The Alpolococci

 

      Dr. Zee, the legendary medical researcher, had just developed a new bacteria strain with promise for the treatment of several dreaded diseases.  The bacteria were kept in an ideal environment to develop properly, and as the culture increased in size, large amounts of food were required.  Unfortunately, this ideal environment had spoiled the bacteria, and they would eat only Alpo dog food.

      “Yikes!” shrieked Dr. Zee.  “These alpolococci are eating us out of house and home.  At 10 a.m. today we had a culture of 100,000 and only 3 hours later it has grown to 500,000.  I wonder how many we’ll have by tomorrow morning at 6 a.m.?”

      “Well,” suggested his talented assistant, “it is often assumed that the rate of increase in a population is proportional to the pop...”

      “I’ve got it!” shouted Dr. Zee.  “The population increases at a rate of proportional to the population.  In other words, if  represents the population count at time t we know that

.

Then    

and

.

Thus

Since  is positive, we can dispense with the absolute value to get

which is equivalent to

,

where .”

 

Letting  correspond to 10 a.m. today we can find K from the equation

We use this value of K to write

.

We still need to find , and we can use the fact that  to do that.

Setting  and  in our expression for  yields

     

Dividing by 100,000 and taking the natural logarithm of each side allows us to solve for  as follows:

We now have .  The population at 6 a.m. tomorrow will be

.”

“That was most impressive sir!” exclaimed Dr. Zee’s assistant.  “Let’s go out and get another case of Alpo!”

 

Copyright (c) 2001 by  Kenneth L. Wiggins

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