If the population grows at the rate $5 \%$ peryear, then the time taken for the population to become double…
If the population grows at the rate $5 \%$ peryear, then the time taken for the population
to become double is $\quad$ (Given $\log 2=0.6912$ )
$13.8275$ years
$13.624$ years
13.725 years
$13.8240$ years
Solution
Let initial population $=P_{0}$ and let $P$ be the population at time $t$ We have $\frac{\mathrm{dP}}{\mathrm{dt}}=\frac{5 \mathrm{P}}{100}$
$\therefore \frac{d P}{d t}=\frac{P}{20} \Rightarrow \int \frac{d P}{P}=\int \frac{1}{20} d t$
$\therefore \log P=\frac{1}{20} t+C$....(1)
When $t=0, P=P_{0}$
$\therefore \log \mathrm{P}_{0}=0+\mathrm{C} \Rightarrow \mathrm{C}=\log \mathrm{P}_{0}$
From (1), $\log \left(\frac{P}{P_{0}}\right)=\frac{t}{20}$....(2)
When $P=2 P_{0}$, we get
$\log 2=\frac{t}{20} \Rightarrow t=20(\log 2)=20 \times 0.6912=13.8240$ years