Greenwood press
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book-20600
EXPECTED VALUE
27 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 . Download 1.81 Mb. Do'stlaringiz bilan baham: |
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