Suppose a population's doubling time is 14 years. Find its annual growth factor 𝑏 and annual percent growth rate 𝑟.

Answer 1

Hello Jenna,

The equation describing the scenario would be: P(t) = P₀ * (2)^t/14

It's t/14 instead of "just" t because it takes 14 years for the population to double, and you have to divide by the doubling time.

Let's rewrite the previous equation so we end up with the typical exp function, i.e y = a*b^x

P(t) = P₀ * (2)^t/14 = P₀ * (2^1/14)^t

Now this looks like y = a*b^x where b, the base, is equal to 2^1/14

Why did we do this step? Well, because the base b represents the annual growth factor.

So... b = 2^1/14 = 1.05075663...

To find the annual growth rate, you would: 1) subtract 1 from the annual growth factor, and 2) multiply that answer by 100 to get it in percentage format

Which gets us: (2^1/14 - 1)*100 = 5.075663% so close to 5.08%

Recap: b = 1.0508; r = 5.08%