Differential Equations Question 283
Question: The population of a country doubles in 40 years. Assuming that the rate of increase is proportional to the number of inhabitants, the number of years in which it would treble itself is
Options:
A) 80 years
B) $ 80\frac{\log 2}{\log 3}years $
C) $ 40\frac{\log 3}{\log 2}years $
D) $ 40\log 2\log 3years $
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Answer:
Correct Answer: C
Solution:
[c] Let the initial population be $ x_0 $ and it is x in t years, then the differential equation is $ \frac{dx}{dt}=kx, $ k is a constant
$ \Rightarrow \frac{dx}{x}=kdt. $ Integrating we get $ \log x+kt+C $
- (i) When $ t=0,x=x_0\Rightarrow c=\log x_0 $ Then from (i) $ \log x=kt+\log x_0 $
$ \Rightarrow \log \frac{x}{x_0}=kt $
- (ii) Now when
$ \Rightarrow \log \frac{x}{x_0}=2\Rightarrow \log 2=k.40\Rightarrow k=\frac{\log 2}{40} $
$ \therefore $ (ii) becomes $ \log \frac{x}{x_0}=\frac{\log 2}{40}.t $ Next put $ \frac{x}{x_0}=3\Rightarrow t=40\frac{\log 3}{\log 2} $
BETA
