Question

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.
the growth model ( a = 7 e^{0.002 t} ) describes the population, ( a ), of a country in millions, ( t ) years after 2003.
a. what is the countrys growth rate?
b. how long will it take the country to double its population?
( square ) years (round to the nearest whole number.)

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.