Question

A countrys population in 1991 was 147 million. In 1998 it was 153 million. Estimate the population in 2017 using exponential growth formula. Round your answer to the nearest million. P=Ae to the power kt

Answer 1

171 million

Explanation:

Given the exponential growth formula expressed as
`P = Ae^(kt)` where P is the countrys population and t is the time.

Initially in 1991, at t= 0, P = 147 million

`P = Ae^(kt)\\147 = Ae^(k(0))\\147 = Ae^0\\147 = A(1)\\A = 147`

If by 1998 it was 153 million, this means that the population is 153 million 7 years later i.e when t = 7, P = 153. On substituting into the formula to get the constant 'k';

`P = Ae^(kt)\\153 = 147e^(k(7))\\153 = 147e^(7k)\\153/147 = e^(7k)\\e^(7k) = 1.041\\Taking \ ln \ of \ both \ sides\\lne^(7k) = ln 1.041\\7k = 0.04018\\k = 0.04018/7\\k = 0.00574`

To estimate the population in 2017, to number of years from 1991 to 2017 is 26 years. Hence we are to find the value of P given t = 27, A = 147 and k = 0.00574

`P = Ae^(kt)\\P = 147e^(0.00574*26)\\P = 147e^0.14924\\P = 147*1.16095\\P = 170.659`

Hence the population in 2017 using exponential growth formula to the nearest million is 171 million.