Question

The bacteria in a dish triples every hour. At the start of the experiment therewere 400 bacteria in the dish. When the students checked again there were32,400 bacteria. How much time had passed?.

Answer 1

Solution:

Given:

$\begin{gathered} At\text{ time (t) = 0, 400 bacteria were present.} \\ \\ \text{The bacteria triples every hour.} \end{gathered}$

Hence, this is an exponential function.

$\begin{gathered} y=ab^x \\ \text{where;} \\ y\text{ is the number of bacteria present} \\ x\text{ is the time} \\ \\ \text{Hence, at the start of the experiment} \\ 400=ab^0 \\ a=400 \\ \\ At\text{ the next hour, the number has tripled} \\ 1200=ab^1 \\ 1200=400(b^1) \\ b=\frac{1200}{400} \\ b=3 \end{gathered}$

Hence, the function can be represented by;

$y=400(3^x)$

The time that had passed when the bacteria was 32,400 will be;

$\begin{gathered} y=400(3^x) \\ 32400=400(3^x) \\ \text{Dividing both sides by 400,} \\ \frac{32400}{400}=3^x \\ 81=3^x \\ 3^4=3^x \\ \\ \text{Equating the exponents since the base are the same,} \\ x=4 \end{gathered}$

Therefore, 4 hours have passed when the bacteria became 32,400