Question

The population of a town increases at a rate proportional to the population at that time. If the population increases from 26,000 to 39,000 in 50 years, then the population in another 25 years will be ______ `(sqrt(3/2) = 1.225)`

Options

• 47757

• 47775

• 47575

• 45777

Answer 1

The population of a town increases at a rate proportional to the population at that time. If the population increases from 26,000 to 39,000 in 50 years, then the population in another 25 years will be `(sqrt(3/2) = 1.225)` 47775.

Explanation:

Let P be the population at time t years.

Then, `(dP)/dt = kP`

⇒ `(dP)/P = kdt`

Integrating on both sides, we get

log P = kt + c

When t = 0, P = 26000

∴ log 26000 = 0 + c ⇒ c = log 26000

∴ log P = kt + log 26000

⇒ `log(P/26000) = kt` .....................(i)

When t = 50 yrs, P = 39000

∴ `log(39000/26000) = 50 k ⇒ k = 1/50 log(3/2)`

∴ `log(P/26000) = t/50log(3/2)` .....[From(i)]

When t = 75 yrs, we have

`log(P/26000) = 75/50 log(3/2)`

⇒ `P/26000 = (3/2)^{3/2} ⇒ P/26000 = 3/2(3/2)^{1/2}`

⇒ P = `26000 xx 3/2 xx 1.225 = 47775`