Question

The population of a country increases at a rate proportional to the number of inhabitants. If the population doubles in 30 years, find after how many years the population will triple–

• A. 48
• B. 47
• C. 46
• D. 49

Answer 1

Answer: (A)

48

Answer 2

Let Population = x, time = t (in years)

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

Where k is a constant of proportionality

or $\frac{dx}{x}$= k dt

Integrating, we get

ln x = kt + ln c

Ž ln $\left( \frac{x}{c} \right)$ = kt Ž $\frac{x}{c}$ = e^kt

or x = ce^kt

If initially i.e., when time t = 0, x = x₀

then x₀ = ce⁰ = c

\ x = x₀e^kt

Given x = 2x₀, t = 30

Then 2x₀ = x₀e^30kŽ 2 = e^30k

\ln 2 = 30 k... (1)

To find t, when it triples, x = 3x₀

\3x₀ = x₀e^kt Ž 3 = e^kt ... (2)

Dividing (2) by (1) then $\frac{t}{30}$ = $\frac{ln3}{ln2}$

or t = 30 × $\frac{ln3}{ln2}$ = 30 × 1.5849 = 48 years