Consider a bacterial population that grows according to the function

Sallie Banks

Sallie Banks

Answered question


Consider a bacterial population that grows according to the function f(t)=500e0.05t measured in minutes. After 4 hours, how many germs are present in the population? When will the population of bacteria reach 100 million?

Answer & Explanation

Jack Maxson

Jack Maxson

Beginner2021-12-31Added 25 answers

PSKt is in minutes
After 4 hours how many bacteria present
f(240)=81,317,395 bacteria
When will be the population of bacteria each 100 million


Expert2022-01-09Added 669 answers

At the starting point t = 0, there are \(f(0) = 500 e^{0.05*0} = 500\) bacteria.
We want to find the nbr of bacteria at t = 4h. As an hour has 60 minutes, we have to calculate \(f(4 \cdot 60) = f(240) = 500 \cdot e^{0.05 \cdot 240} = 500 \cdot e^{12} = 81.377\) Mio
When does it reach 100 Mio? It must be longer than 4h, as after 4h it is “only” 81 Mio.
\(f(t) = 500 \cdot e^{0.05t} = 100000000\); divide both sides by 500
\(e^{0.05t} = 200000\); take natural logarithm on both sides
\(0.05t = \ln(200000) = 12.20607\); divide both sides by 0.05
\(t = 244.1214 = 244\) minutes and 7.287 seconds.
It took 4h to grow to 81 mio and only 4 minutes and 7 seconds later it’s already 100 mio.

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