Math 24 SG 5

Name: ________________________ Class: ___________________ Date: __________ ID: A

1

Worksheet#5 Chapter 6

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

1. True or false?

The solution set of a linear inequality involving two variables is a straight line.

a. Falseb. True

2. Determine graphically the solution set for the system of inequalities and indicate whether the solution set is bounded or unbounded.

5x

x

+

2y

2y

>

>

16

4

a. unbounded

b. unbounded

c. bounded

d. unbounded

e. no solution

3. Determine graphically the solution set for the system of inequalities and indicate whether the solution set is bounded or unbounded.

4x

3x

0

+

3y

y

y

x

12

3

6

0

Name: ________________________ ID: A

2

a. bounded

b. unbounded

c. unbounded

d. bounded

e. no solution

Name: ________________________ ID: A

3

4. Write a system of linear inequalities that describes the shaded region.

a.

x4

5x +6y30

3xy3

Ï

Ì

Ó

ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ

d.

y 4

6x5y5

3x +y3

Ï

Ì

Ó


b.

x <4

5x +6y >30

3xy >3

Ï

Ì

Ó


e.

x4

5x +6y30

3xy3

Ï

Ì

Ó


c.

x4

5x6y30

3x +y3

Ï

Ì

Ó


Name: ________________________ ID: A

4


a.

x y

7x + 5y

x + 4y

13

175

52

Ï

Ì

Ó

ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ

b.

x y

7x + 5y

x + 4y

13

175

52

Ï

Ì

Ó


c.

x y

7x + 5y

x + 4y

13

175

52

Ï

Ì

Ó


d.

x y

7x + 5y

x + 4y

13

175

52

Ï

Ì

Ó


e.

x y

7x + 5y

x + 4y

13

175

52

Ï

Ì

Ó


6. Formulate but do not solve this exercise as a linear programming problem.

Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 4 lb of cast iron and 6 min of labor. To produce each model B grate requires 5 lb of cast iron and 3 min of labor. The profit for each model A grate is $2.00, and the profit for each model B grate is $1.50. If 1160 lb of cast iron and 20 hr of labor are available for the production of grates per day, how many grates of each model should the division produce per day in order to maximize Kane's profits?

Let x denote number of grates model A and let y denote number of grates model B.

a. Minimize P = 2x +32

y

subject to 4x + 5y < 1160 6x + 3y < 1200 x > 0, y > 0

b. Maximize P = 2x +32

y

subject to 4x + 5y 1160 6x + 3y 1200 x 0, y 0

c. Minimize P = 2x +32

y

subject to 4x + 5y 1160 6x + 3y 1200 x 0, y 0

d. Maximize P = 2x +32

y

subject to 4x + 5y 1160 6x + 3y < 1200 x 0, y 0

e. Maximize P = 2x +32

y

subject to 4x + 5y < 1160 6x + 3y 1200 x 0, y 0


Name: ________________________ ID: A

5

AntiFam, a hunger-relief organization, has earmarked between $2 and $2.5 million, inclusive, for aid to two African countries, country A and country B. Country A is to receive between $1 and $1.5 million, inclusive, in aid, and country B is to receive at least $0.75 million in aid. It has been estimated that each dollar spent in country A will yield an effective return of $0.60, whereas a dollar spent in country B will yield an effective return of $0.70. How should the aid be allocated if the money is to be utilized most effectively according to these criteria?

Hint: If x and y denote the amount of money (in million of dollars) to be given to country A and country B, respectively, then the objective function to be maximized is P = 0.6x + 0.7y .

a. Minimize P = 0.6x + 0.7y subject to 2 x + y 2.5 1 x 1.5 y 0.75

b. Minimize P = 0.6x + 0.7y subject to 2 x + y 2.5 1 x 1.5 y > 0.75

c. Maximize P = 0.6x + 0.7y subject to 2 x + y 2.5 1 x 1.5 y 0.75

d. Minimize P = 0.6x + 0.7y subject to 2 < x + y < 2.5 1 < x < 1.5 y > 0.75

e. Maximize P = 0.6x + 0.7y subject to 2 < x + y < 2.5 1 < x < 1.5 y > 0.75


A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $6 on each unit of product A and a profit of $5 on each unit of product B. To manufacture a unit of product A requires 6 min on machine I and 8 min on machine II. To manufacture a unit of product B requires 9 min on machine I and 7 min on machine II. There are 3 hr of machine time available on machine I and 5 hrs of machine time available on machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?

Write the objective function, P (x ,y).

Write the system of inequalities to which the objective function is subjected.

a. P(x,y) =6x +5y , 6x +7y180 , 8x +9y300 , x0 , y0b. P(x,y) =5x +6y , 6x +9y180 , 8x +9y300 , x0 , y0c. P(x,y) =5x +6y , 6x +9y9 , 8x +300y180 , x0 , y0d. P(x,y) =6x +5y , 6x +9y180 , 8x +7y300 , x0 , y0e. P(x,y) =5x +6y , 6x +9y180 , 8x +7y300 , x0 , y0

Name: ________________________ ID: A

6


National Business Machines manufactures two models of fax machines: A and B. Each model A costs $300 to make, and each model B costs $150. The profits are $90 for each model A and $30 for each model B fax machine. If the total number of fax machines demanded per month does not exceed 2500 and the company has earmarked no more than $400000/month for manufacturing costs, how many units of each model should National make each month in order to maximize its monthly profits?



a. P(x,y) =30x +90y ; x +y2500,300x +150y400000 , x0,y0b. P(x,y) =90x +30y ; x +y2500,300x +150y400000 , x0,y0c. P(x,y) =30x +90y ; x +y2500,150x +300y400000 , x0,y0d. P(x,y) =90x +30y ; x +y2500,150x +300y400000 , x0,y0e. P(x,y) =30x +90y ; x +y2500,300x +150y400000 , x0,y0


Madison Finance has a total of $30 million earmarked for homeowner and auto loans. On the average, homeowner loans have a 10% annual rate of return, whereas auto loans yield a 14% annual rate of return. Management has also stipulated that the total amount of homeowner loans should be greater than or equal to four times the total amount of automobile loans. Determine the total amount of loans of each type Madison should extend to each category in order to maximize its returns.



a. P(x,y) =0.14x +0.1y ; x +y30,x4y0,x 0,y0b. P(x,y) =x +0.14y ; 0.1x +y30,x4y0,x 0,y0c. P(x,y) =0.1x +0.14y ; x +y30,0.14x4y0,x 0,y0d. P(x,y) =0.1x +0.14y ; x +y30,x4y0,x 0,y0e. P(x,y) =0.1x +0.14y ; x +y30,x4y0,x 0,y0

Name: ________________________ ID: A

7

11. Find the optimal (maximum and minimum) values of the objective function on the feasible set S .

Z = 8x 3y

a. Z max = 71,Zmin = 3

b. Z max = 73,Zmin = 3

c. Z max = 72,Zmin = 2

d. Z max = 75,Zmin = 6

e. Z max = 71,Zmin = 5

Name: ________________________ ID: A

8

12. Find the optimal (maximum and minimum) values of the objective function on the feasible set S .

Z = 9x + 4y

a. Z max = 75,Z min = 33

b. Z max = 72,Z min = 29

c. Z max = 74,Z min = 32

d. Z max = 71,Z min = 30

e. Z max = 74,Z min = 29

Name: ________________________ ID: A

9

13. Solve the following linear programming problem by the method of corners.

Maximize P = 4x 3y

subject to

x + 5y

4x + y

x

y

24

20

0

0

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Ì

Ó

ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ

a. P 6,4ÊËÁÁ

ˆ¯̃̃ = 12

b. P 4,0ÊËÁÁ

ˆ¯̃̃ = 16

c. P 7,4ÊËÁÁ

ˆ¯̃̃ = 16

d. P 8,0ÊËÁÁ

ˆ¯̃̃ = 32

e. P 5,0ÊËÁÁ

ˆ¯̃̃ = 20

14. Solve the following linear programming problem by the method of corners.

Find the maximum and minimum of P = 3x + 14y

subject to

5x + 2y

x + y

3x + 2y

x

y

63

18

51

0

0

Ï

Ì

Ó

ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ

a. P min(15,3) = 95, P max(6,16.5) = 255

b. P min(15,3) = 87, P max(6,16.5) = 249

c. P min(15,3) = 103, P max(6,16.5) = 258

d. P min(15,3) = 79, P max(6,16.5) = 252

e. P min(15,3) = 111, P max(6,16.5) = 261

Name: ________________________ ID: A

10

15. Solve the linear programming problem by the method of corners.

Minimize C = 5x + 8ysubject to x + y 3

x + 2y 4x 0, y 0

a. x =2 , y =1 , C =18b. x =1 , y =2 , C =21c. x =0 , y =3 , C =24d. x =3 , y =0 , C =10e. x =2 , y =2 , C =26

16. National Business Machines manufactures two models of fax machines: A and B. Each model A costs $84 to make, and each model B costs $100. The profits are $37 for each model A and $46 for each model B fax machine. The total number of fax machines demanded per month does not exceed 2700 and the company has earmarked no more than $720,000/month for manufacturing costs. How many units of each model should National make each month in order to maximize its monthly profits? What is the optimal profit?

a. N type A = 300, N type B = 2,400, P max = $102,600

b. N type A = 200, N type B = 2,500, P max = $101,700

c. N type A = 100, N type B = 2,600, P max = $100,800

d. N type A = 400, N type B = 2,300, P max = $89,700

e. N type A = 0, N type B = 2,700, P max = $124,200

Short Answer

17. Determine whether the statement is true or false.

The solution set of a linear inequality involving two variables is a half plane.

Name: ________________________ ID: A

11


Answer or .

x y

7x + 4y

x + 3y

12

140

36

Ï

Ì

Ó



ID: A

1

Worksheet#5 Chapter 6Answer Section

MULTIPLE CHOICE

1. ANS: A PTS: 1 2. ANS: D PTS: 1 3. ANS: C PTS: 1 4. ANS: A PTS: 1 5. ANS: B PTS: 1 6. ANS: B PTS: 1 7. ANS: C PTS: 1 8. ANS: D PTS: 1 9. ANS: B PTS: 1 10. ANS: E PTS: 1 11. ANS: A PTS: 1 12. ANS: E PTS: 1 13. ANS: E PTS: 1 14. ANS: B PTS: 1 15. ANS: A PTS: 1 16. ANS: E PTS: 1

SHORT ANSWER

17. ANS: True

PTS: 1 18. ANS:

; ;

PTS: 1 19. ANS:

x y 4, x y 0, y 4

PTS: 1

Math 24 SG 5

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