Exponential Decay Formula
The general exponential equation is y = a • bx, where a base number, b, is raised to a variable power, x. There is also a scalar multiplier, a, in front of the exponential expression.
Exponential decay can be modeled with a similar equation:
With exponential decay, b represents the rate of decay, (expressed as a decimal), and is subtracted from 1 to represent decay. We can call (1 – b) the decay factor, since it is multiplied by a (the initial value) x number of times.
Calculating the Amount of a Substance
Many of us take aspirin or ibuprofen when we have a headache, fever, or other ailment. A certain amount of active ingredients enter our bloodstream, but are consumed by our bodies over time, such that only a portion of the drug is still in our blood stream. Suppose we take an 80 mg pill of aspirin, which dissolves at a rate of 60% every hour. How many milligrams is in the blood stream after 4 hours?
From the scenario, we can draw the following values for certain variables in the exponential decay formula:
After 4 hours, only 2.048 mg of the medicine is in the blood stream. Perhaps this is why doctors recommend taking a new dosage every 4 hours.
Solving for Decay Time
How long do you think a world wide Rock Paper Scissors contest would take, if all 7 billion humans on earth entered the tournament? We can answer this question using our exponential decay model.
The first round would have 7 billion contenders, with half of them being eliminated each round. This means that our rate of decay is 50%, since only 50% of the population remains in the tournament each round. This must continue until we have only 1 person left as the world champion in Rock Paper Scissors.
We can now identify knowns and unknowns to use in our formula:
Since x represents the number of rounds in the tournament, we will round up to the nearest whole number. This means the entire world can play a full Rock Paper Scissors tournament in only 33 quick rounds.
There are a couple of things to note when solving these types of problems: