If the population of a country doubles in 60 years, in how many years will it be triple under the assumption that the rate of increase is proportional to the number of inhabitants? [Given: $\log 2 = 0.6912$ and $\log 3 = 1.0986$.]

If the population of a country doubles in 60 years, in how many years will it be triple under the assumption t

Question 151 of 177

\log 2 = 0.6912

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2017Subjective· 3mImportance★★★★★

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Population growth model $\dfrac{dP}{dt}=kP \Rightarrow P=P_0e^{kt}$ ; find $k$ from the doubling data, then solve for tripling time.

Since the rate of increase is proportional to the population, $\dfrac{dP}{dt} = kP \implies P = P_0e^{kt}$ .

Doubling in 60 years: at $t=60$ , $P=2P_0$ :

$2P_0 = P_0e^{60k} \implies e^{60k}=2 \implies 60k = \log 2 = 0.6912 \implies k = \dfrac{0.6912}{60}$

Tripling time $T$ : at $t=T$ , $P=3P_0$ :

$3P_0 = P_0e^{kT} \implies kT = \log 3 = 1.0986$

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