Example The growth of a colony of bacteria is provided by the equation,

                                           Q = Q e^{0.195 t}

If there are at first 500 bacteria exist and t is given in hours find out each of the following.

(a) How several bacteria are there after a half of a day?  

(b) How much time will it take before there are 10000 bacteria in the colony?  

Solution

Following is the equation for this starting amount of bacteria.

                                               Q =500 e^{0.195 t}

(a) How several bacteria are there after a half of a day?

In this case if we desire the number of bacteria after half of a day we will have to use t = 12 as t is in hours.  Thus, to obtain the answer to this part we only need to plug t into the equation.

                         Q = 500 e^0.195(12)  = 500 (10.3812365627 ) =5190.618

Thus, as a fractional population doesn't make any sense we'll say that after half of day there are 5190 of the bacteria present.

 (b) How much time will it take before there are 10000 bacteria in the colony?

Do not make the mistake of supposing that it will be approximately 1 day for this answer depends on the answer to the previous part. Along exponential growth things just don't work that way as we'll illustrate.  To answer this part we will have to solve the following exponential equation.

                                                          10000 = 500 e^{0.195 t}

Let's do that.

 10000/500  =e ^{0.195 t}

20 = e^{0.195 t}

ln 20 = ln e^{0.195 t}

ln 20 = 0.195t  ⇒ t = ln 20 / 0.195 =15.3627

Thus, it only takes approximately 15.4 hours to attain 10000 bacteria and not 24 hours if we only double the time from the first part. In other terms, be careful!