1. Five cards are drawn from a standard deck of 52 cards, without replacement. What is the probability that (a) all of the cards are spades?

Transcription

1 Math 13 Final Exam May 31, 2012 Part I, Long Problems. Name: Wherever applicable, write down the value of each variable used and insert these values into the formula. If you only give the answer I will only credit you one point. 1. Five cards are drawn from a standard deck of 52 cards, without replacement. What is the probability that (a) all of the cards are spades? (b) at least one card is a spade? (c) it is a full house i.e. three of one kind and two of another kind. Examples of a full house: three sevens and two kings, three fours and two tens, three queens and two jacks, etc.

2 2. Abby would like to retire in 25 years. She is planning on making equal payments at the end of each month into a retirement account, to save up the $300, 000 she estimates she will need to retire. Assume the average annual interest rate of the account is 4% and that the account has monthly compounding. (a) Find the amount of each payment. (b) Upon retiring, Abby is planning on making withdrawals of $1700 from her account each month. Assume that Abby succeeded in saving up $300, 000 for her retirement, and that the average annual interest rate of the account is 4% compounded monthly, how long will it take until she runs out of money in her retirement account? Page 2

3 (c) Things did not go exactly as planned with regards to Abby s saving for her retirement. Abby lost her job after 15 years, and although she got a new job, she was forced to take a big pay cut and she could no longer afford to save for her retirement. Assuming she didn t make any withdrawals, how much did she end up with in her retirement account after 25 years. Page 3

4 3. Financial Planning. Oscar has $50, 000 to invest. As his financial planner, you recommend that he diversify into three mutual funds, based on their average annual returns over the past 10 years ending December 31, 2009: Total Return at 5%, High Income at 7%, and Global Bond at 9%. Oscar wants to have an average annual return of $3020 over the next 10 years. Since Oscar is worried that interest rates my go up, resulting in bonds declining, he wants the amount invested in the Total Return Fund to be four times that invested in the Global Bond Fund. Find the amount of each investment. a) Name all variables used, including units. b) Write a system of equations that describes Oscar s financial needs and restrictions. c) Then set up the augmented matrix. Do not solve this problem! Page 4

5 4. Solve the following linear programming problem graphically following all the steps we learned in class. The answer should be the minimum value of z, z min, in addition to the values of x and y at the minimum. Do a nice job graphing. SHOW ALL YOUR WORK!! Minimize z = 2x + 5y subject to x + y 5 x + 3y 9 x 0; y 0 y x z min =, (x, y) = (, ) Page 5

6 5. Solve the following linear programming problem using the simplex method. Write down the initial simplex tableau and the tableau after each pivot operation, even if you use the calculator to do the pivoting. You should also write down what row operations were carried out between each step. The answer should be the maximum value of P, Pmax in addition to the values of x, y and z at the maximum. Maximize P = x + 3y + 2z subject to 2x + y + z 50 2x + 3y + 3z 60 x 0; y 0; z 0 Pmax =, (x, y, z) = (,, ) Page 6

7 Part II, Multiple Choice. Answer only 12 out of the 16 multiple-choice questions. Mark a GIANT X through the 4 problems you do not do. If you answer more than 12 problems, I will only give you credit for the first 12 you do, and X out the rest. 6. A box contains 5 green, 3 blue, and 2 yellow marbles. A marble is randomly drawn. What is the probability that the marble is green or blue? A B. 3 5 C. 5 8 D. 4 5 E Two dice are tossed, and the number of pips on their upper sides are added. What is the probability that the sum is greater than 8? 6. A. 1 3 B C. 2 9 D. P (2, 2) P (6, 2) E The reduced row echelon form of the augmented matrix of a system of equations is given. State the solution. x y z w A. x = 5, y = 8, z = 1, w = 4 B. x = 3w + 5, y = 2z 4w 8, z = any number, w = any number C. x = 8, y = 5, z = 1, w = 4 D. x = 3w 5, y = 2z + 4w, z = any number, w = any number E. no solution 9. Events A and B are mutually exclusive with P (A) = 0.4, and P (B) = 0.3. Find P (A B) A. 0.9 B. 0.1 C. 0.7 D E Page 7

8 10. In a certain art class with 32 students, 14 students liked acrylic painting, 11 students liked sketching, and 9 students liked neither. How many students only liked acrylic painting? A. 21 B. 23 C. 14 D. 12 E Henry and Jill are buying a house in Live Oak. The house cost 425, 000 and they will pay a 20% down payment. The remaining balance will be financed with a 30-year fixed, 3.75% mortgage loan. What is their monthly mortgage payment? 10. A. $1, B. $1, C. $1, D. $ E. $1, Find the APR (annual percentage rate = effective interest rate) in an account paying 4.7% compounded weekly. A. 4.76% B. 4.8% C. 4.81% D. 4.85% E. 5.1% 13. After performing the pivot operation on a simplex tableau, the resulting tableau is shown below. At this point, what is the basic feasible solution? x y s 1 s 2 s 3 P A. x = 0, y = 120, s 1 = 0, s 2 = 1, s 3 = 8, P = 1 B. x = 8, y = 3, s 1 = 5, s 2 = 8, s 3 = 5, P = 120 C. x = 1, y = 0, s 1 = 8, s 2 = 0, s 3 = 120, P = 5 D. x = 0, y = 8, s 1 = 1, s 2 = 5, s 3 = 0, P = 120 E. No solution Page 8

9 14. The cost and revenue functions for a small business are given below, where x is the number of goods produced/sold. Find the break-even quantity. C(x) = 55x R(x) = 80x A. 250 B. 800 C. $5, 500 D. $20, 000 E Find the inverse of matrix A below, when a 0. A = 5/a 2 3 a a 2 a 2 A. A 1 = B. A 3 5/a 1 = 3 5/a a 3 5/a 3 C. A 1 = D. A 2 5a 1 = 2 a E. A does not have an inverse Find BA, given the matrices below. A = B = A E. Undefined B C D Page 9

10 17. In the following linear programming problem, the corner points of the feasible region are: (0, 1), (0, 4), (3, 4), (3, 0), and (2, 0). In which one of the corner points is C minimized? Minimize C = 3x 2y subject to: x + 2y 2 y 4 x 3 A. In (0, 1). B. In (0, 4). C. In (3, 4). D. In (3, 0). E. In (2, 0). 18. Tommy runs a game booth at a school fair. The player pays $1 to draw a card from a standard deck of 52 cards. If it is an ace, he wins $9.75, and if it is not an ace, he wins nothing. What are Tommy s expected earnings per draw? 17. A. $0.25 B. $0.33 C. $0.57 D. $1 E. $ Page 10

11 19. In the following simplex tableau, where is the next pivot element located? x y z s 1 s 2 s 3 P A. row 3, column 2 B. row 1, column 2 C. row 1, column 1 D. row 2, column 1 E. row 3, column Three fair coins were tossed. What is the probability of obtaining exactly 2 heads? 19. A. 2 3 B. 1 4 C. 4 3 D. 1 2 E Solve the following system of equations. 6x + 2y = 12 3x 3y = 20 A. x = 2, y = 7 B. no solution C. x = 2 3, y = any number D. x = 1 3, y = 7 E. none of the above 21. Page 11