use the formula ( t=\frac{ln 2}{k} ) that gives the time for a population, with a growth rate ( k ), to…

use the formula ( t=\frac{ln 2}{k} ) that gives the time for a population, with a growth rate ( k ), to double, to answer the following questions.\nthe growth model ( a = 7 e^{0.002 t} ) describes the population, ( a ), of a country in millions, ( t ) years after 2003.\n a. what is the countrys growth rate?\n b. how long will it take the country to double its population?\n( square ) years (round to the nearest whole number.)

use the formula ( t=\frac{ln 2}{k} ) that gives the time for a population, with a growth rate ( k ), to double, to answer the following questions.\nthe growth model ( a = 7 e^{0.002 t} ) describes the population, ( a ), of a country in millions, ( t ) years after 2003.\n a. what is the countrys growth rate?\n b. how long will it take the country to double its population?\n( square ) years (round to the nearest whole number.)

Answer

Explanation:

Step1: Identify the growth rate formula

The population growth model is (A = A_0e^{kt}). Comparing with (A = 7e^{0.002t}), we can see that (k = 0.002). To convert it to a percentage, we multiply by 100. So the growth rate is (0.002\times100 = 0.2%).

Step2: Use the doubling - time formula

The formula for the time (t) it takes for a population to double is (t=\frac{\ln2}{k}). Given (k = 0.002), we substitute into the formula: (t=\frac{\ln2}{0.002}). Since (\ln2\approx0.693), then (t=\frac{0.693}{0.002}=346.5\approx347) years.

Answer:

a. The country's growth rate is (0.2%). b. It will take approximately (347) years for the country's population to double.