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Question
A population grows at the rate of 5% per year. How long does it take for the population to double?
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Solution
Let P0 be the initial population and P be the population at any time t. Then,
\[\frac{dP}{dt} = \frac{5P}{100}\]
\[ \Rightarrow \frac{dP}{dt} = 0 . 05P\]
\[\Rightarrow \frac{dP}{P} = 0 . 05dt \]
Integrating both sides with respect to t, we get
\[\int\frac{dP}{P} = \int0 . 05dt \]
\[\log P = 0 . 05t + C\]
Now,
\[P = P_0\text{ at }t = 0 \]
\[ \therefore \log P_0 = 0 + C\]
\[ \Rightarrow C = \log P_0 \]
Putting the value of C, we get
\[\log P = 0 . 05t + \log P_0 \]
\[ \Rightarrow \log\frac{P}{P_0} = 0 . 05t\]
To find the time when the population will double, we have
\[P = 2 P_0 \]
\[ \therefore \log\frac{2 P_0}{P_0} = 0 . 05t\]
\[ \Rightarrow \log 2 = 0 . 05t\]
\[ \Rightarrow t = \frac{\log 2}{0 . 05} = 20 \log 2\text{ years }\]
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