Question

A population grows at the rate of 5% per year. Then the population will be doubled in –

• A. 10 log 2 years
• B. 20 log 2 years
• C. 30 log 2 years
• D. None of these

Answer 1

Answer: (B)

20 log 2 years

Answer 2

Let the population be p at any time t. Then,

$\frac{dp}{dt}$ = 5% at P Ž $\frac{dp}{dt}$ = $\frac{p}{20}$ Ž $\frac{dp}{p}$ = $\frac{dt}{20}$

Integrating, we get log p = $\frac{t}{20}$+ c.... (1)

Let the population be p₀ at t = 0.

On putting p = p₀ and t = 0 in equation (1), we get C = log p₀

\ log p = $\frac{t}{20}$ + log p₀ .. (2)

Let p = 2 p₀ at t = t₁.

Then, log (2 p₀) = $\frac{t_{1}}{20}$ + log p₀

Ž log 2 + log p₀ = $\frac{t_{1}}{20}$ + log p₀

Ž log 2 = $\frac{t_{1}}{20}$ Ž t₁ = 20 (log 2),

Which is the required time.