The bacterial race

The model for the exponential growth of a bacterial colony is the famous (and most simple) differential equation:

$\large \frac { dN }{ dt }$ $= \lambda N$

The general solution can be written as:

$N(t)= { N }_{ 0 }{ e }^{ \lambda t }$ ,

in our case ${ N }_{ 0 }=1$ because there is only 1 bacterium at t=0.

Given that there is a 1-hour lag phase, which means no bacterial duplication during that period, the effective exponential growth lasts 23 hours.

Therefore we know (hypothetically) that after 23 hours of duplication , there are 8-million bacteria. So for this statement to be true, all the bacteria considered must have a duplication time smaller than some ${ T }_{ max }$ .

we want ${ T }_{ max }$ in seconds so we need to convert 23 hours in seconds, that's 82800s.

And since $\lambda =\frac { ln2 }{ T }$

${ T }_{ max }$ is given by: $N(82800)=$ $\large { e }^{ \frac { ln2 }{ { T }_{ max } } \times 82800 }$ $=8000000$ .

Solving for ${ T }_{ max }$ actually yields 3610,7s but we can't round it up because a 3611s duplication time would result in a colony with less than 8 million bacteria.

So the maximum duplication time is $\boxed{{ T }_{ max }=3610s}$

Note: This could also be done by using $N(t)={ 2 }^{ n }$ as a model, where n, the number of generation, is given by: $n=\frac { t }{ T }$