Assume that the historical growth rate of the human population was constant at 1.6% per year. For a population of 7 billion in 2012, determine the time in the past when the human population was 2.

Question

Quizplus · Accepted Answer

The Answer is As the annual percentage growth rate is small then we can use the approximation
of equation (2.4) to get the doubling time from R so

t _ { D } = \left( \frac { 100 \ln 2 } { R } ight) = \frac { ( 100 ) imes ( 0.693 ) } { 1.6 } = 43.31 ext { years }

from equation (2.7) the constant

a

can be found to be

a = \left( \frac { \ln 2 } { t _ { D } } ight) = \frac { 0.693 } { 43.31 \mathrm { y } } = 0.016 \mathrm { y } ^ { - 1 }

from equation (2.4) we start with an initial population of

N _ { 0 } = 2

at

t = 0

then

N ( t ) = 6.7 imes 10 ^ { 9 }

then from

N ( t ) = N _ { 0 } \exp ( a t )

so

t = \frac { 1 } { a } \ln \frac { N ( t ) } { N _ { 0 } } = \frac { 1 } { 0.016 } \ln \frac { 7 imes 10 ^ { 9 } } { 2 } = 1374 \mathrm { y }

in the past or at year 2012-1374

= 638

(obviously growth rate was not constant)

Assume That the Historical Growth Rate of the Human Population