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)
