Finite Math Exam 2 Review
Graph the feasible region for the system of inequalities.
1) 3x + y -2
x-y 3
x + 2y < -12
2) 2y + x -2
y + 3x 9
y 0
x 0
3) 2x + 3y 6
x-y 3
x 1
4) 2x + 3y 6
x-y 3
x 1
Use graphical methods to solve the linear programming problem.
z = 6x + 7y
5) Maximize
subject to:
2x + 3y 12
2x + y 8
x 0
y 0
6) Maximize
subject to:
z = 2x + 5y
3x + 2y 6
-2x + 4y 8
x 0
y 0
7) Minimize
subject to:
z = 6x + 8y
2x + 4y 12
2x + y 8
x 0
y 0
8) Minimize
subject to:
z = 2x - y
x-y 8
x + y 20
-3x + y 12
x 0
y 0
1
Solve the problem.
9) The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up
to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2
man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the
company's profit, if the profit on a VIP ring is $40 and on an SST ring is $30?
10) Zach is planning to invest up to $45,000 in corporate and municipal bonds. The least he will invest in corporate
bonds is $9000 and he does not want to invest more than $28,000 in corporate bonds. He also does not want to
invest more than $26,084 in municipal bonds. The interest is 8.4% on corporate bonds and 6.2% on municipal
bonds. This is simple interest for one year. What is the maximum value of his investment after one year?
11) A company makes two kinds of engineering pencils, Type I and Type II (deluxe). Type I needs 2 min of sanding
and 6 min of polishing. Type II needs 5 min of sanding and 3 min of polishing. The sander can run no more than
66 hours per week and the polisher can run no more than 73 hours a week. A $3 profit is made on Type I and $5
profit on Type II. How many of each type should be made to maximize profits?
12) An airline with two types of airplanes, P1 and P2 , has contracted with a tour group to provide transportation
for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip,
airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class
passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class, and 44
economy class passengers. How many of each type of airplane should be used in order to minimize the
operating cost?
13) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average
monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100. The camp can
accommodate up to 35 staff members and needs at least 20 to run properly. They must have at least 10 aides,
and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the
camp hire to minimize cost?
Use slack variables to convert the constraints into linear equations.
14) Maximize z = 3x1 + 5x2
subject to:
with:
15) Maximize
subject to:
with:
5x1 + 4x2
x1 + 2x2
x1
0, x2
30
40
0
z = 4x1 + 12x2 + 7x3
6x1 + 2x2 + x3 12
x1 + 9x2 + x3
x1
0, x2
24
0, x3
0
Introduce slack variables as necessary, and write the initial simplex tableau for the problem.
16) Find x1 0 and x2 0 such that
5x1 + 10x2
10x1 + 15x2
122
139
and z = 2x1 + 5x2 is maximized.
2
Write the solutions that can be read from the simplex tableau.
17) x1 x2 x3 s1 s2 z
1
0
0
18) x1
0
0
1
0
4
1
0
1
3
0
0
1
0
5
2
4
0 6
0 9
1 3
x2 s1 s2 s3 z
3 0 1 1 0 13
4 1 0 1 0 18
5 0 0 1 0 17
-3 0 0 1 1 7
Pivot once about the circled element in the simplex tableau, and read the solution from the result.
19)
20)
21)
The initial tableau of a linear programming problem is given. Use the simplex method to solve the problem.
22) x1 x2 x3 s1 s2 z
3 3 2 1 0 0 12
2 1 2 0 1 0 14
-1 -1 -3 0 0 1 0
23)
x1 x2 x3 s1
2 2 1 1
1 1 4 0
4 2 6 0
-5 -6 -3 0
s2
0
1
0
0
s3
0
0
1
0
z
0 8
0 5
0 12
1 0
3
Use the simplex method to solve the linear programming problem.
24) Maximize z = 7x1 + 2x2 + x3
subject to:
x1 + 5x2 + 7x3
8
x1 + 4x2 + 11x3 9
x1 0, x2 0, x3 0
with
25) Maximize z = x1 + 2x2 + 4x3 + 6x4
subject to:
x1 + 2x2 + 3x3 + x4
100
3x1 + x2 + 2x3 + x4 75
x1 0, x2 0, x3 0, x4 0
with
Solve the problem.
26) Jill Sharp, a fitness trainer, has an exercise regimen that includes running, swimming, and walking. She has no
more than 12 hours per week to devote to exercise, including at most 4 hours running. She wants to walk at
least three times as many hours as she swims. Jill will burn on average 528 calories per hour running, 492
calories per hour swimming, and 348 calories per hour walking. How many hours per week should Jill spend
on each exercise to maximize the number of calories she burns? What is the maximum number of calories she
will burn? (Hint: Write the constraint involving walking and swimming in the
form 0.)
27) An agricultural research scientist is developing three new crop growth supplements -- A, B, and C. Each pound
of each supplement contains four enzymes -- E1 , E2 , E3 , and E4 -- in the amounts (in milligrams) shown in the
table.
A
B
C
E1 E2 E 3 E 4
3
1
2
2
6
3
1
3
1
1
1
5
The cost of E1 is $20/mg, the cost of E2 is $40/mg, the cost of E3 is $10/mg, and the cost of E4 is also $10/mg. The
growth benefit for crops is expected to be proportional to 10 times the amount of A used, 25 times the amount of
B used, and 60 times the amount of C used. However, the total cost of the enzymes used in A, B, and C must be
less than $5000 for each treatment. How many pounds each of A, B, and C should be produced to maximize the
growth effect?
State the dual problem. Use y1 , y2 , y3 and y4 as the variables. Given: y1
28) Maximize z = 3x1 + 2x2
subject to:
x1 + x 2
2x1 + x2
x1
0, x2
23
12
0
29) Maximize z = x1 + 2x2 + 3x3
subject to:
6x1 + 3x2 + x3 30
4x1 + 7x2 + 3x3 45
x1 0, x2 0, x3 0
4
0, y2
0, y3
0, and y4
0.
Use the simplex method to solve the linear programming problem.
30) Minimize w = 4y1 + 2y2
subject to:
3y1 + y2
y1 + 4y2
y1
31) Minimize
subject to:
0, y2
22
26
0
w = y1 + 3y2 + 2y3
y1 + y2 + y3 50
2y1 + y2 25
y1 0, y2 0, y3
0
Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4
units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C.
32) Pill #1 costs 8 cents, pill #2 costs 9 cents, and pill #3 costs 1 cent. How many of each pill must Larry take to
minimize his cost?
A toy making company has at least 300 squares of felt, 700 oz of stuffing, and 230 ft of trim to make dogs and dinosaurs. A
dog uses 1 square of felt, 4 oz of stuffing, and 1 ft of trim. A dinosaur uses 2 squares of felt, 3 oz of stuffing, and 1 ft of
trim.
33) It costs the company $1.44 to make each dog and $1.67 for each dinosaur. The company wants to minimize its
costs. What are the coefficients of the dual objective function?
5
Answer Key
Testname: EXAM 2 REVIEW
1)
2)
3)
6
Answer Key
Testname: EXAM 2 REVIEW
4)
5) Maximum of 32 when x = 3 and y = 2
49
1
9
when x = and y =
6) Maximum of
4
2
4
7) Minimum of
8)
9)
10)
11)
12)
92
10
4
when x =
and y =
3
3
3
Minimum of 18 when x = 8 and y = 0
12 VIP and 12 SST
$48,406
417 Type I, 625 Type II
9 P1 planes and 13 P2 planes
13) 8 counselors and 12 aides
14) 5x1 + 4x2 + s1 = 30
x1 + 2x2 + s2 = 40
15) 6x1 + 2x2 + x3 + s1 = 12
x1 + 9x2 + x3 + s2 = 24
16)
x1 x2 s1 s2 z
5 10 1 0 0 122
10 15 0 1 0 139
-2 -5 0 0 1
0
17) x1 = 6, s1 = 9, z = 3; x2 , x3 , s2 = 0
18) s2 = 13, s1 = 18, x1 = 17, z = 7; x2 , s3 = 0
19) x2 = 3, x1 = -2, z = 6; x3 , s1 , s2 = 0
20) x3 = 14, s2 = -6, s3 = 4, z = 42; x1 , x2 , s1 = 0
21) x2 = 2, s2 = 4, s3 = 2, z = 12; x1 , x3 , s1 = 0
22) Maximum at 18 for x 3 = 6, s2 = 2
23) Maximum at 24 for x 2 = 4, s2 = 1, s3 = 4
24) Maximum is 56 when x1 = 8, x 2 = 0, x3 = 0
25) Maximum is 450 when x1 = 0, x2 = 0, x3 = 0, x4 = 75
26) 4 hr running, 2 hr swimming, and 6 hr walking; 5184 calories burned
27) 0 lb of A, 9.3 lb of B, and 10 lb of C
28) Minimize w = 23y1 + 12y2
subject to:
y1 + 2y2 3
y1 + y2 2
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Answer Key
Testname: EXAM 2 REVIEW
29) Minimize
subject to:
w = 30y1 + 45y2
6y1 + 4y2
3y1 + 7y2
1
2
y1 + 3y2 3
30) 32.7 when y1 = 5.6 and y2 = 5.1
31) 50 when y1 = 50, y2 = 0, and y3 = 0
32) P1 = 0, P2 = 0, P3 = 12
33) 300, 700, 230
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