Maci 
EMAT 4500 

Suppose that at retirement you have saved $500,000, in an account that pays 6% interest, compounded monthly.  In order to cover your living expenses, you plan to withdrawal $4000 per month.  How long will the money last?  How much money do you need to have invested so that the money will last 40 years?

In order to find a solution for this particular problem, I found it much easier to start with determining a spreadsheet solution and then progressing to the algebraic solution.  Just as in class, I was able to set up the spreadsheet in such a way that I could determine precisely the number of months you could live if you had saved $500,000.  I set up the spreadsheet in the following manner: 

Month		Interest		Withdrawal	       Balance
     0						      $500000
     1	   500000(1 + .06/12))      $4000	      $498500.00	
     2        498500(1 + .06/12))      $4000              $$496992.50                                            
      .                      .                             .                          .      
      .                      .                             .                          .   
      			And So on

In order for the spreadsheet to provide accurate results I needed to use all the great properties of excel and allow for each of my rows to be a function of the one before it.  This is very similar to the now/next formulas that we have been looking at in class.  The month column was created by starting at month zero and telling excel to add one to the row before it.  The interest column was created by taking the previous balance and multiplying it buy the interest rate of (1 + .06/12).  I then used the interest and added it to the previous balance and the then subtracted the $4000 dollar withdrawal.  The formula for the new balance looked like this: previous balance + previous balance(1 +.06/12) - $4000.  Since I was able to tell excel to duplicate this process it continued down the columns and I was able to find when the money would run out.  

I was able to determine the number of month that you could live on that $500000, given the specified interest rate and the given withdrawal amount, and determined that his would be about 196 months.  I was able to find this answer by looking at the balance column and finding when the values became negative.  When the negative values appeared I was able to conclude that you had run out of money.  If you look at the month just before the values become negative you will see that at month 196 there will be $2612.62 left, thus meaning that there is not enough to withdrawal the needed $4000 the next month.  In order to find the number of years this would be, I simply divided 196 by 12 to obtain 16.25 years.  

In order to answer the second part of the question, How much money do you need to have invested so that the money will last 40 years?, I will use the same spreadsheet e setup and the same method for solving as before.  

As before we had a starting balance of $500000 in this particular example we will tweak the values of the balance in order to see the balance remaining after 480 years.  In the previous example we only looked up to month 196 and in this example we must extend down to month 480.  After manipulating starting balance we are able to find the balance at 480 months.  This was a tedious process, but I was able to find that the minimum amount that needed to be saved was about $726.990.07.  This will yield a 480 month amount of -$2.93, which is the closest I could get to zero.

In order to find a solution to our problem algebraically it took a little more work.  We all now that in algebra solving equations work best, so I attempted to find an equation that modeled what was happening on my excel sheet.  

Looking at month one this is what I saw:

The second month this is what I saw:

After the third month this is what I saw:

After doing this process a couple of times I was able to discover, or at least I think I was able to discover, an algebraic formula for our problem.  The formula is below, where x is the number of months.

Now that we have this equation we can attempt to find the number of month that we and live off of this money.
We must set the above equation equal to zero because we want to know when the money runs out.  So:

 We can now divide both sides by 4000 and obtain: 

Now that I have gotten to this point I have no idea what to do.  It seems as if I could take the ln of both sides in order for the x and the i to come down but I have no idea how to solve for a summation.  I do know that this formula should we find a result will actual provide us with more accurate results then the spreadsheet did because we will be able to precisely find the number of months and days.  From the spreadsheet we know that the value should be about 16.25, but with this solution it should be even more accurate.  

In order for us to determine the amount of money required to last 40 years we can use our equation, but in this problem we must make a few adjustments.  
 Where x is now the amount invested and 480 is 40 years.  

Using the excel spreadsheet I am able to see that this x value will be close to $726,990.07.  I am not sure how to figure this out, so basically I am not sure how to solve this algebraically but looking at the excel spreadsheet I am pretty convinced.  

Of the examples of exponential growth and decay we have considered, which seemed to be the most interesting and important? Which would be most useful for introducing these concepts to students? Explain.

I felt that the exponential example regarding steroids in the body would be the most interesting for students.  With the recent Olympic games and the several athletes that were suspected of using steroids were tested, I think that students would find this example useful.  I am not sure that this is the most important exponential growth or decay problem that we have done; however, this is an idea that students can relate to and that they don’t need to understand terminology such as compounded monthly and interest rates.  I think that the steroid example would be even more pertinent if you could find information on the steroid that the Olympians were being tested for and determining the effects of this steroid on the body, thus substituting in this steroid into the problem.  This problem as well as several others will provide the students with the necessary knowledge that is needed to develop patterns as well as make the needed connections for understanding.