The population of a country increases at a rate proportional to the number of inhabitants. `f` is the population which doubles in 30 years, then the population will triple in approximately. (a) 30 years (b) 45 years (c) 48 years (d) 54 years

Options

A. 30 years
B. 45 years
C. 48 years
D. 54 years

Answer

C

Explanation

Let population `=x`, at time t years. Given `(dx)/(dt) propto x`
or `(dx)/(dt) = kx,` where k is a constant of proportionality
or `(dx)/x=kdt`
Integrating, we get ln x `=kt+ "ln "c`
or `x/c=e^(kt)` or `x=ce^(kt)`
If initially, i.e., when time t=0, `x=x_(0)` then `x_(0)=ce^(0)=c`
`therefore x=x_(0)e^(kt)`
If initially, i.e., when time t=30, then `2x_(0)=x_(0)e^(30k)` or `2=e^(30k)`.
therefore `"ln "2=30k`..............(1)
To find t, when t triples, `x=3x_(0)`. Thus, `3x_(0)=x_(0)e^(kt)` or `3=e^(kt)`..................(2).
`therefore "ln " 3=kt`
Dividing equation (2) by (1), `t/30 = ("ln"3)/("ln"2)`
or `t=30 xx ("ln"3)/("ln"2) = 30 xx 1.5849=48` years(approx).