Question

A given bacteria culture initially contains 2500 bacteria and doubles every half hour. The number of bacteria p at a given time t is given by the formula p(t)=2500ekt for some constant k. (You will need to find k to answer the following.)
(a) Find the size of the bacterial population after 50 minutes.
(b) Find the size of the bacterial population after 8 hours.

Answer 1

To find k, we need a value of t not equal to zero, say t=0.5, then we know that
\(2500\times 2 = 2500e^{k\times 0.5} \)
simplify
\(2 = e^{0.5k}\)
take log on both sides
\(log_e 2=0.5k\)
\(k=2 log_e 2\)
hence
\(p(t) = 2500e^{log_e(2t)}\)
which simplifies to
\(p(t) = 2500\ 2^{2t}\)

Check:
\(p(0)=2500, p(0.5)=2500e^{log(2)}=5000\) ok
For (a) and (b), put in the necessary values for answer.

In fact, if the quantity \(doubles \) every half hour, the base is 2, and the equation can be assumed to be
\(p(t)=2500\ 2^{qt}\)
where q=2. This is much easier to solve.