Application of Exponentials
Compound Interest
Example: Suppose that you put $2,000 into a bank account that pays 6% interest compounded monthly. How much will you have in 5 years?
Solution 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 2,000(1.005)^{t} 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 = Pe^{rt}
Exercise: 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?
Radioactive Dating If today there is P_{o} grams of a certain radioactive isotope, then after t years there will be P = P_{o}e^{rt}
Example 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 C14 found in a modern skull. Assuming that the half life of C14 is 5730 years, how old is the skull? First we use the fact that after 5730 years, there is half remaining so that 1/2P_{o} = P_{o}e^{rt} 0.5 = e^{r5730} ln 0.5 = r(5730)
ln 0.5 Since today there is .09P_{o} we have 0.09Po = Poe^{.00012t} 0.09 = e^{.00012t} ln0.09 = 0.00012t t = (ln.09)/.00012 = 20,000 years old.
Exercises
For an interactive lesson of finding C and k given the graph of y = Ce^{kt} click here
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