Linear Programming

– set up and solve linear programming problems to find real life minimums and maximums.

Linear Programming Hints To write the constraints and objective

function:– Identify the variables. – Write the objective function– Identify each constraint. Then write an

inequality for each.• Start with x > 0 and y > 0 (usually).

Example 1 The West Hartford Senior Center is trying to establish a

transportation system of small and large vans. It can spend no more than $100,000 for both sizes of vehicles and no more than $500 per month for maintenance. The WHSC can purchase a small van, which carries up to 7 passengers, for $10,000 and maintain it for $100 per month. The large vans, which carry up to 15 passengers, cost $20,000 each and can be maintained for $75 per month. How many of each type of van should they purchase if they want to maximize the number of passengers?

# of small vans # of large vans

sl

00

sl

Initial purchase cost

Monthly maintenance100,000

50010,000 20,000s l100 75s l# of passengers

P = 7s + 15l

Example 1 # of small vans # of large vans

sl

7 15P s l

00

10,000 20,000 100,000100 75 500

sl

s ls l

0 2 4 6 8 10# of small vans, s

0

2

4

6

8

10#

of la

rge

vans

, l (0,5) 7(0) 15(5) 75P

The vertices that should be tested are (0,5), (2,4), and (5,0).

(2,4) 7(2) 15(4) 74P

(5,0) 7(5) 15(0) 35P

The WHSC should buy 5 large vans which can hold 75 passengers.

Example 2 Pancakes Waffles

3 cups Bisquick 2 cups Bisquick

1 cup Milk 2 cups Milk

2 Eggs 2 Eggs

Serves 6 Serves 5 You have 24 cups of Bisquick, 18 cups of milk, and 20

eggs. If you want to feed as many people as possible, how many batches of each should you make?

# of batches of pancakes # of batches of waffles

pw

6 5S p w

00

pw

BisquickMilk

2418

3 2p w2p w

Eggs 202 2p w

Servings

0 2 4 6 8 10

2 2 20

3 2 248

0

2

0

1p wp

w

w

wp

p

Example 2

# of batches of pancakes, p

0

2

4

6

8

10#

of b

atch

es o

f w

affl

es, w

# of batches of pancakes # of batches of waffles

pw

6 5S p w

(0,9) 6(0) 5(9) 45S

The vertices that should be tested are (0,9), (2,8), (4,6), and (8,0).

(2,8) 6(2) 5(8) 52S

(4,6) 6(4) 5(6) 54S

(8,0) 6(8) 5(0) 48S Make 4 batches of pancakes and 6 batches of waffles to feed 54 people.

Example 3 Kayla works no more than 20 hours per

week during the school year. She is paid $10 an hour for tutoring Geometry students and $7 an hour for babysitting. She wants to spend at least 3 hours but no more than 8 hours a week tutoring. Find Kayla’s maximum weekly earnings.

# of hrs spent tutoring # of hrs spent babysitting

tb

10 7E t b

380

ttb

Total hours worked 20t b

Earnings

Example 3380

20

ttbt b

0 2 4 6 8 10# of hrs spent tutoring, t

0

4

8

12

16

20

# of

hrs

spe

nt b

abys

itti

ng, b

# of hrs spent tutoring # of hrs spent babysitting

tb

10 7E t b

(3,17) 10(3) 7(17) 149E

The vertices that should be tested are (3,17) and (8,12).

Kayla should tutor for 8 hours and babysit for 12 hours.

(8,12) 10(8) 7(12) 164E

Example 4 As part of your weight training regimen, you want to

consume lean sources of protein. You want to consume at least 300 Calories a day from at least 48 grams of protein. One ounce of chicken provides 35 Calories and 8.5 g of protein. One ounce of tofu provides 20 Calories and 2.5 g of protein. Your local supermarket charges $5 a pound for chicken and $2.50 a pound for tofu. How much of each food should you eat each day if you want to meet your requirements with the lowest cost? What is this daily cost?

# of lbs of chicken # of lbs of tofu

ct

5 2.50P c t

00

ct

CaloriesProtein

30048

560 320c t136 40c tPrice Paid

560 320 3001 8

0

36 4 40

0cc

tt

ct

Example 4

0 0.2 0.4 0.6 0.8 1.0# of lbs of chicken, c

0

0.2

0.4

0.6

0.8

1.0

# of

lbs

of to

fu, t

# of lbs of chicken # of lbs of tofu

ct

0.5360.9375

ct

0.351.2

ct (0,1.2) 5(0) 2.5(1.2) 3P

The vertices that should be tested are approx. (0, 1.2), (0.16, 0.65), and (.536,0).

(0.16,0.65) 5(0.16) 2.5(0.65) 2.425P

(0.536,0) 5(0.536) 2.5(0) 2.68P

You should eat about 0.16 pounds of chicken and 0.65 pounds of tofu, which will cost about $2.43 per day.

5 2.50P c t