$(40$ points$)$ The four situations that involve money, interest rate, and different amounts of time are described here:

$(1)$ An island was purchased in $1626$ for $$24.$ After $385$ years in $2011,$ at $6\%$ per year compounded interest, the current value must be very large.

$(2)$ At the age of $22,$ if she saved only $$2000$ per year for the next $10$ years $($starting next year$)$ and made a return of $6\%$ per year, by today's standards, she would have accumulated a nice sum at the age of $70.$

$(3)$ A corporation invested $$2$ million in developing and marketing a new product in $1945($just after World War II$,$ this was a lot of money$)$ and has made a steady net cash flow of $$300,000$ per year for some $65$ years. She estimated the annual rate of return must be quite good, especially given that she is lucky to earn $4\%$ per year on her own investments these days.

$(4)$ A friend who is not good with money, went to a pawn shop and borrowed $$200$ for one week and paid $$30$ in interest. She thought this might be a pretty good deal, in case she ever ran low on cash. However, she did not know whether the interest was simple or compounded monthly, and how much may be owed were this loan not paid off for $1$ year.

$($a$)$ What is the annual interest rate for each situation below? Include both the annual simple and the compound rates for the $4^{th}$ situation.

$($b$)$ Calculate and observe the total amount of money involved in each situation at the end of the time periods compared to the starting amount. Is the ending amount larger or smaller than you would expect it to be prior to making any computations?