Math 107 Exam 1 Review—Chapters 1, 2, 3 & 4 Name___________________

1. Assume that each situation can be expressed as a linear cost function. Find the cost function in each case.

a) Fixed cost: $40; 8 items cost $1600 to produce.

b) Marginal cost: $75; 700 items cost $96,000 to produce.

2. Mrs. Alford invested $5000 in securities. Part of the money was invested at 8% and part at 9%. The total annual income from interest earned was $415. How much was invested at each rate? a) Set up the system of equations needed to solve the above problem. Don’t forget to define the variables. b) Solve the system of equations from part a) and write out your answer in a complete sentence. 3. Austin Ave Clothiers pays $57 each for sports coats and has a fixed monthly cost of $770. The store sells the coats for $79 each.

a) Find the cost function b) Find the revenue function c) Find the break even number of sports coats.

4. To manufacture x thousand computer chips requires fixed expenditures of $316 plus $42

per thousand chips. Receipts from the sale of x thousand chips amount to $130 per thousand.

(a) Write a function P(x) for the total profit from the production and sale of x thousand chips.

(b) How many thousand chips must be sold to make a profit of $13,500.

5. Write the linear system represented by the augmented matrix .

−

26017

4310

6301

6. Write

=+

=−

=+

53

0

42

yx

zy

yx

as an augmented matrix.

7. Solve the following system of equations using Gauss-Jordan elimination. List your row

operations. x + y + 2z = 12

2x + 3y − z = − 2

−3x + 4y + z = −8 8. Solve the following system of equations using Gauss-Jordan elimination. List your row

operations and write out the complete solution.

x1 + 2×2 − x3 = −13

2×1 + 5×2 + 3×3 = −3

9. Formulate each of the following as a linear programming problem. Clearly show all three steps. (i) Identify your variables. (ii) Write the objective function. (iii) List the constraints. Do not solve.

(a) A woman wants to design a weekly exercise schedule that involves jogging, handball and aerobic dance. She wants to jog at least 3 hours a week, play handball at least 2 hours per week and dance no more than 5 hours a week. She also wants to devote at least as much time to jogging as to handball. Suppose that jogging, handball and dance consume, respectively, 900, 600 and 800 calories per hour. If she must burn at least 9000 calories per week, how many hours should she devote to each exercise to minimized her exercise time?

(b) The maximum daily production of an oil refinery is 1400 barrels. The refinery can

produce two types of fuel: gasoline and heating oil. The production cost per barrel is $6 for gasoline and $8 for heating oil. The daily production budget is $9600. The profit is $3.50 per barrel on gasoline an $4 per barrel on heating oil. What is the maximum total profit that can be realized daily ?

10. Solve the following linear programming problems graphically. Show your feasible region and label all corner points.

(a) Minimize z = 7x + 5y subject to 9x + 5y > 45 3x + 10y < 60

x > 0, y > 0

(b) Find the maximum of z = 4x + 5y subject to 2x + 3y > 12

3x + y < 11 x > 0, y > 0

11. For the following linear programming problem set up the simplex tableau (include labels). Circle the 1st pivot. Do not solve.

Maximize z = 4×1 + 10×2 + 8×3 , subject to

3×1 + 5×2 + 4×3 < 35 6×1 + x2 + 2×3 < 44 2×1 + 4×2 + x3 + < 50 x1 > 0, x2 > 0, x3 > 0

For problems 12 and 13 give the current solution of the tableaux and circle the next pivot if one is required.

12.

−

−

−

−

2401600016

2000011

60011003

16000104

2

1

2

1

13.

−−

−

981070504

160100301

40010112

480021603

14. For the following tableau perform the next pivot using the simplex method

−− 0100035

8010042

6001021

10000132

15. Use the simplex method to maximize z = 2×1 + 3×2 + 2×3, subject to 2×1 + x2 + 2×3 < 13

x1 + x2 − 3×3 < 8 x1 > 0, x2 > 0, x3 > 0

S09Math 107 Answers for Review –Chapters 1, 2, 3 & 4 1. Let x = number of items and y = total cost a) y = 195x + 40; b) y = 75x + 43,500 2. a) Let x = amount of money invested at 8% and y = amount of money invested at 9%; x + y = 5000, 08x +.09y = 415 b) $3500 was invested at 8% and $1500 at 9%. 3. a) C(x) = 57x +770; b) R(x) = 79x; c) It would take 35 sport coats to break even.

4. (a) P(x) = 88x − 316; (b) 157 thousand chips must be sold to make a profit of $13,500. 5. x1 −3×3 = 6 x2 + 3×3 = 4

17×1 + 6×3 = 2

6.

−

5

0

4

031

110

021

7.

−−

→+

→+−

−

−

−

−

28

26

12

770

510

211

R3R33R1

R2R21R2

8

2

12

143

132

211

R3R3 42

1210

26

38

4200

510

701

R3R3R27

R1R1R2

→

−−

→+−

→+−

5

1

3

5

1

3

100

010

001

R2R23R5

R1R1R37

5

26

38

100

510

701

=

−=

=

−→+

→+−

−−

z

y

x

8.

−− →+−−

−

23

13

5

1

1

2

0

1

R2R2R123

13

3

1

52

21

235

5911

23

59

5

11

1

0

0

1R1R1R22

32

31

=+

−=−−−→+−

xx

xx

(−59 + 11×3, 23 − 5×3, x3 ) 9. (a) (i) Let x1 = number of hours of jogging x2 = number of hours of handball x3 = number of hours of dance z = total exercise time (ii) Minimize z = x1+ x2 + x3 (iii) Subject to: 900 x1 + 600 x2 + 800 x3 > 9000 x1 > 3, x2 > 2, 0 < x3 < 5

x1 − x2 > 0 9. (b) (i) x1 = number of barrels of gasoline x2 = number of barrels of heating oil P = total profit (ii) Maximize P = 3.5×1 + 4×2 (iii) Subject to: x1 + x2 < 1400 6×1 + 8 x2 < 9600 x1 > 0, x2 > 0

10. (a) (0,9) z

(5, 0) 35

(2, 5.4) 41

(20, 0) 140

(0,6) (2, 5.4) Minimum of 35 @ (5, 0) (5,0) (20,0)

(b) (0, 11) z

(0, 4) 20

(0, 11) 55

(3, 2) 22

(0,4) (3,2) Maximum of 55 at (0,11) (11/3, 0) (6,0)

11.

−−− 010008104

500100142

440010216

350001453

12. x1 = 0, x2 = 20, s1 = 160, s2 = 60, s3 = 0, z = 240 pivot is the 3 in Row 2 column 1 13. x1 = 0, x2 = 4, x3 = 0, s1 = 48, s2 = 0, s3 = 16, z = 98 pivot is the 3 in Row 3 column 3 14. pivot on the 2 in Row 3 column 1 giving

−

−−

2010070

400021

201000

2010110

2

5

2

1

2

1

15.

3323

112

0000232

8010311

13001212

RRR

RRR

→+

→+−

−−−

−

335111

22513

201301101

8010311

5011501

RRR

RRR

→+

→+

−

−

−

−

17554110016

55023058

5011501

x1 = 0, x2 = 11, x3 = 1, s1 = 0, s2 = 0, z = 35 z is a maximum of 35 at (0, 11, 1)

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