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MATH 106 Finite Mathematics 2158-OL1-6384-1A

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MATH 106 FINAL EXAMINATION

This is an open-book exam. You may refer to your text and other course materials as you work

on the exam, and you may use a calculator. You must complete the exam individually.

Neither collaboration nor consultation with others is allowed. Use of instructors’ solutions

manuals or online problem solving services in NOT allowed.

Record your answers and work on the separate answer sheet provided.

There are 25 problems.

Problems #1–12 are Multiple Choice.

Problems #13–15 are Short Answer. (Work not required to be shown)

Problems #16–25 are Short Answer with work required to be shown.

MULTIPLE CHOICE

1. Which of the following statements is NOT true:

1. ________

A. If an event cannot possibly occur, then the probability of the event is 0.

B. If events E and F are mutually exclusive events, then 𝑃(𝐸 ∩ 𝐹) = ∅.

C. If only two outcomes are possible for an experiment, then the sum of the probabilities of the outcomes is equal to 1

D. Conditional probability is defined as

𝑃(𝐴|𝐵) = 𝑃(𝐴 ∩ 𝐵)

𝑃(𝐴)

2. – 3. A small company manufactures two types of radios – regular and short-wave. The

manufacturing of each radio requires two operations: Assembly and Finishing. The regular

radios require 1 hour of Assembly and 3 hours of Finishing. The short-wave radios require 3

hours of Assembly and 1 hour of Finishing. The total work-hours available per week in the

Assembly division is 60, and in the Finishing division, 60. A profit of $50 is realized for every

regular radio, and $75 for every short-wave radio. Let x represent number of regular radios and

y represent number of short-wave radios made per week.

2. Identify the weekly production constraint for assembling:

2. _______

A. 𝑥 + 3𝑦 ≥ 60 C. 3𝑥 + 𝑦 ≤ 60

B. 𝑥 + 3𝑦 ≤ 60 D. 3𝑥 + 𝑦 ≥ 60

3. State the objective function.

3. _______

A. 𝑃 = 60𝑥 + 60𝑦 C. 𝑃 = 75𝑥 + 50𝑦

B. 𝑃 = 240𝑥 + 240𝑦 D. 𝑃 = 50𝑥 + 75𝑦

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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4. Students in four different MATH 106 class sections responded to the survey question “How

many hours a day do you work on MATH 106?” as shown below. Which histogram below

accurately reflects the section frequency distribution with the most positive skew?

4. ______

HISTOGRAM A HISTOGRAM C

HISTOGRAM B HISTOGRAM D

5. A jar contains 15 red balls, 12 yellow balls, and 18 green balls. Suppose that each ball has an

equal chance of being picked from the jar. If a single ball is selected at random from the jar,

what is the probability that it is not green?

5. _______

𝐴. 3

5 𝐵.

15 𝐶.

3 𝐷.

6. The Rodriguez family buys a $335,000 home by putting a 20% down payment and financing

the balance with a 30-year fixed mortgage at 4.5%. What is the amount of their monthly loan payment to amortize the loan?

6. _______

A. $1749.44 C. $1697.40

B. $1357.92 D. $2186.81

1 2 3 4 5 6

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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7. Identify the single row operation that transforms the matrix as shown: 7. ________

[ −12 6 3 −1.5

| −4 1

] → [ 0 0 3 −1.5

| 0 1

]

A. 0.25𝑅1 + 𝑅2 → 𝑅1 C. 0𝑅1 → 𝑅1

B. 4𝑅2 + 𝑅1 → 𝑅1 D. 4𝑅1 + 𝑅2 → 𝑅2

8. Find the equation of the line passing through (4, 3) and (– 2 , – 7): 8. _______

A. 3x + 5y = 27 B. 3x – 5y = – 3 C. 5x – 3y = 11 D. 2x – y = 5

9. Determine which region defines the solution region of the following system of linear

inequalities graphed below:

9. ________

2𝑥 − 5𝑦 ≥ −3 −3𝑥 + 𝑦 ≤ −2

A. Region I C. Region III

B. Region II D. Region IV

III

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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10. Which of the corner points for the system of linear inequalities graphed below minimizes the objective function P = 3x + 4y ?

10. _______

A. (3, 0) C. (0, 4)

B. (1, 2) D. (2, 0)

11. The mean time from check-in to completion for a customer served at the Otisburgh branch of the Department of Motor Vehicles is 109 minutes, with a standard deviation of 17 minutes.

Assuming a normal distribution, what is the probability that a randomly chosen customer

experiences service done in greater than 126 minutes?

11. ______

A. 0.6826 C. 0.3413

B. 0.0228 D. 0.1587

12. The amount of money you should deposit in an account paying 8% compounded quarterly in order to receive quarterly payments of $1200 for the next 6 years can be determined using

formula for:

12. _______

A. Single-payment, simple interest

B. Single-payment, compound interest

C. Sequence of payments: present value of an annuity / amortization

D. Sequence of payments: future value of an ordinary annuity

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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SHORT ANSWER (work NOT required to be shown)

13. For the linear equation 3𝑥 + 7𝑦 = 42:

a. Determine the slope: _______________________

b. Determine x – intercept if it exists: _______________________

c. Determine y – intercept if it exists: _______________________

d. Express equation in slope-intercept form: _______________________

14. Let 𝑛(𝐴) = 83, 𝑛(𝐵) = 50, 𝑛(𝐴 ∪ 𝐵) = 105, and 𝑛(𝑈) = 140.

a. Determine 𝑛(𝐴′) : ___________________________________

b. Determine 𝑛(𝐴 ∩ 𝐵) : ___________________________________

c. Determine 𝑛(𝐴′ ∩ 𝐵′): ___________________________________

15. 980 randomly-selected respondents to a marketing survey were asked their age in years (18

to 39 or 40+) and the choice of vacation destination they liked best. Following data obtained:

Preferred Vacation

Destination

Age 18-39 Age 40 + Total

Theme Park 245 17 262

Seashore 135 183 318

Mountains 151 85 236

Spa Resort 51 113 164

Total 582 398 980

(Report your answers as fractions or as decimal values rounded to the nearest hundredth.)

Find the probability that a randomly-selected respondent:

(a) prefers the seashore or is age 40+: Answer: ______________

(b) prefers the seashore given that the respondent is age 40+: Answer: ______________

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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SHORT ANSWER, with work required to be shown, as indicated.

16. Seventeen candidates have filed for the upcoming county council election. 7 are women and

10 are men.

(a) In how many ways can 10 county council members be randomly elected out of the 17

candidates? Show work.

(b) In how many ways can 10 county council members be randomly elected from 17 candidates

if 5 must be women and 5 must be men? Show work.

probability that 5 are women and 5 are men? Round answer to nearest ten-thousandth (4 places

after decimal). Show work.

______________________________________________________________________________

17. Solve the system of equations using substitution, elimination by addition, or augmented

matrix methods (your choice). Show work.

5𝑥 − 3𝑦 = 11

7𝑥 + 4𝑦 = −1

18. Vijay wants to start a retirement account that will have $460,000 in it when he retires in 15

years. How much should he invest every six months in his account to do this if interest is 15%

compounded semiannually? Show work.

A. $4,435.77 C. $4,790.74

B. $4,448.77 D. $17,612.13

______________________________________________________________________________

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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19. In the United States, fossil fuel use in 1985 represented 85% of total energy production. By

2005 that percentage was 79%.

(a) The linear regression model most closely fitting this data (where y represents

percentage of total energy production and x represents years since 1985) is:

Explain/show work.

A. y = – 3.3x + 1985 C. y = – 3.3x + 85

B. y = – 0.3x + 1985 D. y = – 0.3x + 85

(b) Use the equation from part (a) to predict fossil fuel use as a percentage of total

American energy production in the year 2020. Round answer to nearest tenth of a

percent. Show work.

production as a percentage of total American energy production with respect to time is

______________________ per ________________. (Include units of measurement.)

20. You have $5000 to invest in a single payment. Which is the better investment: 9.6%

compounded annually or 9.24% compounded monthly? Show work

______________________________________________________________________________

21. According to a recent report, 0.10 is the probability that an American can identify Janet

Yellen as the Chair of the Federal Reserve Board of the US Government. Ten Americans are

randomly selected. Find the probability that exactly 1 of the 10 Americans selected can identify

Ms Yellen as the Chair of the Federal Reserve Board. Round answer to the nearest thousandth

(three places after decimal). Show work.

______________________________________________________________________________

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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22. The feasible region shown below is bounded by lines 2x – y = 4, x + 3y = 4, and y = 0. Find the coordinates of corner point A. Show work.

23. A network security specialist records the number of incoming e-mails containing links that

six randomly-selected network users receive in a day. Numbers are 51, 66, 58, 43, 51, and 55.

(a) State the mode (if one exists).

(b) Find the median. Show work/explanation.

(d) Using the sample mean found in part (c), and given that the sample standard deviation of

the data set above is 7.75, what percentage of the data set falls within one standard deviation of

the mean? Show work/explanation.

(d) _______

A. 34.2% C. 50.0%

B. 66.7% D. 68.3%

MATH 106 Finite Mathematics 2158-OL1-6384-1A

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24. Air Weegotcha 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 0 1 2 3 4 5

Probability of number overbooked 0.70 0.17 0.06 0.04 0.02 0.01

Find the expected number of overbooked passengers per flight.

25. Mary McMath surveyed 115 students in a finite math class on what they most recently did

for fun. 69 went to the movies, 51 went to the big football game, and 12 went to the movies and

the big football game.

(a) What is the probability that a randomly selected student did neither? Show work.

(b) Let M = {students who went to movies} and G = {students who went to the “big game”}.

Determine the number of attendees belonging to each of the regions I, II, III, IV.

Region I: ________ Region II: __________ Region III: _________ Region IV: __________

______________________________________________________________________________

G M

III I