# How do I do all the math to come up with a final year in which there will be only one square meter for every person on earth?"Given the current populations of 6,994,325,380 (as of 2/14/2012 at...

How do I do all the math to come up with a final year in which there will be only one square meter for every person on earth?

"Given the current populations of 6,994,325,380 (as of 2/14/2012 at 10:49 am) as prvided by the U.S. Census bureau's web site, when will there be only one square meter for every person on earth?" I am taking glaciers and water into account.

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### 3 Answers

P = P_0 e^(rt) is the equation for population growth.

P_0 is the t=0 population (todays population in our problem)

e is Euler's number 2.718281828... it is on scientific calculators.

r is the population growth rate (0.015) the current rate.

t is the time in years.

The ln function, again on most calculators, is the inverse of the e^x function, it undoes e^x so ln(e^x) = x.

So solving for e^(rt) and then taking the ln of both sides of the above equation I get

ln(e^rt) = ln(P/P_0)

Since ln(e^x) = x we get

rt = ln(P/P_0)

and finally

t = (ln(P/P_0))/r

Since we know P, P_0, and r we can calculate the time it will take to reach that population on a calculator, which is what I did in my answer above.

Hope that helps.

I can give you a link that explains exponential population growth.

http://en.wikipedia.org/wiki/Population_growth

The surface area of the earth (including water and glaciers) is `510,072,000 km^2`

This is `510,072,000,000,000 m^2`

Assuming a 1.5% growth rate (world population grew 30% between 1990 and 2010).

Using `P=P_0 e^(rt)` for population growth.

`P_0 = 6,994,325,380`

`P = 510,072,000,000,000` we get

`e^(0.015t) = (510,072,000,000,000)/(6,994,325,380)`

Take the ln of both sides and divide by 0.015 to get

`t = ln((510,072,000,000,000)/(6,994,325,380))/0.015`

I get t = 746.480 years.