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Question

What is the doubling period of a town whose population is increasing exponentially if it takes 20 years to grow from 750 people to 3500?

Answer 1

The equation for an exponential function is the original value (750) x rate of change raised to the time (20). So first we have to find the rate of change.

We have:
3500=750 x^20

Solving this gives us 1.080066110, which is the rate of change. So now we need to find the amount of time it takes the population to double.

First, double the initial value of 750, giving us 1500.

So, 1500=750 * 1.080066110^x

The exponent is the value we don't know, so that is what we solve for. Using a math program to solve, the answer is 8.99 years (rounded, of course).

For a more accurate answer, you can do logs. Divide each side by 750, giving us:
2=1.08006610^x

then:
log 2 = x log 1.08006610
log 2 / log 1.08006610 = x

This answer is more accurate since no rounding is involved, but which you use depends on what exactly is wanted by the problem/teacher.

Answer 2

A Well Known Rule Of Thumb - The Rule Of 70

As a rule of thumb, if you divide the annual percentage increase of human population growth into 70 then you get the projected doubling rate.

Note that the projected population doubling rate should not be confused with the actual population doubling rate. The projected, or future, doubling rate is based upon a sustained annual growth rate as just described, and is easily calculated. The actual, or historical, doubling rate, is measured as how long (in years) it actually took the population to double to its current number (or, how long ago was the population exactly half what it is today?).

Because the current trend is currently toward a decreasing population growth rate, the actual doubling rates these days are typically slower than the projected doubling rates given in the past. If the trend were towards an increasing population growth rate then the actual doubling rates would be typically faster than the projected doubling rates given in the past. None of this invalidates the usefulness or validity of using projected doubling rates to discuss our demographic future.