Question

If the world’s population is increasing at an annual rate of 1.3%, and there were 5 billion people in the year 1986, then in what year will the world’s population be 10 billion?

Answer 1

I forgot, how do I calculate this?

Answer 2

Well, the hard way would be to add 1.3% every year, over and over again, until the result was more than 10 billion. Let's start off that way, and see if any shortcuts make themselves obvious along the way...in science, it's usually a better idea to start off "dumb" and careful, and look for better ways as you go along, then outstmart yourself by trying to do it in one standing jump.

So...here we go...years and billions...

1986: 5.0
1987: 5.0 + 1.3% of 5.0 = 5.0 + 0.013*5.0 = (1 + 0.013)*5.0 = 1.013*5.0
1988: 1.013*1.013*5.0
1989: 1.013*1.013*1.013*5.0
1990: 1.013*1.013*1.013*1.013*5.0

I definitely see a pattern. Looks like:
\[P(y) = P(1986) (1.013)^{y - 1986}\]
What we want to know is what's y, the year, when P(y) = 10 billion:
\[10.0 = 5.0(1.013)^{y - 1986}\]
Now we can just use the rules of exponents to solve for y.

Answer 3

What about using the rule of 70? that would be ln2/0.013=53.3

Therefore, 1986+53 years = 2039

Answer 4

I have no idea what the rule of 70 is, but your equation is simply what you get when you take the log of both sides of mine, so, yeah, that'll work.

Answer 5

http://betterexplained.com/articles/the-rule-of-72/

The rule of 70 is only an approximation for what Carl Pham just derived.