Greenwood press

EXPECTED VALUE
27

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EXPONENTIAL DECAY
Exponential decay can be observed in the depreciation of car values, the half-
life of elements, the decrease of medication in the blood stream, and the cooling
of a hot cup of coffee. The general exponential equations that define exponential
growth, such as the financial model for principal after compound interest is 
applied,
A = P (1 +
r
n
)
nt
, and the general models for exponential growth such 
as
y = ab
x
can be used to describe losses over time for values of 
b that are
between 0 and 1. The changes that are made to the models may involve chang-
ing the base from a number greater than one (growth) to a number less than one
(decay), or leaving the base alone and allowing the power to be negative. 
The term “decay” comes from the use of exponential functions to describe
the decrease of radioactivity in substances over time. The law of radioactive
decay states that each radioactive nuclear substance has a specific time known as
the half-life, during which radioactive activity diminishes by half. Some radioac-
tive substances have half-lives measured in thousands to billions of years (the
half-life of uranium-238 is 4.5 billion years), and some in fractions of a second
(muons have a half-life of 0.00000152 seconds). The way in which radioactivity
is measured varies from substance to substance. Uranium-238 decays into lead,
so the proportions of lead and uranium-238 in a sample can be used to determine 
the amount of decay over time. The law of decay is stated as 
A
R
= A
o
(
1
2
)
t/h
,
where
A
o
is the amount of radioactive substance at the start of the timing, 
h is
the half-life time period, and 
A
R
is the amount remaining after 
t units of time. In
this format, the base of the exponential equation is 
1
2
, clearly a number less than 
one. It can also be stated with a base larger than one if the exponent is negative, 
as in 
A
R
= A
o
(2)
−t/h
. The basic shape of the graph of exponential decay is 
shown in the plot below. One hundred grams of substance with half-life of
24,000 years is followed for 100,000 years. At the end of 24,000 years, 50 grams
of the radioactive substance are left in the sample. At the end of 48,000 years, 25
grams are left, and at the end of 72,000 years, 12.5 grams. The formula that de- 
scribes this model is 
A = 100(
1
2
)
t/24,000
.