Doubling time :)

Question

What is the annual growth rate if a population doubles in size after 22 years?

Answer 1

Hi Chloe,

This question is about continuous growth rate. Luckily, we have an easy to use formula for this problem.

A = Pe^rt

A is the final amount after the formula is used. In this case our A is double what we started with

P is the initial amount. Our starting value. In this case we are not told our P but we are told that A, our final value, is double the amount of P. So we can say:

2P = Pe^rt

By expressing it in this form, we can make our P just 1. x(1) = x So our formula changes again to:

2 = e^rt

e is a special number. We will ignore it. You don't need to know what it means until higher level math. It's about 2.71828......

r is our rate of change. That is what we are looking for

t is time which we know is 22 years. Knowing all this, let's solve:

2 = e^r22

We will take the natural log of both sides

Ln(2) = Ln(e^r22)

One of the special properties of e is that. The natural log of e raised to a number is that number:

Ln(e^b) = b

This means Ln(e^r22) = r22

We now have:

Ln(2) = r22. We divide both sides by 22 to get our r. You can plug this into your calculator and you get

r = Ln(2)/22

r = 0.03150669

Our rate is 3.2 percent continuously.