Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1)
Formulate the following problem as a linear programming problem (DO NOT SOLVE):A
small accounting firm prepares tax returns for two types of customers: individuals and
small businesses. Data is collected during an interview. A computer system is used to
produce the tax return. It takes 2.5 hours to enter data into the computer for an individual
tax return and 3 hours to enter data for a small business tax return. There is a maximum of
40 hours per week for data entry. It takes 20 minutes for the computer to process an
individual tax return and 30 minutes to process a small business tax return. The computer
is available for a maximum of 900 minutes per week. The accounting firm makes a profit of
$125 on each individual tax return processed and a profit of $210 on each small business
tax return processed. How many of each type of tax return should the firm schedule each
week in order to maximize its profit? (Let x1 equal the number of individual tax returns
and x2 the number of small business tax returns.)
1)
2)
Formulate the following problem as a linear programming problem (DO NOT SOLVE).A
company which produces three kinds of spaghetti sauce has two plants. The East plant
produces 3,500 jars of plain sauce, 6,500 jars of sauce with mushrooms, and 3,000 jars of hot
spicy sauce per day. The West plant produces 2,500 jars of plain sauce, 2,000 jars of sauce
with mushrooms, and 1,500 jars of hot spicy sauce per day. The cost to operate the East
plant is $8,500 per day and the cost to operate the West plant is $9,500 per day. How many
days should each plant operate to minimize cost and to fill an order for at least 8,000 jars of
plain sauce, 9,000 jars of sauce with mushrooms, and 6,000 jars of hot spicy sauce? (Let x1
equal the number of days East plant should operate and x2 the number of days West plant
should operate.)
2)
3)
A vineyard produces two special wines a white, and a red. A bottle of the white wine
requires 14 pounds of grapes and 1 hour of processing time. A bottle of red wine requires
25 pounds of grapes and 2 hours of processing time. The vineyard has on hand 2,198
pounds of grapes and can allot 160 hours of processing time to the production of these
wines. A bottle of the white wine sells for $11.00, while a bottle of the red wine sells for
$20.00. How many bottles of each type should the vineyard produce in order to maximize
gross sales? (Solve using the geometric method.)
3)
Give the mathematical formulation of the linear programming problem. Graph the feasible region described by the
constraints and find the corner points. Do not attempt to solve.
4)
Suppose an horse feed to be mixed from soybean meal and oats must contain at least 200 lb
of protein and 40 lb of fat. Each sack of soybean meal costs $20 and contains 60 lb of
protein and 10 lb of fat. Each sack of oats costs $10 and contains 20 lb of protein and 5 lb of
fat. How many sacks of each should be used to satisfy the minimum requirements at
minimum cost?
4)
Provide an appropriate response.
5)
Find the coordinates of the corner points of the solution region for:
3x + 2y 54
4x + 5y 100
x 0
y 0
5)
Solve the problem.
6)
Formulate the following problem as a linear programming problem (DO NOT SOLVE):A
steel company produces two types of machine dies, part A and part B. Part A requires 6
hours of casting time and 4 hours of firing time. Part B requires 8 hours of casting time and
3 hours of firing time. The maximum number of hours per week available for casting and
firing are 85 and 70, respectively. The company makes a $2.00 profit on each part A that it
produces, and a $6.00 profit on each part B that it produces. How many of each type should
the company produce each week in order to maximize its profit? (Let x1 equal the number
of A parts and x2 equal the number of B parts produced each week.)
6)
Give the mathematical formulation of the linear programming problem. Graph the feasible region described by the
constraints and find the corner points. Do not attempt to solve.
7)
A math camp wants to hire counselors and aides to fill its staffing needs at minimum cost.
The monthly salary of a counselor is $2400 and the monthly salary of an aide is $1100. The
camp can accommodate up to 45 staff members and needs at least 30 to run properly. They
must have at least 10 aides, and at most twice as many aides as counselors. How many
counselors and how many aides should the camp hire to minimize cost?
7)
Provide an appropriate response.
8)
Solve the following linear programming problem by determining the feasible region on the
graph below and testing the corner points:
Minimize C =x1+6x2
subject to
3x1+4x2
36
2x1+x2
14
x1, x2 0
x1is shown on the x–axis and x2on the y–axis.
8)
9)
The corner points for the bounded feasible region determined by the system of inequalities:
2x1+5x2
20
x1+x2
7
x1, x2
0
are O = (0, 0), A = (0, 4), B = (5, 2) and C = (7, 0). Find the optimal solution for the objective
profit function: P(x) =3x1+7x2
9)
10)
Refer to the following system of linear inequalities associated with a linear programming
problem:
Maximize P =3x1+7x2
subject to
5x1+x2 28
2x1+x2 13
x1+x2 0
x1, x2 0
(A) Determine the number of slack variable that must be introduced to form a system of
problem constraint equations. (B) Determine the number of basic variables associated with
this system.
10)
11)
The corner points for the bounded feasible region determined by the system of inequalities:
5x1+2x2 40
x1+3x2 21
x1, x2 0
are O = (0, 0), A = (0, 7), B = (6, 5) and C = (8, 0). Find the optimal solution for the objective
profit function: P =5x1+5x2
11)
Solve the problem.
12)
Formulate the following problem as a linear programming problem (DO NOT SOLVE):A
dietitian can purchase an ounce of chicken for $0.25 and an ounce of potatoes for $0.02.
Each ounce of chicken contains 13 units of protein and 24 units of carbohydrates. Each
ounce of potatoes contains 5 units of protein and 35 units of carbohydrates. The minimum
daily requirements for the patients under the dietitian’s care are 45 units of protein and 58
units of carbohydrates. How many ounces of each type of food should the dietitian
purchase for each patient so as to minimize costs and at the same time insure the minimum
daily requirements of protein and carbohydrates? (Let x1 equal the number of ounces of
chicken and x2 the number of ounces of potatoes purchased per patient.)
12)
Provide an appropriate response.
13)
Using a graphing calculator as needed, maximize P =524x1+479x2 subject to
265x1+320x2 3,390
350x1+345x2 3,795
400x1+316x2 4,140
x1, x2 0
Give the answer to two decimal places.
13)
Solve the problem.
14)
The Southern States Ring Company designs and sells two types of rings: the brass and the
aluminum. They can produce up to 24 rings each day using up to 60 total man–hours of
labor per day. It takes 3 man–hours to make one brass ring and 2 man–hours to make one
aluminum ring. How many of each type of ring should be made daily to maximize the
company’s profit, if the profit on a brass ring is $40 and on an aluminum ring is $35?
14)
Provide an appropriate response.
15)
Using a graphing calculator as needed, maximize P =310x1+470x2 subject to
250x1+450x2 4190
301x1+390x2 3700
382x1+289x2 4404
x1, x2 0
Give the answer to two decimal places.
15)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or
unbounded.
16)
16)
A)
Bounded
B)
Unbounded
7
C)
Bounded
D)
Unbounded
Use graphical methods to solve the linear programming problem.
17)
17)
A)
12 counselors and 18 aides
B)
35 counselors and 10 aides
C)
27 counselors and 18 aides
D)
18 counselors and 12 aides
Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or
unbounded.
18)
18)
8
A)
Unbounded
B)
Bounded
C)
Unbounded
D)
Unbounded
Solve the problem.
19)
19)
A)
At least 72.3
B)
At least 62
C)
At least 73.5
D)
At least 63
Use graphical methods to solve the linear programming problem.
20)
20)
A)
Maximum of 120 when x = 3 and y = 8
B)
Maximum of 92 when x = 4 and y = 5
C)
Maximum of 100 when x = 8 and y = 3
D)
Maximum of 96 when x = 9 and y = 2
Solve the problem.
21)
21)
A)
6100 copies
B)
12,000 copies
C)
6000 copies
D)
3000 copies
Graph the inequality.
22)
22)
10
A)
B)
C)
D)
23)
23)
11
A)
B)
C)
D)
Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or
unbounded.
24)
24)
12
A)
Bounded
B)
Unbounded
C)
Bounded
D)
Unbounded