CK12 Calculus - Single Variable

This is a famousexpressionin physicsfor measuringthe half-lifeof a substanceif the decayconstant is
known.It can also be usedto compute if the half-life is known.
Example1:

A radioactivesamplecontains gramsof nobelium.If you knowthat the half-lifeof nobeliumis seconds,
how muchwill remainafter minutes?
Solution:

Beforewe computethe massof nobeliumafter minutes,we needto first knowits decayrate Using
the half-lifeformula,

So the decayrate is The commonunit for the decayrate is the Becquerel :
is equivalentto decayper sec. Sincewe foundk, we are now readyto calculatethe massafter minutes.
We use the radioactivedecayformula.Remember, representsthe initialmass, grams,and
minutes seconds.Thus

So after minutes,the massof the isotopeis approximately grams.
PopulationGrowth

The sameformula can be usedfor populationgrowth,exceptthat sinceit is an increasing
function.
Example2:
A certainpopulationof bacteriaincreasescontinuouslyat a rate that is proportionalto its presentnumber.
The initialpopulationof the bacterialcultureis and jumpedto bacteriain hours.

1. How manywill be therein hours?


2. How long will it take the populationto double?
   Solution: