Find the rate constant r?

Question

Consider a population p of field mice that grows at a rate proportional to the current population, so that dy/dt=rp.

a) Find the rate constant r if the population doubles in 30 days.

b) Find r if the population doubles in N days.

Answer 1

It should be dp/dt = rp.

Separating variables,

dp/p = rdt

lnp = rt + c

p = Ce^(rt)

a) 2C = Ce^(r*30)

r = ln(2) /30

b) 2C = Ce^(r*N)

r = ln(2) /N

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Attn: There is a quick way to get the answer for any exponential growth or decay.

C = C₀ (2)^(r*t)

2C₀ = C₀ (2)^(r*N)

Solve for r,

r = ln(2)/N