Question

If the population of a country doubles in 50 years, in how many years will it treble, assuming that the rate of increase is proportional to the number of inhabitants?

Answer 1

Solution

Number of Inhabitants \( =x, \frac{d y}{d x} \times x \)

\( \frac{ dx }{ dt }= kx \Rightarrow \frac{ dx }{ x }= kdt \quad \)[ k is proportional constant ]

Integration both side, we get

\( \int \frac{d x}{x}=\int k d t \)

\( \Rightarrow \log x=k t+c \)

\( \Rightarrow x=e^{k t+c} \text { at } t=0, x=x_{0} \) \( \Rightarrow x_{0}=e^{c} \)

so \( x = e ^{ c } \cdot e ^{ kt } \Rightarrow x = x _{0} e ^{ kt } \)

Then \( t =50, P =2 x _{0} \)

\( 2 x _{0}= x _{0} e ^{ kt } \Rightarrow 2= e ^{ kt } \)

\( \log 2= k (50) \)

\( k =\frac{\log 2}{50} \)

\( P =3 x _{0}, t =? \)

\( 3 x _{0}= x _{0} e ^{\frac{ log2}{50} t } \Rightarrow 3= e ^{\frac{ log 2}{50} t } \)

Taking log both sides, we get

\( \log 3=\frac{\log 2}{50} t \Rightarrow t =50 \frac{\log 2}{\log 2} \)

\( \implies t=50[\log 3-\log 2]=50log_2(3)=79.24 \)

 

Therefore.\( t=79.24\quad year \)