ASSIGNED PROBLEMS FOR MATH 4400/6400
  1. t
    Time    		y
    exact    		y
    Euler DELt=1    		y
    Euler DELt=0.01
    0 0.00 0.00 0.00
    1 1.96 2.00 1.96
    2 3.84 3.92 3.84
    3 ***** ***** *****
    4 ***** ***** *****
    5 ***** ***** *****
    6 ***** ***** *****
    7 ***** ***** *****
    8 ***** ***** *****
    9 ***** ***** *****
    10 ***** ***** *****
    20 ***** ***** *****
    30 ***** ***** *****
    40 ***** ***** *****
    50 ***** ***** *****
    60 ***** ***** *****
    70 ***** ***** *****
    80 ***** ***** *****
    90 ***** ***** *****
    100 ***** ***** *****
    200 49.98 49.99 49.98
    A tank initially holds 100 gal of pure water. A brine solution runs into the tank at the rate of 4 gal/min. The concentration of salt in this brine ia .5 lb/gal. The mixture is kept stirred and runs out of the tank at the rate of 4 gal/min. Using DE techniques we can show that the solution is

    y = 50 - 50*exp(-.04*t)

    where y is the amount of salt in lbs in the tank at time t min.

    The problem requires that you compute the amount of salt in the tank using Euler's method with step size 1 and then again with step size 0.01 and prepare a table like the one shown . The point of the problem is a comparison of exact, Euler's (DELt =1) and Euler's (DELt = 0.01).




















  2. t
    Time    		x
    Amt in T1    		y
    Amt in T2
    0 0.00 50.00
    1 ****** ******
    2 ****** ******
    3 ****** ******
    4 ****** ******
    5 ****** ******
    6 ****** ******
    7 ****** ******
    8 ****** ******
    9 ****** ******
    10 ****** ******
    20 ****** ******
    30 ****** ******
    40 ****** ******
    50 ****** ******
    60 ****** ******
    70 ****** ******
    80 ****** ******
    90 ****** ******
    100 ****** ******
    200 ****** ******
    300 ****** ******
    Two tanks are linked by pipes. Tank 1 initially contains 100 gal of pure water while tank 2 holds 200 gals and initially has 50 lb of salt dissolved in the water. Brine with a concentration of 1 lb/gal flows into tank 1 at a rate of 5 gal/min from an unspecified source. The solution (kept mixed) flows out of tank 1 into tank 2 at the rate of 6 gal/min while the solution in tank 2 flows through another pipe into tank 1 at the rate of 1 gal/min. The mixture in tank 2 runs out to an unspecified sink at the rate of 5 gal/min. Let t be time in min, x the amount of salt in lbs in tank 1 and y the amount in lbs of salt in tank 2 at time t. Use Euler's method with step size DELt = 0.01 to solve for x and y. put the outcome into a table as shown. Also graph x and y.
























  3. K
    Speed    		 a
    Semimajor Axis    		 e
    Eccentricity    		 P
    Period    		 a3 / P2
    Constant
    28000 6600 0.00 . .
    36000 . . . .
    38000 . . . .
    39000 . . . .
    A spaceraft is initially located at coordinates (0, -6600) and has initial valocity (K, 0) which puts it just above the earth's atmosphere. The center of the earth is at the origin. Note we are assuming for simplicity that space is 2 dimensional. Let k have the 4 values: 28000 km/h, 36000 km/h, 38000 km/h and 39000 km/h resulting in four orbits. Note that the units are kilometers and hours. In these units

    MG = 5.166 E12 km3 / h2 .

    Use Euler's method (DELt = .001)to compute the position of the spacecraft at various times in its orbit. Manage the programming and output as you think appropriate. Make a reasonably accurate drawing of the four orbits putting an appropriate number of time tags on the drawing. Then complete the table. The drawings, time tags and table will demonstrate that Kepler's Laws follow from Newton's work.

    Kepler's Laws

    1. Shape of Orbits: The spacecraft orbits in an ellipse with the origin at one focus.
    2. Speed in Orbit: The speed in orbit is not constant, rather the spacecraft moves faster the nearer it is to the origin.
    3. Relation of Period to Size of Orbit: The semimajor axis a measures the size of the orbit and P is the time required to complete the orbit. Do not expect a3 / P2 to be exactly constant in this simulation because of roundoff and discretization errors. However, Kepler's 3rd law says this should be constant.


  4. Extension of problem 3. Add to your program in problem 3 the gravitational acceleration of the moon. For the moon

    mG = 6.354E10 km3/h2


    We will simplify this model by assuming the moon is stationary at coordinates (0, 384400)km. The orbits already calculated except possibly the last one will not be affected much by the moon's gravity. The moon has only about 1/80 th of the earth's mass. The zero point between the moon and earth has coordinates (0, 346025) km. If the initial speed is k = about 39186 km/h (k has the same meaning as in problem 3), then the spacecraft should go into the figure 8 orbit around the moon. Because of numerical errors this speed might not be exactly right using your software and computer. Keep experimenting until you get the figure 8 orbit. Turn in the initial speed you used together with a drawing of your figure 8 orbit with appropriate time labels. My intuition says that the craft ought to go through the zero point, but my simulation show this not to be true. It needs to pass just in front of the zero point apparently.

  5. A Spacecraft is in circular orbit (green) 6600 km above the center of the earth. It desires to go into circular synchronous orbit (magenta) 42160 km above the center of the earth. To do this, it must increase its velocity (bottom red point) to go into the elliptical transfer orbit (blue) and then increase its velocity again (top red point) at the circular synchronous orbit to go into that orbit. Determine by simulation (this means try various velocities until you find the correct one) needed to enter the transfer orbit and then the velocity needed to enter the circular synchronous orbit. Your work in project 3 should be helpful to this problem. The black half of the transfer orbit is not traveled. A synchronous orbit is 23 hours 56 minutes measured in mean solar time (24 h sidereal time = 23 h 56 m MST).

    Energy in orbit

    This site has a good but short biography of Issac Newton plus links to the giants upon whose shoulders he stood: Descartes, Galileo, Copernicus, Kepler, Brahe, Halley, Euclid, Archimedes
    " If I have been able to see further, it was only because I stood on the shoulders of giants."
    Letter to Robert Hooke

  6. An easy one. A colony of bacteria is growing exponentially. At time 0 there were 1000 of the bacteria, but one hour later they had increased to 1500. How many will the colony contain at time 24? Note: This problem is not meant to be difficult. One page ought to do it.

  7. Another easy one. A wooden stick found by archeologists has only 40% of the C14/C12 ratio of the atmosphere (read 40% of its original amount of C14). How old is this piece of wood? The halflife of C14 is 5600 years.

  8. How long will it take $2500 at 6% interest compounded quarterly to increase to $10000? How long will it take if the money is compounded continuously at 6%? What interest rate compounded quarterly will increase the $2500 to $10000 in 12 years?

    Thomas Malthus ::: Essay on the Principle of Population (1798)

    The International Society of Malthus


  9. An initial population of 100 inhabits an area with a carrying capacity of 100000. In the first year the population increases to 120. Assume the population grows logistically. Determine the population as a function of time. How long will it take the population to reach 95000?

  10. t y
    0 9.6
    1 18.3
    2 29.0
    3 47.2
    4 71.1
    5 119.1
    6 174.6
    7 257.3
    8 350.7
    9 441.0
    10 513.3
    11 559.7
    12 594.3
    13 629.4
    14 640.8
    15 651.1
    16 655.9
    17 659.6
    18 661.3
    Use the three-point-evenly-spaced-fit to fit a logistic curve to the data. In the table t is time in hours and y is the amount (units unspecified) of yeast in a culture. Make a table listing the time, the theoretical amount of yeast, the actual amount, the error and the relative error. Also make a graph showing the fitted logistic curve and the observed data points.

    See the Maple page for information concerning the logistic curve.

  11. Year World Population
    Millions
    1650 545
    1750 728
    1800 906
    1850 1171
    1900 1608
    1950 2509
    1960 3010
    1970 3611
    2000 6000
    Fit an exponential curve to the world population data in the table. Use the method of linearization using logarithms and the analytic method of fit using the formulas for m and b from the lectures. Make the usual observed-predicted-error-relative.error table. The fit is not very good. The data given in the above reference to the International Society of Malthus is probably better data. I cannot remember the source of the data used in this exercise.







  12. t y
    0 92
    1 142
    2 185
    3 213
    4 245
    5 235
    6 279
    7 298
    8 295
    9 367
    10 367
    11 369
    12 415
    13 376
    14 437
    15 446
    16 422
    17 432
    18 448
    19 458
    20 467
    21 466
    22 451
    23 444
    24 431
    25 489
    26 474
    27 466
    28 433
    29 503
    30 481
    31 488
    32 507
    33 500
    34 460
    35 484
    36 452
    *** ***
    *** ***
    Use the "Method of Complete Coverage" to fit a learning curve to the following data. Time t is measured in weeks, while the production y is the number of items produced on the production line each week for 37 weeks. Use a plot of the data to estimate M and y0.Then use one of the points to estimate a. Bracket your estimated values of M, K=M-y0, and a. Try to get each of the three parameters accurate to three significant figures. Make the usual observed-predicted-error-relative-error table and make a plot of the data and fitted curve. The formula for the family of learning curves is:

    y = M - K*e-a*t, *******Parameters: M, K=M-y0, a.

    Fitting a curve to observed data
    Method of least squares

    Linear curve fitting
    Special case

    Exponential curve fitting
    Linearization

    Curve Fitting by
    Method of Complete Coverage

    Software used to fit logistic model to US population: Example of "Complete Coverage"

  13. The parameters and initial conditions for a standard predator/prey situation were determined to be

    • x0 = 10.1
    • y0 = 5.3
    • a = 1.1
    • b = 0.8
    • m = 0.2
    • n = 1.7

    Draw the graph of the orbit putting time tags on the orbit. It is essential that you know where the point(x,y) is in its orbit at certain times t. The problem is not acceptible without the time tags.

    Standard Predator/Prey Model

  14. Using the standard epidemiology model and the parameters given below, calculate (numerically) and graph the
    • S/R orbit
    • S/I orbit
    • S/t orbit
    • R/t orbit
    • I/t orbit.

    Also give the computed values of
    • R_infinity
    • S_infinity
    • I_max
    • I_infinity
    • t_infinity (The approximate length of time the epidemic lasted).

    There are two separate epidemics for which the computations are to be done.
    1. beta = .001, r = .9, S0 = 1000, I0 = 1, R0 = 0 (low infection rate, high removal rate)
    2. beta = .002, r = .7, S0 = 1000, I0 = 1, R0 = 0 (higher infection rate, lower removal rate)


    Linear Programming. Two variable Geometric Method
    Linear Programming. Two variable Geometric Method. The Cheese Problem

  15. A manufacturer makes two kinds of chairs, Type I and Type II. Type I requires for its manufacture 3 hours on Machine A and 2 hours on Machine B. A Type II chair requires 2 hours on Machine A and 4 hours on Machine B. The profit on a Type I is $140 while the profit on a Type II is $120. Each machine is run for 40 hours each week. How many chairs of each type should be made each week to maximize the profit? Solve using the geometric method.

  16. Food M Food N Requirements
    Calcium 30 units/oz 10 units/oz 360 units
    Iron 10 units/oz 10 units/oz 160 units
    Vitamin A 10 units/oz 30 units/oz 240 units
    Cholesterol 8 units/oz 4 units/oz Minimize
    A dietician in a hospital is to arrange a special diet
    composed of two foods, M and N. Each ounce of food M contains 30 units of calcium, 10 units of iron, 10 units of vitamin A and, unfortunantly, 8 units of cholesterol. Each ounce of food N contains 10 units of calcium, 10 units of iron, 30 units of vitamin A and 4 units of cholesterol. If the minimum daily requirements are 360 units of calcium, 160 units of iron and 240 units of vitamin A, how many ounces of each food should be used to meet the minimum requirements of Ca, Fe and vitamin A while at the same time minimizing the cholesterol intake? Solve using the geometric method.

    Fortran program for running linear programing problems
    Instructions for running linprog program

  17. Solve problem 15 using the simplex method.

  18. A boat builder makes 3 kinds of boat, pram, runabout and whaler. His factory has two departments Dept1 and Dept2. A pram requires 1 hour in Dept1 and 2 hours in Dept2. A runabout requires 2 hours in Dept1 and 5 hours in Dept2. A whaler requires 3 hours in Dept1 and 4 hours in Dept2. He has enough fiberglass to make 3000 boats during the coming year. He has available 6240 hours in Dept1 and 10800 hours available in Dept2. He makes a profit of $75 on a pram, $90 on a runabout and $100 on a whaler. How many boats of each kind should he make next year to maximize his profit?

  19. Solve problem 16 using the simplex method.

  20. Shipping Costs
    . Store 1 Store 2
    Warehouse A 5 $/ton 8 $/ton
    Warehouse B 6 $/ton 10 $/ton
    A company has two warehouses A and B and two stores 1 and 2. Of a certain product warehouse A has 100 tons while warehouse B has 150 tons. Store 1 needs 50 tons of the product while store 2 needs 75 tons. The cost of shipping is given in the table. How many tons should be shipped from each warehouse to each store to minimize shipping costs?



    George B Dantzig, Inventor of the Simplex Method

  21. Same as problem 20 except with the a different shipping cost table.
    Shipping Costs
    . Store 1 Store 2
    Warehouse A 8 $/ton 5 $/ton
    Warehouse B 6 $/ton 10 $/ton









  22. Rental cars tend to accumulate in some cities leaving shortages in others. These must then be moved at company cost from where they have accumulated to where they are needed. One such car rental company has a surplus in Cities 1 and 2 and a shortage in Cities 3, 4 and 5. It wishes to move cars from Cities 1 and 2 to Cities 3, 4 and 5 and to minimize the total number of miles driven in doing so. Below are tables giving the surpluses, shortages and miles between cities. Find the number of cars to be moved from each of Cities 1 and 2 to each of the Cities 3, 4 and 5 that minimize the total miles driven.


    City Surplus
    City 1 75 cars
    City 2 50 cars
    		******
    City Shortage
    City 3 11 cars
    City 4 60 cars
    City 5 40 cars
    		*****
    Miles from Cities 1 and 2 to Cities 3,4 and 5
    . City 3 City 4City 5
    City 1 50 miles 140 miles 70 miles
    City 2 30 miles 90 miles 25 miles


  23. Buffon's Needle Problem.Imagine the plane covered with a grid of parallel lines 1 unit apart. Toss a needle of length L onto the plane so that it lands in a random position with a random orientation. What is the probability that the needle lands crossing one of the lines? Clearly the solution is a function of L, the probability being near 1 for a long needle and near zero for a short needle. The answer is 2/Pi for a needle of length 1. Use a random number generator to simulate 10000 tosses of the needle and use this simulation to determine the empirical probability of crossing. The simulation is to be done for three
    needle lengths
    • L = 0.2
    • L = 1
    • L = 4


    Analytic Solution of Buffon's Needle Problem -- Case: L = 1.

  24. A stick is 10 cm long. It is broken at a randomly (and uniformly) selected point and then the longer piece is again broken at a randomly selected place. This results in 3 pieces of length a, b and c. These three pieces can form a triangle if and only if

    a < b + c and
    b < a + c and
    c < a + b

    Use a random number generator to determine the probability by simulation that the three pieces can form a triangle. This approach to probability problems is called "Monte Carlo Simulation".

    Random Number Generator
    Maple

    Random Number Generator
    FORTRAN
    Don't forget that in FORTRAN a number with a decimal point behaves differently from a number with no decimal point. Be sure to copy this subroutine EXACTLY as it is. Also the subroutine needs a seed to get started. Note the part of the main program concerning the seed. You will need to put this part of the main program into your main program.

  25. An insurance company classifies drivers into 3 groups: G0 (no accidents), G1 (one accident) and G2 (more than one accident). The probability that a driver in G0 will stay in G0 at the next step (one year) is .85, that he will become a G1 is .10 and that he will go to G2 is .05. A driver in G1 cannot move to G0, but has a .80 probability of staying in G1 and a .20 probability of becomming a G2. A G2 must remain a G2. Suppose the company accepts 50000 new policyholders all in G0. Find the number in each of G0, G1 and G2:
    1. after 1 year
    2. after 2 years
    3. after 5 years
    4. in the long run

    Solving the toothpaste problem and the gambler's ruin problem by Maple
    Maple source code is not ASCII, but rather something called .mws . Thus you cannot copy the Maple source code from the web. If you would like a copy of this program or any of the other Maple programs posted, bring me a disk and I will copy them for you. Since Maple produces no .exe file, it will only run on computers which have the Maple software.

    A FORTRAN program to compute: P, P**2, P**3, P**4, P**5,....

  26. Transition Probability
    G0 to G0 .85
    G0 to G1 .10
    G0 to G2 .05
    G1 to G0 .15
    G1 to G1 .75
    G1 to G2 .10
    G2 to G0 .10
    G2 to G1 .30
    G2 to G2 .60
    Another insurance company has the three groups G0, G1 and G2, but gives a possibility of moving out of G2. The probabilities of moving from one group to the next in one step (one year) are given in the table. This company accepts 100000 new policies all starting in G0. Find the number of policyholders in each of G0, G1 and G2:
    1. after 1 year
    2. after 2 years
    3. after 5 years
    4. in the long run




  27. A gambler has 500 dollars. He will bet this in increments of 100 dollars on roulette. The probability of winning is 1/38. He will quit if he goes broke or wins a game, which will pay him 3500 dollars (Rich). The states are: Broke, $100, $200, $300, $400, $500, Rich.
    1. What is the probability that he eventually goes broke?
    2. What is the probability that he eventually gets rich?
    3. What is the expected number of steps before he either goes broke or gets rich?


  28. A que will hold eight. At each step one is served and leaves the que (unless the que is empty). At each step 0, 1, 2, or 3 arrive with probabilities .2, .5, .2 and .1 respectively. There are 11 states, the number in the que: 0, 1, 2, 3, 4, 5, 6, 7, 8 and T1, T2. The system is said to be in state T1 if it turned 1 away at that step and in state T2 if it turned 2 away at that step. Set up the one step transition probability matrix P. Many have missed this problem in the past by not thinking clearly what states T1 and T2 mean. T1 is not called 9 because the cue will not hold 9. There are only 8 in the cue in state T1 with one turned away. Similarly there are only 8 in the cue in state T2 with 2 turned away. Thus states 8, T1 and T2 have the same 1-step transition probabilities.
    1. If the que starts empty, how many steps will it take until significant numbers are being turned away (Answer not unique; explain why you gave the answer you did)?
    2. What percent will be turned away in the long run?


  29. Rework the previous problem with all the same assumptions except that now there are two servers, so that two can be served at each step. Servers can be idle meaning only one will be served if only one is in the que and zero served if the que is empty. Do we need state T2?
    1. What effect does having two servers have on the number turned away?
    2. What state is the system is in most often?