a. In 2000, the population of a country was approximately 6.46 million and by 20
ID: 3186229 • Letter: A
Question
a. In 2000, the population of a country was approximately 6.46 million and by 2091 it is projected to grow to 15 million. Use the 12 exponential growth model A=A0 e kt, in which t is the number of years after 2000 and A0 is in millions, to find an exponenta growth function that models the data. By which year will the population be 7 million? Projected 2000 6,460,00 b. 0- 1950 1970 1990 2010 2030 2050 Year a. The exponential growth function that models the data is A- (Simplify your answer. Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) b. The country's population will be 7 million in the year (Use the answer from part a to find this answer. Round to the nearest year as needed.)Explanation / Answer
a. Let the model for the population growth of the country be A = A0 ekt , where A0 is the initial population in 2000, A is the population t years after 2000 and k is the constant of growth. Here, A0 =6.46 million, and also in 2091, when t = 91, we have A = 15 million. Hence 15 = 6.46 e91k or, e91k = 15/6.46. Now, on taking natural log of both the sides, we get 91k ln e = ln 15 – ln 6.46 or, 91k = 2.708050201- 1.865629318 = 0.842420883. Hence k =0.842420883/91 = 0.0092572344. Thus, the required model for the population growth of the country is A = 6.46 e0.0092572344t.
b. Let the population of the country be 7 million t years after 2000. Then 7 = 6.46 e0.0092572344t or, e0.0092572344t= 7/6.46. Now, on taking natural log of both the sides, we get 0.0092572344t = ln 7 – ln 6.46 or, t = 1.945910149-1.865629318 = 0.080280831. Hence, t = 0.080280831/0.0092572344 = 8.67 ( on rounding off to 2 decimal places). Thus, the population of the country will be 7 million in the year 2009.
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