# The rate of growth of a certain town’s population is proportional to the excess of the population over 10 000. If the town initially has 18 000 people and after 4 years the population grows to 25...

The rate of growth of a certain town’s population is proportional to the excess of the population over 10 000. If the town initially has 18 000 people and after 4 years the population grows to 25 000, find how long it would take to reach 40 000

anser: 8.4 years

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### 1 Answer

This fits into the standard form for exponential population growth of the form:

`(A-10000) = (a-10000) e^(kt)` --- (i)

where a is the initial population, A is the new population, k is a constant and t is the time in years.

Here, in the first condition,

a = 18000

A = 25000, t = 4

Putting the values in equation (i),

`(25000-10000) = (18000-10000) e^(k*4)`

`rArr 15000 = 8000 e^(4k)`

Taking log (to the base e) on both sides,

`ln(15000/8000) = 4k `

`rArr k =1/4 ln(15/8)`

Plugging in the value of k in the second condition, we get

`(40000-10000) = (18000-10000) e^(k*t)`

`rArr 30000 = 8000 e^(kt)`

`rArr t = 1/kln(30/8)`

`= ln(30/8)/(1/4 ln(15/8))`

= `4*ln(30/8)/ln(15/8)`

= 8.41 years.

**Therefore, it would take 8.4 years (approximately) to attain a population of 40000**.

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