This document contains instructions for a mathematics exam, including: - The exam consists of multiple choice, true/false, and short answer questions worth a total of 100 points. - No books, notes, or calculators with CAS or QWERTY keyboards are allowed. Cell phones may not be used. - The multiple choice section includes 8 questions worth 5 points each. - The true/false section includes 15 statements worth 15 points total. - Three short answer questions are each worth 15 points.

1. University of Kentucky
Department of Mathematics
MA 162 Exam 2 Fall 2016
Name: Section:
Do not remove this answer page — you will return the whole exam. You will be allowed two hours to
complete this test. No books or notes may be used. You may use a graphing calculator during the exam, but
NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely
no cell phone use during the exam is allowed.
The exam consists of 8 multiple choice questions worth 5 points each, a true/false section worth 15 points,
and 3 short answer questions worth 15 points each. Record your answers to the multiple choice questions
below on this page by filling in the single circle corresponding to the correct answer. All other work must
be done in the body of the exam.
Multiple Choice Responses
Please indicate your answers for the multiple choice questions here by shading in your selections.
1 A
1 B
1 C
1 D
1 E
2 A
2 B
2 C
2 D
2 E
3 A
3 B
3 C
3 D
3 E
4 A
4 B
4 C
4 D
4 E
5 A
5 B
5 C
5 D
5 E
6 A
6 B
6 C
6 D
6 E
7 A
7 B
7 C
7 D
7 E
8 A
8 B
8 C
8 D
8 E
The following table is for administrative purposes only.
MC TF 10 11 12 Total
40 15 15 15 15 100

2. MA 162 Fall 2016 Exam 2
Multiple Choice Questions
Indicate your answer choices by shading in your answers on the cover page.
x
y
x + y = 4
x + 2y = 6
II
IV
III
I
1. (5 points) Consider the feasible region given by the following inequalities, whose boundary lines are
graphed above.
x + 2y ≥ 6, x + y ≥ 4, x ≥ 0, y ≥ 0
Which one of the following labels best indicate the feasible region described above?
A. I
B. III
C. IV
D. II
E. None of these labels adequately or completely indicate the feasible region.
2. (5 points) At which point is the cost function C = x+3y minimized with respect to the feasible region
described in problem 1 above?
A. (6, 0)
B. (0, 4)
C. (2, 2)
D. (4, 0)
E. None of these.
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3. MA 162 Fall 2016 Exam 2
3. (5 points) A linear programming problem has objective function P = 3x + 2y and the following linear
inequality constraints.
x − y ≤ 0, x + y ≤ 3, x ≥ 0, y ≥ 0
How many slack variables are needed for the simplex algorithm?
A. 4
B. 2
C. 1
D. 3
E. None of these.
4. (5 points) When applying the simplex method to the simplex tableau
x y z u v w P RHS
2 −6 2 1 0 0 0 6
6 2 3 0 1 0 0 12
4 4 1 0 0 1 0 20
−8 −12 4 0 0 0 1 0
the column given by variable y should be selected next for pivot operations. Which row should then be
selected?
A. R1
B. R2
C. R3
D. R4
E. None of these.
5. (5 points) The objective function for a linear programming problem is P = 3x + 2y − z. What is the
correct way to arrange this equation in order to enter it into a Simplex Tableau?
A. P = 3x + 2y − z
B. −3x − 2y + P = z
C. −3x − 2y + z + P = 0
D. 3x + 2y − z − P = 0
E. None of these.
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4. MA 162 Fall 2016 Exam 2
6. (5 points) Which of the following matrices has an inverse?
A.
3 4
6 8
B.
0 −4
0 10
C.
4 −10
−2 5
D.
0 0
5 7
E. None of these have an inverse.
7. (5 points) A linear programming problem has been solved by the simplex method and the final simplex
tableau is
x y z u v w P RHS
0 1 0 −2 1 0 0 1
1 0 0 .625 0 −.125 0 2
0 1 1 −.25 0 .25 0 4
0 0 0 .125 0 .375 1 10
Which of the following is an optimal point indicated by the table?
A. (x, y, z) = (2, 1, 4)
B. (x, y, z) = (1, 2, 4)
C. (x, y, z) = (0, 0, 10)
D. (x, y, z) = (2, 0, 4)
E. None of these.
8. (5 points) One constraint for a standard maximization problem is y ≥ .6(x+y). What is the correct way
to arrange this as an equation (including slack variable u) in order to enter it into a simplex tableau?
A. .6x − .4y + u = 0
B. −.6x − 1.6y + u = 0
C. −.6x + .4y + u = 0
D. .6x + 1.6y + u = 0
E. None of these.
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5. MA 162 Fall 2016 Exam 2
True or False Questions
Indicate your responses by circling true or false below.
9. (15 points) .
TRUE False The matrices
6 −7
−4 5
and
2.5 3.5
2 3
are inverses.
True FALSE The point (2, 0) is an appropriate test point for graphing a feasible region given
the constraint y ≥ 2x − 4.
TRUE False If x is the amount of money a company invests in project A and y the amount
invested in project B then the constraint “the amount invested in B should
be no more than 40% of the overall investment” would indicate the inequality
y ≤ .4 (x + y) .
True FALSE The simplex method will always produce an optimal solution at the same unique
point of the feasible region.
True FALSE The point (1, 3) is in the feasible region allowed by the constraints







7x − 3y ≤ 0
5x + 6y ≤ 22
x ≥ 0
y ≥ 0
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6. MA 162 Fall 2016 Exam 2
Short Answer Questions
Show all work to receive credit for the following problems.
If you provide work on the scrap paper, indicate that within the body of the problem.
10. (15 points) SETUP BUT DO NOT SOLVE the following linear programming problem. A complete
answer to this problem must include
• clearly defined variables,
• a clearly labeled objective function, and
• all necessary constraints.
You should NOT setup a simplex table.
As part of a campaign to promote its annual clearance sale, Excelsior Company decided to buy television
advertising time on Station KAOS. Excelsior’s television advertising budget is $100, 000. Morning time
costs $3000 per minute, afternoon time costs $1000 per minute, and evening (prime) time costs $12000
per minute. Because of previous commitments, KAOS cannot offer Excelsior more than 6 minutes of
prime time or more than a total of 25 minutes of advertising time over the two weeks in which the
commercials are to be run. KAOS estimates that morning commercials are seen by 200, 000 people,
afternoon commercials are seen by 100, 000 people and evening commercials are seen by 600, 000 people.
How much morning, afternoon, and evening advertising time should Excelsior buy to maximize exposure
of its commercials?
Solution: Let x be the number of minutes purchased for morning time. Similarly, we select y for
afternoon and z evening (prime) time.
We have the following constraints
3000x + 1000y + 12000z ≤ 100000
z ≤ 6
x + y + z ≤ 25
as well as the non-negativity constraints x ≥ 0, y ≥ 0 and z ≥ 0.
Excelsior would like to maximize the following objective function
E = 200000x + 100000y + 600000z
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7. MA 162 Fall 2016 Exam 2
11. (15 points) Note: A complete answer to this problem must include clearly defined variables, objective
function, and constraints; a clearly drawn and labeled feasible region on the provided axes (please also
label the axes); and the final answer in a complete sentence using the context of the problem.
World renowned ice cream entrepreneurs Sydney and Eden produce two types of premium dairy ice
cream products: Syd n’ Edy’s Chocolate Concussion and Vanilla Brain Freeze. Their chocolate ice
cream requires 6 oz milk and 8 oz of peanuts per pint size container while the vanilla option requires 9
oz milk and 5 oz peanuts. Sydney and Eden currently enjoy a surplus of all other ingredients required
for their ice cream but only have 360 oz of milk and 400 oz of peanuts for this limited production run.
Given that the entrepreneurs charge $5 for each container of Chocolate Concussion and $7 for each
Vanilla Brain Freeze, how many of each type should Syd and Edy produce in order to maximize their
profit and what is the maximum? Feel free to approximate to the nearest tenth of a pint as necessary.
Solution: Let x be the number of pints of chocolate and y the number of pints of vanilla.
Due to the restrictions on the amounts of milk and peanuts we have the following constraints.
Respectively, we get
6x + 9y ≤ 360
8x + 5y ≤ 400
as well as the non-negativity constraints x ≥ 0 and y ≥ 0.
Solving for the points of intersection of certain boundary lines we find the corner points of the
feasible region as follows
(0, 0), (0, 40), (50, 0), and (42.9, 11.4)
Plugging these into the objective function P = 5x + 7y we get
x y P = 5x + 7y
0 0 0
0 40 280
50 0 250
42.85 11.4 294.3
Therefore the profit P is maximized at the optimal point (42.9, 11.4). Eden and Sydney should
make approximately 42.9 pints of chocolate and 11.4 pints of vanilla to achieve a maximum profit
of about $294.3.
x
y
(0,0)
(0,40)
(0,80)
(50,0)
(50,0)
(60,0)
(42.9,11.4)
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8. MA 162 Fall 2016 Exam 2
12. (15 points) Investment banker Donny has decided to make some sensible long term investments in mutual
funds and low-risk stocks. Let x be the amount he will invest in mutual funds and y be the amount he’ll
invest in low-risk stocks. The objective is to maximize P, Donny’s profit.
Donny has started the Simplex Algorithm, but cannot finish it. Finish the algorithm for him.
Explain why you choose any rows and columns that you selected when determining pivots. Show all
necessary work including the row operations you perform. Be sure to interpret your final table by
answering the questions below.
x y u v w P RHS
−1 1 −0.25 −3 0 0 3000
4 0 1 16 0 0 32000
2 0 −0.25 3 1 0 20000
−5 0 0 −4 0 1 0
Solution: The first column should be selected as a pivot since it has the largest negative in the bot-
tom row. Then we consider the ratios of the RHS/(column 1 entries). In row 1 we have a negative
as the column 1 entry. Skipping that we consider rows 2 and 3 and get the ratios 32000/4 = 8000
and 20000/2 = 10000 respectively. Hence the pivot row is row 2 with the smaller ratio.
Now proceeding with pivot operations we have
R2/4
−
−
−
→
x y u v w P RHS
−1 1 −0.25 −3 0 0 3000
1 0 0.25 4 0 0 8000
2 0 −0.25 3 1 0 20000
−5 0 0 −4 0 1 0
R1+R2
−
−
−
−
−
→
R3−2R2
−
−
−
−
−
→
R4+5R2
−
−
−
−
−
→
x y u v w P RHS
0 1 0 1 0 0 11000
1 0 0.25 4 0 0 8000
0 0 −0.75 −5 1 0 4000
0 0 1.25 16 0 1 40000
(a) How much should Donny invest in mutual funds and how much in low-risk stocks? Answers must
correspond to the work shown above.
Solution: Donny should invest x = 8000 in mutual funds and y = 11000 in low-risk stocks.
(b) What is Donny’s maximum profit?
Solution: He’ll get a maximum profit of $40000.
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9. MA 162 Fall 2016 Exam 2
Scrap paper
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10. MA 162 Fall 2016 Exam 2
Scrap paper
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