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I have 8 flavors of ice cream and 4 different toppings. How many different ice cream sundaes can I make if I choose one flavor of ice cream and only one topping?

Discrete Structures Assignment

For all the problems show your work. Giving only the correct answer is insufficient for getting the full credit.

1. Solve the following problems using the multiplication principle for counting.

(a) I have 8 flavors of ice cream and 4 different toppings. How many different ice cream sundaes can I make if I choose one flavor of ice cream and only one topping?

(b) A restaurant is preparing a prix fixe menu that has 4 choices of appetizers, 2 soups, 7 entrees and 4 desserts. How many different dinners can be made from the menu? Assume that each guest chooses one of each dish.

(d) How many 4-letter “words” can I make from the letters of the word COMPUTER? You can re-use letters. For example CCCC, UPCT, etc..

(e) How many 3-letter “words” can I make from the letters in COMPUTER -? This time you cannot re-use letters. For example COM, POT, PUT, etc..

2. I have a bag of marbles with 5 yellow marbles, 11 red marbles and 7 brown marbles. Two marbles are drawn at random. First, draw the tree diagram and then compute the following probabilities – give your answers in fractions, in lowest terms.

With the aid of the probability tree, find the probability of:

(a) choosing 2 red marbles.

(b) Choosing two marbles of the same color.

3. We are studying the success rate of newly opened hotel-casinos in Las Vegas. We measure that (a) 65% of these hotel-casinos generate positive revenue from the gambling business (i.e. the casino), and (b) 45% generate positive revenue from the hospitality business (i.e. the hotel). We are also told that 20% of these hotel-casinos are profitable in both gambling and hospitality.

a. What percentage of the newly opened hotel casinos generate positive revenue either from gambling or from hospitality?

b. What percentage of the newly opened hotel casinos generate positive revenue neither from gambling nor from hospitality?

4. 87% of the people who buy a gas grill are male, and 69% and homeowners. Assuming that the percentage of the buyers that are male homeowners is 59%, what fraction of the buyers either male or homeowners.

5. You make 40% of the basketball shots you take. What is the probability you make exactly 7 out of 12 shots?

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Southern Sporting Good Company makes basketballs and footballs

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Post date July 3, 2021
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1. Southern Sporting Good Company makes basketballs and footballs. Each product is produced from

two resources rubber and leather. Each basketball produced results in a profit of $11 and each

football earns $15 in profit. The resource requirements for each product and the total resources

available are as follows:

Product Resource Requirements per Unit

Rubber (lb.) Leather (ft 2 )

Basketball 2.8 3.7

Football 1.5 5.2

Total resources available 600 900

a. Find the optimal solution.

b. What would be the effect on the optimal solution if the profit for the basketball changed from

$11 to $12?

c. What would be the effect on optimal solution if 400 additional pounds of rubber could be

obtained? What would be the effect if 600 additional square feet of leather could be obtained?

2. A company produces two products, A and B, which have profits of $9 and $7, respectively. Each

unit of product must be processed on two assembly lines, where the required production times are

as follows:

Product Resource Requirements per Unit

Line 1 Line 2

A 11 5

B 6 9

Total Hours 65 40

a. Formulate a linear programming model to determine the optimal product mix that will

maximize profit.

MAT540 Homework Week 7

Page 2 of 3

b. What are the sensitivity ranges for the objective function coefficients?

c. Determine the shadow prices for additional hours of production time on line 1 and line 2 and

indicate whether the company would prefer additional line 1 or line 2 hours.

3. Formulate and solve the model for the following problem:

Irwin Textile Mills produces two types of cotton cloth denim and corduroy. Corduroy is a heavier

grade of cotton cloth and, as such, requires 8 pounds of raw cotton per yard, whereas denim

requires 6 pounds of raw cotton per yard. A yard of corduroy requires 4 hours of processing time;

a yard od denim requires 3.0 hours. Although the demand for denim is practically unlimited, the

maximum demand for corduroy is 510 yards per month. The manufacturer has 6,500 pounds of

cotton and 3,000 hours of processing time available each month. The manufacturer makes a profit

of $2.5 per yards of denim and $3.25 per yard of corduroy. The manufacturer wants to know how

many yards of each type of cloth to produce to maximize profit. Formulate the model and put it

into standard form. Solve it

a. How much extra cotton and processing time are left over at the optimal solution? Is the demand

for corduroy met?

b. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it

select? How much? Explain your answer.

4. The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and

tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs

$210. The Bradleys’ have a budget of $52,500 for next year. The government limits the number

of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the

profit from each acre of tobacco is $520. The Bradleys’ want to know how many acres of each

crop to plant in order to maximize their profit.

a. Formulate the linear programming model for the problem and solve.

b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use

the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The neighbor is

offering the land to them for $110 per acre. Should the Bradleys’ lease the land at that price?

What is the maximum price the Bradleys’ should pay their neighbor for the land, and how

much land should they lease at that price?

MAT540 Homework Week 7

Page 3 of 3

d. The Bradleys’ are considering taking out a loan to increase their budget. For each dollar they

borrow, how much additional profit would they make? If they borrowed an additional $1,000,

would the number of acres of corn and tobacco they plant change?

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What is the order of the graph?

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Graph theory worksheet:

· What is the order of the graph?

· What is the degree of vertex N?

· What is the degree of vertex G?

· How many components does the graph have?

2. Let a graph have vertices D,E,F,G,H,I and edge set {{D,E},{D,F},{D,G},{D,H},{E,I},{H,I}}.

a. Draw the graph.

b. What is the degree of vertex G?

c. What is the degree of vertex D?

d. How many components does the graph have?

3. Which of the following degree sequences are possible for a simple graph?

a. (5,3,3,3,2,2)

b. (9,8,8,7,4,4,4,2,2,1)

c. (8,6,3,3,2,2,2,1)

d. (6,5,4,4,3,3,3)

4. You are a mail deliverer. Consider a graph where the streets are the edges and the intersections are the vertices. You want to deliver mail along each street exactly once without repeating any edges. Would this path be represented by a Euler circuit or a Hamiltonian circuit?

5. A telephone company employee needs to check the telephone lines hanging from telephone poles for a cut in the line over a grid of streets in a city without service. Would the path taken on a graph representing the situation be an Euler circuit or a Hamiltonian circuit?

6. Construct a simple graph with vertices C, D, E, F, G, H whose degrees are 2, 2, 2, 0, 2, 2. What is the edge set? Draw the graph.

7. Which of the following graphs are connected?

8. How many k-cliques are there in K_n?

9. Check the graph below for any of the following: Euler path, Euler circuit, Hamiltonian path, Hamiltonian circuit. Do any apply, if so list them.

Diagram Description automatically generated

10. Check the graph below for any of the following: Euler path, Euler circuit, Hamiltonian path, Hamiltonian circuit. Do any apply, if so list them.

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new work is focused on aspects of logic and logical methods of thinking and problem-solving

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new work is focused on aspects of logic and logical methods of thinking and problem-solving.

One important area is inductive and deductive reasoning. We use these types of reasoning to draw conclusions or make decisions or solve problems.

Chapter 1 section 1-1 in the e-text presents information about these two types of reasoning.

After carefully studying this section respond to the below tasks in the context of Math 105:

Provide an example in the context of Math 105 of a conclusion or decision you made based on inductive reasoning. Explain why inductive reasoning was involved.
Provide an example in the context of Math 105 of when you used deductive reasoning. Explain why this was a use of deductive reasoning.

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The “average” income in the United States can be given by the mean or the median.

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The “average” income in the United States can be given by the mean or the median.

Which measure would be used in anti-U.S. propaganda? Explain your answer.
Which measure would be used in pro-U.S. propaganda? Explain your answer.

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Yearly, students are invited to membership in the honor society for your major if their class rank is at least in the 90thpercentile for all students in that major. In 2019 you ranked 30 out of 128 students in your major

Part 1

Yearly, students are invited to membership in the honor society for your major if their class rank is at least in the 90thpercentile for all students in that major. In 2019 you ranked 30 out of 128 students in your major.

a. Will you be invited to membership?

b. If you had the data on the cumulative gpa of each member of the honor society at the end of the spring semester in 2019, which measure of average would be the most meaningful –mean, median, mode? Explain.

Part 2

A jury pool consists of 9 men and 11 women. The jury will be made up of 12 jurors.

a. If one person from among 6court officers (3 women and 3 men) is chosen at random from to draw the names out of a hat, what is the probability that the court officer chosen to draw the names is a man?

b. How many ways can the jury be formed if there are no restrictions on composition?

c. How many ways can 8women be chosen?

d. How many ways can 4 men be chosen?

e. What is the probability that the random selection of the jury will result in 8 women and 4 men?

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compare quantities are by using percentages, ratios, rates, and unit labels

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Respond to the following in a minimum of 175 words:

We began to examine relationships between quantities. Some ways we can compare quantities are by using percentages, ratios, rates, and unit labels. In your life, there are likely many situations that require you to compare quantities, understand the meaning of those comparisons, and make decisions based on those comparisons. What is an example of a situation from your professional or personal life that requires you to compare, understand, and make decisions based on quantitative comparison? Be sure to describe the types of quantitative comparisons you had to make, what decisions you made, and why. Would your decision be different if you had the benefit of learning this week’s concepts? Why or why not?

RESPOND TO AUDREY AND BRANDON BE CONSTRUCTIVE AND PROFESSIONAL.

Audrey post

In the fall I will start a new position as a testing coordinator for an elementary school. In this role I will be responsible for testing kids on quarterly tests and a yearly state test. I will have to read data comparing the number of students with the best test outcomes to the students with the worst scores and in between. It is important that I master the concepts of week 2 as I will be using percentages and ratios on a daily basis in this new role and will need to have accurate data on each student.

Brandon post

At my current job we focus on rates, rates to my job dictate how well our building is performing to other locations and how well each individual employee is performing against their peers as well. Day to day I would have to track the previous weeks rates for a group of individuals and compare to the week to date rates. By comparing these metrics I can track if those employees a moving in a positive or negative way for the current week. Understanding the comparison I can prepare a way to help those employees but also bridge to my boss what the miss was on the rates for the current week vs the previous week if we fell short on our goal. However, some barriers my interrupt the employees ability to work because of this I have to find the percentage of employees affected by those barriers and compare that percentage to the percentage unaffected and determine if the rates created were on spot or was it off due to the amount to employees affected by operational barriers

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