Question

The population of a region is growing exponentially. There were 20 million people in 1980 (when t=0) and 70 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980.
P(t)=
What population do you predict for the year 2000?
Predicted population in the year 2000 = million people.
What is the doubling time?
Doubling time = years.

Answer 1

Let the exponential model describing the growth in the population of a region be P(t) = a b^t where P(t) is the population(in millions) of the region t years after 1980 and a, b are arbitrary real numbers.

In 1980, when t= 0, P(0) = 20 so at 20= ab⁰ or, a = 20. Then, P(t) = 20*b^t. Further, in 1990, when t = 10, P(10) = 70 so that 70 = 20*b¹⁰ or, b¹⁰ =(70/20) = 7/2 so that b = (7/2)^1/10. Then, P(t) = 20*(7/2)^t/10.

The predicted population in the year 2000, when t = 20, is 20*(7/2)^20/10 = 20*(7/2)² = 245 million.

Let the population double in t years from 1980 . Then 40 = 20*(7/2)^t/10 or, (7/2)^{t /10} = 40/20 = 2. On taking log of both the sides, we get (t/10) log(7/2) = log 2 so that, t = 10*[log 2/ log(7/2)] = 10* 0.301029995/0.544068044 = 5.532947548 = 5.5 years ( on rounding off to the nearest tenth of an year). Thus, the doubling time is 5.5 years.