Mathematical Modeling 2003

First weekend assignment

Do your best on these problems.  If you do not know how to work a problem note in a separate document what you do and do not understand about that problem and ask at least one good question about the problem.

1.  Predict the next number in each of the following sequences:

3               4              6              9              13           ...

1               2              4              8              16           ...

50             40            32            26            22           ...

70             38            22            14            10           ...

1               3              4              7              11            18           ...

13             20            39            76            137         ...

As best you can state the rule that allows you to predict the next number for each sequence.

** 2.  Write each of the sequences of the preceding problem as a data table using Excel, and do a curve fit to each.  Every sequence but one can be exactly modeled by an appropriate function.

3.  Suppose that water depth vs. clock time is given by the function y(t) = .010 (t - 100)^2.  Expand the square of the binomial to obtain the y = a t^2 + b t + c form of this function.  Determine the water depths at t = 0, 10, 20, ..., 100.  Write these depths as a sequence and find the rule for this sequence.

4.  Suppose that the temperature of a certain object in a room gets 1/3 of the way closer to the room temperature every 10 minutes.  If the object starts at a temperature of 6.71 cm on a temperature scale, while room temperature on the same scale is 3.16 cm, then what will the temperatures be after each of the first four 10-minute time intervals?

Write your temperatures as a sequence of numbers and given a rule for calculating the next member of the sequence from the present number.

Sketch a graph of temperature vs. clock time and without doing the calculations for the next three 10-minute intervals, sketch the estimated curve for the graph.

Use Excel to calculate the temperature in excess of the room temperature at each clock time, and do a curve fit for this temperature excess vs. clock time.  One of the functions will fit exactly.

5.  On a certain island live an immortal race of 1000 non-reproducing people. Using a standard psychiatric evaluation developed and refined over a great number of years, every individual can be clearly classified as either sane or demented.  Due to various physical, social and psychological factors, in a period of one year every sane individual has a 10% chance of becoming demented, and every demented individual has a 20% chance of becoming sane.

If on New Year's Day in the year 2000, there are 900 sane and 100 demented individuals, then how many sane and how many demented individuals will there be on New Year's Day in the year 2001?  How many sane and how many demented individuals will there be on New Year's Day in each of the subsequent three years?

Sketch a graph of the number of sane individuals vs. the number of years since the year 2000.  Does your graph seem to share more characteristics with a graph of pendulum frequency vs. length, a graph of water depth vs. length (flow from a uniform cylinder out of a uniform hole), or a graph of temperature vs. clock time (a warm object in a cooler room)?

6.  There are five trees planted in a row.  On the second tree there suddenly appears an infestation of 1000 bugs.  The spacing of the trees and the behavior of the bugs is such that in any given hour, 20% of the bugs on any tree will migrate to each adjacent tree, while 10% of the bugs on an end tree will wander away and starve to death.  Find the number of bugs on each of the five trees after each of the next six hours.

Number the trees, in order, from 1 to 5. For each hour sketch a graph of the number of bugs vs. the number of the tree.  Describe how the shape of the graph changes from hour to hour.

See if you can figure out how to use Excel to solve this problem.  See if you can use Excel to plot the number of bugs on the second tree vs. clock time and see if you can find a way of fitting the curve.

** What does the cardboard model have to do with this problem (if you haven't seen the cardboard model ask about it)?

6.  An mathematical model for an ideal pendulum is T = .2 L^.5, where L is pendulum length in centimeters and T is period in seconds. 

• What would the period be for lengths of 10cm, 20cm, 30cm and 40 cm?
• How many cycles would we count in a minute for each of these lengths?
• How do these results compare with the results we would obtain for the pendulum model f = 251 L^-.46, where L is pendulum length in cm and f is frequency in cycles / minute?
• Challenge problem:  What would the model be for a pendulum whose length is in meters and period in seconds?  What would be the model if length was in feet and period in seconds?

7.  Assuming that T = .2 L^.5 gives us the period in seconds when length L is in cm, what formula would give us the frequency f, in cycles per minute, for pendulum length L in cm?

8.  According to the model T = .2 L^.5 what would be the lengths of the pendulums that give us frequencies of 10, 20, 30, 40, 50 and 60 cycles per minute?

** 9.  Using the frequencies you obtained for pendulum lengths 10, 20, 30 and 40 cm, for each length interval (i.e., 10-20 cm, 20-30 cm, 30-40 cm) determine the average rate at which frequency changes with respect to length, in (cycles/minute) / centimeter of change in length.  Why do we call this an average rate?

10.  Can you observe a discrepancy, using a real pendulum and keeping its swing amplitude less than 10% of its length, between your count and the results of the formula T = .2 L^.5 for any of the lengths 10, 20, 30 or 40 cm?  Recall that you estimated the length of your pendulum as a percent of the width of a tile, the width of a tile is 1 foot, and 1 inch = 2.54 cm.

11.  Can you observe a discrepancy, using a real pendulum and keeping its swing amplitude less than 10% of its length, between your count and the results of the formula f = 251 L^-.46 for any of the lengths 10, 20, 30 or 40 cm?

12.  If on an island we have sane people and demented people, with 500 of each on New Year’s Day 2002, and if every year 10% of the people who were sane at the beginning of the year become demented and 20% of the people who were demented at the beginning of the year become sane, then how many sane and how many demented people will we have on New Year’s Day of each of the subsequent 4 years?

13.  For pendulum lengths of 1, 2, 3 and 4 ergluks, frequencies observed on a certain planet were 19, 14, 9 and 4 clogerks.  Do these results make sense in terms of your observations of pendulum behavior  (other than the fact that you don’t know what ergluks and clogerks are)?

14.  If we flip a coin twice, we could get Heads on the first flip and Tails on the second (call this outcome HT).  Or we could get Heads on the first and Heads on the second (call this outcome HH).  What are the other possibilities?

15.  Suppose that we have a long row of trees with 100 bugs on the center tree and none on the other trees.  The center tree is number 0, the trees to the right are numbered 1, 2, 3, … and the trees to the left are numbered –1, -2, -3, … .   During each transition 10% of the bugs on a given tree migrate to each of the tree’s neighbors.  How many bugs will there be on each tree after each of the first four transitions? 

16.  See how far you can get on this series of questions:

• If a ball requires 5 seconds to accelerate from rest down a uniformly inclined ramp of length 80 cm, then at what average rate does its position change?  This average rate is called the velocity of the ball. 
• If the average velocity is equal to the average of the initial and final velocities of the ball on the ramp, noting that the initial velocity is 0, then what is the final velocity (i.e., what would we have to average with 0 to get the average velocity you calculated in the preceding question)? 
• If there was a little speedometer on the ball, reading velocity in cm/sec, what would it therefore read at the beginning of the run and what would it read at the end? 
• By how much does the velocity of the ball therefore change during its 5-second run? 
• At what average rate does the velocity change during the 5-second run?

17.  Divide 1 by 43, taking your result out to 42 significant figures.  Tally the number 0’s, 1’s, 2’s, etc..  How evenly distributed are the digits of your division?  How could we have predicted beforehand that the digits cannot possibly be evenly distributed?

18.  Take the digits you obtained in the preceding problem and divide them into groups of 2.  For example the first three digits are 2, 3, 2, 5 and 6.  The first two groups would be 2,3 and 2,5.  The first number in the third group would be 6.  Make an ordered pair out of each group.  The first two ordered pairs would be (2,3) and (2,5).   You will get 21 ordered pairs.

• Plot your 21 ordered pairs on a set of x-y axes.  Is there any pattern to your graph?
• If there is a pattern does it tend to support or to refute the contention that the numbers are random?

** 19.  Using the relationship T = .2 sqrt(L), with T in sec when L is in cm, determine the average rate at which T changes with respect to L between L = 30 cm and L = 40 cm.  So the same for the remaining intervals defined by lengths L = 10 cm, 20 cm, 30 cm, 40 cm, 50 cm.  If you did the Galileo experiment, how might this sort of rate information be relevant to the task of adjusting pendulum length to achieve synchronization?