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
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))`
= 8.41 years.
Therefore, it would take 8.4 years (approximately) to attain a population of 40000.