Exponential Word Problems

Many exponential word problems use the growth-decay formula . 

 represents the ending amount of whatever you are dealing with such as radioactive decay of an element, bacteria growth, money, etc.

 represents the beginning amount of that same thing. 

 represents the growth or decay rate. 

  represents time. 

Many times the variables will change from problem to problem but notice that the structure is always the same. 

Example:  A biologist is researching a newly-discovered species of bacteria.  At time  hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium.  Six hours later, he measures 450 bacteria.  Assuming exponential growth, what is the growth constant  for the bacteria? 

                 since the beginning amount  is the amount at time

                 is the ending amount

               

                We are looking to solve for.  Plug in the known values and then solve for the    growth constant.

                               

                               

                               

                Now take the natural log of both sides in order to solve for . 

                               

                               

                               

                And so .  Thus the growth constant is .25 per hour. 

Your turn!!  Answer the following problem.

A certain type of bacteria, given a favorable growth medium doubles in population every 6.5 hours.  Given that there were approximately 100 bacteria to start with, how many bacteria will there be in a day and a half? 

*Examples by purplemath.com.

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