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# The rate at which the population of a bacteria culture grows is proportional to the number of bacteria present. If the number of bacteria grew from $1000$ to $5000$ in $10$ hours find the number of bacteria after $15$ hours

$\begin{array}{1 1} 12200 \\ 11180 \\ 22680 \\ 12130 \end{array}$

Let $B(t)$ = number of bacteria present at time $t \rightarrow \large\frac{dB}{dt}$$\propto B = kB$
Integrating we get, $\log B = kt + C \rightarrow B(t) = e^{kt + c}$
At $t=0$, $B=1000 \rightarrow B(0) = 1000 = e^{k\times0+c} \rightarrow e^c = 1000$
$\Rightarrow B(t) = 1000 e^{kt}$
When $t=10$, $B = 5000 \rightarrow B(10) = 5000 = 1000 e^{k\times10} \rightarrow 5 = e^{10k} \rightarrow 5^{\large\frac{1}{10}}$ $=e^k$
Substituting, we get,$B(t) = 1000\times 5^{\large\frac{t}{10}}$
Whent $t = 15$, $B = 1000 \times 5^{1.5} \approx 11180$ bacteria.