Question

It is predicted that the population of a particular state will double every 25 years

Determine the annual and monthly growth rate.
Determine the continuous growth rate.

Answer 1

For annual (n=1) and monthly (n=12) use the following:

`(1+ (r)/(n))^(25n) = 2`
Where r is rate per period in which 1 will grow to 2 in 25 years.
To solve for r, take reciprocal of exponent and apply to both sides.

`1+ (r)/(n) = 2^{(1)/(25n)} \\ \\ r = n(2^{(1)/(25n)} - 1)`
Now just sub in value for n to get growth rates.

For continuous growth use the following:

`e^(25r) = 2`
Solve using natural log.

`ln(e^(25r)) = ln(2) \\ \\ 25r = ln(2) \\ \\ r = (ln(2))/(25)`