The exponential model A 8506 e0007 t describes the populati

The exponential model A = 850.6 e^0.007 t describes the population, A, of a country millions, t years after 2003. Use the model to determine when the population of the country will be 1028 million. The population of the country will be 1028 million in (Round to the nearest year as needed.)

Solution

The model for the population of a country in millions, t years from 2003, is A = 850.6e^0.007t. When the population of the country is 1028 million, we have 1028 = 850.6 e^0.007t or, e^0.007t = 1028/850.6. Now, on taking natural log of both the sides, we get (0.007t) ln e = ln 1028- ln 850.6 or, 0.007t = 6.935370446-6.745941983= 0.189428463. Hence, t = 0.189428463/0.007 = 27.06120903 = 27 ( on rounding off to the nearest whole number). Thus, the population of the country will be 1028 million after 2003+27 i.e. in the year 2031.

Note:

log(a/b) = log a-log b ; log mⁿ= nlog m and ln e = 1. Also, t = 27.06 ( a number higher than 27). Therefore, the population of the country will be 1028 million after 2030, i.e. in 2031.