States The increase in the population of a country is proportional to that population. The population doubles every 50 years. How quickly does it triple ?


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Question Number 138037 by greg_ed last updated on 09/Apr/21

$\underset{―}{States}$ $The increase in the population of a country is$ $proportional to that population .$ $The population doubles every 50 years .$ $How quickly does it triple ?$
	Answered by MJS_new last updated on 09/Apr/21

	$2^{\frac{y}{50}} = 3$ $\frac{y}{50} \ln 2 = \ln 3$ $y = 50 \frac{\ln 3}{\ln 2} \approx 79.25$
	Commented by greg_ed last updated on 09/Apr/21

	$thank you very much for the promptness,$ $dear friend!$
	Answered by mr W last updated on 09/Apr/21

	$\frac{d P}{d t} = k P$ $\int \frac{d P}{P} = \int k d t$ $\ln P = k t + C_{1}$ $\Rightarrow P = P_{0} e^{k t}$ $2 P_{0} = P_{0} e^{k 50} \Rightarrow 2 = e^{50 k} \Rightarrow \ln 2 = 50 k$ $3 P_{0} = P_{0} e^{k t} \Rightarrow 3 = e^{k t} \Rightarrow \ln 3 = k t$ $\Rightarrow \frac{t}{50} = \frac{\ln 3}{\ln 2}$ $\Rightarrow t = \frac{\ln 3}{\ln 2} \times 50 \approx 79 y e a r s$
	Commented by greg_ed last updated on 09/Apr/21

	$well done!$
	Answered by Dwaipayan Shikari last updated on 09/Apr/21

	$\frac{d N}{d t} = k N$ $\Rightarrow N = N_{0} e^{k t}$ $W h e n 2 N_{0} = N_{0} e^{k (50)} \Rightarrow k = \frac{1}{50} l o g (2)$ $3 N_{0} = N_{0} e^{l o g (2) \frac{t}{50}} \Rightarrow l o g (3) = l o g (2) \frac{t}{50} \Rightarrow t = 50 . \frac{l o g (3)}{l o g (2)} y e a r s$
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