UMUC MATH106 Week 4 Discussion Latest 2019 January Question # 00599590 Subject: Education Due on: 03/19/2019 Posted On: 03/19/2019 06:04 AM Tutorials: 1 Rating: 4.8/5

MATH 106
6383 Finite Mathematics (2192)

Week 4
Discussion

LINEAR PROGRAMMING
(Applied Finite
Mathematics, “Linear Programming: A Geometric Approach”)

For the following
exercises, solve using the graphical method.
Choose your variables, write the objective function and the constraints,
graph the constraints, shade the feasibility region, label all corner points,
and determine the solution that optimizes the objective function.

1.
A farmer has 100 acres of land on which she
plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and
$20 of capital, and each acre of corn requires 16 hours of labor and $40 of
capital. The farmer has at most 800 hours of labor and $2400 of capital
available. If the profit from an acre of wheat is $80 and from an acre of corn
is $100, how many acres of each crop should she plant to maximize her profit?

2.
Mr. Tran has $24,000 to invest, some in bonds
and the rest in stocks. He has decided that the money invested in bonds must be
at least twice as much as that in stocks. But the money invested in bonds must
not be greater than $18,000. If the bonds earn 6%, and the stocks earn 8%, how
much money should he invest in each to maximize profit?

3.
A computer store sells two types of computers,
desktops and laptops. The supplier demands that at least 150 computers be sold
a month. In order to keep profits up, the number of desktops sold must be at
least twice of laptops. The store pays its sales staff a $75 commission for
each desk top, and a $50 commission for each lap top. How many of each type of
computers must be sold to minimize commission to its sales people? What is the
minimum commission?

4.
Mr. Shoemacher has $20,000 to invest in two
types of mutual funds, Coleman High-yield Fund, and Coleman Equity Fund. The
High-yield fund gives an annual yield of 12%, while the Equity fund earns 8%.
Mr. Shoemacher would like to invest at least $3000 in the High-yield fund and
at least $4000 in the Equity fund. How much money should he invest in each to
maximize his annual yield, and what is the maximum yield?

5.
Dr. Lum teaches part-time at two different
community colleges, Hilltop College and Serra College. Dr. Lum can teach up to
5 classes per semester. For every class taught by him at Hilltop College, he
needs to spend 3 hours per week preparing lessons and grading papers, and for
each class at Serra College, he must do 4 hours of work per week. He has
determined that he cannot spend more than 18 hours per week preparing lessons
and grading papers. If he earns $4,000 per class at Hilltop College and $5,000
per class at Serra College, how many classes should he teach at each college to
maximize his income, and what will be his income?

6.
Mr. Shamir employs two part-time typists, Inna
and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an
hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist
must be employed at least 8 hours per week to keep them on the payroll. If Mr.
Shamir has at least 208 pages to be typed, how many hours per week should he
employ each student to minimize his typing costs, and what will be the total
cost?

7.
Mr. Boutros wants to invest up to $20,000 in two
stocks, Cal Computers and Texas Tools. The Cal Computers stock is expected to
yield a 16% annual return, while the Texas Tools stock promises a 12% yield.
Mr. Boutros would like to earn at least $2,880 this year. According to Value
Line Magazine’s safety index (1 highest to 5 lowest), Cal Computers has a
safety number of 3 and Texas Tools has a safety number of 2. How much money
should he invest in each to minimize the safety number? Note: A lower safety
number means less risk.

8.
A department store sells two types of
televisions: Regular and Big Screen. The store can sell up to 90 sets a month.
A Regular television requires 6 cubic feet of storage space, and a Big Screen
television requires 18 cubic feet of space, and a maximum of 1080 cubic feet of
storage space is available. The Regular and Big Screen televisions take up,
respectively, 2 and 3 sales hours of labor, and a maximum of 198 hours of labor
is available. If the profit made from each of these types is $60 and $80,
respectively, how many of each type of television should be sold to maximize
profit, and what is the maximum profit?

9.
A small company manufactures two types of
radios- regular and short-wave. The manufacturing of each radio requires two
operations: Assembly and Finishing. The regular radios require 1 hour of
Assembly and 3 hours of Finishing. The short-wave radios require 3 hours of Assembly
and 1 hour of Finishing. The total work-hours available per week in the
Assembly division is 60, and in the Finishing division, 60. If a profit of $50
is realized for every regular radio, and $75 for every short-wave radio,

a.
how many of each should be manufactured to
maximize profit, and

b.
what is the maximum profit?

10.
A company produces two types of shoes – casual,
and athletic – at its two factories, Factory I and Factory II. Daily production
of each factory for each type of shoe is listed below.

	Factory I	Factory II
Casual	100	200
Athletic	300	100

The company must produce at least
8000 pairs of casual shoes, and 9000 pairs of athletic shoes. If the cost of
operating Factory I is $1500 per day and the cost of operating Factory II is
$2000,

a.
how many days should each factory operate to
complete the order at a minimum cost, and

b.
what is the minimum cost?

SETS AND COUNTING
(Applied Finite Mathematics, “Sets and Counting”)

List the elements of the following sets:

11.
Let Universal set=U={a,b,c,d,e,f,g,h,i,j},
V={a,e,i,f,h}, and W={a,c,e,g,i}.

List the members of the following
sets.

V?W

V?W

V?W ‘

V’ ?W’

12.
Let Universal set=U={1,2,3,4,5,6,7,8,9,10},
A={1,2,3,4,5}, B={1,3,4,6}, and C={2,4,6}.

List the members of the following sets.

A?B

A?C

(A?B’)?C

(A’?B)?C’

Find the number of
elements in the following sets.

13.
In Mrs. Yamamoto’s class of 35 students, 12
students are taking history, 18 are taking English, and 4 are taking both. Draw
a Venn diagram and determine how many students are taking neither history nor
English?

14.
In the County of Santa Clara 700,000 people read
the San Jose Mercury News, 400,000 people read the San Francisco Examiner, and
100,000 read both newspapers. How many read either the Mercury News or the
Examiner?

15.
A survey of athletes revealed that for their
minor aches and pains, 30 used aspirin, 50 used ibuprofen, and 15 used both. No
athletes reported using neither. How
many athletes were surveyed?

16.
In a survey of computer users, it was found that
50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and
IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use
at least one of these Brands?

17.
This quarter, a survey of 100 students at De
Anza College finds that 50 take math, 40 take English, and 30 take history. Of
these 15 take English and math, 10 take English and history, 10 take math and
history, and 5 take all three subjects. Draw a Venn diagram and determine the
following.

a. The
number of students taking math but not the other two subjects.

b. The
number of students taking English or math but not history.

c. The
number of students taking none of these subjects.

18.
In a survey of investors it was found that 100
invested in stocks, 60 in mutual funds, and 50 in bonds. Of these, 35 invested
in stocks and mutual funds, 30 in mutual funds and bonds, 28 in stocks and
bonds, and 20 in all three. Determine the following.

a. The
number of investors that participated in the survey.

b. How
many invested in stocks or mutual funds but not in bonds?

c. How
many invested in exactly one type of investment?

TREE DIAGRAMS AND THE MULTIPLICATION AXIOM

Do the following problems using a tree diagram or the
multiplication axiom.

19.
In a city election, there are 2 candidates for
mayor, and 3 for supervisor. Use a tree diagram to find the number of ways to
fill the two offices.

20.
Brown Home Construction offers a selection of 3
floor plans, 2 roof types, and 2 exterior wall types. Use a tree diagram to
determine the number of possible homes available.

21.
A Virginia license plate consists of three
letters followed by four digits. How many such plates are possible?

22.
How many different 4-letter radio station call
letters can be made if the first letter must be K or W and none of the letters
may be repeated?

23.
How many seven-digit telephone numbers are
possible if the first two digits cannot be ones or zeros?

24.
How many 3-letter word sequences can be formed
using the letters {a,b,c,d} if no letter is to be repeated?

25.
A family has two children, use a tree diagram to
determine all four possibilities.

26.
A coin is tossed three times and the sequence of
heads and tails is recorded. Use a tree diagram to determine the different
possibilities.

27.
A combination lock is opened by first turning to
the left, then to the right, and then to the left again. If there are 30 digits
on the dial, how many possible combinations are there?

28.
How many different answers are possible for a
multiple-choice test with 10 questions and five possible answers for each
question?

PERMUTATIONS

Do the following problems using permutations.

29.
How many three-letter words can be made using the
letters {a,b,c,d,e} if no repetitions are allowed?

30.
A grocery store has five checkout counters, and
seven clerks. How many different ways can the clerks be assigned to the
counters?

31.
A group of fifteen people who are members of an
investment club wish to choose a president, and a secretary. How many different
ways can this be done?

32.
In how many different ways can five people be
seated in a row if two of them insist on sitting next to each other?

33.
In how many different ways can five people be
seated in a row if two of them insist on not sitting next to each other?

34.
In how many ways can 3 English, 3 history, and 2
math books be set on a shelf, if they are grouped by subject?

35.
You have 5 math books and 6 history books to put
on a shelf with five slots. In how many ways can you put the books on the shelf
if the first two slots are to be filled with the books of one subject and the
next three slots are to be filled with the books of the other subject?

COMBINATIONS

Do the following problems using combinations.

36.
How many different 5-player teams can be chosen
from eight players?

37.
In how many ways can a person choose to vote for
three out of five candidates on a ballot for a school board election?

38.
How many 13-card bridge hands can be chosen from
a deck of cards?

39.
There are twelve people at a party. If they all
shake hands, how many different hand-shakes are there?

40.
In how many ways can a student choose to do four
questions out of five on a test?

41.
There are five teams in a league. How many total
league games are played if every team plays each other twice?

42.
A team plays 15 games a season. How many ways are there to end up with 8 wins
and 7 losses for the season (for example: win the first 8, lose the last 7; win
4 then lose 7 then win 4, etc.)?

43.
In how many different ways can a 4-child family
have 2 boys and 2 girls?

44.
A coin is tossed five times. In how many ways
can it fall three heads and two tails?

COMBINATIONS INVOLVING SEVERAL SETS

Following problems involve combinations from several
different sets.

45.
How many 5-people committees consisting of three
boys and two girls can be chosen from a group of four boys and four girls?

46.
How many 4-people committees chosen from four
men and six women will have at least three men?

47.
A batch contains 10 transistors of which three
are defective. If three are chosen, in how many ways can one get two defective?

48.
In how many ways can five counters labeled A, B,
C, D and E at a store be staffed by two men and three women chosen from a group
of four men and six women?

49.
Three marbles are chosen from an urn that
contains 5 red, 4 white, and 3 blue marbles. How many samples of the following
type are possible?

a. All
three white.

b. Two
blue and one white.

c. One
of each color.

d. All
three of the same color.

e. At
least two red.

f.
None red.

50.
Five coins are chosen from a bag that contains 4
dimes, 5 nickels, and 6 pennies. How many samples of five of the following type
are possible?

a.
At least four nickels.

b.
No pennies.

c.
Five of a kind.

d.
Four of a kind.

e.
Two of one kind and two of another kind.

f.
Three of one kind and two of another kind.

51.
Find the number of different ways to draw a
5-card hand from a deck to have the following combinations.

a.
Three face cards.

b.
A heart flush(all hearts).

c.
Two hearts and three diamonds.

d.
Two cards of one suit, and three of another
suit.

e.
Two kings and three queens.

f.
Two cards of one value and three of another
value.