Each of the above four quantities (5 through 8) is a function of the form ce−kt (k > 0). (See Figure
N9–7b.)
FIGURE N9–7aFIGURE N9–7b
EXAMPLE 15
At a yearly rate of 5% compounded continuously, how long does it take (to the nearest year) for
an investment to triple?
SOLUTION: If P dollars are invested for t yr at 5%, the amount will grow to A = Pe0.05t in t yr.
We seek t when A = 3P:EXAMPLE 16
One important method of dating fossil remains is to determine what portion of the carbon content
of a fossil is the radioactive isotope carbon-14. During life, any organism exchanges carbon with
its environment. Upon death this circulation ceases, and the^14 C in the organism then decays at a
rate proportional to the amount present. The proportionality factor is 0.012% per year.
When did an animal die, if an archaeologist determines that only 25% of the original amount of^14
C is still present in its fossil remains?
SOLUTION: The quantity Q of^14 C present at time t satisfies the equationwith solution