What is exponential decay? Whenever something is decreasing or shrinking rapidly as a result of a constant rate of decay applied to it, that thing is experiencing exponential decay.
For example suppose that the population of a city was 100,000 in 1980. Then every year, the population has decreased by 3% of a result of heavy pollution. This is an example of exponential decay.
Notice that the rate of decay is 1% or 0.01 and it is constant. This is important since the rate of decay cannot change.
Let us find the exponential function.
Year 1981 or 1 year after:
100,000 - 100,000 x 0.03 = 100,000 (1 - 0.03 ) = 100,000(0.97)
= 100,000(0.97)^{1}
Year 1982 or 2 years after:
100,000(0.97) - 100,000(0.97) x 0.03 = 100,000(0.97) [1 - 0.03]
= 100,000(0.97)(0.97)
= 100,000(0.97)^{2}
Following this pattern, suppose that
Then y = 100,000(0.97)^{x}
General rule for modeling exponential decay
Exponential decay can be modeled with the function
y = ab^{x} for a > 0 and 0 < b < 1
y = ab^{x}
x is the exponent
a is the starting amount when x = 0
b is the base, rate, or decay factor and it is a constant and it is smaller than 1.
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