Exponential growth and decay please help!

Question

Given that a colony of ants in an experiment follows the law of uninhibited growth for

the first 7 months. The colony has tripled in population after 6 months, how many months

did it take for the colony to double its initial population?

Answer 1

Population (P) as a function of time (t) in months for exponential growth would fit this general equation:

P(t) = P₀(1 + r)^t where R is thr growth rate in decimal form, and P₀ is the initial population.

In this case P(t)/P₀ = 3 when t = 6 so:

3 = (1 + r)⁶

3^1/6 = 1 + r

r = 3^1/6 - 1 = 0.200937

So, the equation becomes P(t) = P₀(1.200937)^t

To find when the population doubled, make P(t)/P₀ = 2

2 = (1.200937)^t

log(2) = log(1.200937)^t

log(2) = t•log(1.200937)

t = log(2)/log(1.200937)

t = 3.79 months