Question

The number of bacteria in a certain population increases according to a continuous exponential growth rate parameter of 5.4% per hour. How many hours does it take for the size of the sample to double? Round your answer to the nearest hundredth.

Answer 1

We can express the groth of the bacterias by the following expression:

$g(t)=A\times(1.054)^t$

Where A is the initial amount of bacteria, we want to know how many hours we need to double the initial amount. So we want to solve:

$2\times A=A\times(1.054)^t$

Lets use natural logaritm to help to solve the question:

$2\times A=A\times(1.054)^t\rightarrow2=(1.054)^t\rightarrow\ln 2=\ln (1.054)^t\rightarrow\ln 2=t\times\ln 1.054\rightarrow t=\frac{\ln 2}{\ln 1.054}\cong13.179594$

So, we need 13.18 hours to double our initial amount.