Question

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after $2\frac{1}{2}$ hours. [Take $\sqrt{2}=1.414$]

Answer 1

Given,

The rate of growth is proportional to the number present.

$\left(\frac{dN}{dt}\right)\propto N$

$\Rightarrow \left(\frac{dN}{dt}\right)=kN$

$\Rightarrow \left(\frac{dN}{N}\right)=kdt$

$\Rightarrow logN=kt+logC$

$\Rightarrow N=c{e}^{kt}$

At $t=0$

$N=C=1000$

$\therefore N=1000e^{kt}$

At $t=1$hr

$N=2000$

$\therefore 2000=1000e^k$

So $N=1000e^{0.693t}$

At $t=2.5$hr

$N=1000e^{(0.693\times 2.5)}$

$N=5654.77$.