Friday, October 1, 2010

Lesson # 15: Exponential Growth and Decay

There are two different formulas:

The most common formula to use is:
F= IR^ t/p                    F: final amount       I: initial amount     R: rate of growth ( 5% = 0.05 +1 R>1)   growth decay ( 5%  R= 0.95)     t: time       p: time for R to occur (days, weeks, months,etc)  
* note: t and p must be in same units

The second formula is for continuous interest rates:
 P= Po e^ Kt          P: final amount       Po: initial amount      e: calculator function   K: growth/decay (no 1)  
t: time 


Example 1:
What will $3500 grow to, if invested at 6% interest for 10 years, compounded annually?
F= IR ^ t/p               F=?     I= 3500   R= 1.06  t= 10    p=1 
(*note: if p is a fraction take the reciprocal and multiply it to the top number) 
F= 3500(1.06)^10/1
F= 3500(1.06)^10
F= $6267.97

Example 2:
What amount of money would grow to $ 4000 if invested at 91/4%, compounded annually for 4 years?
F= IR ^ t/p              F= 4000   I= ?   R= 1.0925   t= 4     p=1
4000   =     I  (1.0925) ^ 4/ 1
(1.0925)^4     (1.0925)^4
I= $2807.85

Example 3:
In April. the atmosphere at Chernobyl was contaminated with radioactive iodine-131, which has a half-life of 8.1 days.  How long did it take for the level of radiation to reduce to 1% of the level immediately after the accident?
*Half-life  R= 0.5     P= half-life
1/100 = 100(0.5)^t/8.1
0.01 = 0.5^t/8.1
log0.01 = t/8.1 log0.5
8.1 log0.01/log0.5 = t

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