The rate of change of population of a community is known to increase at a rate p

Question

The rate of change of population of a community is known to increase at a rate proportional to the number of people present at time t the model is (dp/dT) = kP.

a) Solve the ODE for the generl solution subject to the initial condition P(0) = Po.

b) Using the ODE in part a, if the initial population doubled in 6 years then find the constant K.

c) Using the ODE in part a, what will the population be in 8 years if the initial population was Po?

i know the answers to part a and b but am a little confused on part c

Explanation / Answer

dp/dt = kP

dp/P = kdt

integrating both sides

ln(P) = kt + c (where c = constant)

at t = 0

a) ln(P0) = c

b) ln(p) = kt + ln(Po)

given

p = 2Po

ln(2Po) = 6k + ln(Po)

ln(2) = 6k

k = ln(2)/6 = 0.116

c) ln(p) = kt + ln(Po)

ln(p) = [ln(2)/6]*8 + ln(Po)

ln(p) = 4ln(2)/3 + ln(Po)

ln(p/Po) = ln(2)^(4/3)

p = Po*(2)^(4/3) = 2.52 Po

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