CK-12-Calculus

6.4. Exponential Growth and Decay http://www.ck12.org

So the solution to the equation(dy/dt) =kyhas the formy=Cekt.The box below summarizes the details of this

function.

The Law of Exponential Growth and Decay

The functiony=Cektis a model for exponential growth or decay, depending on the value ofk.

• Ifk>0: The function represents exponential growth (increase).


• Ifk<0: The function represents exponential decay (decrease).
  Wheretis the time,Cis the initial population att= 0 ,andyis the population after timet.

Applications of Growth and Decay

Radioactive Decay

In physics, radioactive decay is a process in which an unstable atomic nucleus loses energy by emitting radiation in

the form of electromagnetic radiation (like gamma rays) or particles (such as beta and alpha particles). During this

process, the nucleus will continue to decay, in a chain of decays, until a new stable nucleus is reached (called an

isotope). Physicists measure the rate of decay by the time it takes a sample to lose half of its nuclei due to radioactive

decay. Initially, as the nuclei begins to decay, the rate starts very fast and furious, but it slows down over time as more

and more of the available nuclei have decayed. The figure below shows a typical radioactive decay of a nucleus. As

you can see, the graph has the shape of an exponential function withk< 0.

The equation that is used for radioactive decay isy=Cekt.We want to find an expression for the half-life of an
isotope. Since half-life is defined as the time it takes for a sample to lose half of its nuclei, then if we starting with
an initial massC(measured in grams), then after some timet,ywill become half the amount that we started with,
C/ 2 .Substituting this into the exponential decay model,

y=Cekt

C

2 =Ce

kt.