Question

The number of bacteria in a culture, N, at time t is attached belowWhen is the rate of change of the number of bacteria equal to zero?

Answer 1

Given that the number of bacteria in a culture, N, at time t is expressed as

The rate of change is expressed by taking the derivative of the above relation with respect to t.

This gives:

`N^(\prime)(t)=(dN)/(dt)=2000\left(-\frac{te^{-(t)/(20)}}{20}+e^{-(t)/(20)}\right)`

Thus, when the rate of change equals zero, we have

`\begin{gathered} N^(\prime)(t)=0 \\ \Rightarrow2000\left(-\frac{te^{-(t)/(20)}}{20}+e^{-(t)/(20)}\right)=0 \\ thus\text{ we have} \\ -\frac{te^{-(t)/(20)}}{20}=-e^{-(t)/(20)} \\ multiply\text{ both sides by 20} \\ -te^{-(t)/(20)}=-20e^{-(t)/(20)} \\ cancel\text{ out similar factors,} \\ \therefore \\ t=20 \end{gathered}`

Hence, the rate of change of of the number of bacteria equals zero when

`t=20`