Search any question & find its solution
Question:
Answered & Verified by Expert
If the population grows at the rate of $8 \%$ per year, then the time taken for the population to be doubled is (Given $\log 2=0.6912$ )
Options:
Solution:
1633 Upvotes
Verified Answer
The correct answer is:
8.64 years
Let the initial population be $\mathrm{P}$ and rate of increase is $8 \%$ per year.
$$
\begin{aligned}
& \therefore \frac{\mathrm{dP}}{\mathrm{dt}}=\frac{8}{100} \mathrm{P} \\
& \therefore \int \frac{\mathrm{dP}}{\mathrm{P}}=\int 0.08 \mathrm{t} \\
& \therefore \log |\mathrm{P}|=0.08 \mathrm{t}+\mathrm{c}
\end{aligned}
$$
When $\mathrm{t}=0$, we get $\mathrm{c}=\log \mathrm{P}$
$$
\therefore \log \mathrm{P}=0.08 \mathrm{t}+\log \mathrm{P}
$$
When ' $\mathrm{P}$ ' doubles, we write
$$
\begin{aligned}
& \log 2 p=0.08 \mathrm{t}+\log \mathrm{P} \\
& \therefore \log \left(\frac{2 \mathrm{P}}{\mathrm{P}}\right)=\log 2=0.6921=0.08 \mathrm{t} \\
& \therefore \mathrm{t}=\frac{0.6912}{0.08}=8.64 \text { years }
\end{aligned}
$$
$$
\begin{aligned}
& \therefore \frac{\mathrm{dP}}{\mathrm{dt}}=\frac{8}{100} \mathrm{P} \\
& \therefore \int \frac{\mathrm{dP}}{\mathrm{P}}=\int 0.08 \mathrm{t} \\
& \therefore \log |\mathrm{P}|=0.08 \mathrm{t}+\mathrm{c}
\end{aligned}
$$
When $\mathrm{t}=0$, we get $\mathrm{c}=\log \mathrm{P}$
$$
\therefore \log \mathrm{P}=0.08 \mathrm{t}+\log \mathrm{P}
$$
When ' $\mathrm{P}$ ' doubles, we write
$$
\begin{aligned}
& \log 2 p=0.08 \mathrm{t}+\log \mathrm{P} \\
& \therefore \log \left(\frac{2 \mathrm{P}}{\mathrm{P}}\right)=\log 2=0.6921=0.08 \mathrm{t} \\
& \therefore \mathrm{t}=\frac{0.6912}{0.08}=8.64 \text { years }
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.