Fall 2015
Math 141:505
Final Exam
Form A
Last Name:
Exam Seat #:
First Name:
UIN:
On my honor, as an Aggie, I have neither given nor received unauthorized aid on
this academic work.
Signature:
INSTRUCTIONS
• Part 1: Multiple Choice and True-False (Problems 1-11) Each multiple choice problem is worth 3 points for
a total of 30 points. Each true-false problem is worth 2 points for a total of 10 points. Answers should be written
in the boxes provided. No partial credit will be given.
• Part 2: Work Out (Problems 12-27) The number of points for each problem is indicated next to the problem
for a total of 60 points. Partial credit will be given. All steps must be written clearly and neatly to receive credit.
If you use your calculator for anything beyond an arithmetic calculation, please indicate how at the appropriate
step. Box your final answer.
“An Aggie does not lie, cheat or steal or tolerate those who do.”
c
Yeong-Chyuan
Chung, December 14, 2015
Part 1: Multiple Choice and True-False. Each multiple choice problem is worth 3 points for a total of
30 points. Each true-false problem is worth 2 points for a total of 10 points. No partial credit will be given.
Answers should be written in the boxes provided.
1. Given the equation 2x + y = 5, what is the change in y if x decreases by 2 units?
(a) y increases by 4 units
(b) y decreases by 4 units
(c) y increases by 9 units
(d) y decreases by 9 units
a
2. Suppose that A is a 2 × 3 matrix, B is a 3 × 3 matrix, and C is a 2 × 2 matrix. Which of the following operations
can be performed?
(a) A + B
(b) ACT
(c) AB
(d) B −C
c
3. Which is the correct description of the shaded region in the following Venn diagram?
(a) (A ∩ B ∩C) ∪ (Ac ∩ B ∩C)
(b) (A ∩ Bc ∩C) ∪ (A ∩ B ∩C)
(c) (A ∩ Bc ∩Cc ) ∪ (Ac ∩ B ∩Cc )
(d) (A ∩ Bc ∩C) ∪ (Ac ∩ B ∩C)
d
2
c
Yeong-Chyuan
Chung, December 14, 2015
4. You need to make up a five-digit code, to be chosen from the numbers 0 through 9, and you have decided that
the code should not have 0 as the first digit, should have an even number as the third digit, and should have an
odd number as the last digit. How many such codes are possible? (Repetition of digits is allowed.)
(a) 18,000
(b) 20,000
(c) 22,500
(d) 25,000
c
5. How many distinguishable permutations can be formed from the 11 letters of the word EXAMINATION?
(a) 4,989,600
(b) 9,979,200
(c) 19,958,400
(d) 39,916,800
a
6. An optician estimates that the probability that a customer coming into his store will purchase a pair of glasses
but not contact lenses is .4, and the probability that a customer will purchase a pair of contact lenses is .3. What
is the probability that a customer will purchase neither a pair of glasses nor a pair of contact lenses?
(a) .25
(b) .3
(c) .35
(d) .4
b
7. Two cards are selected at random without replacement from a well-shuffled deck of 52 playing cards. Find the
probability that the two cards are of the same suit. (Recall that there are 4 suits, each having 13 cards.)
(a)
(b)
(c)
(d)
3
26
3
104
1
17
4
17
d
3
c
Yeong-Chyuan
Chung, December 14, 2015
8. A pair of fair dice is rolled. What is the probability that the number landing uppermost on the first die is 2 if it is
known that the sum of the numbers landing uppermost is 4?
(a)
(b)
(c)
(d)
1
3
1
6
1
9
1
12
a
9. What are the odds against an event E occurring if P(E) = 13 ?
(a)
(b)
(c)
(d)
2 to 1
1 to 2
2 to 3
3 to 2
a
10. Let X be the number of successes in ten independent trials in a binomial experiment in which the probability of
success is .4. Which of the following gives the mean, variance, and standard deviation of X in the correct order?
(The values below are rounded to 2 decimal places if necessary.)
Mean Variance Standard Deviation
(a)
(b)
(c)
(d)
4
4
2.40
2.40
2.40
2.40
4
4
2
1.55
2
1.55
b
11. (2pts each) State whether each of the following is true or false.
(i) 0/ ⊆ A for every set A.
T
(ii) If A = {a, b, c}, then {a} ∈ A.
F
(iii) n(A ∪ B) = n(A) + n(B) for every pair of sets A and B.
F

1 2 0 0
(iv) The augmented matrix  0 0 1 0  is in row-reduced form.
0 0 0 1
(v) The number of times a person takes a driving test before passing is a continuous random variable.

T
F
4
c
Yeong-Chyuan
Chung, December 14, 2015
Part 2: Work Out. The number of points for each problem is indicated next to the problem. The total number of
points available is 60. Partial credit will be given. All steps must be written clearly and neatly to receive credit.
If you use your calculator for anything beyond an arithmetic calculation, please indicate how at the appropriate
step. Box your final answer.
12. A manufacturer of 24-hr variable timers has a monthly fixed cost of $48,000 and a production cost of $8 for each
timer manufactured. The units sell for $14 each.
(a) (4pts) Write down the cost function C(x), the revenue function R(x), and the profit function P(x) (assuming
they are linear).
Solution:
C(x) = 8x + 48000
R(x) = 14x
P(x) = R(x) −C(x) = 14x − (8x + 48000) = 6x − 48000
(b) (2pts) Find the break-even point.
Solution:
Set P(x) = 0 (or set C(x) = R(x)) and solve for x.
We get 6x − 48000 = 0 so x = 8000. Plug this value into either the cost function or the revenue function. If
we use the revenue function, then we get R(8000) = 14(8000) = 112, 000. Hence the break-even point is
(8000 timers, $112, 000).
13. (3pts) A deli selling sandwiches found that the demand for their sandwiches is related to the price of their
sandwiches by the equation p = −0.0025x + 9.6 where x is the number of sandwiches sold and p is the price of
each sandwich in dollars. Find the revenue function R(x), and determine the number of sandwiches that should
be sold to maximize the revenue.
Solution:
R(x) = (−0.0025x + 9.6)x = −0.0025x2 + 9.6x.
We need to find the vertex (because that is where the revenue is maximized) so we use the formula x =
−9.6
2(−0.0025) = 1920. This means that 1920 sandwiches should be sold to maximize the revenue.
−b
2a
=
14. (3pts) Solve the following system of linear equations:
2x + 3y = 2
x + 3y = −2
x−y = 3
Solution:




2 3
2
1 0 0
f
 1 3 −2  rre
→ 0 1 0 
1 −1 3
0 0 1
Converting the row-reduced form back into equations, we get x = 0, y = 0, 0 = 1. The last equation is always
false, so the system has no solutions.
5
c
Yeong-Chyuan
Chung, December 14, 2015
1 0
1 x
15. (3pts) Let A =
and B =
where x is a constant. Compute (2A + B)2 .
0 2
0 2
Solution:
(2A + B)2
2 2 2 0
1 x
3 x
3 x 3 x
3(3) + x(0) 3(x) + x(6)
9 9x
=
+
=
=
=
=
0 4
0 2
0 6
0 6 0 6
0(3) + 6(0) 0(x) + 6(6)
0 36
16. (3pts) A pack of candies contains 10 red candies and 10 yellow candies. A sample of 6 candies is selected. How
many ways are there to select such a sample if we want at least one red candy and at least one yellow candy?
Solution:
The complement of the set of desired samples consists of samples with no red candies or no yellow candies. We
will subtract the number of such samples from the total number of samples to get the answer.
C(20, 6) − (C(10, 6) +C(10, 6)) = 38760 − (210 + 210) = 38340
17. (3pts) Consider the following word problem.
“A furniture factory manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3
labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $45, and
the profit for each chair is $20. In a certain week, the factory has 3200 board feet of wood available and 520
labor-hours. How many tables and chairs should they manufacture to maximize their profits?”
Clearly define the appropriate variables, and set up (but do not solve) a linear programming problem that would
be used to answer the question.
Solution:
Let x be the number of tables manufactured. Let y be the number of chairs manufactured.
Maximize P = 45x + 20y
subject to 40x + 16y ≤ 3200
3x + 4y ≤ 520
x ≥ 0, y ≥ 0
6
c
Yeong-Chyuan
Chung, December 14, 2015
18. (6pts) Solve the following linear programming problem using the method of corners:
Maximize P = 2x + 5y
subject to y ≤ −x + 3
y ≤ −2x + 4
x ≥ 0, y ≥ 0
Solution:
(Diagram omitted. The feasible set has four corners.)
Corner
(0,0)
(0,3)
(1,2)
(2,0)
P=2x+5y
0
15
12
4
A maximum value of 15 occurs at the point (0, 3).
19. (3pts) An experiment consists of rolling a standard six-sided die and observing whether the number that falls
uppermost is a large number (at least 4) or a small number (less than 4), and then selecting a card from a
standard deck of 52 playing cards and observing its color (red or black). Write down the sample space, S, for
this experiment.
Solution:
Let l represent large, s represent small, r represent red, and b represent black.
S = {(l, r), (l, b), (s, r), (s, b)}
20. (3pts) Let E and F be two events such that P(E) = .3, P(F) = .4, and P(E ∪F) = .6. Determine, with appropriate
justification, whether the events E and F are mutually exclusive.
Solution:
We need to determine whether E ∩ F = 0.
/ In terms of probability, this is equivalent to determining whether
P(E ∩ F) = 0.
We have the formula P(E ∪F) = P(E)+P(F)−P(E ∩F) so P(E ∩F) = P(E)+P(F)−P(E ∪F) = .3+.4−.6 =
.1. Hence E and F are not mutually exclusive.
7
c
Yeong-Chyuan
Chung, December 14, 2015
21. (4pts) In a survey of 200 adults 50 years old and older of whom 60% were retired and 40% were pre-retired, the
following question was asked: Do you expect your income needs to vary from year to year in retirement? Of
those who were retired, 33% answered no, and 67% answered yes. Of those who were pre-retired, 28% answered
no, and 72% answered yes. If a respondent in the survey was selected at random and had answered yes to the
question, what is the probability that he or she was retired?
Solution:
(Tree diagram omitted.)
P(R|Y ) =
P(R∩Y )
P(Y )
=
(.6)(.67)
(.6)(.67)+(.4)(.72)
=
67
115
≈ .583.
22. (3pts) The management of a company has projected the sales of its products (in millions of dollars) for the
upcoming year, with the associated probabilities shown in the following table:
Sales
20 22 24 26 28 30
Probability .05 .10 .35 .30 .15 .05
What are the expected sales for next year?
Solution:
20(.05) + 22(.10) + 24(.35) + 26(.30) + 28(.15) + 30(.05) = 25.1 million dollars
23. The scores on a mathematics examination are normally distributed with a mean of 70 and a standard deviation
of 15. Suppose that the instructor assigns a grade of A to 15% of the class and a grade of B to 20% of the class.
(a) (2pts) What is the probability that a student selected at random has a score greater than 60?
Solution:
Compute normalcd f (60, 100, 70, 15) = .7248 or normalcd f (60, 1E99, 70, 15) = .7475
(b) (3pts) What is the lowest score a student needs to obtain a B?
Solution:
35% of the class will get at least a B so the other 65% will have lower scores. In other words, the probability
that a student gets less than a B grade is .65.
The lowest score required for a B grade is invnorm(.65, 70, 15) = 75.7798.
8
c
Yeong-Chyuan
Chung, December 14, 2015
24. (3pts) Determine the simple interest rate at which $1200 will grow to $1250 in 8 months. Give your answer in
percent per year.
Solution:
8
The amount of interest earned is $50. Using the formula I = Prt for simple interest, we get 50 = 1200r( 12
) so
50 12
r = 1200 8 = .0625, which is equivalent to 6.25%.
25. (3pts) Five and a half years ago, Chris invested $10,000 in a retirement fund that grew at the rate of 6.82%/year
compounded quarterly. What is his account worth today?
Solution:
N = 5.5 × 4
I = 6.82
PV = −10000
PMT = 0
FV =?
P/Y = C/Y = 4
[Interest is compounded quarterly so P/Y = C/Y = 4 and we are considering a period of 5.5 years so N = 5.5×4.]
His account is worth $14,505.43 today.
26. (3pts) Investment A offers an 8%/year return compounded semiannually, and Investment B offers a 7.5%/year
return compounded monthly. Which investment has a higher rate of return over a 5-year period? Justify your
answer.
Solution:
Investment A: E f f (8, 2) = 8.16%
Investment B: E f f (7.5, 12) = 7.7633%
Investment A has a higher rate of return.
9
c
Yeong-Chyuan
Chung, December 14, 2015
27. Suppose that you borrow $100,000 from a bank to help finance the purchase of a house, and the bank charges
interest at a rate of 5% per year on the unpaid balance, with interest computations made at the end of each month.
(a) (3pts) If you decide to repay the loan in equal monthly installments over 30 years, then how much will each
payment be?
Solution:
N = 30 × 12
I=5
PV = 100000
PMT =?
FV = 0
P/Y = C/Y = 12
[Interest is compounded monthly so P/Y = C/Y = 12 and we are considering a period of 30 years so
N = 30 × 12.]
Each payment will be $536.82.
(b) (3pts) What is the total amount of interest you will be paying?
Solution:
This is computed by subtracting the amount owed from the total amount paid.
(536.82)(12)(30) − 100000 = $93, 255.20.
Problem Part 1
12 - 14
15 - 17
18 - 20
Score
10
21 - 23
24 - 26
27
Total