Bacteria colony as an hourly function measured in seconds

Text

The bacterium responsible for cattle plague triples in number every 20 minutes. At $t=0$ the colony has $100\,000$ members. Write a function $f(t)=a\cdot b^{t}$, with $t$ measured in seconds, that describes the evolution of the colony.

Solution

$a=f(0)=100\,000$. It triples every $20$ min $=1200$ s: $b^{1200}=3\Rightarrow b=3^{1/1200}\approx1.000916$. Hence $f(t)=100\,000\cdot 3^{t/1200}$.

$\boxed{f(t)=100\,000\cdot 3^{t/1200},\quad b=3^{1/1200}\approx1.000916}$