Population growth

Question

Given that the initial population, $N_0$ of bacteria is $12$ and the population doubles every $20$ minutes. I wish to find a general formula for the population if one unit of time is $50$ minutes for $t=0,1,2,3,\ldots$.

If one unit of time is $20$ minutes, then I can 'easily' find $N(t)=12\cdot 2^t,~t=0,1,2,\ldots$.

How do I solve the original problem? Is there a general method for approaching these types of questions?

Answer 1

The "doubling time" becomes $\dfrac{20}{50} = \dfrac{2}{5}$ units, so $$N(t) = 12\cdot 2^{t/(2/5)} = 12 \cdot 2^{\Large\left(\frac{5t}{2}\right)} = 12\cdot2^{(2.5t)}$$

Answer 2

The doubling time is now $20/50 = 0.4$ units, so $N(t) = 12 \cdot 2^{t/0.4}$, with $t$ in units of time.