Rate of increase using a differential equation

Question

If a population doubles every 8 years and the rate of increase is proportional to the number of people find.

a) the number of people in 10years

b)how long it will take for the population to treble

Answer 1

dP/dt = kP

 

dP/P = kdt

 

ln(P) = kt + ln(C), C = unknown constant

 

P = Ce^kt

 

where C = P(0)

 

The population is described by a growing exponential function of the form

 

P(t) = P(0)e^kt

 

P(0) = initial population at time t = 0

 

To find k,

 

P(t + 8) = 2 P(t)

 

P(0)e^kte^k(8) = 2 P(0)e^kt ⇒ e^8k = 2

 

k = ln(2) / 8 = 0.086643 years^-1

 

Then

 

P(t) = P(0)e^0.086643t

 

with t in years.

 

(a)

 

After 10 years,

 

P(10) = P(0)e^0.086643(10) = 2.378 P(0)

 

[P(0) was not specified in the problem.]

 

(b)

 

Time to triple = T

 

3 = 1 e^0.086643T

 

T = ln(3) / 0.086643 = 12.680 years