Greenwood press

28
EXPONENTIAL DECAY
A graph that describes an exponential decay
of radioactive substance as a function of
time: Quantity remaining of 100 grams of 
a radioactive substance with half-life of
24,000 years.

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Exponential decay models are also written using base 
e. The equation A =
100e
−kt
, where 
k =
ln2
24,000
is the same equation plotted in the graph.
Radiocarbon dating of animal or plant remains that are thousands of years old
is based on the radioactive isotope carbon-14, which has a half-life of 5,700
years. Carbon-14 is constantly produced in the earth’s atmosphere through the
absorption of radiation from the sun. When living organisms breathe or eat, they
ingest some carbon-14 along with ordinary carbon. After an organism dies, no
more carbon-14 is ingested, so the age of its remains can be calculated by deter-
mining how much carbon-14 is left.
Exponential decay in prices is called depreciation. Some types of deprecia-
tion used in accounting are linear. For example, tax law permits a business to
depreciate 20 percent of the original cost of computer equipment for each of five
years. Market prices, however, do not follow a linear pattern. Automobiles typi-
cally depreciate rapidly during the first year, and then less rapidly during each
subsequent year. The used-car prices for one popular automobile that sold for
$27,000 when new are given by 
P = 27, 000(0.83)
t
, where 
t is the number of
years after purchase. In this case, the automobile lost 17 percent of its value each
year. 
Inflation problems can be viewed as growth problems (increases in prices) or
as drops in the value of currency. For example, the purchasing power of the dol-
lar dropped by 7.2 percent per year during the 1970s. The purchasing power of
$100 is given by 
P = 100(1 − 0.072)
t
= 100(0.928)
t
, where 
t is the number of
years after 1970.
Concentrations of a medication that are carried in the bloodstream often fol-
low an exponential decay model. Such drugs are said to have half-lives. Each day
you replace about 25 percent of the fluids in your blood. If you are taking a med-
ication that depends on the bloodstream for circulation, then 25 percent of the
dose is lost as you replace fluids. A person who takes one pill containing 20 mg
of medicine will have about 15 mg (75 percent of 20 mg) in his or her body one
day later, and 11.25 mg (75 percent of 15 mg) two days later, and so on. The half-
life for this drug can be found by solving the equation 
1
2
= 0.75
t
, or
t ≈ 2.4 days.
Some drugs do not follow an exponential decay pattern. Because alcohol is
metabolized by humans, the quantity of alcohol in the bloodstream after inges-
tion will show a linear decrease rather than exponential decay.
For the many drugs and steroids that have half-lives, the drop off in drug con-
centration decreases less rapidly over time. Therefore it is possible to measure
the quantity of the drug in the body long after ingestion. This means that users of
illegal or dangerous drugs will have traces of the drugs remaining in their blood-
streams for many days. Sensitive drug tests, such as those used on Olympic ath-
letes, can pick up indications of banned drugs used within two weeks or more of
the testing, depending on the half-life of the substance.
If you pour a cup of hot coffee, the temperature will drop off quickly, then
the coffee will remain lukewarm for a long while. Newton’s law of cooling states
that the rate at which the temperature drops is proportional to the difference