Michael has a box of colored balls. It contains two red balls, three green balls, one purple ball, two yellow balls, and five blue balls. Michael will perform an experiment which goes as follows.

First, a ball is drawn from the box at random, the color of the ball is noted (R for red, G for green, etc.), and the ball is set aside (i.e. not replaced into the box). The next stage of the experiment depends on the color of the ball Michael draws. If the ball is red, he will draw another ball and note its color. If the ball he draws at the beginning is green, he will draw five more balls, simultaneously and at random, and note how many of the balls he has drawn are red. Otherwise (if the ball drawn at the beginning is neither green nor red), he will flip a coin and note the result (H for heads, T for tails). Thus, for example, BH, RR, and G2 are three possible outcomes of the experiment.

Let S denote the sample space of the experiment, and let E denote the event that the ball drawn at the beginning is blue.
What is n(S)?

What is n(E′)?

Andre's dog Fifi knows fifteen tricks, five of which are interesting. Fifi does a show by performing either two or three different tricks one after another. If the first two tricks are both interesting, she will perform a third trick; otherwise, she only performs two tricks.
How many different shows could Fifi do?

Proximity measures are defined in terms of pairs of objects. Assume that our pairs are actually groups of objects which are not vectors. How might we define similarity in this case?

Determine if the following set W spans R^3 .

W = {(1, 3, 1),(2, 1, 1),(−1, 0, 1)}

Reflect on the concept of composite and inverse functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest composite and inverse functions you can imagine? In your day to day, is there any occurring fact that can be interpreted as composite and inverse functions? What strategy are you using to get the graph of composite and inverse functions?

3) You are an experienced machinist with a small tool shop. You have been asked to submit a bid to produce twenty (20) special clamps for a new die that is under construction. You estimate that the material costs (metal and fasteners) for each clamp are $150.00. You also estimate that 10.00 labor hours are needed to produce the first clamp. The value of your labor is $60/hour.

a. From similar orders in the past, you estimate that there is a 80% learning curve for this type of work. Using this information, what is the lowest total bid you should submit if you charge a 25% markup on material and you wish to recoup your labor costs? (Hint: if you know how, it will save time to set up formulas in Excel. Just attach a print-out of Excel work) (3 POINTS)

b. Your potential customer is receptive to your bid from part A but is hesitant to award you the order because the average cost per clamp that you bid is 5% higher than his target. You respond to the customer that if they would increase their order from 20 clamps to X clamps, you could beat his target price. What is X? (Hint: again, while you can do it by hand, it is tedious so setting it up in Excel saves time. Just attach a print-out of Excel work) (3 POINTS)

Find y as a function of x if y^(4)-6y'''+9y''=-200e^(-2x) y(0)=17 y'(0)=7 y''(0)=1 y'''(0)=16

Please provide proofs for parts i.)-iii.)

(i) Refer to the sequence in 1(ii). Show that with respect to the supremum norm on ?[0,1] this is a bounded sequence that has no convergent subsequence. (hint: What is the value of ‖?? − ??‖∞ if ? ≠ ??)

(ii) Refer to the sequence in 1(v). Show that this is a bounded sequence with respect to the 1-norm on ?[0,1] that has no convergent subsequence.

(iii) Let ℎ?(?) = sin??. Show that with respect to the 2-norm ?[0,2?], (ℎ?) is a bounded sequence that has no convergent subsequence. (This exercise shows that the Bolzano-Weierstrass Theorem does not generalise to ?[?,?] with any of the 3 “natural” norms on ?[?,?])

Note: sequences from 1ii.) and 1v.) are pointwise functions and are defined respectively below:

1ii.) For ? ≥ 2, define the function ?? on [0,1] by: ??(?) =( ??, if 0 ≤ ? ≤ 1/?)
(2-??, if 1/?< ? ≤ 2/n)

(0, if 2/?< ? ≤ 1)

1v.) Hn=n?? (and fn is defined as above)
  

Formulate the situation as a linear programming problem by identifying the variables, the objective function, and the constraints. Be sure to state clearly the meaning of each variable. Determine whether a solution exists, and if it does, find it. State your final answer in terms of the original question. A rancher raises goats and llamas on his 400-acre ranch. Each goat needs 2 acres of land and requires $100 of veterinary care per year, and each llama needs 5 acres of land and requires $80 of veterinary care per year. The rancher can afford no more than $13,200 for veterinary care this year. If the expected profit is $42 for each goat and $63 for each llama, how many of each animal should he raise to obtain the greatest possible profit? The rancher should raise goats and llamas for a maximum profit of $ .

If A is a set, write A={(x,x):x∈A}.

Prove: If S is a strict partial order on A, then S∗=S∪A is a partial order on A .

1.

1. A house was listed for sale for $218,600. When it didn't sell, the owners reduced its price by 10%.

   1. What is the percent change in the house price? (Remember that we use negative numbers to represent percentage decreases.

   2. By what number can we multiply the original listing price to get its new price? (Hint: What portion of the original price remains?) We can multiply the listing price by

   3. What is the new price after the reduction?

2. A bookstore is having a 30% off sale. A new hardcover book normally retails for $28.99.

   1. What is the percent change in the book price because of the sale?

   2. By what number can we multiply the original retail price to get its sale price? (Hint: What portion of the original price remains?) We can multiply the retail price by

   3. What is the price of the book on sale?

3. A furniture store is going bankrupt and liquidating their inventory at prices that are 65% off of the retail price. One sofa retails for $1589.99.

   1. What is the percent change in the sofa price because of the sale?

   2. By what number can we multiply the original retail price to get its sale price? (Hint: What portion of the original price remains?) We can multiply the retail price by   

   3. What is the price of the sofa on sale?

2.

The population of a town (Town #1) increased by 1.5% per year from 1995 to 2015. The town's population at the beginning of 1995 was 16,112.

1. Write a formula for function ff that models the town's population in terms of the number of years since the beginning of 1995, t

2. Over the same time period, a second town's (Town #2's) population increased by 2.3% per year. The town's population at the beginning of 1995 was 23,914. Write a formula for function g that models the town's population in terms of the number of years since the beginning of 1995, t

3.

A city's population was 18,400 people at the beginning of 2000 and grew by 4% per year. Use this information to complete the following. You may round off all growth factors to two decimal places, percent change values to the nearest percent, and population values to the nearest person.

1. The 1-year growth factor for the town's population is answer: 1.04   

2. The 5-year growth factor for the town's population is    answer: 1.21

3. The town's population 5 years after the beginning of 2000 was

4. The 5-year percent change for the town's population is

5. The 10-year growth factor for the town's population is    answer:1.48

6. The town's population 10 years after the beginning of 2000 was

7. The 10-year percent change for the town's population is

4.

1. If a function's 1-unit growth factor is 1.22, then:

   1. the 3-unit growth factor is answer: 1.81

   2. the 3-unit percent change is

2. If a function's 1-unit growth factor is 1.07, then:

   1. the 8-unit growth factor is answer: 1.71

   2. the 8-unit percent change is

3. If a function's 1-unit growth factor is 1.42, then:

   1. the 5-unit growth factor is answer:5.77

   2. the 5-unit percent change is

Hello, I am having difficulty with these questions. Please provide answers to all parts and show some work so I can learn. Thank you so much in advance! :)

1. Suppose that you began a one-year study of tuberculosis (TB) in a subsidized housing community in the Lower East Side of New York City on January 1^st, 2016. You enrolled 500 residents in your study and checked on their TB status on a monthly basis. At the start of your study on January 1^st, you screened all 500 residents. Upon screening, you found that 20 of the healthy residents were immigrants who were vaccinated for TB and so were not at risk. Another 30 residents already had existing cases of TB on January 1^st. On February 1^st, five residents developed TB. On April 1^st, five more residents developed TB. On June 1^st, 10 healthy residents moved away from New York City were lost to follow-up. On July 1st, 10 of the residents who had existing TB on January 1^st died from their disease. The study ended on December 31, 2016. Assume that once a person gets TB, they have it for the duration of the study, and assume that all remaining residents stayed healthy and were not lost to follow-up.

Is the subsidized housing community in the Lower East Side of New York City a dynamic or fixed population? Briefly explain the rationale for your answer.

a) What was the prevalence of TB in the screened community on January 1^st? Format: (x) / (y) = (z), where X is the numerator for prevalence, Y is the denominator for prevalence and Z is the prevalence %

b) What was the cumulative incidence of TB over the year? Format: (x) / (y) =(z) where X is the numerator for incidence, Y is the denominator for incidence, and Z is the incidence as a %

c) Suppose that you wanted to calculate the incidence rate of TB in the study population. Calculate the amount of person-time that would go in the denominator of this incidence rate. Be sure to show your work.

d) What was the case-fatality rate among residents with TB over the course of the year? Format: (x) / (y) = (z) where X is the numerator for case-fatality, Y is the denominator for case-fatality, and Z is the case-fatality as a %.

Which of the following graphs are Eulerian and semi-Eulerian? Please provide short
justifications, written in sentences and drawings.
(a) The complete graph K _5,5 .
(b) The complete graph K _6,6 .
(c) The complete bipartite graph K _2,3 .
(d) The Petersen graph (Google it or look in your book)

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G), u =v such
that there is a trail from u to v which goes over every edge of G. The following
sequence of questions is towards a proof of the following:
Theorem 1. A connected graph G is semi-Eulerian (but not Eulerian) if and only
if it has exactly two vertices of odd degree.
Let G be semi-Eulerian with a trail t starting at a vertex u₀ and ending at a vertex
v₀ Let G ⁰ be a graph obtained by adding an edge e₀ joining u₀ and v₀ , so that
G ⁰ − e = G.
(a) Prove that given a semi-Eulerian trail t in G from v₀ to u₀ , it is possible to
construct a Eulerian trail in G₀.
(b) Prove that given an Eulerian trail in G 0 it is possible to construct a semi-Eulerian
trail in G.
(c) Prove Theorem 1.

Task 2

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The Story of Archimedes

About 2,200 years ago, one of the most famous mathematicians in history made a major discovery. The mathematician was Archimedes, who was presented with a problem by King Hieron II of Syracuse in Sicily.

The king had given a bar of gold to a jeweler and ordered him to make it into a crown. After receiving the crown, the king suspected that the jeweler had replaced some of the gold with an equal weight of a cheaper metal like silver and kept the remainder of the gold for himself. The king had no way to determine whether this was true, so he gave the crown to Archimedes and asked him to devise a way to find out.

At the time, Archimedes knew that gold was more dense than silver. So, if the volume of the crown was greater than the volume of a bar of gold, they would have proof that the jeweler had stolen some of the gold. However, Archimedes had no way to find the volume of an irregular shape like a crown.

Archimedes struggled with this problem for a long time. One day, he went to take a bath. As he lowered himself into the bath, he noticed that the water level rose. As he continued to lower himself into the water, the water level continued to rise. He was so excited at his discovery that he jumped out of the bath and, before he could remember to put his pants on, went running through the streets yelling “Eureka! Eureka!” which in Greek means “I’ve found it! I’ve found it!” He now had a way to measure the volume of the irregularly shaped crown.

In Archimedes’s case, the jeweler had indeed stolen gold from the king. This did not end well for the jeweler.

Part A

A key component to this story is that the jeweler deceptively replaced “an equal weight” (or equal mass) of gold with silver. How does this action result in increasing the volume of the crown? Explain using one or more equations.

Part B

What did Archimedes discover in the bath? Why was he so excited? Describe the discovery using geometric terms. What do you predict Archimedes did next with his new discoveries? Explain.

why can't we use bisection methods or newton's method for nonconvex functions? x^4+x^3-2x^2-2x especially for this function?