Question

A culture of bacteria contains 1,500 bacteria initially and doubles every 30 minutes.

1. Write a function that models the number of bacteria after t time.
2. Find the number of bacteria after 2 hours.
3. After how many minutes will there be 12,000 bacteria

Answer 1

if the bacterial is doubling, that means the growth rate is 100%, or 100% plus whatever it was before, so

`\textit{Periodic/Cyclical Exponential Growth} \\\\ A=P(1 + r)^{(t)/(c)}\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &1500\\ r=rate\to 100\%\to (100)/(100)\dotfill &1\\ t=minutes\\ c=period\dotfill &30 \end{cases} \\\\\\ A=1500(1 + 1)^{(t)/(30)}\implies A=1500(2)^{(t)/(30)} \\\\\\ \stackrel{\textit{after two hours, or 120 minutes, t = 120}~\hfill }{A=1500(2)^{(120)/(30)}\implies A=1500(2)^4\implies A=24000}`

`~\dotfill\\\\ \stackrel{\textit{current amount being 12000}}{12000=1500(2)^{(t)/(30)}}\implies \cfrac{12000}{1500}=(2)^{(t)/(30)}\implies 8=(2)^{(t)/(30)}\implies 8=\sqrt[30]{2^t} \\\\\\ 8^(30)=2^t\implies (2^3)^(30)=2^t\implies 2^(90)=2^t\implies 90=t`