The doubling time of any quantity n is related to the contin

The doubling time of any quantity n is related to the continuous growth rate R_cont by the relation 2 = exp(n*R_cont). Derive an expression for n in terms of the annual growth rate and then estimate what the doubling time for a population growth rate is averaging about 2% per year.

Solution

The relation is

2=exp(n*R_cont)

for annual growth rate

2= b^n where b= 1+r and r is annual growth rate

2 = (1+r)^n

taking log both sides

ln 2 = n ln(1+r)

therefore n= (ln 2) / ln(1+r)

The above expression is for n for annual growth rate.

doubling time = (ln 2) / growth rate

doubling time = (ln 2) / 0.02 = 34.65 years

Hence doubling time is approximately 35 years.