The population of a community is known to increase at a rate proportional to the number of people present at timet. If an initial population P0 has doubled in 5 years, how long will it take to triple? To quadruple?
We are given that,
dP/dt = K.P
i.e. int dP/P = K * int dt
So, ln(P/A) = K.t, where ln(1/A) is the const of integration
Then, P = A.e^(Kt)
at t = 0, P = P0 = A.1 -> A = P0
Thus, P = P0.e^(Kt) (P0 is the initial population)
For P = 2*P0 (population doubles, after 5 yrs)
Then 2*P0 = P0.e^(5K)
2 = e^(5K) = [e^K]^5
2^(1/5) = e^K
2^(t/5) = e^(Kt)
Therefore P = P0.2^(t/5)
For P = 3*P0 (popularion triples)
Then 3*P0 = P0.2^(t/5)
3 = 2^(t/5)
ln(3) = (t/5).ln(2)
t = 5*ln(3)/ln(2) = 5*(1.0986)/(0.6931) = 5*1.58496 = 7.9248 yrs
Time to triple = 7 yrs, 11 mths, 3 days
For P = 4*P0 (popularion quadruples)
Then 4*P0 = P0.2^(t/5)
4 = 2^(t/5)
i.e. t/5 = 2 -> t = 10 yrs
Time to quadruple = 10 yrs