Q95 / 180 · KCET 2009 · Mathematics
A population grows at the rate of $10 \%$ of the population per year. How long does it take for the population to double?
- A$20 \log 2 \mathrm{yr}$
- B$10 \log 2 \mathrm{yr}$
- C$5 \log 2 \mathrm{yr}$
- Correct answerDNone of the Above
Solution
Given, $r=10 \%$
Let population is $P$.
According to the question
$$
\begin{aligned}
2 \mathrm{P} &=\mathrm{P}\left(1+\frac{10}{100}\right)^{\mathrm{n}} \\
\Rightarrow \quad 2 &=\left(1+\frac{1}{10}\right)^{\mathrm{n}}
\end{aligned}
$$
$$
\begin{aligned}
\Rightarrow \quad \log 2 &=\mathrm{n}(\log 11-\log 10) \\
\Rightarrow \quad \mathrm{n} &=\frac{\log 2}{1.0414-1}=\frac{\log 2}{0.0414} \\
&=24.15 \log 2
\end{aligned}
$$
Hence, none of the given option is correct.
Hence, the correct answer is option: 'D'.