MATH 106 QUIZ 4

quiz_4_-_math106_-sol x

MATH 106 QUIZ 4 Due: by 11:59 PM, Sunday, September 22, 2013,

(take-home part) via the Assignment Folder

NAME: _______________________________

I have completed this assignment myself, working independently and not consulting anyone except the instructor.

INSTRUCTIONS

· The take-home part of Quiz 4 is worth 75 points. There are 10 problems (5 pages), some with multiple parts. This quiz is
open book
and
open notes
. This means that you may refer to your textbook, notes, and online classroom materials, but
you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your quiz no later than Sunday, September 22, 2013.

·
Show work/explanation where indicated. Answers without any work may earn little, if any, credit.
You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work.

· General quiz tips and instructions for submitting work are posted in the Quizzes conference.

· If you have any questions, please contact me via Private Message in WebTycho.

1. (4 pts) Determine how many six-character codes can be formed if the first, second, third, and fourth characters are letters, the fifth character is a nonzero digit, the sixth character is an odd digit, and repetition of letters and digits are allowed. (A digit is 0, 1, 2, .., or 9.)
Show your work . 1. ____D__

D. 20,563,920

Number of possible codes

2. (4 pts) Suppose that a multiple choice exam has seven questions and each question has five choices. In how many ways can the exam be completed?
Show your work. 2. ___D___

D. 78,125

Number of possible ways exam can be completed=

3. (4 pts) Given the feasible region shown to the right, find the values of x and y that minimize the objective function 8x + 7y.
Show your Work. 3. ___D____
D. (x, y) = (1, 4)

At (6,0) 8(6) +7(0)=48
At (3,2) 8(3) +7(2)=24+14=38
At (1,4) 8(1) + 7(4)= 36
At (0,7) 8(0)+7(7)= 49

The minimum is at (1,4)

4. (4 pts) Six customers in a grocery store are lining up at the check-out. In how many different orders can the customers line up?
Show your work. 4. __C_____

C. 720

5. (4 pts) A restaurant’s menu has six appetizers, four entrees, and five beverages. To order dinner, a customer must choose one entrée and one beverage, and may choose one appetizer. (That is, a dinner must include one entrée and one beverage, but not necessarily an appetizer. An appetizer is optional.) How many different dinners can be ordered?
Show your work. 5. __A_____

The number of possible dinners without appetizers= (4)(5)= 20

The number of possible dinners including appetizer= (6)(5)(4)=120

Total number of possible dinners = 120+20=140

A. 140

HINT: There are several ways to solve the problem. Here is something to think about, in general: Declining an item actually is a choice. For example, suppose that a person has an option of choosing a sweetener for a coffee drink. The person might use sugar or a particular sugar-free substitute, or decline a sweetener. Declining a sweetener means choosing no sweetener. (choices are sugar, sugar-free, or none).

6. (10 pts) Let U = {10, 20, 30, 40, 50, 60, 70, 80, 90}, A = {30, 50, 60, 90} and B = {10, 20, 50, 80, 90}.

List the elements of the indicated sets. (No work/explanation required).

(a)
{50, 90}

(b)
(Be sure to notice the complement symbol applied to A)
{10, 20, 80}

(c)
(Be sure to notice the complement symbol applied to B)
{30, 40, 50, 60, 70, 90}
7. (7 pts) Use the given information to complete the following table.
n(U) = 80 , n(A) = 22, n(B) = 35, n(A B) = 15. (No work/explanation required)

A

A

Totals

B

15

30

45

B

7

28

35

Totals

22

58

80

8. (9 pts) 200 baseball fans in a Maryland county have been surveyed about the baseball teams they watch on TV. 103 fans watch the Washington Nationals. 90 fans watch the Baltimore Orioles. 170 watch the Washington Nationals or the Baltimore Orioles (or both).
(a) How many of the fans watch both the Washington Nationals and the Baltimore Orioles? Show work.

Lets set of fans who watch Baltimore orioles = B, set Fans who watch the Washington nationals =W

n(WB)=n(W)+n(B)-n(WB)= 103+90-170 = 23 Fans

(b) How many of the fans watch the Baltimore Orioles but not the Washington Nationals? Show work.

Fans

(c) Complete the following Venn diagram, filling in the number of fans belonging in each of the four regions. Circle W = {fans who watch the Washington Nationals} and Circle B = {fans who watch the Baltimore Orioles}. (no explanation required)
(
U
B
W
_23___
__
30
__
___67_
__80___
)

9. (9 pts) A panel of 7 politicians is to be chosen from a group of 15 politicians.
(a) In how many ways can the panel be chosen? Show work/explanation.

The number of ways we can choose 7 politicians out of 15=

(b) Now suppose that the group of politicians consists of 5 Democrats, 7 Republicans, and 3 Independents. In how many ways can the 7-person panel be chosen if it must consist of 3 Democrats, 3 Republicans, and 1 Independent? Show some work/explanation.

The number of ways we can choose 3 out of 5 Democrats :

The number of ways we can choose 3 out of 7 Republicans

The number of ways we can choose1 out of 3 Independents

The total number of ways we can choose 7politicans = 10(35)(3)=1050

10. (21 points)Two kinds of cargo, A and B, are to be shipped by a truck. Each crate of cargo A is 25 cubic feet in volume and weighs 100 pounds, whereas each crate of cargo B is 40 cubic feet in volume and weighs 120 pounds. The shipping company collects $180 per crate for cargo A and $220 per crate for cargo B. The truck has a maximum load limit of 1,200 cubic feet and 4,200 pounds. The shipping company would like to earn the highest revenue possible.
(a) Fill in the chart below as appropriate.

Cargo A
(per crate)

Cargo B
(per crate)

Truck Load Limit

Volume

25

40

1200

Weight

100

120

4200

Revenue

180

220

Let x be the number of crates of cargo A and y the number of crates of cargo B shipped by one truck.
(b) State an expression for the total revenue R earned from shipping x crates of cargo A and y crates of cargo B.

(c) Using the chart in (a), state two inequalities that x and y must satisfy because of the truck’s load limits.

(d) State two inequalities that x and y must satisfy because they cannot be negative.

(e) State the linear programming problem which corresponds to the situation described. Be sure to indicate whether you have a maximization problem or a minimization problem, and state the objective function and all the inequalities. (This part is mostly a summary of the previous parts)

Maximize
Subject to

(f) Solve the linear programming problem. You will need to find the feasible region and determine the corner points. You do
not
have to submit your graph, and you do
not
have to show algebraic work in finding the corner points, but you must list the corner points of the feasible region and the corresponding values of the objective function.

Corner Point (x, y)

Value of Objective Function

(24,15)

$7620 Maximum

(0,30)

$6600

(42,0)

$7560

(0,0)

0

(g) Write your conclusion with regard to the word problem. State how many crates of cargo A and how many crates of cargo B should be shipped in the truck, in order to earn the highest total revenue possible. State the value of that maximum revenue.
The revenue is maximum at (24, 15), so when 24 crates of A and 15 Crates of B are shipped the Revenue is a maximum value of $7620.
1
B
A
Ç
¢
B
A
¢
È
B
A
Ç