Exponential decay models apply to any situation where the decay (decrease) is proportional to the current size of the quantity of interest. Such situations are encountered in biology, business, chemistry and the social sciences.
Exponential decay models are also used very commonly, especially for radioactive decay, drug concentration in the bloodstream, of depreciation of value.
Radioactive decay problems are often given in terms of half-life of a radioactive element. This is modeled by the equation:
where is the initial amount of the element, is the amount remaining after years, and is the half-life.
If you start with a quantity of the unstable element Potassium- , it takes billion years for half of it to decay into Argon- . So the half-life of Potassium- is billion years.Write an exponential decay model to find the number of Potassium- atoms remaining after years, if you start with Potassium- atoms.
Here, and . So the model is:
For drug concentration problems, you may be given the fraction of the original amount of the drug left in the bloodstream after a unit of time. In this case, the situation is modeled by the equation
where is the concentration remaining after time , and is the initial amount .
If a person takes milligrams of a drug at time , then gives the concentration left in the bloodstream after hours. If the initial dose is mg, what is the concentration of the drug in the bloodstream after hours?
You might want a calculator!
So there are about milligrams of the drug left in the bloodstream after four hours.
If the value of some article (for example, a car), originally , depreciates per year, then the value after years is given by the formula:
The original value of a car is . If it depreciates by each year, find its value in years.
So after four years, the car is worth about .