Question

A culture started with 4000 bacteria after 8 hours it grew to 4400 bacteria. Predict how many bacteria will be present after 9 hours.

Answer 1

After 9 hours there will be 4453 bacteria

Answer 2

$\bf \qquad \textit{Amount for Exponential change}\\\\ A=P(1\pm r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{starting amount}\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed period}\\ \end{cases}\\\\ -------------------------------$

$\bf A=P(1+r)^t\qquad \begin{cases} \textit{hour 0, starting point}\\ t=0\qquad A=4000 \end{cases}\implies 182=ae^{k0} \\\\\\ 4000=P(1+r)^0\implies 4000=P \\\\\\ thus\qquad A=4000(1+r)^t\\\\ -------------------------------$

$\bf A=P(1+r)^t\qquad \begin{cases} \textit{8 hours later}\\ t=8\qquad A=4400 \end{cases}\implies 4400=4000(1+r)^8 \\\\\\ \cfrac{4400}{4000}=(1+r)^8\implies \cfrac{11}{10}=(1+r)^8\implies \sqrt[8]{\cfrac{11}{10}}=1+r \\\\\\ \sqrt[8]{\cfrac{11}{10}}-1=r\implies 0.012\approx r\qquad thus\qquad \boxed{A=4000(1+0.012)^t}$

how many bacteria after 9 hours? well, 9hours later is just t = 9, plug that in, to get A