1. A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue.

a) Transcribe the elements needed to establish the constraints.

b) Transcribe the elements needed to establish the function to be optimized.

1. A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue.

a) Transcribe the elements needed to establish the constraints.

b) Transcribe the elements needed to establish the function to be optimized.

1. A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue.

a) Transcribe the elements needed to establish the constraints.

b) Transcribe the elements needed to establish the function to be optimized.

MTH-5101 Pretest A SolutionsDIMENSION 1

2. The community center activities of the municipality wants to publish a guide advertising the sports and cultural activities for the summer. The publisher asks $0.20 for the printing of a page containing photographs and $0.15 for a page without photographs. The guide will have between 25 and 40 pages with a minimum of 10 pages containing photographs.

How many pages will the guide contain for its price to be minimum?

a) Identify the variables.

b) Translate the constraints into a system of inequalities.

c) Express the function to be optimized.

Let x = the number of pages with photographs

Let y = the number of pages without photographs

0.2x + 0.15y = Z

DIMENSION 2

x + y ≥ 25

x + y ≤ 40

x ≥ 25

x ≥ 0

y ≥ 0

3. Draw the polygon of constraints associated with the following system of inequalities.

C1: x ≥ 0

DIMENSION 3

C2: y ≥ 0

C3: y ≤ -2x + 16

824 x

y:C

C5: y ≥ 2

C3: y = -2x + 16

x y

0 16

8 0

x y

0 8

16 16

824 x

y:C

4. Verify algebraically whether the point (250,100) belongs to the polygon of constraints. Show all steps to your solution.

DIMENSION 3 continued

x + y ≥ 180

250 + 100 ≥ 180

350 ≥ 180 True

x + y ≤ 300

250 + 100 ≤ 300

350 ≤ 300 False

y ≤ x + 60

100 ≤ 250 + 60

100 ≤ 310 True

x ≥ 100

250 ≥ 100

250 ≥ 100 True

x ≥ 0

250 ≥ 0 True

y ≥ 0

100 ≥ 0 True

The point (250,100) does not belong to the polygon of constraints because it makes the constraint x + y ≤ 300, false.

200 400 800600

200

400

800

600

5. Determine algebraically the coordinates of the vertices of the polygon of constraints.

C1: x ≥ 0

DIMENSION 4

C2: y ≥ 0

C4: x + y ≥ 450

C5: 4x + 10y ≤ 3600C3: x ≤ 2y

C

A

B

D

x + y = 450y = 0

x + 0 = 450x = 450

x + y = 450x = 2y

2y + y = 450

3y = 450y = 150x = 2y

x = 2(150)x = 300

4x + 10y = 3600y = 0

4x + 10(0) = 3600

4x = 3600x = 900

A x + y = 450y = 0 (450,0)

B x + y = 450x = 2y (300,150

)C 4x + 10y = 3600

x = 2y(400,200

)D 4x + 10y = 3600y = 0

(900,0)

x = 2y4x + 10y = 3600

4(2y) + 10y = 3600

18y = 3600y = 200x = 2y

x = 2(200)x = 400

100 200 400300

100

200

400

300

6. Alain is the owner of a clothing store. He wishes to sell coats that cost him $25 each and pants at $15 each. He is able to store a maximum of 500 units. He hopes to sell at least 250 pairs of pants and a least four times more pants than coats.. If Alain charges $25 for pants and $45 for coats, how many of each must he sell to maximize his profits? Show all steps of your work.

x ≥ 250y ≥ 0

10x + 20y = Z

x + y = 500

x y

0 500

500 0

x ≥ 4y

x y

0 0

400 100

x = 4y

Let x = the number of pairs of pants sold

Let y = the number of coats sold

x + y ≤ 500

x ≥ 0

A

B

x = 250y = 0

x = 250x = 4y

250 = 4yy = 62.5

x = 4yx + y = 500

4y + y = 5005y = 500y = 100

x = 4yx = 4(100)

x = 400y = 0

x + y = 500x + 0 = 500

x = 500

DIMENSION 5

D

C

100 200 400300

100

200

400

300

10x + 20y = ZLet x = the number of pairs of pants sold

Let y = the number of coats sold

C

A

B

D

A x = 250y = 0 (250,0)

B x = 250x = 4y (250,62.5

)C x + y = 500x = 4y (400,100)

D x + y = 500y = 0 (500,0)

A(250,0)

B(250,62.5)C(400,100)

D(500,0)

10x + 20y = 10(250) + 20(0) = 2500 + 0 = $250010x + 20y = 10(250) + 20(62.5) = 2500 + 1250 = $375010x + 20y = 10(400) + 20(100) = 4000 + 2000 = $600010x + 20y = 10(500) + 20(0) = 5000 + 0 = $5000

To maximize profits he must sell 400 pairs of pants and 100 coats.

50 100 200150

50

100

200

150

7. In a hunting and fishing camp, we offer rifle hunting and bow hunting excursions. In order to preserve the wildlife, excursions are carried out according to certain constraints. There is a maximum of 150 registrants There must be at least 30 registrants for rifle hunting and 50 registrants for bow hunting. If $500 is charged for each rifle hunting excursion and $400 for each bow hunting excursion, how many excursions for rifle hunting and bow hunting are necessary to maximize revenue?

x ≥ 30y ≥ 0

500x + 400y = Z

x + y = 500

x y

0 150

150 0

y ≥ 50

Let x = the number of rifle hunting excursions

Let y = the number of bow hunting excursions

x + y ≤ 150

x ≥ 0

C

B

x = 30y = 50

x = 30x + y = 150

30 + y = 150

y = 120y = 50

x + y = 150x + 50 = 150

x = 100

AA x = 30

y = 50 (30,50)

B x + y = 150x =30 (30,120)

C x + y = 150y = 50 (100,50)

500x + 400y = Z

Let x = the number of pairs of pants sold

Let y = the number of coats sold

A(30,50)

B(30,120)

C(100,50)

500x + 400y = 500(30) + 400(50) = 15000 + 20000

= $35000500x + 400y = 500(30) + 400(120) = 15000 +

48000= $63000500x + 400y = 500(100) + 400(50) = 50000 +

20000= $70000

100 rifle excursions and 50 bow excursions are necessary to maximize revenue.

A x = 30y = 50 (30,50)

B x + y = 150x =30 (30,120)

C x + y = 150y = 50 (100,50)

50 100 200150

50

100

200

150

C

B

A

8. Alex wants to buy himself some pants and sweaters. The saleswoman tells him that a pair of pants costs $50 and a sweater costs $25. His father offers to pay for his purchases as a gift for his admission to CEGEP.

The polygon of constraints is presented in the graph on the right where:

25x + 50y = 25(2) + 50(2) = 50 + 100

= $150

25x + 50y = 25(5) + 50(2) = 125 + 100

= $225

25x + 50y = 25(6) + 50(3) = 150 + 150

= $300

According to the polygon of constraints presented, the minimal cost to pay would be $150. With the additional consideration, the minimal cost would have to be higher because with 2 sweaters and 2 pants Alex will NOT have 2 times more sweaters than pants.

2 4 86

2

4

8

6

(2,2)

(6,3)

(5,2)

(4,4)

x = the number of sweaters

y = the number of pantsAlex wants at least 2 times more sweaters than pants. What will be the impact of this additional consideration on the minimal cost that his father will have to pay?Function of Optimization: 25x + 50y =

Z(2,2)

(5,2)

(6,3)

(4,4) 25x + 50y = 25(4) + 50(4) = 100 + 300

= $300

x ≥ 2y

25x + 50y = 25(4) + 50(2) = 100 + 100

= $200

(4,2)

To be consistent with the additional consideration, the father would have to buy 4 sweaters and 2 pants at a cost of $200.

DIMENSION 6

x y

0 0

10

5