In 2000 the population of a country was approximately 629 mi

In 2000, the population of a country was approximately 6.29 million and by 2091 it is projected to grow to 15 million. Use the exponential growth model A=A_0 e^kt, in which t is the number of years after 2000, to find an exponential growth function that models the data. By which year will the population be 12 million? The exponential growth function that models the data is A=. (Simplify your answer. Use integers or decimals for any numbers in the expressions. Round to two decimal places as needed.)

Solution

(a) We have A = A₀e^kt , where A₀ is the initial population and A is the population after t years. Here, A₀ is the population in the year 2000, i.e. 6.29 million. Also, when t = 2091-2000 = 91, the population is projected to be 15 million. Therefore, 15 = 6.29 e^91k or, e^91k = 15/6.29 = 2.384737679. On taking natural logarithms of both the sides, we have 91k = ln 2.384737679 = 0.86908913 (as ln e = 1). Therefore, k = 0.86908913/91 = 0.00955043. Thus, the required population growth function whichmodels the given data is A = 6.29e^0.00955043t.

(b) If A = 12million, then 12 = 6.29 e^0.00955043t or, e^0.00955043t = 12/6.29 = 1.907790143. On taking natural logarithms of both the sides, we have 0.00955043t = ln 1.907790143 = 0.645945579 so that t = 0.645945579 /0.00955043 = 64.635 Thus, the population of the country will be 12 million in the year 2065.