Half-Life and Exponential Decay: Concepts, Formulas, and Examples, Study notes of Russian

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Math 1030 # 12 b

Doubling Time and Half-Life

Half-Life

Dou blin g Ha lf- life Exp one nti al Gr ow th Ex po ne nti al^ De ca y

Half-life is the time it takes for half of a population to vanish if it decreases by the same percent each time period. EX 1 : If a city of 1000 people is decreasing by 10 % each year, when will there be half as many people? Year Number of people

EX 2 : Radioactive carbon- 14 has a half-life of about 5700 years. It collects in organisms only while they are alive. Once they are dead, it only decays. What fraction of carbon- 14 in an animal bone still remains 800 years after the animal has died? EX 3 : A clean-up project is reducing the concentration of a pollutant in the water supply, with a 3. 5 % decrease per week. a) What is the approximate half-life of the concentration of the pollutant? b) What fraction of the original will remain after one year?

Exact half-life formula: Thalf = - where r is a decimal and negative_._ Note: The units of time for r and T must be the same (per month, year, etc.) log 10 ( 2 ) log 10 ( 1 +r) EX 4 : Suppose the Russian ruble is falling in value against the dollar at 11 % per year. a) Approximately how long will it take the ruble to lose half its value? b) Exactly how long will it take the ruble to lose half its value?