STATEMENT OF PROBLEM

Investment Strategies - first order DDS

We go to our investment analyst and she says she has a new investment program. Basically we invest our money in an interest bearing account at the rate of $100 per year with an initial investment of $100. The account pays interest at the end of each year. We plan to invest for 12 years.


She says we can pick our annual interest rate(!!) - up to a maximum rate of 20% per year or r = 0.20.


There is one caveat. At the end of 12 years if we select annual interest rate r (e.g., r = .05 is 5% annual interest) then we have to pay a penalty of 90,000 r^2. Thus for example if we invest our money at r = .05 then we shall have a penalty of 90,000(.05)^2 or $225.00. She says think of it as a "brokerage fee" for getting the higher rate.


(a) Set up a discrete dynamical system to model our money in the bank after year n, a(n), in terms of the undetermined annual interest rate r.


(b) Solve the above discrete dynamical system to determine a(n).


(c) Let n = 12 years and ascertain (1) lowest interest rate which will give a return of $1,400 after 12 years and (2) the best interest rate you should pick in order to get the highest balance in your account after n = 12 years.


(d) What will your brokerage fee be for this rate?

Polymer production problem and optimization

We have two polymers - A and B. In a given period of time, say 1 sec, 90% of polymer A stays as polymer A and 10% of polymer A transforms to polymer B, while 80% of polymer B stays as polymer B and 20% of polymer B becomes polymer A.


Suppose we start with 1 g of each of Material A and Material B. A rule of thumb when combining these particular polymers is this:


If we add x g to the initial amount of Material A we need to subtract off x^2 g from the initial amount of Material B.


What are the physical restrictions on x?


For our initial conditions, how many g of initial material A, x, should we add to 1 g of Material A (and hence subtract off x^2 g from an initial amount of 1 g of Material B) so that we maximize the amount of Material B at time n = 10 sec?


What is this maximum value for Material B?


You can maximize any function of x by examining its plot and estimating its highest value, perhaps zooming in on its maximum value. OR use first derivative information to detemine optimimum value.


Determine the eventual distribution of polymers in ANY solution containing one or the other of Material A or Material B.