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Course: AP®︎/College Environmental science>Unit 2

Lesson 3: Human populations

Rule of 70 to approximate population doubling time

Doubling time is the period it takes for a population growing at a certain rate to double its size. One way to approximate doubling time is known as the rule of 70. This rule is used in various fields, including finance and environmental science. However, the rule of 70 is an approximation, not a precise calculation.  Created by Sal Khan.

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• Is the rate here the annual growth rate? And is it computed by dividing the increase in population (increased value minus initial value) by 100? And finally, are we assuming the population growth rate is stable in projections? Is it really the case that population growth rate is actually stable? Or does it vary? Or does it remain more or less stable to make rounding it off to one rate reliable?
• How come '70' can always be used to approximate the doubling time for a population? Is there some mathematical explanation for this phenomenon?
(1 vote)
• This formula is derived from a really well-known general formula to describe exponential change for a function.

The formula is
``Final Value = Initial Value * e^(Rate * Time)``

Let's take rate = 1% which is 0.01 to test Rule of 70.

Let initial population be x.
Since final population is initial population doubled, then final population = 2x.

2x = x * e^(0.01t)
2x / x = x * e^(0.01t) / x
2 = e^(0.01t)
ln2 = 0.01t
t = ln2 / 0.01
t ≈ 69.3147
t ≈ 70

Notice when the rate is for example doubled, simply multiply 0.01 by 2, then it means divide t by half.

This equation is also used in various aspect, such as describing rate of nucleus in a radioactive decay.