FM Dept Final Exam Review Sp06
Finite Mathematics--Departmental Review Sheet
Spring 2006
(Finite Mathematics, an Applied Approach, 3rd Edition. Young, Lee, Long and Graening. Pearson, Addison Wesley,
2004.)
Unless otherwise stated, all work should be done algebraically, all work should be shown, and exact
answers should be given. Only departmental formula sheets will be allowed for use during the final exam.
[Objective 1, Section 4.3]
1.
Graphically solve the following:
2.
x  y  8

Maximize f  6x  y , subject to:  3x  y  0
y  0

First formulate the linear program, and then solve it by the graphical method.
Nutrition: A dietitian is to prepare two foods to meet certain requirements. Each pound of Food I
contains 100 units of vitamin C, 40 units of vitamin D, and 10 units of vitamin E and costs 20
cents. Each pound of Food II contains 10 units of vitamin C, 80 units of vitamin D, and 5 units of
vitamin E and costs 15 cents. The mixture of the two foods is to contain at least 260 units of vitamin
C, at least 320 units of vitamin D, and at least 50 units of vitamin E. How many pounds of each type
of food should be used to minimize the cost?
[Objective 1, Section 5.2]
3.
a.
Use the simplex algorithm to solve the given linear programming problems.
Maximize f  2x  y
Subject to:
4x  y  20
x  2y  16
3x  2y  20
x  0, y  0
b.
Maximize f  2x  3y  3z Subject to:
x  2z  8
 y  3z  10
 x  y  z  12
x  0, y  0, z  0
c.
Advertising: An advertising agency has developed radio newspaper, and television ads for a
particular business. Each radio ad costs $200, each newspaper ad costs $100, and each television
ad costs $500 to run. The business does not want the television ad to run more than 20 times, and
the sum of the numbers of times the radio and newspaper ads can be run is to be no more than 110.
The agency estimates that each airing of the radio ad will reach 1000 people, each printing of the
newspaper ad will reach 800 people, and each airing of the television ad will reach 1500 people. If
the total amount to be spent on ads is not to exceed $15,000, how many times should each type of ad
be run so that the total number of people reached is a maximum?
[Objective 2, Section 5.3]
4.
a.
Use Crown’s pivoting rules and the simplex method to solve:
Use Crown’s method to maximize f  4x  2y
xy5
Subject to: 3x  y  12
x  0, y  0
1
FM Dept Final Exam Review Sp06
x  2y  40
x  2y  3z  60
subject to:
y  z  30
x  0, y  0, z  0
b.
Minimize f  2x  y  2z
c.
The InfoAge Communication Store stocks fax machines, computers, and portable CD players.
Space restrictions dictate that it stock no more than a total of 100 of these three machines. Past sales
patterns indicate that it should stock an equal number of fax machines and computers and at least 20
CD players. If each fax machine sells for $500, each computer for $1800, and each CD player for
$1000, how many of each should be stocked and sold for maximum revenues?
[Objective 3, Sections 6.1 and 6.2]
5.
Find the amount that will be accumulated in each given account under the conditions set forth.
A principal of $800 is accumulated for 12 years:
a.
b.
c.
at 7% simple interest
at 7% compounded quarterly
at 7% compounded monthly
6.
Find the principal P required to achieve a future amount A=$5000 with an interest rate of 6%
compounded quarterly for 5 years.
7.
Ramero plans to buy a new car three years from now. Rather than borrow at that time, he plans to
invest part of a small inheritance at 7.5% compounded semiannually to cover the estimated $6000
trade-in difference. How much does he need to invest if he starts investing now?
8.
Investing for Retirement: Jan is a 35-year-old individual who plans to retire at age 65. Between
now and then $4000 is paid annually into her IRA account, which is anticipated to pay 5%
compounded annually. How much will be in the account upon Jan’s retirement?
9.
The Brewsters are saving for their daughter’s college days. They would like to be able to withdraw
$800 each month from their account for five years once their daughter starts college. Assuming that
their account will earn interest at the rate of 9% compounded monthly, what sum of money should
the Brewsters have in the account when their daughter starts college?
[Objective 4, Section 6.2]
10.
Business Investment: A freight-hauling firm estimates that it will need a new forklift in six years.
The estimated cost of the vehicle is $40,000. The company sets up a sinking fund that pays 8%
compounded semiannually, into which it will make semiannual payments to achieve the goal.
Calculate the size of the payments.
[Objectives 5 and 6, Section 6.3]
11.
Convert the given interest rates to the APR. Round your answer to the nearest thousandth of a
percent.
a.
b.
c.
6.5% compounded quarterly
12% compounded daily
7% compounded monthly
12.
Suppose that you borrow $12,500 for the purchase of a new car at 9.8% APR for 48 months. What
is the approximate amount of your monthly payment?
2
FM Dept Final Exam Review Sp06
13.
Jared buys a new motorcycle for $8500 from The Fastrack Motorcycle Co., which agrees to finance
80% of the purchase at 9% APR for a period of 60 months. What is the approximate size of Jared’s
monthly payment?
[Objective 7, Section 7.3]
14.
Use the following information to answer parts a-c.
U={a, b, c, d, e, f, g, h, i, j,}; A={a, c, e, g, i}; B={b, d, f, h, j}; C={a, b, d,}
a.
b.
c.
A  B'
C–A
(A  B' )'
[Objective 7, Section 7.4]
15.
In a particular school district, 90 families were asked these two questions:
Q1: Do you have children attending public kindergarten?
Q2: Do you have children in grades 1 through 5 attending public school?
Thirty answered “yes” to Question 1, 50 answered “yes” to Question 2, and 10 answered “yes” to
both questions.
a.
b.
c.
Draw and label a Venn diagram that numerically represents this survey then use the Venn diagram to
answer parts b and c.
How many answered “yes” to at least one of these questions.
How many have no children in grades 1 through 5 attending public school.
16.
Mary McMath surveyed 200 students in a finite class. Thirty went to the movies, 60 went to football
games, 40 went to the theater, 10 went to the movies and football games, 25 went to the movies and
the theater, 20 went to football games and the theater, and 10 went to all three. (Use a Venn diagram).
a.
b.
How many of the finite students did not go to the theater?
How many of the students went to exactly two of the different events?
[Objective 8, Section 7.5, 7.6, 7.7]
17.
In how many ways can eight books be arranged on a shelf if:
a.
b.
there are no restrictions
one of the books, Of Mice and Men, must be displayed on the left end?
18.
How many different car license plates can be made using two letters followed by four digits if:
a.
b.
c.
there are no restrictions
no letter can be repeated
the last digit must be a 4, and no digit can be repeated
19.
How many permutations are there of the word college?
20.
Nine people are to travel to dinner in a five-passenger van and a four-passenger car. How many
different groups of five and four are possible for the trip?
21.
In how many ways can four couples be seated in a row of eight seats at a theater if each couple is
seated together?
22.
In how many ways can a five card hand be drawn, without replacement from a standard deck of
fifty-two if exactly three of the cards are to be clubs?
3
FM Dept Final Exam Review Sp06
23.
The eight-person board of directors of the Acme Corporation is to elect a president, a vice-president,
and a treasurer, and the remaining members will be a committee to study future expansion. In how
many ways can these officers be elected?
24.
A psychology experiment observes groups of 5 individuals. In how many ways can the experiment
take 15 people and group them into three groups of five?
25.
Five freshmen, 4 sophomores, and 3 juniors are present at a meeting. In how many ways can a
committee of 3 be selected consisting of:
a.
b.
c.
exactly 2 freshmen and 1 junior .
all of one class.
least 2 juniors.
26.
In how many ways can the names of six candidates be arranged on a ballot if Joe Shmo, must be at
the top?
27.
A new employee is offered a choice of 3 health care plans and 8 retirement plans. In how many ways
can she choose to set up her employee benefit package?
28.
How many different "words" can be formed from the letters in the word MATHEMATICS?
29.
Seven pictures are to be arranged on a wall. Four are by Van Gogh and three are by Monet. In how
many different ways can the pictures be arranged on the wall if:
a.
b.
c.
there are no restrictions on the placement?
Starry Night by Van Gogh must be in the middle position?
there must be Monet pictures on each end?
[Objective 9, Sections 8.1, 8.2, and 8.3]
30.
A three card hand is dealt.
a.
b.
c.
d.
Find the probability that all three cards are spades
Find the probability that all three cards are from the same suit
Find the probability that the hand contains at least one king.
Find the odds that all three cards are spades.
31.
A pair of standard dice is to be rolled. Find the probability that:
a.
b.
c.
d.
a sum greater than 1 is rolled.
a sum less than 5 is rolled.
a sum less than 3 or greater than 8 is rolled.
a sum less than 3 and greater than 8 is rolled.
32.
Find the expected winnings for the game of chance described below.
In one form of the game KENO, the house has a pot containing 80 balls, each marked with a
different number from 1 to 80. You buy a ticket for $1 and mark one of the 80 numbers on it. The
house then selects 20 numbers at random. If your winning number is among the 20, you get $3.20
(for a net winning of $2.20).
33.
Air Fayetteville overbooks as many as five passengers per flight because some passengers with
reservations do not show up for the flight. The airline's records indicate the following probabilities
that it will be overbooked at flight time.
Number overbooked at flight time
Probability of number overbooked
0
0.75
1
0.10
2
0.06
3
0.04
4
0.04
5
0.01
Find the expected number of overbooked passengers per flight.
4
FM Dept Final Exam Review Sp06
34.
A single card is to be selected from a deck of 52 cards. Find the odds.
a.
b.
c.
for the card being a king
for the card being red
against the card being a club.
[Objective 9, Sections 9.1 – 9.5]
35.
A summary of whether graduating students have received one or more job offers is given in the
following table:
BA
MBA
Job Offer
140
40
No Job Offer
50
20
a.
b.
c.
What is the probability that a graduate receives a job offer?
What is the probability that a graduate is an MBA or receives a job offer?
What is the probability that a person does not receive a job offer given that the person earned a BA?
36.
A seed company claims that 90% of its bean seeds will germinate. If ten of these seeds are planted
in warm, moist, soil, what is the probability that:
a.
b.
c.
exactly 9 of them will germinate?
at least 9 of them will germinate?
at least on of them will germinate?
37.
Claire interviews with two companies, Tyson Foods and Wal-Mart. The probability of receiving an
offer from Tyson's is 0.35 from Wal-Mart is 0.48 and from both is 0.15. Find the probability that
Claire will receive an offer from Wal-Mart given she receives an offer from Tyson's.
38.
A bag contains 5 red balls, 4 white balls, and 3 black balls. Three balls are drawn without
replacement. What is the probability that:
a.
b.
c.
d.
at least two balls are red?
the first ball is white, the second ball is black,, and the third is red?
the second ball is black given that the first ball was white?
If 4 balls are drawn what is the probability that the first is red, the second black, the third white and
the fourth red?
39.
A baseball player has a batting average of 0.245. Assume that each turn at bat is not affected by the
previous turns and that no walks occur on the times at bat considered. What is the probability that,
for the next three times at bat, the player will:
a.
b.
c.
get a hit all three times?
get exactly two hits?
get at least two hits?
40.
A shipment of computer chips contains 12 that are good and three that are defective. Three of these
chips are selected, in succession, with replacement, and checked for a defect. Find the probability that:
a.
b.
c.
the first chip will be defective, the second defective, and the third good.
none of the chips will be defective.
all of the chips will be defective.
5
FM Dept Final Exam Review Sp06
41.
The probability that a customer of a local department store will be a “slow pay” is .02. The
probability that a “slow pay” will make a large down payment when buying a refrigerator is .14.
The probability that a person who is not a “slow pay” will make a large down payment when buying
a refrigerator is .50. Find the probabilities that: (Show all work including your tree diagram):
a.
b.
c.
a customer makes a large down payment and is a “slow pay”
a customer made a large down payment given that he is a “slow pay”
a customer is not a “slow pay” given that he made a large down payment.
[Objective 10, Sections 10.1 and 10.2]
42.
A random sample of 6 students revealed the following number of books they were carrying:
1, 8, 4, 3, 2, 4.
a.
b.
c.
d.
Calculate the median.
Calculate the mode.
Calculate the mean.
What is the sample standard deviation?
43.
Expand the table to include the relative frequency and the cumulative frequency. Then draw
frequency and relative frequency histograms.
Class
150 to 179
180 to 209
210 to 239
240 to 269
270 to 300
Frequency
30
50
80
60
40
44.
Compute the following for the given population: 101, 86, 137, 96, 120
a.
b.
c.
d.
mean
median
mode
standard deviation
[Objective 11, Section 10.4]
45.
Suppose that the lifetime of a particular type of light bulb are normally distributed with a mean life
of 1200 hours and a standard deviation of 160 hours. Find the probability that a light bulb will burn
out in less than 1100 hours. Show your z-score.
46.
Production samples from a bolt factory indicate that a certain type of bolt has lengths that are a
normally distributed random variable with a mean of 8cm and a standard deviation of 0.06 cm.
From a day’s production population of 2000 bolts, find the following: (Show z-scores)
a.
b.
How many would have a length less than 7.5 cm?
How many would have a length between 7.9 and 8.2 cm?
6
FM Dept Final Exam Review Sp06
Answers
1.
Max f of 48 when x = 8 and y = 0.
2.
x = lbs of Food I y = lbs of Food II
Minimize cost, C = .2x + .15y, subject to:
100x + 10y ≥ 260
40x + 80y ≥ 320
10x + 5y ≥ 50
x≥0 y≥0
Minimum C = $1.10 when x = 4 lbs of Food I and y = 2 lbs of Food II
3a.
Maximum f = 12 when x = 4, y = 4, s1 = 0, s2 = 4, s3 = 0.
3b.
Maximum f = 76 when x = 8, y = 20, z = 0, s1 = 0, s2 = 30, s3 = 0.
3c.
x = # of times radio ads are run y = # of times newspaper ads are run z = # of times TV ads are run
Maximize number of people reached, N = 1000x + 800y + 1500z, subject to:
200x + 100y + 500z ≤ 15000
z ≤ 20
x + y ≤ 110
x≥0
y≥0
z≥0
Maximum N = 100,000 when radio ads are run 0 times, newspaper ads are run 110 times, and
TV ads are run 8 times. If this is done, the number of TV ads will be 12 under the limit of 20 and
the entire advertising budget of $15,000 will be used.
[x = 0, y = 110, z = 8, s1 = 0, s2 = 12, s3 = 0]
4a.
Max value of f = 24 when x = 0, y = 12, s1  7 , s 2  0
4b.
The function f has no minimum value over the feasible region.
4c.
x = # fax machines stocked & sold, y = # computers stocked & sold, z = # CD players stocked & sold
Maximize revenue, R = 500x + 1800y + 1000z, subject to:
x + y + z ≤ 100
x=y
z ≥ 20
x≥0
y≥0
Maximum revenue, R = $112,000, will be generated if InfoAge stock and sell 40 fax machines,
40 computers, and 20 portable CD players.
5a.
$1472.00
6.
$3712.35
7.
$4810.86
8.
$265,755.39
9.
$38,538.70
10.
$2662.09
11a.
6.66%
12.
$315.83
13.
$141.16
14a.
A  B  a, c, e, g, i  A
15a.
b.
$1839.68
c.
$1848.58
b.
12.75%
c.
7.23%
b.
C  A  b, d
c.
A  B  b, d, f, h, j  B
b.
70
c.
40
7
FM Dept Final Exam Review Sp06
16a.
160
b.
25
17a.
40,320
b.
5,040
18a.
6,760,000
b.
6,500,000
18c.
340,704
19.
1260 distinguishable permutations
20.
126
21.
384
22.
211,296
23.
336
24.
3003
25a.
30
b.
15
c.
28
26.
120
27.
24
28.
4,989,600
29a.
5040
b.
720
c.
720
30a.
.01294
b.
.05176
c.
.21738
d.
11 to 839
31a.
1
b.
1
6
c.
11
36
d.
0
32.
-.20
33.
.55
34a.
1 to 12
b.
1 to 1
c.
3 to 1
35a.
.72
b.
.8
c.
.26316
36a.
.387420489
b.
.7360989291
c.
.9999999999
37.
c.
 
pw  t  .15 3

  .42857
If the events are not independent, pw t  

If the events are independent, p w t  pw   .48
pt
.35
7
38a.
4
11
b.
1
22
c.
3
11
39a.
.01471
b.
.1360
c.
.15071
40a.
.032
b.
.512
c.
.008
41a.
.0028
b.
.14
c.
.99432
d.
2
99
8
FM Dept Final Exam Review Sp06
42a.
3.5
43.
b.
4
c.
11
3
Class
Frequency
Relative Freq
Cumulative Freq
150 to 179
30
.11538
30
180 to 209
50
.19231
80
210 to 239
80
.30769
160
240 to 269
60
.23077
220
270 to 300
40
.15385
260
44a.
108
b.
45.
x ≤ 1100 
46.
a.
x ≤ 7.5 
b.
7.9 ≤ x ≤ 8.2 
101
z ≤ –.625 
c.
no mode
d.
2.42212
d.
18.23184
A = .2643
z ≤ –8.33 
none of the bolts will have length less than 7.5
–1.67 ≤ z ≤ 3.33 
1905 bolts will have length between 7.9 and 8.2
9