Suppose that the concentration of a bacteria sample is 30,000 bacteria per milliliter. If the concentration triples in 4 days, how long will it take for the concentration to reach 51,000 bacteria per milliliter?

Question

Razor · Answer

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AndrewArni · Answer

Ur right

SemiDefinite · Answer

Where $t_n$ is total number of bacteria per  mm on day $n$, the difference equation is:

$\qquad t_{n+4} = 3 t_n \qquad \circ$

Solution is exponential:

$\qquad t_n = t_0 \lambda^n, \qquad t_0 = 30,000 \qquad \square$

$\circ ~ \& ~\square \implies   t_0 \lambda^{n+4} = 3 t_0 \lambda^ n $

Ignoring trivial solutions:

$  \qquad \lambda^4 = 3, \qquad \lambda = \sqrt[4]3 $

To find day $N$ such that $ ~ ~ t_N = 51,000$, set:

$\qquad 30,000 ( \sqrt[4]{3} )^N = 51,000$

ie:

$N = \log_{\sqrt[4]3} \left( \frac{51}{30}\right ) = 1.932 $ (to 4 s.f.), using any decent calculator

So day 0: 30,000 /mm $\to $ day 1.932: 51,000 /mm