Finite Math

MATH 106 Finite Mathematics Fall, 2012, 4.1

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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. 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. Tom purchases a car for $27,000, makes a down payment of 20%, and finances the rest with a 4-year car loan at an annual interest rate of 3.3% compounded monthly. What is the amount of his monthly loan payment? 1. _______ A. $480.97 B. $509.40 C. $601.21 D. $636.75

2. Find the result of performing the row operation (5)R1 + R2 → R2 2. _______

�2 −13 9 � 1

−5 �

A. �10 −53 9 � 5

−5 � B. � 2 −113 9 �

1 −5

�

C. � 2 −113 4 � 1 0 � D. � 2 −117 44 �

1 −24

�

MATH 106 Finite Mathematics Fall, 2012, 4.1

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3. Find the values of x and y that maximize the objective function 3x + 5y for the feasible

region shown below. 3. _______

A. (x, y) = (0, 20) B. (x, y) = (5, 15) C. (x, y) = (8, 10) D. (x, y) = (10, 0)

4. Adult American have normally distributed heights with a mean of 5.8 feet and a standard deviation of 0.2 feet. What is the probability that a randomly chosen adult American male will have a height between 5.6 feet and 6.0 feet? 4. ______

A. 0.9544 B. 0.7580 C. 0.6826 D. 0.5000

MATH 106 Finite Mathematics Fall, 2012, 4.1

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5. Determine which shaded region corresponds to the solution region of the system of linear inequalities

x + 2y ≥ 4

4x + y ≥ 4

x ≥ 0

y ≥ 0 5. _______

GRAPH A.

GRAPH B.

GRAPH C.

GRAPH D.

MATH 106 Finite Mathematics Fall, 2012, 4.1

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For #6 and #7: A merchant makes two raisin nut mixtures. Each box of mixture A contains 10 ounces of peanuts and 3 ounces of raisins, and sells for $2.80. Each box of mixture B contains 14 ounces of peanuts and 5 ounces of raisins, and sells for $4.00. The company has available 5,200 ounces of peanuts and 1,800 ounces of raisins. The merchant will try to sell the amount of each mixture that maximizes income. Let x be the number of boxes of mixture A and let y be the number of boxes of mixture B. 6. Since the merchant has 1,800 ounces of raisins available, one inequality that must be satisfied is: 6. _______

A. 2.8x + 4y ≤ 1,800

B. 3x + 5y ≤ 1,800

C. 5x + 3y ≥ 1,800

D. 10x + 3y ≤ 1,800 7. State the objective function. 7. _______ A. 5,200x + 1,800y

B. 10x + 14y C. 10x + 3y D. 2.8x + 4y 8. A jar contains 22 red jelly beans, 18 yellow jelly beans, and 14 orange jelly beans. Suppose that each jelly bean has an equal chance of being picked from the jar. If a jelly bean is selected at random from the jar, what is the probability that it is not red? 8. _______

A. 8

3 B.

27

11 C.

27

16 D.

8

5

MATH 106 Finite Mathematics Fall, 2012, 4.1

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9. When solving a system of linear equations with the unknowns x1 and x2 the following reduced augmented matrix was obtained. 9. _______

� 1 4 0 0 � 6 0 �

What can be concluded about the solution of the system?

A. The unique solution to the system is x1 = 4 and x2 = 6.

B. There are infinitely many solutions. The solution is x1 = − 4t + 6 and x2 = t, for any real number t.

C. There are infinitely many solutions. The solution is x1 = 4t + 6 and x2 = t, for any real number t.

D. There is no solution. 10. Which of the following is NOT true? 10. ______

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

the outcomes is equal to 1. 11. In a certain manufacturing process, the probability of a type I defect is 0.07, the probability of a type II defect is 0.09, and the probability of having both types of defects is 0.04. Find the probability that neither defect occurs. 11. ______

A. 0.96 B. 0.88 C. 0.84

D. 0.80 12. Which of the following statements is NOT true? 12. ______

A. The standard deviation is the square root of the variance. B. The variance is a measure of the dispersion or spread of a distribution about its mean.

C. The variance can be a negative number. D. If all of the data values in a data set are identical, then the standard deviation is 0.

MATH 106 Finite Mathematics Fall, 2012, 4.1

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SHORT ANSWER: 13. Let the universal set U = {1, 2, 3, 4, 5, 6}. Let A = {1, 3, 4} and B = {2, 3, 5}.

Determine the set A′ ∩ B . Answer: ______________ (Be sure to notice the complement symbol applied to A.)

14. Consider the following graph of a line.

(a) State the x-intercept. Answer: ______________ (b) State the y-intercept. Answer: ______________ (c) Determine the slope. Answer: ______________ (d) Find the slope-intercept form of the equation of the line. Answer: ____________________ (e) Write the equation of the line in the form Ax + By = C where A, B, and C are integers. Answer: ____________________

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15. A company compiled information about the gender and working status of its 420 employees,

as shown below.

Full-time Part-time Totals

Male 210 40 250 Female 90 80 170 Totals 300 120 420

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

Find the probability that a randomly selected employee is:

(a) a male employee or a full-time employee. Answer: ______________ (b) a male full-time employee. Answer: ______________ (c) male, given that the employee is full-time. Answer: ______________

SHORT ANSWER, with work required to be shown, as indicated. 16. For a six year period, Amanda deposited $700 each quarter into an account paying 5.6% annual interest compounded quarterly. (Round your answers to the nearest cent.) (a) How much money was in the account at the end of 6 years? Show work. (b) How much interest was earned during the 6 year period? Show work.

Amanda then made no more deposits or withdrawals, and the money in the account continued to earn 5.6% annual interest compounded quarterly, for 3 more years. (c) How much money was in the account after the 3 year period? Show work. (d) How much interest was earned during the 3 year period? Show work.

17. Twenty five athletes are competing in an Olympic event. The outcome of the event will be the awarding of three medals, where one athlete wins the Gold Medal, one wins the Silver Medal, and one wins the Bronze Medal. How many different outcomes are possible? Show work.

MATH 106 Finite Mathematics Fall, 2012, 4.1

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18. A recreational club has 14 members. 4 of the members are men and 10 are women. (a) In how many ways can the club choose 6 members to form a volleyball team? Show work. (b) In how many ways can the club choose 6 members to form the volleyball team, if 3 team members must be men and 3 team members must be women? Show work.

(c) If a 6-person volleyball team is selected at random from the 14 club members, what is the probability the team consists of 3 men and 3 women? Show work.

19. In 2006, there were 750 students enrolled in an online training program, and in 2010, there

were 974 students enrolled in the program. Let y be enrollment in the year x, where x = 0

represents the year 2006.

(a) Which of the following linear equations could be used to predict the enrollment y in a given year x, where x = 0 represents the year 2006? Explain/show work. A. y = 750x + 56 B. y = 56x + 224 C. y = 224x + 750 D. y = 56x + 750 (b) Use the equation from part (a) to predict the enrollment for the year 2013. Show work. (c) Fill in the blanks to interpret the slope of the equation: The rate of change of enrollment with respect to time is ______________________ per ________________. (Include units of measurement.)

20. Solve the system of equations using elimination by addition or by augmented matrix methods

(your choice). Show work.

3x + 4y = 2

x − 2y = 4

MATH 106 Finite Mathematics Fall, 2012, 4.1

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

22. During the holidays, 42 people attended a holiday event. 26 of the attendees ate a piece of

cake. 14 of the attendees ate a piece of pie. 32 of the attendees ate a piece of cake or a piece of

pie (or both).

(a) How many of the attendees ate both a piece of cake and a piece of pie? Show work. (b) Let C = {attendees eating a piece of cake} and P = {attendees eating a piece of pie}. Determine the number of attendees belonging to each of the regions I, II, III, IV.

U

P C

II

IV

III I

MATH 106 Finite Mathematics Fall, 2012, 4.1

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23. Use the sample data 52, 22, 4, 22, 90, 48, 70. (a) State the mode. (b) Find the median. Show work/explanation. (c) State the mean. (d) The sample standard deviation is 30.1. What percentage of the data falls within one standard deviation of the mean? Show work/explanation. (d) _______ A. 71% B. 68% C. 57% D. 50%

24. If the probability distribution for the random variable X is given in the table, what is the expected value of X? Show work.

xi – 70 20 40 60

pi 0.20 0.25 0.45 0.10

25. The probability that an eligible voter voted in the November, 2012 presidential election was 0.58. Five eligible voters were randomly selected. Find the probability that exactly 3 of the 5 eligible voters voted in the election. Show work.