FINAL EXAM
Finite Mathematics
06/27/2006Name______
PLEASE TURN OFF YOUR CELL PHONES AND ALL
OTHER ELECTRONIC DEVICES NOW!
PLEASE PLACE YOUR PHOTO ID ON YOUR DESK SO THAT IT CAN BE CHECKED.
For this exam, you are permitted the following:
- A TI 83, 84 or 86 calculator
- A 8 ½” X 11” sheet of handwritten notes
- Pencil, Eraser & Ruler
Do not write below this line
Part I
Problem # / Points1
2
3
4
5
Total
Part II
Number of questions correct: ______
Points:______
Grand Total:______
Part I, Long Problems
SHOW EVERY STEP WHEN YOU SOLVE THE EQUATION. WRITE DOWN THE VALUE OF EVERY VARIABLE USED, AND INSERT THESE VALUES INTO THE FORMULA! If you only give the answer I will only credit you 1 point.
1. Julia works as a nurse, and earns $4,300 per month after taxes. She wants to open an account into which she intends to deposit 8% of her monthly take-home pay, at the end of each month over the next 5 years. She finds an account with 3.7% annual interest, compounded monthly.
a) What is the APR of that account, in percent?
b) How much money will Julia have in her account after five years?
c) Suppose Julia gets a pay raise after working for 2 years, increasing her monthly take-home pay by 5%, and suppose that she continues to deposit 8% of her monthly take-home pay into the account. How much money will she have saved in the account after five years?
2. One box, A, contains three red and four green balls, while a second box, B, contains eight red and two green balls. A box is selected at random, then a ball is drawn from that box.
a) What is the probability that the ball is red, given that it came from box A?
b)What is the probability that the ball is green and came from box B?
c) What is the probability that the ball is red?
d) What is the probability that the ball came from box B, given that it was red?
e) What is the expected number of red balls?
3.The friends of Owls club is planning a picnic. The committee members in charge estimate that they will spend a total of $400 on rental of a pavilion, decorations and so on. They also agree that they will spend $7.50 per person for food and drinks. They plan to charge each person who attends $10. Let x be the number of people who attend the picnic.
a)Find linear functions for the cost, C(x), revenue, R(x), and profit, P(x).
b) Find the number of people needed to attend the picnic in order to break even.
c) Graph the functions in a), marking the break-even point and other points of interest in your graph.
4.Solve the following linear programming problem graphically, carefully following the steps we have learned in class. The answer should be the minimum value of z, in addition to the values of x and y at the minimum.
Minimizez = 2x + 4y
Subject to:x + 2y 10
3x + y 10
x 0
y 0
5.Solve the linear programming problem below, using the simplex method. The answer should be the maximum value of P, and the values of the variables for which f reaches its maximum value.
(Write down the initial simplex tableau, and the tableau after each pivoting step, even if you use the calculator to do the pivoting. You should also write down what row operations were carried out between each step.)
Maximize f = 2x + 5y
Subject to x + 2y 8
x 8
x 0, y 0
Part II, Multiple Choice
Directions: 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 grade only the first 12 you do, and “X” out the rest.
Read each question carefully at least twice before answering it. Choose the BEST of the answers given by circling the letter of your choice. Only one answer is correct for each question.
------
6.If the matrices A and B are defined as follows:, find AB
a) AB is not definedb) c) d) e)
------
7.A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting exactly five twos.
a) .0003b) .129c) .083d) .921e) .598
------
8.Solve the following matrix equation for :
a) b) c) d) no solutione)
------
9.Three people are running a foot race. In how many ways can they finish the race if there are no ties?
a) 3 waysb) 6 waysc) 9 waysd) 10 wayse) 12 ways
------
10.If 2 cards are drawn from a deck of 52 cards, what is the expected number of spades?
a) .50b) .47c) .25d) .75e) .63
------
11.Perform the row operation (-2)R1+ R2 R2 on the matrix:
a) b) c) d)
e)
------
12.Suppose the feasible region is shaded below, and the objective function to maximize is P = 2x + y. The figure shows the lines P = 20 and P = 10. At what point (x,y) does the maximum value of P occur?
a) (0,0)b) (10,0)c) (0,20)d) (10,10)e) (20,0)
------
13.Mark wants to start an IRA that will have $200,000 when he retires in 25 years. How much should he invest quarterly in his IRA if the interest is 6% compounded quarterly?
a) $431.33b) $2,000c) $874.11d) $3,874.11e) $2,554.37
------
14.Solve the following system of equations:
x + 2y = 6
3x + 6y = 19
a) (2,2)b) (1, )c) x = -2y+6, y = any real numberd) no solutione) (0,0)
------
15.The linear programming problem below is on standard form.
Minimizez = 49x + 69y
Subject to:x + 2y 2
3x + 4x 3
x 0, y 0
a) Trueb) False
------
16.Jerry Highland sells used books. He pays $2.50 for each book. He has startup costs of $4,000 for the first year. He sells each book for $5.25. Express the cost function C(x) as a function of x, where x represents the number of books sold in the first year.
a) C(x) = 4000x + 2.50b) C(x) = 2.50x + 4000 c) C(x) = 2.50x - 5.25
d) C(x) = 5.25x + 2.50e) C(x) = 2.75x – 4000
------
17.If 2 cards are drawn from a deck of 52 cards, find the probability that the second card is red, given that the first card was a heart.
a) b) c) d) e)
------
18.An 8-sided die is rolled. The sides contain the numbers 1,2,3,4,5,6,7,8. List the sample space of rolling one die.
a) {1,8}b) {1,2,3,4,5,6,7,8}c) {1,64}d) {8}e) {2,4,8}
------
19.A $124,000 trust is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%, annually. The sum of the amount invested in bonds and the amount invested in CDs must equal the mortgage investment. To earn an $11,400 annual income from the investments, how much should the bank invest in each?
Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount invested in mortgages.
a)b)c)
d)
------
20.Which of the following is the best investment, and which is the worst:
- Investment A pays 10% annual interest compounded daily
- Investment B pays 10.5 % annual interest compounded monthly
- Investment C pays 10.55 % annual interest compounded quarterly
a) Best is A, worst is Bb) Best is B, worst is Cc) Best is A, worst is C
d) Best is C, worst is Ae) Best is B, worst is A
------
21.Perform the pivot operation on the pivot element in the following simplex tableau. After the pivot operation has been completed,
x y s1 s2 f
a) x = 60, y= 0, f = 98 b) x = 20, y = 60, f = 38c) x = 0, y = 60, f = 38
d) x = 20, y= 60, f = 98e) x = 20 , y= 40, f = 58
------