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Homework Assignments
Mathematical Models – Spring 2010
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Midterm Practice (Not to be turned in)
Exam 2 of 2: May 10, 2010
  • The exam covers all the material from the class since the first exam. (See detail on class notes page.)
  • In preparation for the midterm, I am compiling some practice problems.
    • Book:  1.5.2, 4.2.1, 4.2.2, 4.2.3, 4.2.7 (setup only)
    • Q1.  Create the linear program that will answer question 4.2.19 (p. 264). If you feel inspired, use Mathematica to solve the linear program once you have found it.
    • Q2.  Answer the questions from Section 4 of Mathematica Tutorial 5.
    • Q3.  Explain what types of errors may occur when your calculator calculates e10.
    • Q4.  (Modified from this page)
      In the Dark Ages, Harvard and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard and the rest went to Yale, and 60 percent of the sons of Yale men went to Yale and the rest went to Harvard.
      First, set up a transition matrix for the Markov Chain applying to the above situation. Next, determine the equilibrium distribution of students. Last, determine the probability that a grandson of a Harvard graduate goes to Harvard.
    • Q5.  Give the definitions of state space and random variable.
  • Here are some additional Mathematica questions that would be fair game:
    • Write an If statement that outputs "Heads" with probabilty one third, and "Tails" with probability two thirds.
    • How would you generate a list of 100 trials of Heads and Tails?
    • What does % represent in Mathematica? What would happen if you evaluated it by itself?
Homework 8
Due on Wednesday, May 5, 2010. To be turned in by Monday, May 10, 2010.
  • Read Sections 4.2 and 4.4. Then answer the following two questions.
    • 8-1.  (10 pts) Create the linear program that will answer question 4.2.13 (p. 262). Then use Mathematica to plot the feasible region (Hint: use RegionPlot) and solve the linear program.
    • 8-2.  (10 pts) Perform a sensitivity analysis on the fertilizer example on page 253 of the book and page 112 of the notes. That is, determine the equilibrium cost of both a unit of phosphate and a unit of nitrate. (Hint: Increase the amount of phosphates available and see how much the profit increases.)
Homework 7
To be turned in by Wednesday, April 21, 2010
  • Read Section 5.1 and complete Mathematica Tutorial 5 (from class, Monday, 4/19). Then answer the following three questions.
    • 7-1.  (5 pts) Use a computer simulation to approximate the area inside the curve x4+y4=1. How much larger is this area than the area of the unit circle (x2+y2=1)?
    • 7-2.  (5 pts) Give a pseudocode description of how you might use a computer to simulate the situation in homework question 6-2(a) in order to calculate the expected reliability of the system.
    • 7-3.  (10 pts) Modify the waiting room algorithm from the notes and tutorial in order to simulate that patients are most likely to arrive in the 9:00 hour, somewhat likely to arrive in the 10:00 hour, and somewhat unlikely to arrive in the 11:00 hour. While you can choose the relative probabilities as you wish, you should continue to ensure that the expected number of patients that arrive in any day is 13.5. (Make sure that you justify that your model ensures this restriction.) Run your simulation 1000 times to determine if your modification increases, decreases, or keeps the same the expected number of patients in the waiting room at noon. Discuss whether the answer you find is what you expected to find.
Homework 6
To be turned in on Wednesday, April 14, 2010
  • Read Section 2.1. You may find it useful to read this webpage on Markov chains.
    • 6-1.  Determine and justify the category of error involved in each of the Sidelight stories in Section 2.1 (pages 76 and 77). After this, write a paragraph discussing the difference between the two errors encountered. Last, write a few sentences explaining the morals of these errors and how it relates in general to the modeling process in real-life.
    • 6-2.  Here are two unrelated probability questions.
      (a) Determine the system reliability of the following communication system. There are two different methods of communicating; at least one method must succeed in order for the system to succeed. The first method is by an FM radio, which has reliability 75%. The second method is by a satellite radio; both the physical radio must transmit the signal (with 90% reliability) AND the satellite must retransmit the signal (with 95% reliability) in order for the satellite radio to succeed.
      (b) Suppose that you are modeling the arrival of patients at a doctor's office in the first five minutes, starting at 9am. Suppose that at every minute (9:00, 9:01, 9:02, 9:03, 9:04), there is a probability of 0.075 that a patient arrives. What is the probability that the first patient arrives at 9:01? the first patient arrives at 9:02? the first patient arrives at 9:03? the first patient arrives at 9:04? What is the probability that no patients arrive in these first five minutes?
    • 6-3.  Suppose that you are setting up a pizza delivery business with three stores, Alpha Pizza in Flushing, Beta Pizza in Long Island City, and Gamma Pizza in Jamaica. When a customer calls a store, that store sends out a delivery person, who delivers the pizza and then returns to the closest store. Because of this, the delivery people end up transitioning from store to store throughout the night. Suppose that when Alpha Pizza is called, then 1/2 of the time the delivery person returns to Alpha, 1/5 of the time the delivery person goes to Beta, and 3/10 of the time goes to Gamma. When Beta Pizza is called, then 3/5 of the time the delivery person returns to Beta and with proababilty 1/5 the person goes to each of Alpha or Gamma. When Gamma Pizza is called, then 3/5 of the time the delivery person returns to Gamma, 4/15 of the time goes to Alpha, and 2/15 of the time goes to Beta. Use a Markov Chain to determine the equilibrium distribution of the delivery people at the end of the day.
    • 6-4.  This question involves Mathematica.
      (a) Use the method of Notes p. 94 to simulate flipping a fair coin 1000 times and calculate how many heads appear.
      (b) Use a Table command to repeat this experiment 500 times. The result will be a list of 500 numbers, each representing the number of heads that appeared out of 1000. Take the average of this list by using the command Mean.
      (c) Input the list from part (b) into the Histogram command to see a visualization of the 500 trials.
      (d) Discuss what you see in the answers to part (b) and (c) and how they relate to the real-life coin flipping situation.
Homework 5
To be turned in on Wednesday, March 24, 2010
  • Read Section 1.5, especially the end of the section "Sidelight",
  • Read Section 5.3A (pp.377–390); you do not need to know about standard deviation.
    • 5-1.  Let A be the matrix 
      A = [  2  3  6] 
    [ -1  0  4].
      (a) Give a matrix B such that both matrix products AB and BA make sense.
      (b) Do these multiplications. Does AB=BA?
      (c) Is it possible for there to be some matrix B where AB=BA? If so, give it; if not, explain why not.
    • 5-2.  Problem 1.5.1 abc. (p. 50) There is no need to do part (d).
    • 5-3.  Problem 5.3A.5 (p. 391) Answer parts a through d, and then answer part (e), given here:
      (e) Are the rollings of the die flippings of the coin independent events? Justify your answer.
    • 5-4.  Suppose you decide to visit Las Vegas (or Atlantic City, your choice) with exactly $255 to spend. You decide to spend your time at the roulette wheel and bet either on red or black. As you may or may not know, a single-zero roulette wheel has eighteen red number spaces, eighteen black number spaces, and one green zero space, each as likely as one another. When you bet on red (or black), you win the amount of your bet if the ball lands on one of the red (or black) spaces, and lose the amount of your bet if not. Note that because of the green space, you will lose money on average.
      (a) What is the expected value of betting red (or black)? If you bet one dollar each of 255 times, how much money can you expect to have at the end of the day?
      (b) Now consider the following strategy. You play the first game and bet one dollar. If you win, you walk away having won one dollar. If you lose, you play again and this time bet two dollars. Again, if you win, you walk away having won one dollar. If you lose, you play again and this time bet four dollars. Continue in this fashion, walking away if you win at any step, or betting twice as much in each game following a loss, until you have no more money. Determine the expected value of playing this strategy; how much money do you expect to have at the end of the day?
      (c) Do you prefer either of the strategies from part (a) or part (b), or would you spend your money in a different way?
Midterm Practice (Not to be turned in)
Exam 1: March 10, 2010
  • I ask that you post a question (and/or a response) on the course discussion board before class on Monday, 3/8.
  • The exam covers all the material from the class up through Wednesday, 3/3/10. (See detail on class notes page.)
  • In preparation for the midterm, I have compiled some practice problems.
    • Book:  1.4.2 (p. 42), 3.1.5, 3.1.13, 3.1.14 (p. 149), 3.2.3, 3.2.5 (p. 167–168), 3.3.3 (p. 179),
    • P-1.  3.4.9 (p. 196) [Practice calculating R-squared.]
    • P-2.  What are the steps in the modeling process? What is done at each step?
    • P-3.  The following data is assumed to fit a logarithmic model, y=a+b*ln[x]. Determine the best values for a and b using the least squares criterion. [You may use Mathematica, but that is not required.]
       x 1.11.92.83.85.16.27.28.0 8.79.8
       y 3.25.26.47.38.08.69.19.5 9.910.2
  • Here are some Mathematica questions that would be fair game:
    • Explain the difference between the following two lines of Mathematica code:
      a=Table[3i,{i,1,5}];
      a=Table[3i,{i,1,5}]
    • Write down the Mathematica code which would determine the numerical value of the expression Sqrt[10]+Log[2].
    • What are the differences between (), [], and {}?
    • What do you expect when you evaluate the code Table[2^k,{k,1,10}]?
    • When would you use the TableForm command?
    • Explain the difference between the Fit and the FindFit commands.
    • You need to know how to plot a list, how to plot a function, and how to incorporate two plots together.
  • Also to keep in mind: Your project topic is due on Wednesday, March 17.
Homework 4
To be turned in on Wednesday, March 3, 2010
  • Read Sections 3.1 and 3.3.
    • 4-1.  Question 3.1.2 (p149). Explain some advantages and disadvantages in complete sentences; give at least four total (advantages + disadvantages).
    • 4-2.  Question 3.1.7. DO give examples of such models and explain why they satisfy the conditions you claim.
    • 4-3.  Suppose that you are inviting friends to a party from your online virtual network. You have 120 friends who are women and 100 friends who are men. For various reasons, you expect 1/6 of your women friends to come to your party and 1/4 of your men friends to come to your party. You anticipate a 20% error in these estimates.
      (a) Create a table like that on page 146 that accounts for higher or lower predicted values.
      (b) Use this table to determine the percentage error in the output. Does error in equal error out?
      (c) Discuss the robustness of your model.
    • 4-4.  (No problem 4-4 to cut down on homework length; see email sent Friday.)
    • 4-5.  (a) 3.3.1(a)
      (b) 3.3.1(b); use Mathematica to calculate the regression line between density and speed.
      (c) Suppose that your model was commissioned by a city planner who was concerned about the traffic in the area. Discuss the implications of the formula you determine and use your model to justify some recommendations for the city planner.
Homework 3
To be turned in on Wednesday, February 24, 2010
  • Read Section 2.3.3 and 2.3.4 (pp. 110-119), and finish reading Section 3.2.
    • 3-1.  (a) 1.4.10 (p43) (b) Suppose the bacteria in part (a) are expanding inside a petri dish. Suppose further that they start expanding at noon on Monday and completely fill the petri dish at noon on Tuesday. At what time is the petri dish half full?
    • 3-2.  Complete all of Problem 3.2.1 (page 167) by hand, with the simplification that the height (h) is proportional to the time (t). [You'll be fitting the data to a line of the form h=ct for some c.]
    • 3-3.  (a) In the price-demand model of Example 3, assume that the best-fitting curve instead has the form d=c/p2. Determine the best value of c using the method of least squares. This will give you a model for demand in terms of price.
      (b) Suppose that it costs $6 to produce the product. The profit on one item is the difference between the selling price and this $6. Given the demand curve determined in part (a), write down (with explanation!) an equation for the total profit the company will make as a function of the price p.
      (c) Use calculus on the equation from part (b) to determine the selling price that results in the largest total profit for the company.
    • 3-4.  In this problem, you will show three individual plots and then combine them in a fourth plot. First create a plot of the sine function from 0 to 2 pi that has a thick blue line. Next, create a plot of the cosine function from 0 to 2 pi that has a thin dashed line and is not blue. Third, create a plot of large bright fluorescent points from the tangent function every pi/5 from 0 to 2 pi. [To clarify, there should be 11 points in all, at coordinates (0,0), (pi/5,tan(pi/5)), etc. to (2pi,0).] Last, combine all three graphs on one set of axes.
    • 3-5.  Recall the US population data from the years 1900-2009 (see Homework 2-4). Use Mathematica to fit the data to a curve of the form y=Crt. Discuss your method and determine what your model predicts will be the population in 2015 and 2050. In which prediction do you have more confidence?
  • Problems 3-4 and 3-5 should be completed in the same Mathematica notebook with your name and the date.
  • Recall that all homework should be explained in complete sentences with justification. Even your Mathematica solutions should be answered using complete sentences and paragraphs to explain your work.
  • Important: Class on Monday 2/22/10 will be held in the computer lab in I-building Room 212 (with overflow in Rm 201).
Homework 2
To be turned in on Thursday, February 18, 2010
  • Read Section 1.3 to page 25, all of Section 1.4, and Section 3.2 to the middle of page 154.
    • 2-1.  (a) Problem 3.2.11 (page 169). [A ``regression line'' is a line of best fit.]
      (b) Describe why the situation in part (a) is not descriptively realistic as stated, and give an argument that you can use to make the model descriptively realistic.
    • 2-2.  Here is some data that fits well to a curve of the form y=Cxk.
       x 0.50.751.01.52.02.53.03.54.0
       y 0.090.270.622.155.010.317.627.240.9
      Calculate and plot the values of log(x) versus log(y) to determine appropriate approximate values for C and k. Explain your method.
    • 2-3.  on pages 42–43, complete problems (a) 1.4.1 (b) 1.4.3 (c) 1.4.9a (do not do b or c)
    • 2-4.  (a) Include recent data on population from the years 1980, 1990, 2000, and 2009 to update the chart on page 42.
      (b) Does the original model appear to still be accurate? Explain.
      (c) If there are any discrepancies in the model, try to explain them (you may have to do some research).
      (d) Modify the model to try to model the population of the United States for the years up through 2050, and explain your reasoning.
    • 2-5.  Create and print out a Mathematica file (also called a notebook) with your name, the date, and which solves the following questions
      (a) Create a list of the positive multiples of 7 that are no larger than 100.
      (b) Have Mathematica determine how many integers are in the list from Part (a), and also determine the sum of the entries of this list.
      (c) The function Prime[n] gives the nth prime number. For example, Prime[1] is 2, Prime[2] is 3, Prime[3] is 5, etc. Create a list which has as its entries 30 ordered pairs {x,y}, where x ranges from 1 to 30 and y is the x-th prime. For example, the first of the thirty ordered pairs will be {1,2}, the second will be {2,3}, etc. Then use TableForm to display this list of ordered pairs nicely.
Homework 1
To be turned in on Monday, February 8, 2010
  • Thoroughly read the class web page including the syllabus and schedule. This should answer all the questions that you may have about the class. Read Sections 1.1 and 1.2 to page 18.
  • Problem 1-1 must be completed online before class on Monday 2/8 for credit.
    • 1-1.  (a) Email me at chanusa@qc.cuny.edu with the email address where you are best contacted. (Make sure to include your name in the message as well!)
      (b) Take the syllabus quiz on Blackboard. Retake the quiz as necessary to earn a score of 100%.
      (c) On the class Discussion Board, introduce yourself on the "Class Introductions" thread.
  • Problems 1-2 through 1-5 should be written up and handed in in class before class starts on Monday:
    • 1-2.  Read Section 1.1 from the textbook; it gives six different models that describe real-life situations. Write a paragraph or two explaining where you have experience with mathematical models in real life. (Do not use the examples from the book unless you feel you have a unique perspective.) Explain how the real-life situation and the model differ. (Are there simplifying assumptions? Does the model truly describe what happens in real life or are there limits to the model's effectiveness? ... )
    • 1-3.  Here are three different vague scenarios. Choose one of the three and identify a precise problem statement that you would like to study. Determine eight variables that affect your proposed problem and highlight those that are the most important.
      1. Walmart is building and illuminating a new parking lot.
      2. Queens College wants to redesign a lecture hall for a large class.
      3. A toy manufacturer needs to determine how many toys to make and at what price to sell them.
    • 1-4.  1.2.3abc (Chapter 1, Section 2, problem 3, page 21)
    • 1-5.  Here is some data that represents an independent variable x and a dependent variable y. It is thought that y satisfies a linear function of x. Plot the data on a graph and give a rough estimate for this function. Explain how you found your answer.
      x 1 3 4 7 9 10 13 15 18 20
      y 3 10 2 20 31 27 33 39 41 50
  • Remember what is expected:
    • Write your solutions using complete sentences.
    • Follow the guidelines for turning in homework.
  • Important: Class on Monday 2/8/10 will be held in the computer lab in I-building Room 212 (with overflow in Rm 201). If you wish to have help installing the mathematical software Mathematica on your laptop, bring it to class on Monday.

Back to the Mathematical Models Home Page.
Christopher Hanusa – Queens College – Mathematics Department.