Application of Exponentials
Suppose that you put $2,000 into a bank account that pays 6% interest compounded monthly. How much will you have in 5 years?
After the first month, the new balance will be
A = P(1 + rt) = 2,000(1 + (.06)(1/12))
the next month's balance is
2,000(1 + .06/12)(1 + .06/12) = 2,000(1 + .06/12)2 = 2,000(1.005)2
The third month, the balance will be
2,000(1.005)2(1.005) = 2,000(1.005)3
After t months, the balance will be
Five years is 60 months so the final balance will be
2,000(1.005)60 = $2697.70
In general for an account that initially has P dollars in it and is left for t years in an account that pays interest at a rate of r and compounds m times per year we have
A = P(1 + r/m)mt
For continuous compounding such as inflation, the formula is
A = Pert
If health care costs $300 per month for the average family, how much will health care cost in the year 1050 if the inflation rate is 8% per year?
If today there is Po grams of a certain radioactive isotope, then after t years there will be
P = Poert
You find a skull in a nearby Native American ancient burial site and with the help of a spectrometer, discover that the skull contains 9% of the C-14 found in a modern skull. Assuming that the half life of C-14 is 5730 years, how old is the skull?
First we use the fact that after 5730 years, there is half remaining so that
1/2Po = Poert
0.5 = er5730
ln 0.5 = r(5730)
Since today there is .09Po we have
0.09Po = Poe-.00012t
0.09 = e-.00012t
ln0.09 = -0.00012t
t = (ln.09)/-.00012 = 20,000 years old.
For an interactive lesson of finding C and k given the graph of y = Cekt click here