# 3

A diet is to include at least 140 milligrams of Vitamin A and at least145 milligrams of Vitamin B. These requirements can be obtained from types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound.If type X food costs $12 per pound and type Y costs $8 per pound how many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost?

X = type xY= type y

1451520

1403010

0

0

yx

yx

y

x

(14, 0 ), (0, 4.3)

f(x,y)=12x+8yf(0, 9.7)=12(0)+8(9.7)

=77.6f(5,3)=12(5)+8(3)

=84f(14,0)=12(14)+0(0)

=168

1451520

1403010.

0

0

yx

yxa

y

x

(14, 0)

(0, 4.3)

(7.25, 0), (0, 9.7)

(7.25, 0)

(5, 3)

( 0, 9.7)

Minimum cost is $77.60 pounds of X and 9.7 pounds of Y

# 4

The Cruiser Bicycles Company makes two styles of bicycles: the Traveler, which sells for $200, and the Tourester, which sells for $600. Each bicycle has the same frame and tires, but the assembly and painting time required for the Traveler is only 1 hour,while it is 3 hours for the Trourister. There are 300 frames and 360 hours of labor available for production. How many bicycles of Each model should be produced to maximize revenue?

X = Traveler Y= Tourister

3603

300

0

0

yx

yx

y

x

(300, 0 ), (0, 300)

f(x,y)=200x+600yf(0,120)=200(0)+600(120)

=72000f(300,0)=200(300)+600(0)

=60000f(270,30)=200(270)+600(30)

=72000

(360, 0), (0, 120)

alternate optional 0 Traveler and 120 Tourister270 traveler and 30 Tourister

3603

300

0

0

yx

yx

y

x

(300, 0)

(0, 300)

(360, 0)

(0, 120)

(270, 30)

#12

Dr. Chen told Miranda that her new puppy needs a diet that includes at least 1.54 ounces of protein and 0.56 ounces of fat each day to grow into a healthy dog. Each cup of Good Start puppy food contains 0.84 ounce of protein and 0.21 ounce of fat. Each cup of Sirius puppy food contains 0.56 ounce of protein and 0.49 ounce of fat. If Good Start puppy food costs 36c per cup and Sirius puppy food costs 22c per cup how much of each food should Miranda use in order to satisfy the dietary requirements at the minimum cost?

a).X = the cups of Good StartY= the cups of Sirius

56.049.021.0

54.156.084.0.

0

0

yx

yxb

y

x

56.049.021.0

54.156.084.0

0

0

yx

yx

y

x

(1.83, 0 ), (0, 2.75)

(2.7, 0 ), (0, 1.14)

(1.83, 0 )

(0, 2.75)

(2.7, 0 )

(0, 1.14)

(1.5, 0.5)

c. F (x ,y) = 0.36x + 0.22 y

d). f(0, 2.75) = 0.36(0) + 0.22(2.75) =0.605f(1.5, 0.5) = 0.36(1.5) +0.22 (0.5) = 0.65 f(2.7, 0) = 0.36(2.7) +0.22 (0)= 0.97

f. F (0, 2.75)=0.250 cups of Good Start and 2.75 cups of Sirius

#13

Angela’s Pizza is open from noon to midnight each day. Employees work 8 –hour shifts from noon to 8 p.m. or 4p.m. to midnight. The store manager estimates that she needs at least 5 employees from noon to 4 p.m.., at least 14 employees from 4 p.m. to 8 p.m., and 6 employees from 8p.m. to midnight. Employees are paid $ 5,50 per hour for hours worked between noon and 4 p.m. The hourly pay between 4p.m. and midnight is $7.50.

X = the day – shift workersY= the night – shift workers

14.

6

5

yxa

y

x

(14, 0 ), (0, 14)

c. Day - shift per day 5.50(4) + 7.50 (4)= 52Night- shift per day 7.50 (8)= 60F (x ,y) = 52x + 60y

d. F(8, 6) = 52( 8) + 60 (6) =776F (5,9) = 52(5) + 60 (9) =800

8 day – shift workers6 night – shift workers

14.

6

5

yxa

y

x

(14, 0)

(0, 14)

(0,6)

(5, 0)

(8, 6)(5, 9)

(5, 6)

b.

e. Min = $776.00

#14

The county officials in Chang Qing County used linear programming to aid the farmers in their choices of crops and other forms of agricultural production. This led to a 12% increase in crops profits, a 54% increase in animal husbandry profits, while improving the region’s ecology. Suppose that an American farmer has 180 acres on which to grow corn and soybeans. He is planning at least40 acres of corn and 20 acres of soybeans.Based on his calculations, he can earn $150 per acre of corn and $250 per acre of soybeans.a. If the farmer plants at least 2 acres of corn for every acre

of soybeans, how many acres of each should he plant to earn the grates profit?

b. What is the farmer’s maximum profit?

x = acres of corny = acres of soybeans

xy

yxa

y

x

2

180.

20

40

(180 , 0), (0, 180)

(0, 0), (80, 160)

F (x ,y) = 150x + 250 y

F(40, 20) = 150(40) +250(20) =11000

F(40, 80) = 150( 40) +250 (80) =26000

F (60, 120) = 150(60) +250(120) = 39000

F (160, 20) = 150(60) + 250(20)= 14000

60 acres of corn and 120 acres of soybeans

b. 39000 is the max profit

xy

yxa

y

x

2

180.

20

40

(40, 0)

(0, 20)

(80, 160)

(180, 0)

(0, 180)

(160, 20)

(60, 120)

(40, 20)

(40, 80)

#15

Ms. Carlyle has written a final exam for her class that contains two different sections. Questions in section I are worth 10 points each, and questions in section II are worth 15 points each. Her students will have 90 minutes to complete the exam. From pastexperience, she knows that on average questions from section I take 6 minutes to complete and questions from section II take 15 minutes.Ms. Carlyle requires her students to answer at least two questions from section II. Assuming they answer correctly, how manyquestions from each section will her students need to answer to getthe highest possible grade?

x = the questions from section Iy = the questions from section II

90156

2

0

yx

y

x

(15, 0 ), (0, 6)

F (x ,y) = 10x + 15y

F(0, 6) = 10( 0) + 15 (6) =90

F (0, 2) = 10(0) + 15(2) =30

F(10, 2) =10(10) +15(2)= 130

10 questions from section I2 questions from section II

90156

2

0

yx

y

x

(0, 0)

(0, 2)

(15, 0)

(0, 6) (10, 2)

#16

Newline Recyclers processes used aluminum into food or drinkcontainers. The recycling plant processes up to 12000 tons ofaluminum per week. At least 300 tons must be processed for food containers, while at least 450 tons must be processed for drinkcontainers. The profit is $17.50 per ton for processing food containersand $20 per ton for processing drink containers. What is the profit if the plant maximizes processing?

x = food containersy = drink containers

1200

450

300

yx

y

x

(1200, 0 ), (0, 1200)

F (x ,y) = 17.50x + 20y

F(700, 450) = 17.5(700) + 20(450) = 21250

F (300, 450) = 17.5(300) + 20(450) = 14250

F(200, 900) = 17.5(200) +20(900)= 21500

$21500 is the profit if the plant maximizes processing

1200

450

300

yx

y

x

(300, 0)

(0, 450)

(1200, 0)

(0, 1200)

(700, 450)

(200, 900)

(300, 450)

#17

Diego wants to invest up to $11000 in certificates of deposit at First Bank and City Bank. He doesn’t want to deposit more than$7500 at First Bank. He will deposit at least $1000 but not more than $7000 at City Bank. First Bank offers 6% simple interest ondeposit, while City Bank offers 6.5% simple interest. How much should Diego deposit into each account so he can earnthe most interest possible in one year?

x = the deposit at First Banky = the deposit at City Bank

11000

70001000

7500

yx

y

x

(11000, 0 ), (0, 11000)

F (x ,y) = 0.06x + 0.065y

F(7500, 3500)= .06(7500) +.065(3500) = 677.50

F(4000, 7000)= .06(4000) +.065(7000) = 695.00

F(0, 7000) = .06(0) +.065(7000)= 455.00

F(0, 1000) = .06(0)+.065(1000)=65.00F(7500, 1000) =.06(7500)+.065(1000)

= 515.00

(0, 7000)

11000

70001000

75000

yx

y

x

(7500, 0)

(0, 1000)

(11000, 0)

(0, 11000)

(4000, 7000)

(7500, 3500)

(7500, 1000)

$4000 at First Bank$7000 at City Bank

19

A potato chip company makes chips to fill snack-size bags and family-size

bags. In one week, production cannot exceed 2,400 units, of which at least 600

units must be for snacks-size bags and at least 900 units must be for family-

size. The profit on a unit of snack-size bags is $12, and the profit on a unit of

family-size bags is $18. How much of each type of bag must be processed to

maximize profit?

x = snack-size bags

y = family-size bagsx + y ≤2400

x≥600

y≥900

(600, 1800)

(600, 900)

(1500, 900)

f(x, y)=12x+18y

f(600, 900)=23400

f(600, 1800)=39600

f(1500, 900)=34200

x = snack-size bags

y = family-size bags

x + y ≤2400

x≥600

y≥900