = number of knit suits made per day
y = number of worsted suits made per day
cutting
2x + 4y < 20
x-int (10,0)
y-int (0,5)
sewing
4x + 2y < 16
x-int (4,0)
y-int (0,8)
A
B
C
D
8
(0,5)
(2,4)
(4,0)
(0,0)
6
A
B
4
2
D
C
x
0
2
4
6
8
3. Bead Counter stores sell necklaces that you make yourself. Customers create jewelry by selecting beads from
various bins. Grace wants to design her own Halloween necklace from orange and black beads. She wants to
make a necklace that is at least 12 inches long but no more than 24 inches long. Grace also wants her necklace
to contain black beads that are least twice the length of orange beads. Finally, she wants her necklace to have
at least 5 inches of black beads. Write a system of linear inequalities to represent the constraints in the problem
and then graph the feasible region.
y
x > 0, y > 0
0
x = inches of orange beads
y = inches of black beads
30
common sense
let
x + y > 12
x + y < 24
x-int (12,0)
y-int (0,12)
x-int (24,0)
y-int (0,24)
y > 2x
y>5
common sense
x > 0, y > 0
A
B
C
D
(0,24)
(8,16)
(4,8)
(0,12)
A
20
B
D
10
0
0
C
10
20
30
x
(1) The profit on a hamburger is $0.33 and on a hotdog is $0.21. How many of each item should be sold in
order to make the maximum profit? What is the maximum profit?
P = .33x + .21y
A
B
C
D
E
(10,70)
(20,70)
(40,50)
(40,30)
(10,30)
= .33(10) + .21(70)
= .33(20) + .21(70)
= .33(40) + .21(50)
= .33(40) + .21(30)
= .33(10) + .21(30)
= $18.00
= $21.30
= $23.70 *max profit
= $19.50 40 hamburgers
50 hot dogs
= $9.60
(2) The profit on a knit suit is $34 and on a worsted suit is $31. How many of each suit should be made in order
to maximize the profit? What is the maximum profit?
P = 34x + 31y
A
B
C
D
(0,5)
(2,4)
(4,0)
(0,0)
= 34(0) + 31(5)
= 34(2) + 31(4)
= 34(4) + 31(0)
= 34(0) + 31(0)
= $155
= $192 *max profit
= $136 2 knitted suits
4 worsted suits
= $0
(3) Orange beads cost $2/inch and black beads cost $1.50/inch. Write an objective function for the cost and
find the solution which minimizes the cost.
C = 2x + 1.5y
A
B
C
D
(0,24)
(8,16)
(4,8)
(0,12)
= 2(0) + 1.5(24)
= 2(8) + 1.5(16)
= 2(4) + 1.5(8)
= 2(0) + 1.5(12)
= $36
= $40
= $20
= $18 *min cost
0 inches orange beads
12 inches black beads