Finite math problems and solutions

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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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. Amalgamated Furniture Company makes dining room tables and chairs. A table requires 8 labor-hours for assembling and 2 labor-hours for finishing. A chair requires 2 labor-hours for assembly and 1 labor-hour for finishing. The maximum labor-hours available per day for assembling and finishing are 400 and 120, respectively. Production costs are $600 per table and $150 per chair. Let x represent number of tables and y represent number of chairs made per day. Identify the daily production constraint for assembling:

1. _______ A. 8𝑥 + 2𝑦 ≥ 400 C. 8𝑥 + 2𝑦 ≤ 400

B. 2𝑥 + 𝑦 ≥ 120 D. 2𝑥 + 𝑦 ≤ 120

2. Which of the following statements is NOT true: 2. ________

A. A probability must be less than or equal to 1. B. If an event cannot possibly occur, then the probability of the event is a negative number. C. If events G and H are mutually exclusive events, then P (G ∩ H) = 0. D. If only two outcomes are possible for an experiment, then the sum of the probabilities of

the outcomes = 1.

3. Raoul buys an Amalgamated dining room furniture set for $12,000, makes a down payment of 20%, and finances the rest with 60-month store financing at an annual interest rate of 6.0% compounded monthly. What is the amount of his monthly loan payment to amortize the loan?

3. _______ A. $185.59 C. $231.99

B. $208.00 D. $260.00

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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4. “Vegas Insider” lists the National Football League’s Jacksonville Jaguars as the team least likely to win Super Bowl 50 in 2016, with “odds of winning” listed as “300 to 1” (correctly written as 1:300). What is the probability that the Jaguars actually win? 4. ________ A. 300 𝑡𝑜 1 C. 1 300 (0.003333)⁄ B. 1 𝑖𝑛 300 (0.003344) D. 1 301⁄ (0.003322) 5. Which histogram below accurately reflects 15 survey responses presented in the following frequency table?

Value Frequency 1 X 2 XX 3 XXXXXX 4 XXX 5 XX 6 X

5. ______ HISTOGRAM A HISTOGRAM C

HISTOGRAM B HISTOGRAM D

0

2

4

6

1 2 3 4 5 6

0

2

4

6

1 2 3 4 5 6

0

2

4

6

8

1 2 3 4 5 6

0

2

4

6

8

1 2 3 4 5 6

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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6. Determine which graph shows the correct solution region of the system of linear inequalities:

𝑥 + 𝑦 ≥ 2 𝑥 ≥ 0 𝑥 + 3𝑦 ≤ 3 𝑦 ≥ 0

6. _______ GRAPH A. GRAPH B.

GRAPH C. GRAPH D.

7. Find the equation of the line passing through (2, 3) and (– 3, 7): 7. _______ a. 5x – 4y = – 2 b. 5x + 4y = 22 c. 4x + 5y = 23 d. 4x – 5y = – 7

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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8. The total amount of money you should deposit in an account paying 8% compounded quarterly in order to receive quarterly payments of $1000 for the next 4 years can be determined using formula for: 8. _______

A. Sequence of payments: present value of an annuity / amortization B. Sequence of payments: future value of an ordinary annuity C. Single-payment, compound interest D. Single-payment, simple interest

9. Which of the corner points for the system of linear inequalities graphed below minimizes the objective function P = 7x + 6y ?

9. _______ A. (3, 0) C. (2, 0)

B. (1, 2) D. (0, 4)

10. The mean time from check-in to completion for a customer service task at the Townsburg branch of the DMV is 112 minutes, with a standard deviation of 18 minutes. Assuming a normal distribution, what is the probability that a randomly chosen customer experiences service done between 94 and 130 minutes? 10. ______

A. 0.3413 C. 0.9544

B. 0.4772 D. 0.6826

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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11. Identify the single row operation that transforms the matrix as shown: 11. ________

[ −12 6 3 −1.5

| −4 1

] → [ 0 0 3 −1.5

| 0 1

]

A. 0.25𝑅1 + 𝑅2 → 𝑅1 C. 4𝑅2 + 𝑅1 → 𝑅1

B. 0𝑅1 → 𝑅1 D. 4𝑅1 + 𝑅2 → 𝑅2

12. Freestate Power & Light, an electricity producer, and Galactic Gas, a natural-gas producer, have a business relationship. Each needs some of the other’s output in order to meet overall consumer demand in the city of Megalopolis. Following graphic shows flow of total monthly product output (what flows OUT of the company) for each energy company (in $ millions) to its consumers and each other. Let a = Freestate Power & Light total monthly output, and b = Galactic Gas total monthly output.

What is the linear equation in two variables that correctly describes the total monthly output of Galactic Gas (output = b)? 12. _______

A. 0.6𝑎 − 𝑏 = −7 C. 0.6𝑎 + 𝑏 = 7

B. 𝑎 + 0.8𝑏 = 10 D. 𝑎 − 0.8𝑏 = −10

* * * * * * * * * * * * *

Megalopolis Consumers

Freestate Power & Light (Output = a)

Galactic Gas (Output = b)

0.8b

10

0.6a

7

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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SHORT ANSWER (work NOT required to be shown) 13. For the linear equation −4𝑥 + 7𝑦 = 56:

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) = 74, 𝑛(𝐵) = 69, 𝑛(𝐴 ∪ 𝐵) = 99, and 𝑛(𝑈) = 125.

a. Determine 𝑛(𝐵′) : ___________________________________

b. Determine 𝑛(𝐴 ∩ 𝐵) : ___________________________________

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

15. 1000 randomly-selected respondents to a TV marketing survey were asked their age in years (18 to 39 or 40+) and the kind of TV show most often watched. Following table was obtained.

Show Most

Watched on TV

Viewer Age

18 - 39

Viewer Age

40 +

Total

Comedy 208 88 296

Drama 150 74 224

News 32 140 172

Sports 228 80 308

Total 618 382 1000

(Report your answers as fractions or as decimal values rounded to the nearest hundredth.) Find the probability that a randomly-selected respondent: (a) most often watches news and is age 18 – 39: Answer: ______________ (b) most often watches news given that viewer is age 18 – 39: Answer: ______________ (c) most often watches news or is age 18 – 39: Answer: ______________

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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SHORT ANSWER, with work required to be shown, as indicated. 16. Seventeen people are summoned to jury duty. 11 are women and 6 are men. (a) In how many ways can 12 jurists be randomly selected out of the 17 people? Show work. (b) In how many ways can 12 jurists be chosen, if 7 must be women and 5 must be men? Show work. (c) If 12 jurists are randomly selected from the 17 people, what is the probability that 5 are men and 7 are women? 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.

7𝑥 + 5𝑦 = 3

2𝑥 + 3𝑦 = −7

18. Cara needs $9,000 in 11 years. What amount can she deposit at the end of each quarter at 8% annual interest compounded quarterly so she will have her $9,000? Show work.

A. $540.69 C. $129.49

B. $134.01 D. $125.19

______________________________________________________________________________

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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19. According to the US Center for Disease Control and Prevention National Center for Health Statistics, 10.0 percent of adolescent (ages 12 – 19) Americans were obese in 1995; an amount which climbed to 18.0 percent in 2008.

(a) Which of the following linear equations could be used to predict percent of adolescent Americans who are obese “y” in a given year “x”, where x = 0 represents the year 1995? Explain/show work.

𝐴. 𝑦 = 13

8 𝑥 + 1995 𝐶. 𝑦 =

13

8 𝑥 + 10

𝐵. 𝑦 = 8

13 𝑥 + 10 𝐷. 𝑦 =

8

13 𝑥 + 1995

(b) Use the equation from part (a) to predict the percent of adolescent Americans who are obese in the year 2034. Round answer to nearest tenth of a percent. Show work.

(c) Fill in the blanks to interpret the slope of the equation: The rate of change of number of Americans age 65 or older with respect to time is ______________________ per ________________. (Include units of measurement.)

20. Tamika’s savings account has a balance of $6200. If she makes no deposits or withdrawals for 5 years, and the account accrues interest at 4% compounded quarterly, what will be the total amount of money in her savings account at the end of the 5-year period? Round answer to nearest cent. Show work

______________________________________________________________________________

21. According to a recent Pew Research Center report, 0.8 is the probability that a randomly selected American adult knows what Twitter™ is. Ten American adults are randomly selected.

Find the probability that exactly 8 of the 10 American adults randomly selected know what Twitter™ is. Round answer to nearest ten-thousandth (4 places after decimal). Show work ______________________________________________________________________________

MATH 106 Finite Mathematics 2152-OL4-7980-V1

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22. The feasible region shown below is bounded by lines – x + 2y = 4, 3x + y = 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 15, 36, 15, 45, 44, and 49.

(a) State the mode (if one exists). (b) Find the median. Show work/explanation. (c) Determine the sample mean. Show work (d) Using the sample mean found in part (c), and given that the sample standard deviation of the data set above is 15.3, what percentage of the data set falls within one standard deviation of the mean? Show work/explanation. (d) _______ A. 34.2% C. 68.3%

B. 66.7% D. 83.3%

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24. A local car rental agency charted daily demand as shown in the following table:

Number of customers 6 8 10 12 14 Probability 0.15 0.20 0.25 0.30 0.10

Find the expected number of customers. Show work.

25. A mall developer surveyed 500 customers yesterday to learn what they go shopping for at the mall. 300 customers said they shopped for clothes. 350 customers said they went shopping for electronics. 180 customers said they shopped for both clothes and electronics.

(a) What is the probability that a single randomly-selected customer shopped for neither clothing nor electronics yesterday? Show work. (b) Let C = {customers shopping for clothes} and E = {customers shopping for electronics}. Determine the number of customers belonging to each of the regions I, II, III, IV.