MATH 106 6383 Finite Mathematics (2192)

Week 8 Discussion

Linear Programming

1. Graphically solve the following:

Maximize , subject to:

2. Solve the given linear programming problem graphically:

Nutrition: A dietitian is to prepare two foods to meet certain requirements. Each pound of Food Icontains 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?

3. Solve the given linear programming problem graphically:

Maximize Subject to:

4. Solve the given linear programming problem graphically:

Maximize Subject to:

5. Solve the given linear programming problem graphically:

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?

6. Solve the following linear programming problem graphically:

Maximize Subject to:

7. Solve the following linear programming problem graphically:

Minimize subject to:

8. Solve the following linear programming problem graphically:

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?

Mathematics of Finance

9. Find the amount that aprincipal of $800will accumulate in 12 years for each given account:

a. 7% simple interest

b. 7% compounded quarterly

c. 7% compounded monthly

10. Find the principal P required to achieve a future amount A=$5000 with an interest rate of 6% compounded quarterly for 5 years.

11. Antonio 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?

12. 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?

13. 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?

14. 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.

15. A small business borrows $80,000 at 8.4% interest compounded monthly for 8 years.

a. What is the monthly payment?

b. What is the unpaid balance at the end of the first year?

c. How much interest was paid in the first year?

16. The Washingtons decide to buy a $400,000 home by putting a 20% down payment and financing the balancewith a 30-year fixed mortgage at 4.2%.

What is the amount of the monthly payment for their mortgage?

What is the total interest paid on the mortgage at the end of 30 years?

17. Jared buys a new motorcycle for $8500 from The Fastrack Motorcycle Co., which agrees to finance 80% of the purchase at 9.6% interest for a period of 60 months.

What is Jared’s monthly payment?

What is the total interest paid on Jared’s loan at the end of 60 months?

Sets and Counting

18. 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’

A U C

A’ ∩ B’

19. 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.

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?

20. 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).

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?

21. In how many ways can eight books be arranged on a shelf if:

there are no restrictions

one of the books, Of Mice and Men, must be displayed on the left end?

22. How many different car license plates can be made using two letters followed by four digits if:

there are no restrictions

no letter can be repeated

the last digit must be a 4, and no digit can be repeated

23. How many permutations are there of the word COLLEGE?

24. Nine people are to travel to dinner in a five-passenger van and a four-passenger car. How manydifferent groups of five and four are possible for the trip?

25. In how many ways can four couples be seated in a row of eight seats at a theater if each couple is seated together?

26. 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?

27. 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?

28. 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?

29. 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:

exactly 2 freshmen and 1 junior .

all of one class.

least 2 juniors.

30. In how many ways can the names of six candidates be arranged on a ballot if Joe Shmo, must be at the top?

31. 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?

32. How many different “words” can be formed from the letters in the word MATHEMATICS?

33. 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:

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?

Probability

34. A three card hand is dealt from a standard deck of 52 playing cards.

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.

35. A pair of standard 6-sided dice is to be rolled. Find the probability that:

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 rolle

36. Find the expected winnings for the game of chance described as follows: 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).

37. 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.

.