Question

If the population of a country doubles in 50 years in how many years will it triple under the assumption that the rate of increase is proportional to the number of inhabitants –

• A. 79 years
• B. 78 years
• C. 76 years
• D. 77 years

Answer 1

Answer: (A)

79 years

Answer 2

Let x denote the population at a time t in years.

then $\frac{dx}{dt}$µ x Ž $\frac{dx}{dt}$ = kx

when k is a constant of proportionality.

Solving $\frac{dx}{dt}$ = kx, we get

$\int_{}^{}\frac{dx}{x}$ = $\int_{}^{}k$dt Ž log x = kt + c Ž x = e^{kt + c}

Ž x = x₀ e^kt

Where x₀ is the population at time t = 0.

Since it doubles in 50 years, at t = 50, we must have x = 2x₀.

Hence 2x₀ = x₀e^{50 k} Ž 50 k = log 2

Ž k = $\frac{\log 2}{50}$so that x = x₀$e^{\frac{\log 2}{50}t}$

To find t, when it triples, x = 3x₀ Ž 3x₀ = x₀$e^{\frac{\log 2}{50}t}$ Ž log 3 = $\frac{\log 2}{50}$t

Ž t = $\frac{50\log 3}{\log 2}$= 79 years