Under ideal conditions a certain bacteria population is know to double every three hours. Suppose that there are initially 100 bacteria. How would you go about formulating a function for this?

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Raphael Singleton · Accepted Answer

The general approach described by Todd Wilcox is the best way to go about things: It is important to become thoroughly familiar with the function       e          r      t      When doubling time is given, there is a quick formula that works. Let d be the doubling time. Then if P(t) is the population at time t, we have  P  (  t  )  =  P  (  0  )        2          t              /            d        .So in our case,                              (                      1                    )                            P        (        t        )        =        (        100        )                  2                      t                          /                        3                          .            The general approach, which you should carry out, will give you after some manipulation  P  (  t  )  =  (  100  )      e          t      log      ⁡      2              /            3        ,where by log we mean logarithm to the base e (ln on your calculator). This is the same as the answer (1), since      e          x      log      ⁡      2        =            (              e                  log          ⁡          2                    )        x    =      2    x    ,since       e          log      ⁡      2        =  2Remark: As long as one is willing to believe that the answer has shape   P  (  t  )  =  A        2          k      t      , the formula is easy to prove. For put t=0. Then   P  (  0  )  =  A        2          (      k      )      (      0      )        =  A, so A=P(0). Also, since doubling time is d, we have P(d)=2P(0). But   P  (  t  )  =  P  (  0  )        2          k      t      . Put t=d. We get   2  P  (  0  )  =  P  (  0  )        2          k      d      , and therefore kd=1, meaning that k=1/d. This yields the formula   P  (  t  )  =  P  (  0  )        2          t              /            d       mentioned above.

Under ideal conditions a certain bacteria population is know to double every three hours. Suppose that there are initially 100 bacteria. How would you go about formulating a function for this?

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