Calculating population and how fast its growing

Question

The population of a town grows at a rate proportional to the population present at time t. The initial population of 10000 increases by 20% in 5 years . What will be the population in 50 years? How fast is the population growing at t = 50 ?

Answer 1

When we are told that the rate (which is dP/dt) is proportional to the population (P) then it means:

dP/dt = kP where "k" is the constant of proportionality.

If dP/dt = kP then dP = dt•kP or dP/P = kdt

Then we can integrate both sides to get:

∫dP/P = ∫kdt

lnP = kt + C₁

e^(lnP) = e^{(kt + C1)}

P = e^kt• e^C1 but since e^C1 is just a number, we can call it C₂ so

P(t) = C₂e^kt

We are told the initial population is 10000 so we can say:

10000 = C₂e^k(0)

10000 = C₂(1) or C₂ = 10000 so:

P(t) = 10000e^kt

We are also told the initial population increase by 20% in 5 years so:

10000(1.2) = 10000e^k(5)

1.2 = e^5k

ln(1.2) = ln(e^5k)

ln(1.2) = (5k)ln(e)

ln(1.2) = (5k)(1)

(using a calculator) k = ln(1.2)/5 ≈ 0.0364643

So P(t) = 10000e^0.0364643t

Using this, P(50) = 61917

Using P(t), we can take the derivative dP/dt

dP/dt = 0.0364643•10000e^0.0364643t

dP/dt = 364.643e^0.0364643t

P'(50) = 364.643e^{0.0364643(50)}

P'(50) = 2258 people/year