Question

If the population of a country doubles in $50$ years in how many years will it become thrice the original, assume the rate of increase is proportional to the number of inhabitants.

• A. $75$
• B. $50lo{g}_{2}3$
• C. $50lo{g}_{3}2$
• D. $100$

Answer 1

The correct option is B $50lo{g}_{2}3$

Number of Inhabitants $=x,\frac{dy}{dx}\times x$

$\frac{dx}{dt}=kx\Rightarrow \frac{dx}{x}=kdt\left[kisproportionalconstant\right]$

Integration both side, we get

$\begin{array}{c}\int \frac{dx}{x}=\int kdt\\ \Rightarrow \log x=kt+c\\ \Rightarrow x=e^{kt+c}att=0,x=x_0\\ \Rightarrow x_0=e^c\\ So,\\ x=e^c.e^{kt}\Rightarrow x=x_0{e}^{kt}\\ So,t=50,P=2x_0\\ 2x_0=x_0{e}^{kt}\Rightarrow 2=e^{kt}\\ \log 2=k\left(50\right)\\ k=\frac{\log 2}{50}\\ P=3x_0,t=?\\ 3x_0=x_0{e}^{\frac{\log 2}{50}t}\Rightarrow 3=e^{{}^{\frac{\log 2}{50}t}}\end{array}$

Taking log both sides, we get

$\begin{array}{c}\log 3=\frac{\log 2}{50}t\Rightarrow t=50\frac{\log 2}{\log 2}\\ t=50\left[\log 3-\log 2\right]\\ Thisisalsowritten\\ 50{\log }_3^2\end{array}$